International Handbooks on Information Systems
Series Editors Peter Bernus, Jacek Błażewicz, Günter J. Schmidt, Michae...

This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. Start by pressing the button below!

International Handbooks on Information Systems

Series Editors Peter Bernus, Jacek Błażewicz, Günter J. Schmidt, Michael J. Shaw

For further volumes: http://www.springer.com/series/3795

Titles in the Series M. Shaw, R. Blanning, T. Strader and A. Whinston (Eds.) Handbook on Electronic Commerce ISBN 978-3-540-65882-1

P. Bernus, K. Merlins and G. Schmidt (Eds.) Handbook on Architectures of Information Systems ISBN 978-3-540-25472-0, 2nd Edition

J. Błaz˙ewicz, K. Ecker, B. Plateau and D. Trystram (Eds.) Handbook on Parallel and Distributed Processing ISBN 978-3-540-66441-3

S. Kirn, O. Herzog, P. Lockemann and O. Spaniol (Eds.) Multiagent Engineering ISBN 978-3-540-31406-6

H.H. Adelsberger, Kinshuk, J.M. Pawlowski and D. Sampson (Eds.) Handbook on Information Technologies for Education and Training ISBN 978-3-540-74154-1, 2nd Edition C.W. Holsapple (Ed.) Handbook on Knowledge Management 1 Knowledge Matters ISBN 978-3-540-43527-3 Handbook on Knowledge Management 2 Knowledge Directions ISBN 978-3-540-43848-9 J. Błaz˙ewicz, W. Kubiak, I. Morzy and M. Rusinkiewicz (Eds.) Handbook on Data Management in Information Systems ISBN 978-3-540-43893-9 P. Bernus, P. Nemes and G. Schmidt (Eds.) Handbook on Enterprise Architecture ISBN 978-3-540-00343-4 S. Staab and R. Studer (Eds.) Handbook on Ontologies ISBN 978-3-540-70999-2, 2nd Edition S.O. Kimbrough and D.J. Wu (Eds.) Formal Modelling in Electronic Commerce ISBN 978-3-540-21431-1

J. Błaz˙ewicz, K. Ecker, E. Pesch, G. Schmidt and J. Weglarz (Eds.) Handbook on Scheduling ISBN 978-3-540-28046-0 F. Burstein and C.W. Holsapple (Eds.) Handbook on Decision Support Systems 1 ISBN 978-3-540-48712-8 F. Burstein and C.W. Holsapple (Eds.) Handbook on Decision Support Systems 2 ISBN 978-3-540-48715-9 D. Seese, Ch. Weinhardt and F. Schlottmann (Eds.) Handbook on Information Technology in Finance ISBN 978-3-540-49486-7 T.C. Edwin Cheng and Tsan-Ming Choi (Eds.) Innovative Quick Response Programs in Logistics and Supply Chain Management ISBN 978-3-642-04312-3 J. vom Brocke and M. Rosemann (Eds.) Handbook on Business Process Management 1 ISBN 978-3-642-00415-5 Handbook on Business Process Management 2 ISBN 978-3-642-01981-4 T.-M. Choi and T.C. Edwin Cheng Supply Chain Coordination under Uncertainty ISBN 978-3-642-19256-2

Tsan-Ming Choi

l

T.C. Edwin Cheng

Editors

Supply Chain Coordination under Uncertainty

Editors Tsan-Ming Choi The Hong Kong Polytechnic University Business Division, Institute of Textiles and Clothing Hung Hom, Kowloon Hong Kong SAR [email protected]

T.C. Edwin Cheng The Hong Kong Polytechnic University Department of Logistics and Maritime Studies Hung Hom, Kowloon Hong Kong SAR [email protected]

ISBN 978-3-642-19256-2 e-ISBN 978-3-642-19257-9 DOI 10.1007/978-3-642-19257-9 Springer Heidelberg Dordrecht London New York Library of Congress Control Number: 2011935633 # Springer-Verlag Berlin Heidelberg 2011 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

Preface

Channel coordination is a core subject of supply chain management. It is well-known that a stochastic multi-echelon supply chain system usually fails to be optimal owing to the presence of the bullwhip effect and the double marginalization issue. Motivated by the importance of the topic, over the past decade, much research effort has been devoted to exploring the detailed mechanisms (such as incentive alignment schemes) for achieving supply chain coordination under uncertainty and has generated many fruitful analytical and empirical results. Despite the abundance of research results, there is an absence of a comprehensive reference source that provides state-of-the-art findings on both theoretical and applied research on the subject “under one roof”. In addition, many new topics and innovative measures for supply chain coordination under uncertainty have appeared in recent years and many new challenges have emerged. As a result, we believe it is significant to put together all these interesting works and the respective insights into an edited volume. In view of the above, we co-edit this Springer handbook. The handbook contains five parts, covering (1) introductory materials and review of supply chain coordination; (2) analytical models for innovative coordination under uncertainty; (3) channel power, bargaining, and coordination; (4) technological advancements and applications in coordination; and (5) empirical analysis and case studies. The specific topics covered include the following: – Coordination of Supply Chains with Risk-Averse Agents – A Timely Review on Supply Chain Coordination – A Review of Control Policies for Multi-Echelon Inventory Systems with Stochastic Demand – Supply Chain Models with Active Acquisition and Remanufacturing – Facilitating Demand Risk-sharing with the Innovative Percent Deviation Contract – Value-added Retailer in a Mixed Channel under Asymmetric Information – Capacity Management and Price Discrimination under Demand Uncertainty using Options – Dynamic Procurement and Quantity Discounts in Supply Chains – Coordination in a Multi-period Setting: The Additional Ordering Cost Contract

v

vi

Preface

– Use of Supply Chain Contract to Motivate Selling Effort – Price and Warranty Competition in a Duopoly Supply Chain – Supply Chain Coordination for Newsvendor-type Products with Two Ordering Opportunities – Bargaining in a Two-Stage Supply Chain through Revenue-Sharing Contract – Should a Stackelberg-dominated Supply-chain Player Help her Dominant Opponent to Obtain Better Information – Supply Chain Coordination under Demand Uncertainty using Credit Option – Supply Chain Coordination under Consignment Contract – A Heuristic Approach for Collaborative Planning in Detailed Scheduling – RFID Technology Adoption and Supply Chain Coordination – Possibilistic Mixed Integer Programming Approach for Supply Chain Network Problems – Coordination of Converging Material Flows in Supply Chains under Uncertainty – Bioenergy Systems and Supply Chains in Europe: Conditions, Capacity, and Coordination – Benefits of Involving Contract Manufacturers in Collaborative Planning for Three-Echelon Supply Networks – A Capability-Based Approach for Managing IT Suppliers – Methodology for Assessing Collaboration Strategies and Incentives in the Pulp and Paper Industry We are very pleased to see that this research handbook has generated a lot of new analytical and empirical results with precious insights, which will not only help supply chain agents to understand more about the latest measures for supply chain coordination under uncertainty, but also help practitioners and researchers to know how to improve supply chain performance based on innovative methods. This will be especially meaningful to industries such as fashion apparel and consumer electronics, in which effective supply chain management has been known to be the key to success. We would like to take this opportunity to show our gratitude to Werner A. Mueller and Christian Rauscher for their kind support and advice along the course of carrying out this project. We sincerely thank all the authors who have contributed their decent research to this handbook. We are grateful to the professional reviewers who reviewed the submitted papers and provided us with timely comments and constructive recommendations. We are indebted to our student Pui-Sze Chow for her editorial assistance. We also acknowledge the funding support of the Research Grants Council of Hong Kong under grant number PolyU 5143/07E (General Research Fund) and The Hong Kong Polytechnic University under grant number J-BB6U. Last but not least, we are grateful to our families, colleagues, friends, and students, who have been supporting us during the development of this important research handbook. Tsan-Ming Choi, T.C.E. Cheng The Hong Kong Polytechnic University

Contents

Part I

Introduction and Review

Coordination of Supply Chains with Risk-Averse Agents . . . . . . . . . . . . . . . . . . 3 Xianghua Gan, Suresh P. Sethi, and Houmin Yan Addendum to “Coordination of Supply Chains with Risk-Averse Agents” by Gan, Sethi, and Yan (2004) . . . . . . . . . . . . . . . . . 33 Xianghua Gan, Suresh P. Sethi, and Houmin Yan A Review on Supply Chain Coordination: Coordination Mechanisms, Managing Uncertainty and Research Directions . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 Kaur Arshinder, Arun Kanda, and S.G. Deshmukh Control Policies for Multi-echelon Inventory Systems with Stochastic Demand . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 Qinan Wang Supply Chain Models with Active Acquisition and Remanufacturing . . . 109 Xiang Li and Yongjian Li Part II

Analytical Models for Innovative Coordination under Uncertainty

Facilitating Demand Risk-Sharing with the Percent Deviation Contract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 Matthew J. Drake and Julie L. Swann

vii

viii

Contents

Value-Added Retailer in a Mixed Channel: Asymmetric Information and Contract Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 Samar K. Mukhopadhyay, Xiaowei Zhu, and Xiaohang Yue Capacity Management and Price Discrimination under Demand Uncertainty Using Option Contracts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 Fang Fang and Andrew Whinston Dynamic Procurement, Quantity Discounts, and Supply Chain Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 Feryal Erhun, Pinar Keskinocak, and Sridhar Tayur Coordination of the Supplier–Retailer Relationship in a Multi-period Setting: The Additional Ordering Cost Contract . . . . . 235 Nicola Bellantuono, Ilaria Giannoccaro, and Pierpaolo Pontrandolfo Use of Supply Chain Contract to Motivate Selling Effort . . . . . . . . . . . . . . . . 255 Samar K. Mukhopadhyay and Xuemei Su Price and Warranty Competition in a Duopoly Supply Chain . . . . . . . . . . . 281 Santanu Sinha and S.P. Sarmah Supply Chain Coordination for Newsvendor-Type Products with Two Ordering Opportunities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315 Yong-Wu Zhou and Sheng-Dong Wang Part III

Channel Power, Bargaining and Coordination

Bargaining in a Two-Stage Supply Chain Through Revenue-Sharing Contract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347 Jing Hou and Amy Z. Zeng Should a Stackelberg-Dominated Supply-Chain Player Help Her Dominant Opponent to Obtain Better System-Parameter Knowledge? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379 Jian-Cai Wang, Amy Hing-Ling Lau, and Hon-Shiang Lau Supply Chain Coordination Under Demand Uncertainty Using Credit Option . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403 S. Kamal Chaharsooghi and Jafar Heydari Supply Chain Coordination Under Consignment Contract with Revenue Sharing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 427 Sijie Li, Jia Shu, and Lindu Zhao

Contents

Part IV

ix

Technological Advancements and Applications in Supply Chain Coordination

DEAL: A Heuristic Approach for Collaborative Planning in Detailed Scheduling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 457 J. Benedikt Scheckenbach Inventory Record Inaccuracy, RFID Technology Adoption and Supply Chain Coordination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483 H. Sebastian Heese Possibilistic Mixed Integer Linear Programming Approach for Production Allocation and Distribution Supply Chain Network Problem in the Consumer Goods Industry . . . . . . . . . . . . . . . . . . . . . . 505 Bilge Bilgen Coordination of Converging Material Flows Under Conditions of Uncertainty in Supply Chains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 525 Liesje De Boeck and Nico Vandaele Part V

Empirical Analysis and Case Studies

Bioenergy Systems and Supply Chains in Europe: Conditions, Capacity and Coordination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 545 Kes McCormick Three Is a Crowd? On the Benefits of Involving Contract Manufacturers in Collaborative Planning for Three-Echelon Supply Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563 Henk Akkermans, Kim van Oorschot, and Winfried Peeters Managing IT Suppliers: A Capability-Based Approach . . . . . . . . . . . . . . . . . . 599 Carlos Brito and Mafalda Nogueira Methodology for Assessing Collaboration Strategies and Incentives in the Pulp and Paper Industry . . . . . . . . . . . . . . . . . . . . . . . . . . . . 625 Nadia Lehoux, Sophie D’Amours, and Andre´ Langevin Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 651

.

Contributors

Henk Akkermans Supply Network Dynamics, Department of Information Management, Tilburg University, Warandelaan 2, P.O. Box 90153, 5000 LE, Tilburg, The Netherlands, [email protected] Nicola Bellantuono Dipartimento di Ingegneria dell’Ambiente e per lo Sviluppo Sostenibile, Politecnico di Bari, via De Gasperi s.n, 74100 Taranto, Italy, [email protected] Bilge Bilgen Department of Industrial Engineering, Dokuz Eylul University, 35160 Izmir, Turkey, [email protected] Carlos Brito Faculty of Economics, University of Porto, Rua Roberto Frias, 4200-464 Porto, Portugal, [email protected] S. Kamal Chaharsooghi Industrial Engineering Department, Tarbiat Modares University, Tehran, Iran, [email protected] Sophie D’Amours FORAC, Department of Mechanical Engineering, Pavillon Adrien-Pouliot, Universite´ Laval, Que´bec, Canada, G1V 0A6, sophie. [email protected] Liesje De Boeck Centre for Modeling and Simulation, HUBrussel, Stormstraat 2, 1000 Brussels, Belgium; Research Centre for Operations Management, K.U.Leuven, Naamsestraat 69, 3000 Leuven, Belgium, [email protected] S.G. Deshmukh Department of Mechanical Engineering, Indian Institute of Technology Delhi, New Delhi, 110016, India, [email protected] Matthew J. Drake Palumbo-Donahue Schools of Business, Duquesne University, Pittsburgh, PA 15282, USA, [email protected]

xi

xii

Contributors

Feryal Erhun Department of Management Science and Engineering, Stanford University, Stanford, CA, USA, [email protected] Fang Fang Department of ISOM, College of Business Administration, California State University at San Marcos, 333 S. Twin Oaks Valley Road, San Marcos, CA 92096, USA, [email protected] Xianghua Gan Department of Logistics and Maritime Studies, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong, China, [email protected] edu.hk Ilaria Giannoccaro Dipartimento di Ingegneria Meccanica e Gestionale, Politecnico di Bari, viale Japigia 182, 70125 Bari, Italy, [email protected] H. Sebastian Heese Kelley School of Business, Indiana University, 1309 East Tenth Street, Bloomington, IN 47405, USA, [email protected] Jafar Heydari Industrial Engineering Department, Shiraz University of Technology, Shiraz, Iran, [email protected] Jing Hou Business School, Hohai University, Nanjing, Jiangsu 211100, China, [email protected] hotmail.com Arun Kanda Department of Mechanical Engineering, Indian Institute of Technology Delhi, New Delhi 110016, India, [email protected] Arshinder Kaur Department of Management Studies, Indian Institute of Technology Madras, Chennai 600036, India, [email protected] Pinar Keskinocak School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, GA, USA, [email protected] Andre´ Langevin CIRRELT, Department of Mathematics and Industrial Engineering, E´cole Polytechnique de Montre´al, C.P. 6079, succ. Centre-ville, Montre´al, Canada, H3C 3A, [email protected] Amy Hing Ling Lau School of Business, University of Hong Kong, Pokfulam, Hong Kong, [email protected] Hon-Shiang Lau Department of Management Sciences, City University of Hong Kong, Kowloon Tong, Hong Kong, [email protected] Nadia Lehoux FORAC, Department of Mechanical Engineering, Pavillon AdrienPouliot, Universite´ Laval, Que´bec, Canada, G1V 0A6, [email protected]

Contributors

xiii

Sijie Li Institute of Systems and Engineering, Southeast University, Nanjing, Jiangsu People’s Republic of China, [email protected] Xiang Li Research Centre of Logistics, College of Economic and Social Development, Nankai University, Tianjian 300071, P.R. China, [email protected] Yongjian Li Business School, Nankai University, Tianjin 300071, P.R. China, [email protected] Kes McCormick International Institute for Industrial Environmental Economics (IIIEE), Lund University, Lund, Sweden, [email protected] Samar K. Mukhopadhyay Graduate School of Business, Sungkyunkwan University, Jongno-Gu, Seoul 110–745, South Korea, [email protected] Mafalda Nogueira Management School, Lancaster University, Lancaster, LA1-4YX, UK, [email protected] Winfried Peeters BU HPMS, NXP Semiconductors, High Tech Campus 60, 5656 AG Eindhoven, The Netherlands, [email protected] Pierpaolo Pontrandolfo Dipartimento di Ingegneria dell’Ambiente e per lo Sviluppo Sostenibile, Politecnico di Bari, via De Gasperi s.n, 74100 Taranto, Italy, [email protected] S.P. Sarmah Department of Industrial Engineering and Management, Indian Institute of Technology, Kharagpur 721302, India, [email protected] J. Benedikt Scheckenbach [email protected]

Cranachstr. 16, 50733 Koeln, Germany, benedikt.

Suresh P. Sethi School of Management, SM30, The University of Texas at Dallas, 800W Campbell Road, Richardson, TX 75080-3021, USA, [email protected] Jia Shu Department of Management Science and Engineering, School of Economics and Management, Southeast University, Nanjing, Jiangsu P.R. China, [email protected] Santanu Sinha Complex Decision Support Systems, Tata Consultancy Services, Akruti Trade Centre, MIDC, Andheri (E), Mumbai 400093, India, [email protected] yahoo.com Xuemei Su College of Business Administration, California State University Long Beach, 1250 Bellflower Blvd, Long Beach, CA 90840, USA, [email protected]

xiv

Contributors

Julie L. Swann H. Milton Stewart School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, GA 30332-0205, USA, [email protected] gatech.edu Sridhar Tayur Tepper School of Business, Carnegie Mellon University, Pittsburgh, PA, USA, [email protected] Kim van Oorschot Department of Leadership and Organizational Behaviour, BI Norwegian School of Business, NO-0442 Oslo, Norway, [email protected] Nico Vandaele Research Centre for Operations Management, K.U.Leuven, Naamsestraat 69, 3000 Leuven, Belgium; Faculty of Business and Economics, K.U. Leuven-Campus Kortrijk, Etienne Sabbelaan 53-bus 0000, 8500 Kortrijk, Belgium, [email protected] Jian-Cai Wang School of Business, University of Hong Kong, Pokfulam, Hong Kong; School of Management and Economics, Beijing Institute of Technology, Beijing, China, [email protected] Qinan Wang Nanyang Business School, Nanyang Technological University, Singapore, Singapore 639798, [email protected] Sheng-Dong Wang Department of Mathematics, Hefei Electronic Engineering Institute, Hefei, Anhui, P.R. China, [email protected] Andrew B. Whinston Department of IROM, McCombs School of Business, The University of Texas at Austin, 1 University Station B6000, Austin, TX 78712, USA, [email protected] Houmin Yan Department of Systems Engineering and Engineering Management, The Chinese University of Hong Kong, Shatin, NT, Hong Kong, [email protected] Xiaohang Yue Sheldon B. Lubar School of Business, University of WisconsinMilwaukee, P.O. Box 742, Milwaukee, WI 53201, USA, [email protected] Amy Z. Zeng School of Business, Worcester Polytechnic Institute, Worcester, MA 01609, USA, [email protected] Lindu Zhao Institute of Systems and Engineering, Southeast University, Nanjing, Jiangsu People’s Republic of China, [email protected] Yong-Wu Zhou School of Business Administration, South China University of Technology, Guangzhou, Guangdong, P.R. China, [email protected] Xiaowei Zhu College of Business and Public Affairs, West Chester University of Pennsylvania, West Chester, PA 19383, USA, [email protected]

.

Part I

Introduction and Review

.

Coordination of Supply Chains with Risk-Averse Agents Xianghua Gan, Suresh P. Sethi, and Houmin Yan

Abstract The extant supply chain management literature has not addressed the issue of coordination in supply chains involving risk-averse agents. We take up this issue and begin with defining a coordinating contract as one that results in a Paretooptimal solution acceptable to each agent. Our definition generalizes the standard one in the risk-neutral case. We then develop coordinating contracts in three specific cases (1) the supplier is risk neutral and the retailer maximizes his expected profit subject to a downside risk constraint, (2) the supplier and the retailer each maximizes his own mean-variance trade-off, and (3) the supplier and the retailer each maximizes his own expected utility. Moreover, in case (3) we show that our contract yields the Nash Bargaining solution. In each case, we show how we can find the set of Pareto-optimal solutions, and then design a contract to achieve the solutions. We also exhibit a case in which we obtain Pareto-optimal sharing rules explicitly, and outline a procedure to obtain Pareto-optimal solutions. Keywords Capacity • Coordination • Nash bargaining • Pareto-optimality • Risk averse • Supply chain management

X. Gan (*) Department of Logistics and Maritime Studies, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong e-mail: [email protected] S.P. Sethi School of Management, SM30, The University of Texas at Dallas, 800W. Campbell Road, Richardson, TX 75080-3021, USA e-mail: [email protected] H. Yan Department of Systems Engineering and Engineering Management, The Chinese University of Hong Kong, Shatin, NT, Hong Kong e-mail: [email protected] T.-M. Choi and T.C. Edwin Cheng (eds.), Supply Chain Coordination under Uncertainty, International Handbooks on Information Systems, DOI 10.1007/978-3-642-19257-9_1, # Springer-Verlag Berlin Heidelberg 2011

3

4

X. Gan et al.

1 Introduction Much of the research on decision making in a supply chain has assumed that the agents in the supply chain are risk neutral, i.e., they maximize their respective expected profits. An important focus of this research has been the design of supply contracts that coordinate the supply chain. When each of the agents maximizes his expected profit, the objective of the supply chain considered as a single entity is unambiguously to maximize its total expected profit. This fact alone makes it natural to define a supply chain to be coordinated if the chain’s expected profit is maximized and each agent’s reservation profit is met. A similar argument holds if each agent’s objective is to minimize his expected cost. In this paper we consider supply chains with risk-averse agents. Simply put, an agent is risk averse if the agent prefers a certain profit p to a risky profit, whose expected value equals p. In the literature, there are many measures of risk aversion; see Szeg€o (2004) for examples. Regardless of the measure used, when one or more agents in the supply chain are risk averse, it is no longer obvious as to what the objective function of the supply chain entity should be. Not surprisingly, the issue of coordination of supply chain consisting of risk-averse agents has not been studied in the supply chain management literature. That is not to say that the literature does not realize the importance of the risk-averse criteria. Indeed, there are a number of papers devoted to the study of inventory decisions of a single riskaverse agent. These include Lau (1980), Bouakiz and Sobel (1992), Eeckhoudt et al. (1995), Chen and Federgruen (2000), Agrawal and Seshadri (2000a), Buzacott et al. (2002), Chen et al. (2007), and Gaur and Seshadri (2005). There also have been a few studies of supply chains consisting of one or more risk-averse agents. Lau and Lau (1999) and Tsay (2002) consider decision making by a risk-averse supplier and a risk-averse retailer constituting a supply chain. Agrawal and Seshadri (2000b) introduce a risk-neutral intermediary to make ordering decisions for risk-averse retailers, whose respective profits are side payments from the intermediary. Van Mieghem (2003) has reviewed the literature that incorporates risk aversion in capacity investment decisions. While these papers consider risk-averse decision makers by themselves or as agents in a supply chain, they do not deal with the issue of the supply chain coordination involving risk-averse agents. It is this issue of coordination of supply chains consisting of one or more riskaverse agents that is the focus of this paper. That many decision makers are riskaverse has been amply documented in the finance and economics literature; see, for example, Van Neumann and Morgenstern (1944), Markowitz (1959), Jorion (2006), and Szeg€o (2004). We shall therefore develop the concept of what we mean by coordination of a supply chain, and then design explicit contracts that achieve the defined coordination. For this purpose we use the Pareto-optimality criterion, used widely in the group decision theory, to evaluate a supply chain’s performance. We define each agent’s payoff to be a real-valued function of a random variable representing his profit, and propose that a supply chain can be treated as coordinated if no agent’s payoff can be

Coordination of Supply Chains with Risk-Averse Agents

5

improved without impairing someone else’s payoff and each agent receives at least his reservation payoff. We consider three specific cases of a supply chain (1) the supplier is risk neutral and the retailer maximizes his expected profit subject to a downside risk constraint, (2) the supplier and the retailer each maximizes his own mean-variance trade-off, and (3) the supplier and the retailer each maximizes his own expected utility. We show how we can coordinate the supply chain in each case according to our definition. In each case we do this by finding the set of Paretooptimal solutions acceptable to each agent, and then constructing a flexible contract that can attain any of these solutions. Moreover, the concept we develop and the contracts we obtain generalize the same known for supply chains involving risk-neutral agents. The remainder of the paper is organized as the follows. In Sect. 2 we review the related literature in supply chain management and group decision theory. In Sect. 3 we introduce a definition of coordination of a supply chain consisting of risk-averse agents. In Sect. 4 we characterize the Pareto-optimal solutions and find coordinating contracts for the supply chains listed as the first two cases. In Sect. 5 we first take up the third case using exponential utility functions for the agents, and design coordinating contracts as well as obtain the Nash Bargaining solution. Then we examine a case in which the supplier has an exponential utility followed by a linear utility. Section 6 provides a discussion of our results. The paper concludes in Sect. 7 with suggestions for future research.

2 Literature Review There is a considerable literature devoted to contracts that coordinate a supply chain involving risk-neutral agents. This literature has been surveyed by Cachon (2003). In addition, the book by Tayur et al. (1999) contains a number of chapters addressing supply contracts. In light of these, we limit ourselves to reviewing papers studying inventory and supply chain decisions by risk-averse agents. First we review papers dealing with a single risk-averse agent’s optimal inventory decision. Then we review articles dealing with decision making by risk-averse agents in a supply chain. Chen and Federgruen (2000) re-visit a number of basic inventory models using a mean-variance approach. They exhibit how a systematic mean-variance trade-off analysis can be carried out efficiently, and how the resulting strategies differ from those obtained in the standard analyses. Agrawal and Seshadri (2000a) consider how a risk-averse retailer, whose utility function is increasing and concave in wealth, chooses the order quantity and the selling price in a single-period inventory model. They consider two different ways in which the price affects the distribution of demand. In the first model, they assume that a change in the price affects the scale of the distribution. In the second model, a change in the price only affects the location of the distribution. They show that in comparison to a risk-neutral retailer, a risk-averse retailer will charge a higher price

6

X. Gan et al.

and order less in the first model, whereas he will charge a lower price in the second model. Buzacott et al. (2002) model a commitment and option contract for a risk-averse newsvendor with a mean-variance objective. The contract, also known as a takeor-pay contract, belongs to a class of volume flexible contracts, where the newsvendor reserves a capacity with initial information and adjusts the purchase at a later stage when some new information becomes available. They compare the performance of strategies developed for risk-averse and risk-neutral objectives. They conclude that the risk-averse objective can be an effective approach when the quality of information revision is not high. Their study indicates that it is possible to reduce the risk (measured by the variance of the profit) by six- to eightfold, while the loss in the expected profit is almost invisible. On the other hand, the strategy developed for the expected profit objective can only be considered when the quality of information revision is high. They show furthermore that these findings continue to hold in the expected utility framework. The paper points out a need for modeling approaches that deal with downside risk considerations. Lau and Lau (1999) study a supply chain consisting of a monopolistic supplier and a retailer. The supplier and the retailer employ a return policy, and each of them has a mean-variance objective function. Lau and Lau obtain the optimal wholesale price and return credit for the supplier to maximize his utility. However, they do not consider the issue of improving the supply chain’s performance, i.e., improving both players’ utilities. Agrawal and Seshadri (2000b) consider a single-period model in which multiple risk-averse retailers purchase a single product from a common supplier. They introduce a risk neutral intermediary into the channel, who purchases goods from the vendor and sells them to the retailers. They demonstrate that the intermediary, referred to as the distributor, orders the optimal newsvendor quantity from the supplier and offers a menu of mutually beneficial contracts to the retailers. In every contract in the menu, the retailer receives a fixed side payment, while the distributor is responsible for the ordering decisions of the retailers and receives all their revenues. The menu of contracts simultaneously (1) induces every risk-averse agent to select a unique contract from it; (2) maximizes the distributor’s profit; and (3) raises the order quantities of the retailers to the expected value maximizing (newsvendor) quantities. Tsay (2002) studies how risk aversion affects both sides of the supplier–retailer relationship under various scenario of relative strategic power, and how these dynamics are altered by the introduction of a return policy. The sequence of play is as follows: first the supplier announces a return policy, and then the retailer chooses order quantity without knowing the demand. After observing the demand, the retailer chooses the price and executes on any relevant terms of the distribution policy as appropriate (e.g., returning any overstock as allowed). Tsay shows that the behavior under risk aversion is qualitatively different from that under risk neutrality. He also show that the penalty for errors in estimating a channel partner’s risk aversion can be substantial.

Coordination of Supply Chains with Risk-Averse Agents

7

In a companion paper (Gan et al. 2005), we examine coordinating contracts for a supply chain consisting of one risk-neutral supplier and one risk-averse retailer. There we design an easy-to-implement risk-sharing contract that accomplishes the coordination as defined in this paper. Among these supply chain papers, Lau and Lau (1999) and Tsay (2002) consider the situation in which both the retailer and the supplier in the channel are risk averse. However, neither considers the issue of the Pareto-optimality of the actions of the agents. The aim of Agrawal and Seshadri (2000b) is to design a contract that increases the channel’s order quantity to the optimal level in the risk-neutral case by having the risk-neutral agent assume all the risk. Once again, they do not mention the Pareto-optimality aspect of the decision they obtain. Finally since our definition of coordination is based on the concepts used in the group decision theory, we briefly review this stream of literature. From the early fifties to the early eighties, a number of papers and books appeared that deal with situations in which a group faces intertwined external and internal problems. The external problem involves the choice of an action to be taken by the group, and the internal problem involves the distribution of the group payoff among the members. Arrow (1951) conducted one of the earliest studies on the group decision theory, and showed that given an ordering of consequences by a number of individuals, no group ordering of these consequences exists that satisfies a set of seemingly reasonable behavioral assumptions. Harsanyi (1955) presented conditions under which the total group utility can be expressed as a linear combination of individuals’ cardinal utilities. Wilson (1968) used Pareto-optimality as the decision criterion and constructed a group utility function to find Pareto-optimal solutions. Raiffa (1970) illustrates the criterion of Pareto-optimality quite lucidly, and discusses how to choose a Pareto-optimal solution in bargaining and arbitration problems. LaValle (1978) uses an allocation function to define Pareto-optimality. Eliashberg and Winkler (1981) investigate properties of sharing rules and the group utility functions in additive and multilinear cases.

3 Definition of Coordination of a Supply Chain with Risk-Neutral or Risk-Averse Agents In this section we define coordination of a supply chain consisting of agents that are risk neutral or risk averse. We use concepts developed in group decision theory that deals with situations in which a group faces intertwined external and internal problems. The external problem involves the choice of an action to be taken by the group, and the internal problem involves the distribution of the group payoff among the members. In group decision problems, a joint action of the group members is said to be Pareto-optimal if there does not exist an alternative action that is at least as acceptable to all and definitely preferred by some. In other words, a joint action is Pareto-optimal if it is not possible to make one agent better off without making

8

X. Gan et al.

another one worse off. We call the collection of all Pareto-optimal actions as the Pareto-optimal set. It would not be reasonable for the group of agents to choose a joint action that is not Pareto-optimal. Raiffa (1970) and LaValle (1978) illustrate this idea quite lucidly with a series of examples. A supply chain problem is obviously a group decision problem. The channel faces an external problem and an internal problem. External problems include decisions regarding order/production quantities, item prices, etc. The internal problem is to allocate profit by setting the wholesale price, deciding the amount of a side payment if any, refund on the returned units, etc. Naturally, we can adopt the Pareto-optimality criterion of the group decision theory for making decisions in a supply chain. Indeed, in the risk-neutral case, the optimal action under a coordinating contract is clearly Pareto-optimal. In general, since the agents in the channel would not choose an action that is not in the Pareto-optimal set, the first step to coordinate a channel is to characterize the set. Following the ideas of Raiffa (1970) and LaValle (1978), we formalize below the definition of Pareto-optimality. Let (O; F ; P) denote the probability space and N denote the number of agents in the supply chain, N r2. Let Si be the external action space of agent i; i ¼ 1; . . . ; N, and S ¼ S1 SN . For any given external joint action s ¼ ðs1 ; . . . ; sN Þ 2 S, the channel’s total profit is a random variable Pðs; oÞ; o 2 O. Let E and V denote the expectation and variance defined on (O; F ; P), respectively. Now we define a sharing rule that governs the splitting of the channel profit among the agents. Let Y be the set of all functions from S O to RN . P Definition 1. A function uðs; vÞ 2 Q is called a sharing rule if i ui ðs; vÞ ¼ 1 almost surely. Under the sharing rule uðs; oÞ, agent i’s profit is represented by Pi ðs; v; uðs; vÞÞ ¼ ui ðs; vÞPðs; vÞ; i ¼ 1; . . . ; N: Often, when there is no confusion, we write Pðs; vÞ simply as PðsÞ, uðs; vÞ as uðsÞ, and Pi ðs; v; uðs; vÞÞ as Pi ðs; uðsÞÞ. A supply chain’s external problem is to choose an s 2 S and its internal problem is to choose a function uðsÞ 2 Y. Thus the channel’s total problem is to choose a pair ðs; uðsÞÞ 2 S Y. Now we define the preferences of the agents over their random profits. Let G denote the space of all random variables defined on ðO; F ;PÞ. For X; X0 2 G, the agent i’s preference will be denoted by a real-valued payoff function ui ðÞ defined on G. The relation ui ðXÞ>ui ðX0 Þ, ui ðXÞ0. Example 2. Assume that agent i maximizes his expected profit under the constraint that the probability of his profit being less than his target profit level a does not exceed a given level b; 0

Coordination of Supply Chains with Risk-Averse Agents

ui ðXÞ ¼

9

EðXÞ; if PðXbaÞbb; 1; if PðXbaÞ>b:

Example 3. Suppose agent i has a concave increasing utility function gi : R1 ! R1 of wealth and wants to maximize his expected utility. Then the agent’s payoff function is ui ðXÞ ¼ E½gi ðXÞ; X 2 G. Remark 1. In Raiffa (1970) and LaValle (1978), each agent is assumed to have a cardinal utility function of profit, and his objective is to maximize his expected utility. However, some preferences, such as the one in Example 2, cannot be represented by a cardinal utility function. A point a 2 RN is said to be Pareto-inferior to or Pareto-dominated by another point b 2 RN , if each component of a is no greater than the corresponding component of b and at least one component of a is less than the corresponding component of b. In other words, we say b is Pareto-superior to a or b Pareto-dominates a. A point is said to be a Pareto-optimal point of a subset of RN , if it is not Paretoinferior to any other point in the subset. With these concepts, we can now define Pareto-optimality of a sharing rule uðsÞ and an action pair ðs; uðsÞÞ. Definition 2. Given an external action s of the supply chain, u ðsÞ is a Paretooptimal sharing rule, if ðu1 ðP1 ðs; u ðsÞÞÞ; ; uN ðPN ðs; u ðsÞÞÞÞ is a Pareto-optimal point of the set fðu1 ðP1 ðs; uðsÞÞÞ; ; uN ðPN ðs; uðsÞÞÞÞ; u 2 Yg; where ui ðPi ðs; uðsÞÞÞ is the payoff of the ith agent. Definition 3. ðs ; u ðs ÞÞ is a Pareto-optimal action pair if the agents’ payoffs ðu1 ðP1 ðs ; u ðs ÞÞÞ; ; uN ðPN ðs ; u ðs ÞÞÞÞ is a Pareto-optimal point of the set fðu1 ðP1 ðs; uðsÞÞÞ; ; uN ðPN ðs; uðsÞÞÞÞ; ðs; uðsÞÞ 2 S Yg: Clearly if ðs ; u ðs ÞÞ is a Pareto-optimal action pair, then u ðs Þ is a Paretooptimal sharing rule given s . We begin now with an examination of the Pareto-optimal set in a supply chain consisting of risk-neutral agents. If an external action maximizes the supply chain’s expected profit, then it is not possible to make one agent get more expected profit without making another agent get less. More specifically, we have the following proposition.

10

X. Gan et al.

Proposition 1. If the agents in a supply chain are all risk neutral, then an action pair ðs; uðsÞÞ is Pareto-optimal if and only if the channel’s external action s maximizes the channel’s expected profit. Proof. The proof follows from the fact that in the risk-neutral case, for each s, X

ui ðPi ðs; uðsÞÞÞ ¼

X

EPi ðs; uðsÞÞ ¼ E

X

Pi ðs; uðsÞÞ ¼ EPðsÞ:

Thus, every ðs ; uðs ÞÞ 2 S Y is Pareto-optimal provided s maximizes EPðs Þ. □ Since agents in a supply chain maximize their respective objectives, the agents’ payoffs might not be Pareto-optimal if their objectives are not aligned properly. In this case, it is possible to improve the chain’s performance, i.e., achieve Paretosuperior payoffs. The agents can enter into an appropriately designed contract, under which their respective optimizing actions leads to a Pareto-superior payoff. In the supply chain management literature, a contract is defined to coordinate a supply chain consisting of risk-neutral agents if their respective optimizing external actions under the contract maximize the chain’s expected profit. Then, according to Proposition 1, a coordinating contract is equivalent to a Pareto-optimal action in the riskneutral case. It is therefore reasonable to use the notion of Pareto-optimality to define supply chain coordination in the general case. Definition 4. Supply Chain Coordination. A contract agreed upon by the agents of a supply chain is said to coordinate the supply chain if the optimizing actions of the agents under the contract 1. Satisfy each agent’s reservation payoff constraint. 2. Lead to an action pair ðs ; u ðs ÞÞ that is Pareto-optimal. Besides Pareto-optimality of a contract, we have introduced the individualrationality or the participation constraints as part of the definition of coordination. The constraints ensure that each agent is willing to participate in the contract by requiring that each gets at least his reservation payoff. It is clear that each agent’s reservation payoff will not be less than his status-quo payoff, which is defined to be his best payoff in the absence of the contract. Thus, we need consider only the subset of Pareto-optimal actions that satisfy these participating constraints. The reservation payoff of an agent plays an important role in bargaining, as we shall see in the next section. Now we illustrate the introduced concept of coordination by an example. Example 4. Consider a supply chain consisting of one supplier and one retailer who faces a newsvendor problem. Before the demand realizes, the supplier decides on his capacity first, and the retailer then prices the product and chooses an order quantity. The supplier and the retailer may enter into a contract that specifies the retailer’s committed order quantity and the supplier’s refund policy for returned items. In this channel, the external actions are the supplier’s capacity selection and the retailer’s pricing and ordering decisions. These are denoted as s. The internal

Coordination of Supply Chains with Risk-Averse Agents

11

actions include decision on the quantity of commitment, the refundable quantity, and the refund credit per item. These internal actions together lead to a sharing rule denoted by uðsÞ. Once the contract parameters are determined, the agents in the supply chain choose their respective external actions that maximize their respective payoffs. If ðs; uðsÞÞ satisfies the agents’ reservation payoffs and is Pareto-optimal, then the channel is coordinated by the contract. The definition of coordination proposed here allows agents to have any kind of preference that can be represented by a payoff function satisfying the complete and transitive axioms specified earlier. For example, all of the seven kinds of preferences listed in Schweitzer and Cachon (2000), including risk-seeking preferences, are allowed. Since often in practice, an agent is either risk neutral or risk averse, we restrict our attention to only these two types. Remark 2. Our definition applies also to a T-period case. For this, we define the payoff function of player i as ui ðP1i ðs ; u ðs ÞÞ; P2i ðs ; u ðs ÞÞ; ; PTi ðs ; u ðs ÞÞÞ : GT ! R1 ; where Pti ðs ; u ðs ÞÞ is agent i’s profit in period t.

4 Coordinating Supply Chains Each Pareto-optimal action pair ðs; uðsÞÞ results in a vector of payoffs ðu1 ðP1 ðs; uðsÞÞÞ; ; uN ðPN ðs; uðsÞÞÞÞ; where ui ðPi ðs; uðsÞÞÞ is the payoff of the ith agent. Let C ¼ fðu1 ðP1 ðs; uðsÞÞÞ; ; uN ðPN ðs; uðsÞÞÞÞjðs; uðsÞÞ is Pareto - optimal; ðs; uðsÞÞ 2 S Yg; denote the set of all Pareto-optimal payoffs, and let F C be the subset of Paretooptimal payoffs that satisfy all of the participation constraints. We shall refer to F as Pareto-optimal frontier. We will assume that F is not empty. To coordinate a supply chain, the first step is to obtain the Pareto-optimal frontier F. If F is not a singleton, then agents bargain to arrive at an element in F to which they agree. A coordinating contract is one with a specific set of parameters that achieves the selected solution. A contract is appealing if it has sufficient flexibility. In Cachon (2003), a coordinating contract is said to be flexible if the contract, by adjustment of some parameters, allows for any division of the supply chain’s expected profit among the risk-neutral agents. This concept can be extended to the general case as follows.

12

X. Gan et al.

Definition 5. A coordinating contract is flexible if, by adjustment of some parameters, the contract can lead to any point in F: We shall now develop coordinating contracts in supply chains consisting of two agents: a supplier and a retailer. We shall consider three different cases. In each of these cases, we assume that agents have complete information. In Case 1, the supplier is risk neutral and the retailer has a payoff function in Example 2, i.e., the retailer maximizes his expected profit subject to a downside constraint. In Case 2, the supplier and the retailer are both risk averse and each maximizes his own meanvariance trade-off. In Case 3, the supplier and the retailer are both risk averse and each maximizes his own expected concave utility. We consider the first two cases in this section and the third case in Sect. 5. In each case, let us denote the retailer’s and the supplier’s reservation payoffs as pr r0 and ps r0, respectively. We first obtain F and then design a flexible contract that can lead to any point in F by adjusting the parameters of the contract.

4.1

Case 1: Risk Neutral Supplier and Retailer Averse to Downside Risk

We consider the supplier to be risk neutral and the retailer to maximize his expected profit subject to a downside risk constraint. This downside risk constraint requires that the probability of the retailer’s profit to be higher than a specified level is not too small. The risk neutrality assumption on the part of the supplier is reasonable when he is able to diversify his risk by serving a number of independent retailers, which is quite often the case in practice. When the retailers are independent, the supply chain can be divided into a number of sub-chains, each consisting of one supplier and one retailer. This situation, therefore, could be studied as a supply chain consisting of one risk-neutral supplier and one risk-averse retailer. We say that an action pair ðs; uðsÞÞ is feasible if the pair satisfies the retailer’s downside risk constraint. We do not need to consider a pair ðs; uðsÞÞ that is not feasible since under the pair the retailer’s payoff is 1 and he would not enter the contract. We denote PðsÞ, Pr ðs; uðsÞÞ, and Ps ðs; uðsÞÞ as the profits of the supply chain, the retailer, and the supplier, respectively. Other quantities of interest will be subscripted in the same way throughout the chapter, i.e., subscript r will denote the retailer and subscript s will denote the supplier. Then we have the following result. Theorem 1. If the supplier is risk neutral and the retailer maximizes his expected profit subject to a downside risk constraint, then a feasible action pair ðs; uðsÞÞ is Pareto-optimal if and only if the supply chain’s expected profit is maximized over the feasible set. Proof. ONLY IF: It is sufficient to show that if EPðsÞ is not maximal over the feasible set, then ðs; uðsÞÞ is not Pareto-optimal.

Coordination of Supply Chains with Risk-Averse Agents

13

If EPðsÞ is not the maximal channel profit, then there exists an s0 such that EPðs0 Þ>EPðsÞ. Consider the pair ðs0 ; u0 ðs0 ÞÞ in which Pr ðs0 ; u0 ðs0 ÞÞ ¼ Pr ðs; uðsÞÞ and Ps ðs0 ; u0 ðs0 ÞÞ ¼ Pðs0 Þ Pr ðs; uðsÞÞ, we then get ur ðPr ðs0 ; u0 ðs0 ÞÞÞ ¼ EPr ðs; uðsÞ) and us ðPs ðs0 ; u0 ðs0 ÞÞÞ ¼ EPðs0 Þ EPr ðs; uðsÞÞ: We can see that ur ðPr ðs0 ; u0 ðs0 ÞÞÞ ¼ ur ðPr ðs; uðsÞÞ) and us ðPs ðs0 ; u0 ðs0 ÞÞÞ>us ðPs ðs; uðsÞÞÞ: This means that ðs; uðsÞÞ is Pareto-inferior to ðs0 ; u0 ðs0 ÞÞ, which contradicts with the Pareto-optimality of ðs; uðsÞÞ. IF: Suppose the supply chain’s expected profit is maximized. If ðs; uðsÞÞ is not Pareto-optimal, then according to the definition of a Pareto-optimal action pair, there exists a feasible pair ðs00 ; u0 ðs00 ÞÞ that is Pareto-superior to ðs; uÞ. Since it is Paretosuperior to ðs; uðsÞÞ, it is also feasible. Thus, us ðPs ðs00 ; u00 ðs00 ÞÞÞ þ ur ðPr ðs00 ; u0 ðs00 ÞÞÞ ¼ EPðs00 Þ>EPðsÞ ¼ ur ðPr ðs; uðsÞÞÞ þ us ðPs ðs; uðsÞÞÞ; which contradicts the fact that EPðsÞ is the maximal expected channel profit. □ Let s be an action of the channel that maximizes the channel’s expected profit, and let EPr ðs Þ be the retailer’s payoff. Since the retailer’s and the supplier’s reservation payoffs are pr and ps , respectively, we must impose the participating constraints of the agents on the solutions in C. Thus, EPr ðs Þrpr and EPðs Þ EPr ðs Þrps :

(1)

Together with Theorem 1, we get F ¼ fðEPr ðs Þ; EPðs Þ EPr ðs ÞÞjEPðs Þ ps rEPr ðs Þrpr g: Clearly, if EPðs Þ ps rpr , then F is not empty. In Gan et al. (2005), we show that a retailer, who is subject to a downside risk constraint, may order a lower quantity from a supplier than that desired by the channel under a wholesale, buy-back or revenue-sharing contract. Based on an initial contract, a risk-sharing contract is designed, which stipulates the supplier to offer a full refund on unsold items up to a limited quantity. The contract coordinates the supply chain, and requires that both the supplier and the retailer share the risk. Another coordinating contract is possible when EPr ðs Þ exceeds the retailer’s target profit a, where s is the channel’s optimal action. In this case, a contract that provides a payoff of EPr ðs Þ to the retailer and remainder to the supplier coordinates the supply chain. This contract is of two-part tariffs type as defined, for example, in Chopra and Meindl (2001, p. 160). However, if EPr ðs Þ is less than the retailer’s target profit a, then the contract does not work since the downside risk constraint of the retailer is not satisfied. But the risk-sharing contract in

14

X. Gan et al.

Gan et al. (2005) still works, since the retailer’s downside risk constraint P ðXbaÞbb is always satisfied under that contract.

4.2

Case 2: Mean-Variance Suppliers and Retailers

In this case, both the supplier and the retailer maximize their respective meanvariance trade-offs. First we consider a two-agent scenario and then extend it to the case of N agents. Let the retailer’s payoff function be EPr ðs; uðsÞÞ lr VðPr ðs; uðsÞÞÞ;

(2)

and the supplier’s payoff function be EPs ðs; uðsÞÞ ls VðPs ðs; uðsÞÞÞ:

(3)

We first find all Pareto-optimal sharing rules for any given channel’s external action s. We show that regardless of the selected external action s, the optimal sharing rule has the same specific form. Under this form of a sharing rule, we obtain optimal external actions. This procedure results in a Pareto-optimal ðs; uðsÞÞ. We now solve for the Pareto-optimal set for a supply chain consisting of N agents, and then specialize it for supply chains with two agents. We assume that the ith agent’s payoff function is EPi ðs; uðsÞÞ li VðPi ðs; uðsÞÞÞ:

(4)

To obtain Pareto-optimal sharing rules, we solve max u2Y

s.t: X

X

EPi ðs; uðsÞÞ

i

X

li VðPi ðs; uðsÞÞÞ;

(5)

i

Pi ðs; uðsÞÞ ¼ PðsÞ:

(6)

i

The solution of this problem is given in the following proposition. Proposition 2. A sharing rule u is a solution of the problem (5)–(6) if and only if 1=li PðsÞ þ pi ; i ¼ 1; . . . N; Pi ðs; uðsÞÞ ¼ P j 1==lj almost surely, where

P i

pi ¼ 0.

(7)

Coordination of Supply Chains with Risk-Averse Agents

Proof. Because

P i

15

EPi ðs; uðsÞÞ ¼ EPðsÞ, the problem is equivalent to X

min u2Y

s.t: X

li VðPi ðs; uðsÞÞÞ;

(8)

i

Pi ðs; uðsÞÞ ¼ PðsÞ:

(9)

i

It is easy to see that X

li VðPi ðs; uðsÞÞÞ

i

¼

X

h i X 2 E PðsÞ PðsÞ P ðs; uðsÞÞ i i j 1=lj

li VðPi ðs; uðsÞÞÞ þ P

i

" # X 1 1=li ¼P VðPðs; uðsÞÞÞ þ li V Pi ðs; uðsÞÞ P PðsÞ : j 1=lj j 1=lj i

(10)

Since the second term on the RHS of (10) is nonnegative, we have shown that X

1 VðPðs; uðsÞÞÞ j 1=lj

li VðPi ðs; uðsÞÞÞr P

i

(11)

for any feasible Pi ðs; uðsÞÞ; i ¼ 1; . . . N. Thus, P 11=l VðPðs; uðsÞÞÞ provides a j

j

lower bound for the objective function (8). Note that a u satisfies (7) if and only if 1=li PðsÞ ¼ 0; i ¼ 1; . . . N: Pi ðs; uðsÞÞ P j 1=lj This means that X

li VðPi ðs; uðsÞÞÞ ¼ P

i

1 VðPðsÞÞ j 1=lj

(12)

and X

1 VðPðsÞÞ j 1=lj

li VðPi ðs; uðsÞÞÞ> P

i

for any u not satisfying (7).

□

16

X. Gan et al.

For any optimal sharing rule u given in Proposition 2, the sum of the agents’ payoffs equals 1 VðPðsÞÞ: j 1=lj

EPðsÞ P

By adjusting p0 , the sharing rule u allows for any division of the total payoff among the agents. Therefore, an optimal external action, given u , has to maximize the total payoff, i.e., it must be an action pair of "

# 1 VðPðsÞÞ : max EPðsÞ P s2S j 1=lj

(13)

Next we characterize the set of Pareto-optimal actions by summarizing the results we have got. Theorem 2. An action pair ðs ; u Þ is Pareto-optimal if and only if "

#

1 VðPðsÞÞ s ¼ arg max EPðsÞ P s2S j 1=lj

(14)

1=li PðsÞ þ pi ; i ¼ 1; . . . N; Pi ðs; u ðsÞÞ ¼ P j 1=lj

(15)

and

almost surely. Clearly, if a contract can allocate the channel profit among the N agents proportionally, then the contract along with a side payment scheme can coordinate the supply chain. Moreover, this contract is flexible by adjusting the amounts of side payment. Theorem 3, as a special result of Theorem 2, characterizes the set of Paretooptimal actions for supply chains consisting of one supplier and one retailer. Theorem 3. An action pair ðs ; u ðs ÞÞ is Pareto-optimal if and only if lr ls s ¼ arg max EPðsÞ VðPðsÞÞ s2S lr þ ls

(16)

and Pr ðs ; u ðs ÞÞ ¼

ls Pðs Þ þ p0 ; lr þ ls

(17)

Coordination of Supply Chains with Risk-Averse Agents

Ps ðs ; u ðs ÞÞ ¼

lr Pðs Þ p0 ; lr þ ls

17

(18)

almost surely. It follows from Theorem 3 that under any Pareto-optimal solution, the retailer gets a fixed proportion ls =ðlr þ ls Þ of the channel profit plus p0 and the supplier gets the remaining profit, i.e., lr =ðlr þ ls Þ of the channel profit minus p0 . If lr >ls , i.e., the retailer is more risk-averse than the supplier, then the supplier takes a greater proportion of the channel profit. In other words, the agent with a lower risk aversion takes a higher proportion of the total channel profit than the other one does. The side payment, which is determined by the respective bargaining powers of the agents, determines the agents’ final payoffs. According to Theorem 3, C¼

ls lr 1 uðs Þ þ p0 ; uðs Þ p0 jp0 2 R ; lr þ l s lr þ l s

where uðs Þ represents EPðs Þ lVðPðs ÞÞ: Since the retailer’s and the supplier’s reservation payoffs are pr and ps , respectively, p0 has to satisfy the participating constraints of the agents. Thus, ls lr uðs Þ þ p0 rpr and uðs Þ p0 rps : lr þ ls lr þ l s Then F can be represented by

lr ls lr ls uðs Þ þ p0 ; uðs Þ p0

uðs Þ ps rp0 rpr uðs Þ : l r þ ls lr þ l s lr þ l s l r þ ls

Furthermore, if lr ls uðs Þ ps rpr uðs Þ; lr þ ls lr þ l s i.e., if pr þ ps buðs Þ; then F is not empty. The problem considered thus far is quite general, in the sense that the external action s is rather an abstract one that can include such decisions as order quantity, item price, etc. We next consider a special case. Here the retailer faces a newsvendor problem and makes a single purchase order of a product from the supplier at the beginning of a period, who in turn produces and delivers the order to the retailer before the selling season commences. Let p denote the price per unit,

18

X. Gan et al.

c the supplier’s production cost, v the salvage value, and q the retailer’s order quantity. In this problem, the supply chain’s external action is the retailer’s order quantity q. According to Theorem 3, the coordinating contract should allocate the profit in the same proportion for every realization of the channel profit in the absence of any side payment. We shall call such a sharing rule a proportional sharing rule. Here we only examine buy-back and revenue-sharing contracts. With a buy-back contract, the supplier charges the retailer a wholesale price per unit, but he pays the retailer a credit for every unsold unit at the end of the season. With a revenuesharing contract, the supplier charges a wholesale price per unit purchased, and the retailer gives the supplier a percentage of his revenue. See Pasternack (1985) and Cachon and Lariviere (2005) for details on these contracts. In the following, we see that both buy-back and revenue-sharing contracts allocate the channel profit proportionally. Proposition 3. A revenue-sharing contract allocates the channel profit (a random variable) proportionally. If w ¼ fc;

(19)

the retailer’s share is f and the supplier’s share is 1 f. Proof. Let D denote the demand faced by the retailer. Then the supply chain’s profit is

pD þ ðq DÞv cq pq cq;

if Dbq; if D>q:

(20)

On the other hand, the retailer’s profit is

fpD þ fðq DÞv wq if Dbq; fpq wq; if D>q:

(21)

By using w ¼ fc into (21), we can see that the retailer gets the proportion f of the supply chain’s profit for every realization of the demand. □ Cachon and Lariviere (2005) prove that for each coordinating revenue-sharing contract, there exists a unique buy-back contract that provides the same profit as in the revenue-sharing contract for every demand realization. They show that the buyback contract’s parameters have the form b ¼ ð1 fÞðp vÞ;

(22)

w ¼ pð1 fÞ þ fc;

(23)

where b is the refund to the retailer for each unsold unit, and f is the retailer’s share of the channel profit in the revenue-sharing contract. It is easy to see that the same result holds here as well.

Coordination of Supply Chains with Risk-Averse Agents

19

Proposition 4. A buy-back contract allocates the channel profit (a random variable) proportionally. If the contract parameters satisfy (22) and (23), then the retailer’s fractional share is f and the supplier’s is 1 f under this contract. Nagarajan and Bassok (2008) obtain the Nash Bargaining solution in the riskneutral case. According to their results, if both the retailer and the supplier are risk neutral, the retailer’s share of profit is fPðsÞ ¼ ½PðsÞ ps þ pr =2:

(24)

Under a buy-back or a revenue-sharing contract, the retailer’s problem is max p0 þ s2S

ls ½EPðsÞ lV ðPðsÞÞ: lr þ l s

(25)

For a given p0 , the retailer’s problem (25) becomes max EPðsÞ lVðPðsÞÞ; s2S

(26)

which has been solved by Lau (1980) and Chen and Federgruen (2000). Since the solution is s , the retailer would choose the optimal external action voluntarily. So we can state the following two theorems. Theorem 4. If the parameters of a revenue-sharing contract satisfy w¼

ls c; lr þ l s

(27)

then the revenue-sharing contract along with a side payment p0 to the retailer coordinates the supply chain. The profit allocation is given in (17)–(18). Theorem 5. If the parameters of a buy-back contract satisfy b¼ w¼p

lr ðp vÞ; lr þ ls

lr ls þ c; lr þ ls lr þ ls

(28)

(29)

then the buy-back contract along with a side payment p0 to the retailer coordinates the supply chain. The profit allocation is given in (17)–(18). Note that by adjusting the side payment p0 , the revenue-sharing as well as the buy-back contract can lead to any point in F. Thus, both contracts are flexible. The contracts obtained in Theorems 5 and 4, when lr ¼ ls ¼ 0, reduce to the standard contracts obtained in the risk-neutral case, because the fraction lr =ðlr þ ls Þ can take any value in ½0; 1. In particular, if the supplier is risk neutral and the retailer is risk averse, i.e., ls ¼ 0, the fraction lr =ðlr þ ls Þ ¼ 1, which

20

X. Gan et al.

means that the supplier takes the entire channel profit and gives a side payment to the retailer. In this case, it is Pareto-optimal for the supplier to bear all of the risk. Since the retailer’s profit is a side payment from the supplier, the supplier’s expected profit is the channel’s profit minus that payment. Therefore, the supplier’s payoff is maximized when the channel’s expected profit is maximized. Thus, we have a coordinating contract under which the supplier and the retailer execute s , the retailer gets a constant profit p0 , and the supplier gets the remaining profit. In the following example, we design a coordinating contract according to Theorem 5. We also obtain the optimal ordering quantity and determine the required side payment. Example 5. Consider a supply chain consisting of one retailer and one supplier. The retailer faces a newsvendor problem and makes a single purchase order of a product from the supplier at the beginning of a period, who in turn produces and delivers the order to the retailer before the selling season. Suppose that the demand D is uniformly distributed on some interval, which without loss of generality, can be taken as interval [0,1]. Thus, the distribution function FðxÞ ¼ x for 0bxb1 and FðxÞ ¼ 1 for xr1. We confine the ordering quantity q to be in [0,1]. Let the unit price p be 100, the supplier’s production cost c be 60, and the salvage value v be 20. Let the retailer’s and the supplier’s payoff functions be, respectively, EPr lr VðPr Þ and EPs ls VðPs Þ;

(30)

where lr ¼ 0:05 and ls ¼ 0:01. We assume that the agents have equal bargaining powers in the sense that their payoffs are equal. According to Theorem 3, the retailer’s payoff is ls lr ls EPðqÞ VðPðqÞÞ þ p0 ; lr þ ls lr þ ls

(31)

where PðqÞ is the channel’s profit when the retailer’s ordering quantity is q, 0bqb1. Thus, the retailer’s optimal order quantity is a q that maximizes (31). From Chen and Federgruen (2000), we have EPðqÞ ¼ 40q 80q2 and VðPðqÞÞ ¼ 6400ðq3 =3 q4 =4Þ:

(32)

With this, the retailer’s problem is 160 3 40 4 q þ q ; max 40q 80q2 0b qb 1 9 3

(33)

and q ¼ 0:236. According to Theorem 5, the retailer’s and the supplier’s payoffs are 0:799 þ p0 and 3:99 p0 , respectively. It is easy to see that p0 ¼ 1:598 equalizes their payoffs, as has been assumed.

Coordination of Supply Chains with Risk-Averse Agents

21

5 Coordinating a Supply Chain Consisting of Agents with Concave Utility Functions In this case, we assume that agent i has an increasing concave utility function gi ðÞ of his profit, and wants to maximize his expected utility, i ¼ r; s. Then his payoff function is E½gi ðÞ . To compute the set of Pareto-optimal actions, we first find the Pareto-optimal sharing rules given an external action s. According to the group decision theory literature (Wilson 1968; Raiffa 1970), the problem can be formulated as follows: max ar Egr ðPr ðs; uðsÞÞÞ þ as Egs ðPs ðs; uðsÞÞÞ;

(34)

s.t: Pr ðs; uðsÞÞ þ Ps ðs; uðsÞÞ ¼ PðsÞ;

(35)

u2Y

where ar ; as >0, ar þ as ¼ 1. The specification of ðar ; as Þ is derived from their respective bargaining powers. By varying ar and as , we can get all possible Pareto-optimal sharing rules Cs , denoted as fður ðPr ðs; uðsÞÞÞ; us ðPs ðs; uðsÞÞÞÞju is Pareto - optimal; u 2 Yg:

(36)

Clearly, each point in Cs represents, given s, the agents’ payoffs under a Paretooptimal sharing rule. Then we can get C, which is the set of Pareto-optimal points S of the set s2S Cs . According to Definition 3, any action pair that leads to a point in C is Pareto-optimal. It is well known that the problem of maximizing the expected quadratic utility can be reduced to one of maximizing a mean-variance trade-off. Therefore, when both agents’ utility functions are quadratic, we can coordinate the channel with the contracts developed in Sect. 4.2. Levi and Markowitz (1979) show that a utility function exhibiting constant risk aversion, particularly of the form log x or xa ; 0

5.1

Characterizing the Pareto-Optimal Set

Let the retailer’s and the supplier’s utility functions be, respectively,

22

X. Gan et al.

gr ðxÞ ¼ 1 elr x and gs ðxÞ ¼ 1 els x :

(37)

Then the absolute risk aversion measure for the retailer and the supplier are lr and ls , respectively; see Pratt (1964). We want to find a Pareto-optimal sharing rule for any given channel’s external action s. Raiffa (1970) solves the problem (34)–(35), which implies the following result. Proposition 5. For a given external action s for the channel under consideration, a sharing rule u ðsÞ is a Pareto-optimal sharing rule if and only if Pr ðs; u ðsÞÞ ¼

ls as ls PðsÞ l ln ; lr þ ls ar lr

(38)

Ps ðs; u ðsÞÞ ¼

lr as ls PðsÞ þ l ln ; lr þ ls ar lr

(39)

1 almost surely, where ar ; as >0, ar þ as ¼ 1, and l ¼ lr þl . s Thus we can get

Cs ¼ fður ðPr ðs; u ðsÞÞÞ; us ðPs ðs; u ðsÞÞÞÞjar ; as >0; ar þ as ¼ 1g; where lr as ls lr ls PðsÞ E exp ; ln ur ðPr ðs; u ðsÞÞÞ ¼ 1 exp lr þ ls ar lr lr þ ls

ls ar ls lr ls PðsÞ us ðPs ðs; u ðsÞÞÞ ¼ 1 exp ln E exp : lr þ ls as lr lr þ ls

(40)

(41)

Since both the retailer and the supplier’s payoff functions decrease with lr ls PðsÞ ; E exp lr þ ls it is easy to check that C ¼ fður ðPr ðs ; u ÞÞ; us ðPs ðs; u ÞÞÞjar ; as >0; ar þ as ¼ 1g;

(42)

where s is the solution of the problem lr ls PðsÞ : min E exp s2S l r þ ls

(43)

Coordination of Supply Chains with Risk-Averse Agents

23

Now the supply chain’s external problem has been transformed to problem (43). This problem has been studied in the literature in some special situations. Bouakiz and Sobel (1992) have shown that a base-stock policy is optimal in a multi-period newsvendor problem, when the newsvendor has an exponential utility function. Eeckhoudt et al. (1995) discuss the situation in which the entity faces a newsvendor problem, and they prove that the newsvendor orders less than that in the risk-neutral case. Agrawal and Seshadri (2000a) consider the entity’s price and inventory decision jointly in a newsvendor framework. Remark 3. Although we have got proportional sharing rules for the above case and the second case in Sect. 4, the Pareto-optimal sharing rules usually are not proportional for any two utility functions (Raiffa 1970). Moreover, the Pareto-sharing rules may depend on the channel’s external action. See Wilson (1968), Raiffa (1970), and LaValle (1978) for further details on Pareto-optimal sharing rules. Now we summarize the results in the following theorem. Theorem 6. An action pair ðs ; u ðs ÞÞ is Pareto-optimal if and only if lr ls PðsÞ ; s ¼ arg minE exp s2S lr þ ls

(44)

Pr ðs ; u ðs ÞÞ ¼

ls as ls Pðs Þ l ln ; lr þ ls ar lr

(45)

Ps ðs ; u ðs ÞÞ ¼

lr as ls Pðs Þ þ l ln ; lr þ ls ar lr

(46)

almost surely, where ar ; as >0, ar þ as ¼ 1. It follows from Theorem 6 that under any Pareto-optimal solution, the retailer and the supplier get fixed proportions of channel profit minus/plus a side payment. If lr >ls , i.e., if the retailer is more risk averse than the supplier, then the supplier takes a greater proportion of the channel profit if we ignore the side payment.

5.2

Bargaining Issue

We have got Pareto-optimal payoffs set C in (42). Since ur <1 and us <1 from (40) and (41), we assume that pr <1 and ps <1. Participating constraints of the agents are ur ðPr ðs ; u ðs ÞÞÞrpr and us ðPr ðs ; u ðs ÞÞÞrps :

(47)

Conditions (47) are equivalent to

lr lr ls Pðs Þ as ls lr þls l l Pðs Þ ð1 pr Þ=E exp =ð1 ps Þ: r rE exp r s ar lr lr þ ls lr þ ls (48)

24

X. Gan et al.

Then F can be represented by fður ðPr ðs ;u ðs ÞÞÞ;us ðPs ðs ;u ðs ÞÞÞÞjar ;as >0;ar þ as ¼ 1; and ð48Þ is satisfied:g: (49)

Pðs Þ If ð1 pr Þ=E exp lrllrsþl s

Pðs Þ =ð1 ps Þ, i.e., if rE exp lrllrsþl s

ð1 pr Þð1 ps Þr

lr ls Pðs Þ 2 E exp ; lr þ ls

then F is not empty. Nagarajan and Bassok (2008) use Nash Bargaining concept to deal with bargaining issue in the risk-neutral case. Here we use the same concept to deal with the bargaining issue in a risk-averse case. The approach of Nash (1950) requires that a bargaining solution satisfy the following eight axioms. 1. An agent offered two possible anticipations can decide which is preferable or that they are equally desirable. 2. The ordering thus produced is transitive; if A is better than B and B is better than C then A is better than C. 3. Any probability combination of equally desirable states is just as desirable as either. 4. If A, B, and C, are as in Axiom 2, then there is a probability combination of A and C which is just as desirable as B. This amounts to an assumption of continuity. 5. If 0bpb1 and A and B are equally desirable, then pA þ ð1 pÞC and pB þ ð1 pÞC are equally desirable. Also, A may be substituted for B in any desirability ordering relationship satisfied by B. 6. Let S, which is compact and convex, be the agents’ payoffs set and let cðSÞ denote the bargaining solution point in this set. If a is a point in S such that there exists another point b in S with the property ur ðbÞ>ur ðaÞ and us ðbÞ>us ðaÞ, then a2 = cðSÞ. 7. If the set T contains the set S and cðTÞ is in S, then cðTÞ ¼ cðSÞ: 8. If S is symmetrical with respect to the line ur ¼ us , and ur and us display this, then cðSÞ is a point on the line ur ¼ us . Clearly, exponential utilities in (37) satisfy the first five axioms. The agents’ payoff set S is fður ðPr ðs; uðsÞÞÞ; us ðPs ðs; uðsÞÞÞÞjðs; uðsÞÞ 2 S Yg: We assume that PðsÞ be continuous in s, so that S is compact. Now we prove the convexity of S by showing that its frontier is a concave curve. The Pareto-frontier of this set is given in (49), where

Coordination of Supply Chains with Risk-Averse Agents

25

lr as ls lr ls Pðs Þ E exp ; ur ðPr ðs ; u ðs ÞÞÞ ¼ 1 exp ln lr þ ls ar lr lr þ ls

us ðPs ðs ; u ðs ÞÞÞ ¼ 1 exp

ls ar lr lr ls Pðs Þ E exp : ln lr þ ls as ls lr þ ls

(50)

(51)

From (50) and (51), lr þls lr

ð1 u r Þ

ð1 us Þ

lr þls ls

¼

lr ls Pðs Þ E exp lr þ ls

ðlr þls Þ2 =ðlr ls Þ3

:

(52)

Clearly, the curve represented by (52) is concave since the right side is a constant. Axiom 6 assures that the solution is Pareto-optimal, Axiom 8 expresses the equality of bargaining skills. Nash (1950) shows that the solution point is the point that is the solution of the problem max ður pr Þðus ps Þ:

(53)

ður ;us Þ2S

Since the solution has to be Pareto-optimal, (53) is equivalent to max ður pr Þðus ps Þ:

(54)

ður ;us Þ2F

Pðs Þ : Then, Now we solve the problem (54). Let EðuÞ represent E exp lrllrsþl s ður pr Þðus ps Þ ¼ ð1 pr Þð1 ps Þ þ ½EðuÞ2 " # ls lr as ls lr þls as ls lr þls ð 1 pr Þ EðuÞ þ ð1 ps Þ EðuÞ : ar lr ar l r Thus, problem (54) is equivalent to min ð1 pr Þ a1 ;a2

as ls ar lr

llþls r

s

EðuÞ þ ð1 ps Þ

as ls ar lr

l lþlr r

s

EðuÞ:

(55)

Thus, when as ls ð1 pr Þls ¼ ; ar lr ð1 ps Þlr

(56)

26

X. Gan et al.

ður pr Þðus ps Þ is maximized. Therefore, the Nash Bargaining solution is represented by the payoffs given in (50) and (51) with parameters ar; as that satisfy ar >0; as >0, ar þ as ¼ 1; and (56). From (56), we can see that as ls =ar lr decreases in the retailer’s reservation payoff pr and increases in the supplier’s reservation payoff ps . Thus, the retailer’s payoff increases in pr and decreases in ps . The same property holds for the supplier. These properties imply that in the Nash Bargaining solution, each agent’s payoff increases with his own reservation payoff and decreases with the other agent’s reservation payoff. When pr ¼ ps , l lnðas ls =ar lr Þ ¼ 0, the side payment disappears. Finally, we can see that the Nash bargaining solution assigns a player a higher payoff when the other player becomes more risk averse.

5.3

Designing a Coordinating Contract

For the special supply chain considered in Sect. 4.2, we can use either a buy-back or a revenue-sharing contract to allocate the channel profit. Under either of these contracts, the retailer’s problem is: lr ls PðsÞ max 1 E exp : s2S lr þ ls This problem is equivalent to problem (43), which implies that the retailer would voluntarily choose the optimal external action s . So we have the following two results. Theorem 7. If the parameters of a buy-back contract satisfy b¼ w¼p

lr ðp vÞ; lr þ ls

lr ls þ c; lr þ ls lr þ ls

then the buy-back contract along with the side payment l lnðas ls =ar lr Þ to the supplier coordinates the supply chain. The profit allocation is given in (45)–(46). Theorem 8. If the parameters of a revenue-sharing contract satisfy w¼

ls c; lr þ l s

then the revenue-sharing contract along with the side payment l lnðas ls =ar lr Þ to the supplier coordinates the supply chain. The profit allocation is given in (45)–(46).

Coordination of Supply Chains with Risk-Averse Agents

27

Note that if ar and as satisfy condition (56), then both the revenue-sharing and the buy-back contracts achieve the Nash Bargaining solution. By adjusting the bargaining coefficients ar and as , one can attain any point in F. Thus, both these contracts are flexible. We should note that in general, Pareto-optimal sharing rules are not proportional as in the case with exponential utility functions. Wilson (1968) provides a necessary and sufficient condition for Pareto-optimality of a sharing rule in a channel with N agents. The condition is stated in the following theorem. Theorem 9. Given an external action s, a necessary and sufficient condition for Pareto-optimality of a sharing rule is that there exists nonnegative weights a ¼ ða1 ; a2 ; . . . ; aN Þ and a function m : R1 ! R1 ; such that X

Pi ðs; uðsÞÞ ¼ PðsÞ;

(57)

i

almost surely, and for each i ai g0i ðPi ðs; uðsÞÞÞ ¼ mðPðsÞÞ;

(58)

almost surely. In what follows, we give an example in which the sharing rule is not of the form of (45)–(46). Here we see that the Pareto-optimal sharing rule depends on the realized channel profit, i.e., it depends on the chosen external action as well as the realized random event. Example 6. Let gr ðxÞ ¼ 1 elr x ; gs ðxÞ ¼

1 els x ; xbx0 ; 1 ex0 ls þ l1s ex0 ls ðx x0 Þ; x>x0 :

(59) (60)

In this example, the retailer’s utility function is the same as in (37), but the supplier’s utility is changed in a way that his risk attitude is the same as in (37) at low profit levels and he is risk neutral at higher profit levels. Proposition 6. For a given external action s for the channel under consideration, a sharing rule u is a Pareto-optimal sharing rule if and only if as ls s PðsÞ l ln aars llsr ; PðsÞb lrlþl x l ln 0 ar lr r ; Pr ðs; u ðsÞÞ ¼ lr þls as ls : ls x0 l ln as ls ; PðsÞ> x l ln 0 lr ar lr lr ar lr 8 <

Ps ðs; u ðsÞÞ ¼

8 <

ls lr þls

lr lr þls PðsÞ

þ l ln aars llsr ;

: PðsÞ ls x0 þ l ln as ls ; lr

ar lr

as ls s PðsÞb lrlþl x l ln 0 ar lr r ; lr þls PðsÞ> lr x0 l ln aars llsr

(61)

(62)

28

X. Gan et al.

Fig. 1 Example of a Paretooptimal frontier

u2 Pareto-optimal Frontier

Ψs5 Ψs4

Ψs3

r2

Ψs1

Ψs2

r1

u1

1 almost surely, where ar ; as >0, ar þ as ¼ 1, and l ¼ lr þl . s Proof. Let

8 lr þls as ls < ðar lr Þlls ðas ls Þllr exp lr ls t ; t b x l ln 0 lr ar lr ; h lr þl s i mðtÞ ¼ : ðar lr Þlls ðas ls Þllr exp ls x0 l ln as ls ; t> lr þls x0 l ln as ls : ar lr lr ar lr Then, according to Theorem 9, (61) and (62) are Pareto-optimal since conditions (57) and (58) are satisfied. □ We can see that under the Pareto-optimal sharing rule, the retailer’s profit increases linearly with the channel’s realized profit when the latter is below a certain level, and remains unchanged thereafter. This is not a proportional sharing rule, and consequently, neither a buy-back nor a revenue-sharing contract along with side payments would coordinate the channel. It appears that new contract forms need to be designed to achieve coordination in such cases. In order to obtain s , we outline the following procedure. First, we compute Cs for each S s 2 S according to (36), and then we find the Pareto-optimal frontier of the set s2S Cs . Any action pair ðs ; u ðs ÞÞ that leads to a point on this frontier is Pareto-optimal. Note that s may not be unique. To illustrate this procedure, let us assume S ¼ fs1 ; s2 ; s3 ; s4 ; s5 g for convenience in exposition. Suppose that the sets Cs for s 2 S are as shown in Fig. 1. Then the frontier consisting of Pareto-optimal solutions is shown as the bold-faced boundary in the figure. Construction of such a frontier in general would require development of numerical procedures. This is not the focus of this chapter, and it is a topic for future research.

6 Discussion One of our main findings is that in any Pareto-optimal joint action, the retailer and the supplier must share the risk appropriately. Specifically, the less risk averse an agent is, the more risk he assumes by taking a larger portion of the channel’s

Coordination of Supply Chains with Risk-Averse Agents

29

random profit. The agents’ final payoffs can be adjusted by a side payment depending on their respective bargaining powers. In the extreme case when one of the agents is risk neutral, then that agent may assume all of the risk. Owing to the risk-sharing effect, the supply chain, when considered as a single entity, is less risk averse than either the risk-averse retailer or the risk-averse supplier if considered as the single owner of the whole channel. For example, in Case 2, the channel’s problem according to (25) is equivalent to solving the problem max ½EðPðsÞÞ lVðPðsÞÞ: s2S

If the retailer or the supplier were to own the channel, he would solve the problem max ½EPðsÞ lr VðPðsÞÞ or max ½EPðsÞ ls VðPðsÞÞ: s2S

s2S

Since l

N X i¼1

ai ui ðPðsÞÞ; ai 2 ½0; 1;

N X

ai ¼ 1;

i¼1

where the agent i’s utility function is denoted as ui ðPðsÞÞ; i ¼ 1; 2; . . . N. In this case, each agent’s identity is not preserved and the critical issue is how to determine the weights ai ; i ¼ 1; 2; . . . N. Once the optimal action for the supply chain is obtained by maximizing uðPðsÞÞ; the profit could be allocated according to some weighting scheme, possibly different from ai ; i ¼ 1; 2; . . . N: It is clear that this method generalizes the risk-neutral case. However, we do not follow this method because it does not guarantee Pareto-optimality of the final outcome.

30

X. Gan et al.

7 Conclusion and Further Research We have proposed a definition of coordination of a supply chain consisting of riskaverse agents. We show that to coordinate such a chain, the first step is to characterize the set of Pareto-optimal solutions and select a solution from this set based on the agents’ respective bargaining powers. The second step is to design a contract to achieve the selected solution. In the risk-neutral case, it is easy to see that an action pair is Pareto-optimal if and only if the supply chain’s expected profit is maximized. But in the risk-averse case, it is more difficult to find Pareto-optimal actions. We characterize Paretooptimal solutions in three specific cases of a supply chain involving one supplier and one retailer, and in each case we design a flexible contract to coordinate the channel. Furthermore, we discuss the bargaining issue in one of the cases. We provide answers to the following questions: What is the optimal external action of the supply chain and what is the optimal sharing rule? In the specific cases that we have considered, we are able to obtain Paretooptimal actions by first obtaining a Pareto-optimal sharing rule that can be used with any external action. This property allows us to obtain an objective function for the supply chain, whose optimization yields an external action, which together with the sharing rule provides us with Pareto-optimal solutions. In more general cases, however, we do not have the above property, and therefore, the sequential procedure used in the special cases does not work. In such cases, we show by a specially constructed example, that obtaining Paretooptimal solutions requires finding first the Pareto-optimal sets corresponding to external actions, and then identifying the Pareto-optimal frontier of the union of these sets. Moreover, the standard contract forms that work for risk-neutral cases do no longer coordinate, and research is required to find new coordinating contract forms.

References Agrawal V, Seshadri S (2000a) Impact of uncertainty and risk aversion on price and order quantity in the newsvendor problem. Manuf Serv Oper Manage 4:410–423 Agrawal V, Seshadri S (2000b) Risk intermediation in supply chains. IIE Trans 32:819–831 Arrow KJ (1951) Social choice and individual values. Wiley, New York Bouakiz M, Sobel MJ (1992) Inventory control with an expected utility criterion. Oper Res 40:603–608 Buzacott J, Yan H, Zhang H (2002) Optimality criteria and risk analysis in inventory models with demand forecast updating. Working paper, The Chinese University of Hong Kong, Shatin Cachon GP (2003) Supply coordination with contracts. In: Kok T, Graves S (eds) Handbooks in operations research and management science. North-Holland, Amsterdam Cachon GP, Lariviere M (2005) Supply chain coordination with revenue-sharing contracts: strengths and limitations. Manage Sci 51:30–44

Coordination of Supply Chains with Risk-Averse Agents

31

Chen F, Federgruen A (2000) Mean-variance analysis of basic inventory models. Working paper, Columbia University, New York Chen X, Sim M, Simchi-Levi D, Sun P (2007) Risk aversion in inventory management. Oper Res 55:828–842 Chopra S, Meindl P (2001) Supply chain management. Prentice-Hall, New Jersey, NJ Eeckhoudt L, Gollier C, Schlesinger H (1995) The risk averse (and prudent) newsboy. Manage Sci 41:786–794 Eliashberg J, Winkler RL (1981) Risk sharing and group decision making. Manage Sci 27:1221–1535 Gan X, Sethi SP, Yan H (2005) Coordination of a supply chain with a risk-averse retailer and a risk-neutral supplier. Prod Oper Manage 14:80–89 Gaur V, Seshadri S (2005) Hedging inventory risk through market instruments. Manuf Serv Oper Manage 7(2):103–120 Harsanyi JC (1955) Cardinal welfare, individualistic ethics, and interpersonal comparisons of utility. J Polit Econ 63:309–321 Jorion P (2006) Value at risk. McGraw-Hill, New York, NY Lau HS (1980) The newsboy problem under alternative optimization objectives. J Oper Res Soc 31:393–403 Lau HS, Lau AHL (1999) Manufacturer’s pricing strategy and return policy for a single-period commodity. Eur J Oper Res 116:291–304 Lavalle IH (1978) Fundamentals of decision analysis. Holt, Rinehart, and Winston, New York Levi H, Markowitz H (1979) Approximating expected utility by a function of mean and variance. Am Econ Rev 69:308–317 Markowitz H (1959) Portfolio selection: efficient diversification of investment. Cowles foundation monograph 16, Yale University Press, New Haven, CT Nagarajan M, Bassok Y (2008) A bargaining framework in supply chains: the assembly problem. Manage Sci 54:1482–1496 Nash JF (1950) The bargaining problem. Econometrica 18:155–162 Pasternack BA (1985) Optimal pricing and returns policies for perishable commodities. Mark Sci 4:166–176 Pratt WJ (1964) Risk aversion in the small and in the large. Econometrica 32:122–136 Raiffa H (1970) Decision analysis. Addison-Wesley, Reading, MA Schweitzer ME, Cachon GP (2000) Decision bias in the newsvendor problem with a known demand distribution: experimental evidence. Manage Sci 46:404–420 Szeg€o G (ed) (2004) New risk measures for the 21st century. Wiley, West Sussex Tayur S, Ganeshan R, Magazine M (eds) (1999) Quantitative models for supply chain management. Kluwer, Boston, MA Tsay A (2002) Risk sensitivity in distribution channel partnerships: implications for manufacturer return policies. J Retailing 78:147–160 Van Mieghem JA (2003) Capacity management, investment, and hedging: review and recent developments. Manuf Serv Oper Manage 5:269–302 Van Neumann J, Morgenstern O (1944) Theory of games and economic behavior. Princeton University Press, Princeton, NJ Wilson R (1968) The theory of syndicates. Econometrica 18:155–162

.

Addendum to “Coordination of Supply Chains with Risk-Averse Agents” by Gan, Sethi, and Yan (2004) Xianghua Gan, Suresh P. Sethi, and Houmin Yan

Abstract In “Coordination of Supply Chains with Risk-Averse Agents” (POMS, Volume 13, 2004), we study the issue of coordination in supply chains involving risk-averse agents, and define a coordinating contract as one that results in a Paretooptimal solution acceptable to each agent. We then develop coordinating contracts in various cases. In the case where the supplier and the retailer each maximizes his own expected utility, we also show that our contract yields the Nash Bargaining solution. In this addendum, we first review some related works that have appeared since the publication of the paper, and then discuss directions for future research. Keywords Supply chain management • Risk aversion • Pareto-optimality • Coordination • Nash bargaining

1 Introduction In Gan et al. (2004), we study the issue of coordination in supply chains involving risk-averse agents, and define a coordinating contract as one that results in a Paretooptimal solution acceptable to each agent. We develop coordinating contracts in three specific cases (1) the supplier is risk neutral and the retailer maximizes his expected profit subject to a downside risk constraint, (2) the supplier and the retailer

X. Gan (*) Department of Logistics and Maritime Studies, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong S.P. Sethi School of Management, SM30, The University of Texas at Dallas, 800W. Campbell Road, Richardson, TX 75080-3021, USA H. Yan Department of Systems Engineering and Engineering Management, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong T.-M. Choi and T.C. Edwin Cheng (eds.), Supply Chain Coordination under Uncertainty, International Handbooks on Information Systems, DOI 10.1007/978-3-642-19257-9_2, # Springer-Verlag Berlin Heidelberg 2011

33

34

X. Gan et al.

each maximizes his own mean-variance trade-off, and (3) the supplier and the retailer each maximizes his own expected utility. Moreover, in case (3) we show that our contract yields the Nash Bargaining solution. In each case, we demonstrate how to find the set of Pareto-optimal solutions, and then design a contract to achieve the solutions. We also exhibit a case in which we obtain Pareto-optimal sharing rules explicitly, and outline a procedure to obtain Pareto-optimal solutions. Gan et al. (2004) was published in Productions and Operations Management and won the Wickham-Skinner Best Paper Award at the Production and Operations Management Society Cancun meeting in 2004. This paper and Gan et al. (2005) are identified as part of a knowledge cluster by means of factor analysis in the paper entitled “The evolution of the intellectual structure of operations management1980–2006: A citation/co-citation analysis” by Pilkington and Meredith (2009). Specifically, the clusters in the general knowledge structure in the 2000s appear on Fig 7 on p.196 of their paper.

2 Related Literature since the Publication of Gan et al. (2004) Since the publication of Gan et al. (2004), there has been a considerable amount of literature devoted to contracts that coordinate supply chains involving risk-averse agents. Some papers study coordination of supply chains with agents who maximize their own mean-variance objectives, for example, Choi et al. (2008) and Wei and Choi (2010). Some papers study management of disruption risk faced by supply chains, for example, Tomlin (2006) and Bakshi and Kleindorfer (2009). Chen and Seshadri (2006) use optimal control theory to show that the contract menu proposed in Agrawal and Seshadri (2000) is optimal. Shi and Chen (2007) focus on Pareto€ u et al. (2007) study optimal contracts for supply chains with risk-averse agents. Ulk€ the optimal allocation of risk between two agents in a supply chain. Sobel and Turcic (2008) employ the Nash bargaining solution to determine the value of an opportunity to negotiate a contract in an archetypal supply chain game. Another stream of literature focuses on inventory management for risk-averse agents. Some papers incorporate financial instruments into inventory management, for example, Gaur and Seshadri (2005), Martı´nez-de-Albe´niz and Simchi-Levi (2005), and Chen et al. (2008). Choi et al. (2009) study a multi-product newsvendor under lawinvariant coherent risk measures. In the following, we limit our review to papers that focus on supply chain coordination. Gan et al. (2005) examine coordinating contracts for a supply chain consisting of one risk-neutral supplier and one risk-averse retailer. They design an easy-toimplement risk-sharing contract that accomplishes the coordination as defined in Gan et al. (2004). Martı´nez-de-Albe´niz and Simchi-Levi (2005) develop a framework that provides buyers with the ability to select multiple contracts at the same time in order to optimize their expected profit. For this purpose, they define a new type of contract, called a portfolio contract, which is in fact a combination of many

Addendum to “Coordination of Supply Chains with Risk-Averse Agents”

35

traditional contracts, such as long-term, option, and flexibility contracts. Then they specialize the model to the case of a portfolio consisting of option contracts, and show that a modified base-stock policy is optimal. Chen and Seshadri (2006) revisit the problem in Agrawal and Seshadri (2000), and use optimal control theory to prove that the contract menu proposed in that paper is optimal. They also show that the menu is optimal among nearly all contracts. Shi and Chen (2007) study a supply chain with two agents, each maximizing his own probability of achieving a predetermined target profit. They investigate two types of contracts: linear tariff contracts and buy-back contracts. They first identify the Pareto-optimal contract(s) for each type, and then evaluate the performance of those contracts. They also show that in some cases, a wholesale price contract can coordinate the supply chain whereas a buy-back contract cannot. € u et al. (2007) consider a supply chain with a contract manufacturer (CM) Ulk€ and a number of original equipment manufacturers (OEMs). Since investment into productive resources is made before demand realization, the supply chain faces the € u et al. (2007) investigate two scenarios in risk of under- or over-investment. Ulk€ which this risk is borne by the OEM and CM, respectively. They show that premium-based schemes are effective in inducing the best party to bear the risk, and conclude that they function well despite information asymmetry when double marginalization is not very high. Note that premium-based schemes are similar to the side payment scheme in Gan et al. (2004). Choi et al. (2008) study channel coordination in a supply chain consisting of a retailer and a supplier, each maximizing his mean-variance trade-off. They find that incorporation of risk aversion substantially affects the achievability of channel coordination. They also show that channel coordination depends on the difference between the risk preferences of the retailer and the supplier. Sobel and Turcic (2008) study a supply chain that faces a newsvendor problem. They characterize the Nash-optimal contracts, and contrast the risk-neutral case with the risk-averse one. They show that risk aversion has a significant impact on the contract terms and one firm’s risk aversion may be advantageous to the other. They suggest that a firm should consider its own and its prospective partner’s sensitivities to risk when it looks for a supply chain partner. Bakshi and Kleindorfer (2009) consider disruption risk management in a global supply chain with two participants who face interdependent losses resulting from supply chain disruptions. They use the Harsanyi-Selten-Nash Bargaining framework to model the supply chain participants’ choice of risk mitigation investments. The bargaining approach allows a framing of both joint financing of mitigation activities before a disruption and loss-sharing after the disruption. Wei and Choi (2010) study coordination of a supply chain with a wholesale pricing and a profit sharing scheme (WPPS) under the mean-variance decision framework. They show that there exists a unique equilibrium of the Stackelberg game with WPPS in the decentralized case. They also show that the manufacturer may benefit from pretending to be more risk-averse than he actually is. Finally, they propose a method to prevent this from happening.

36

X. Gan et al.

3 Future Research The issue of supply chain coordination with risk-averse agents has drawn much attention in recent years and has been studied in a number of research papers. Future research on this issue might be conducted in the following directions. First, optimization problems for a single firm can be revisited from the supply chain’s point of view. In the literature, numerous papers study a single risk-averse firm’s optimal decisions regarding order/production quantities, item prices, etc. Usually, those papers assume that the wholesale price, the amount of a side payment if any, refund on the returned units, etc., are exogenously given. More often than not, the single firm’s optimal decisions do not lead to a Pareto-optimal solution for the whole supply chain. In this case, it is worthwhile to study those problems from a system-wide view. As suggested in Gan et al. (2004), all of the agents in a supply chain can negotiate wholesale price, side payment, and refund policy, and come up with Pareto-optimal decisions on order/production quantities, item prices, etc. Recently, asymmetric information problems have been widely studied in the OR/MS literature. In these studies, the agents are usually assumed to be riskneutral. When those agents are risk-averse, the results in risk-neutral case may not apply and future research is warranted. For example, when the production cost of the supplier is asymmetric, C¸akanyldrm et al. (2010) show that a Pareto-optimal solution is achievable. However, when the agents in a supply chain are risk-averse, the achievability of a Pareto-optimal solution is not clear. The decisions of the agents in a supply chain depend crucially on customer demand. However, in the OR/MS literature, customer demand is usually assumed, for simplification, to be a function of the selling price, and the issue of customers’ risk-aversion is ignored. When the customer demand depends on risk attitudes of the customers, it might be interesting to investigate how the agents in supply chain share risk with customers.

References Agrawal V, Seshadri S (2000) Risk intermediation in supply chains. IIE Trans 32:819–831 Bakshi N, Kleindorfer P (2009) Co-opetition and investment for supply-chain resilience. Prod Oper Manage 2009:583–603 C¸akanyldrm M, Feng Q, Gan X, Sethi SP (2010) Contracting and coordination under asymmetric production cost information. Working paper, University of Texas at Dallas, TX Chen YJ, Seshadri S (2006) Supply chain structure and demand risk. Automatica 2006:1291–1299 Chen F, Gao F, Chao X (2008) Joint optimal ordering and weather hedging decisions: a newsvendor model. Working paper, Chinese University of Hong Kong, Hong Kong Choi S, Ruszczynski AR, Zhao Y (2009) A multi-product risk-averse newsvendor with law invariant coherent measures of risk. Working paper, Rutgers University, New Jersey Choi TM, Li D, Yan H, Chiu CH (2008) Channel coordination in supply chains with agents having mean-variance objectives. Omega 36:565–576

Addendum to “Coordination of Supply Chains with Risk-Averse Agents”

37

Gan X, Sethi SP, Yan H (2004) Coordination of supply chains with risk-averse agents. Prod Oper Manage 13:135–149 Gan X, Sethi SP, Yan H (2005) Coordination of a supply chain with a risk-averse retailer and a risk-neutral supplier. Prod Oper Manage 14:80–89 Gaur V, Seshadri S (2005) Hedging inventory risk through market instruments. Manuf Serv Oper Manage 7:103–120 Martı´nez-de-Albe´niz V, Simchi-Levi D (2005) A portfolio approach to procurement contracts. Prod Oper Manage 14:90–114 Pilkington A, Meredith J (2009) The evolution of the intellectual structure of operations management-1980–2006: a citation/co-citation analysis. J Oper Manage 27:185–202 Shi C, Chen B (2007) Pareto-optimal contracts for a supply chain with satisfying objectives. J Oper Res 58:751–759 Sobel M, Turcic D (2008) Risk aversion and supply chain contract negotiation. Working paper, Case Western Reserve University, Cleveland, OH Tomlin B (2006) On the value of mitigation and contingency strategies for managing supply chain disruption risks. Manage Sci 52:637–659 € u S, Toktay LB, Y€ Ulk€ ucesan E (2007) Risk ownership in contract manufacturing. Manuf Serv Oper Manage 9:225–241 Wei Y, Choi TM (2010) Mean-variance analysis of supply chains under wholesale pricing and profit sharing schemes. Eur J Oper Res 204:255–262

.

A Review on Supply Chain Coordination: Coordination Mechanisms, Managing Uncertainty and Research Directions Kaur Arshinder, Arun Kanda, and S.G. Deshmukh

Abstract The Supply Chain (SC) members are dependent on each other for resources and information, and this dependency has been increasing in recent times due to outsourcing, globalization and rapid innovations in information technologies. This increase in dependency brings some extent of risk and uncertainty too along with benefits. To meet these challenges, SC members must work towards a unified system and coordinate with each other. There is a need to identify the coordination mechanisms which helps in addressing the uncertainty in supply chain and achieving supply chain coordination. A systematic literature review is presented in this paper to throw light on the importance of SC coordination. The objectives of this paper are to: Report and review various perspectives on SC coordination issues, understand and appreciate various mechanisms available for coordination and managing SC uncertainty and identify the gaps existing in the literature. Perspectives on various surrogate measures of supply chain coordination have been discussed followed by the scope for further research. Keywords Coordination mechanisms • Supply chain coordination • Supply chain coordination index • Supply chain uncertainty

This paper is based on earlier version of the following paper: Arshinder K, Kanda A, Deshmukh SG (2008) Supply chain coordination: perspectives, empirical studies and research directions. Int J Prod Econ 115(2):316–335. This paper is also based on the doctoral research work done by Arshinder (2008) at Indian Institute of Technology Delhi, India. K. Arshinder (*) Department of Management Studies, Indian Institute of Technology Madras, Chennai 600036, India e-mail: [email protected] A. Kanda • S.G. Deshmukh Department of Mechanical Engineering, Indian Institute of Technology Delhi, New Delhi 110016, India e-mail: [email protected]; [email protected] T.-M. Choi and T.C. Edwin Cheng (eds.), Supply Chain Coordination under Uncertainty, International Handbooks on Information Systems, DOI 10.1007/978-3-642-19257-9_3, # Springer-Verlag Berlin Heidelberg 2011

39

40

K. Arshinder et al.

1 Introduction Supply chain has evolved very rapidly since 1990s showing an exponential growth in papers in different journals of interest to academics and practitioners (Burgess et al. 2006). The rise in papers on supply chain (SC) as well as the case studies in different areas in different industries motivates to study SC issues further. Supply chains are generally complex with numerous activities (logistics, inventory, purchasing and procurement, production planning, intra-and inter-organizational relationships and performance measures) usually spread over multiple functions or organizations and sometimes over lengthy time horizons. Supply chains tend to increase in complexity and the involvement of numerous suppliers, service providers, and end consumers in a network of relationships causes risks and vulnerability for everyone (Pfohl et al. 2010). The continuous evolving dynamic structure of the supply chain poses many interesting challenges for effective system coordination. Supply chain members cannot compete as independent members. The product used by the end customer passes through a number of entities contributed in the value addition of the product before its consumption. Also, the practices like globalization, outsourcing and reduction in supply base have exacerbated the uncertainty and risk exposure as well as more prone to supply chain disruption. Earlier literature considers risks in relation to supply lead time reliability, price uncertainty, and demand volatility which lead to the need for safety stock, inventory pooling strategy, order split to suppliers, and various contract and hedging strategies (Tang 2006). But today’s supply networks have become very complex and vulnerable to various supply chain risks hence these issues have pulled attention of various academics and practitioners for the last few years (Oke and Gopalakrishnan 2009). Uncertainty relates to the situation in which there is a total absence of information or awareness of a potential event occurrence, irrespective of whether the outcome is positive or negative. The terms risk and uncertainty are frequently used interchangeably (Ritchie and Brindley 2007). As firms move to leaner operating models and increasingly leverage global sourcing models, uncertainty in both supply and demand is growing along with supply chain complexity. To improve the overall performance of supply chain, the members of supply chain may behave as a part of a unified system and coordinate with each other. Thus “coordination” comes into focus. There seems to be a general lack of managerial ability to integrate and coordinate the intricate network of business relationships among supply chain members (Lambert and Cooper 2000). Stank et al. (1999) studied inter-firm coordination processes characterized by effective communication, information exchange, partnering, and performance monitoring. Lee (2000) proposes supply chain coordination as a vehicle to redesign decision rights, workflow, and resources between chain members to leverage better performance such as higher profit margins, improved customer service performance, and faster response time.

A Review on Supply Chain Coordination

41

Though, there are efforts in literature regarding coordination of different functions of the supply chain, the study of coordinating functions in isolation may not help to coordinate the whole supply chain. It appears that the study of supply chain coordination (SCC) is still in its infancy. Though, the need for coordination is realized, a little effort has been reported in the literature to develop a holistic view of coordination. It is interesting to note the following perspectives on supply chain coordination as reported in the literature: • Collaborative working for joint planning, joint product development, mutual exchange information and integrated information systems, cross coordination on several levels in the companies on the network, long term cooperation and fair sharing of risks and benefits (Larsen 2000). • A collaborative supply chain simply means that two or more independent companies work jointly to plan to execute supply chain operations with greater success than when acting in isolation (Simatupang and Sridharan 2002). • Kleindorfer and Saad (2005) asserted that continuous coordination, cooperation, and coordination among supply chain partners are imperative for risk avoidance, reduction, management and mitigation such that the value and benefits created are maximized and shared fairly. • Supply chain coordination is a strategic response to the challenges that arise from the dependencies supply chain members (Xu and Beamon 2006). • Supply chain coordination can be defined as identifying interdependent supply chain activities between supply chain members and devise mechanisms for manage those interdependencies. It is the measure of extent of implementation of such aggregated coordination mechanisms, which helps in improving the performance of supply chain in the best interests of participating members (Arshinder 2008). Various perspectives have been presented in the literature for coordinating supply chain (discussed in Sect. 2). These perspectives and classification of coordination literature has been adopted from the review paper by Arshinder et al. (2008a), however, the authors are motivated to revise the paper with view of incorporating uncertainty in SCC and up gradation of coordination mechanisms. The following developments have motivated the authors to upgrade the current review paper. • Growth in reporting of coordination mechanisms in supply chain. • Managing uncertainty has become more and more challenging, which can be tackled with SCC. • Information technology has been evolving and playing an important role in making global supply chain seamless. To develop a better understanding of the coordination issues in supply chain, a systematic literature review is required to throw light on the importance of SCC and specifically to address the objectives as: to understand and appreciate SCC in different processes of supply chain, to explore various coordination mechanisms to coordinate the supply chain, to understand the role of SCC in managing SC uncertainty and to relate surrogate measures of SCC with supply

42

K. Arshinder et al.

chain performance. The last objective is to identify the gaps existing in the literature followed by few research directions. The terms like integration, collaboration, cooperation and coordination are at times complementary and at times contradictory to each other and when used in the context of supply chain can easily be considered as a part of SCC. This assumption can be followed without loss of generality as the elements like integration (combining to an integral whole); collaboration (working jointly) and cooperation (joint operation) are the elements of coordination.

2 Supply Chain Coordination Literature Classification and Observations The papers related to supply chain coordination were searched using library databases covering a broad range of journals (Appendix). The papers were selected based on the issues addressed by these papers: How to define supply chain coordination and the imperatives of SCC? How to achieve supply chain coordination? Will coordinated supply chain be beneficial to all the individual members of the supply chain? What is the impact of SCC on the performance of various activities and processes of a supply chain? How SCC can help in mitigating supply chain uncertainties? The papers in response to the above mentioned questions were gathered and classified in categories presented in the following sections. To capture each and every aspect of SCC an attempt has been made to classify the literature on SCC as follows: • Perspectives and conceptual models on supply chain coordination. • Joint consideration of functions or processes by supply chain members at different levels to coordinate the supply chain. • Various supply chain coordination mechanisms adopted in the supply chain. • Supply chain coordination to manage uncertainties in the supply chain. • Empirical case studies in supply chain coordination. A schematic overview of hierarchical classification of literature is shown in Fig. 1 which shows that how the different categories of coordination will help in understanding the importance of SCC, utility of coordination mechanisms and the application of SCC on real life problems.

2.1

2.1.1

Perspectives and Conceptual Models on Supply Chain Coordination Challenges in Coordinating the Supply Chain

In any system, the smooth functioning of entities is the result of well-coordinated entities. It may be very difficult to define “coordination” precisely, but the lack of

A Review on Supply Chain Coordination

43

Supply Chain Coordination

Perspectives and conceptual models on SCC

Joint consideration of functions/ processes by various SC members

Coordination across functions of the supply chain

Supply chain coordination mechanisms

Integrated ProcurementProductionDistribution processes

Contracts

Supply uncertainty

Information technology and Information sharing

Supply chain coordination to manage uncertainty

Production disruptions

Demand uncertainty

Other collaborative initiatives

Fig. 1 Overview of the literature classification scheme

coordination can be easily articulated through a variety of surrogate measures. The supply chain members have conflicting goals or objectives and disagreements over domain of supply chain decisions and actions. It must be noted that a typical supply chain also deals with human systems, and hence, which may pose following challenges and difficulties in coordinating supply chain members. • The individual interest, local perspective and opportunistic behavior of supply chain members results in mismatch of supply and demand (Fisher et al. 1994). • The traditional performance measures based on the individual performance may be irrelevant to the maximization of supply chain profit in a coordinated manner. Similarly, the traditional policies, particularly rules and procedures, may not be relevant to the new conditions of inter organizational relationship. There has been over reliance on technology in trying to implement IT (Lee et al. 1997; McCarthy and Golocic 2002). • According to Piplani and Fu (2005), supply chain “plug and play” misalignment is associated with the difficulties involved in dynamically interchanging products (with short life cycle) and partners in the fast changing business environment. • The organizations want to reach to the best suppliers regardless their location globally, which brings many risks and uncertainties in managing cross border supply chains.

44

K. Arshinder et al.

• The benefits accrued by the whole supply chain after joint determination of supply chain performance indicators by supply chain members has no value in the absence of fair share mechanisms. There are multiple benefits accruing from effective SCC. Some of these include: elimination of excess inventory, reduction of lead times, increased sales, improved customer service, efficient product developments efforts, low manufacturing costs, increased flexibility to cope with high demand uncertainty, increased customer retention, and revenue enhancements (Fisher et al. 1994; Lee et al. 1997). 2.1.2

Various Perspectives and Conceptual Models on SCC

The literature reviewed by Burgess et al. (2006) showed that there is relative paucity of strong multi-theoretic approaches in supply chain. By looking at the problems of managing relationships between supply chain members, a need arises to tackle this problem using coordination theory. The most commonly accepted definition of coordination in the literature is “the act of managing dependencies between entities and the joint effort of entities working together towards mutually defined goals” (Malone and Crowston 1994). Coordination is perceived as a prerequisite to integrate operations of supply chain entities to achieve common goals. Various perspectives are reported in the literature regarding SCC. The researchers have described SCC either in the context of the application of coordination in different activities of supply chain or they are derived from other disciplines, summarized in Table 1. Several coordination strategies have been developed to align supply chain processes and activities to ensure better supply chain performance. The papers addressing various forms of coordination are Buyer–Vendor coordination by coordinating Procurement–Inventory–Production–Distribution processes (Goyal and Deshmukh 1992; Thomas and Griffin 1996; Sarmiento and Nagi 1999; Sarmah et al. 2006). Hoyt and Huq (2000) presented a literature review on the buyer-supplier relationship from the perspective of transaction cost theory, strategy structure theory and resource-based theory of the firm. There is abundant literature on conceptual based supply chain partnership but the testing of these concepts is required by utilization of operations research in supply chain (Maloni and Benton 1997). Various models have been discussed presenting various form of coordination such as price changes, quantity discounts (Sharafali and Co 2000), and partial deliveries and establishing their joint policies in context of manufacturing firms (Sarmah et al. 2007), information sharing and decision-making coordination (Sahin and Robinson 2002). Some of the coordination forms can be seen in Table 2. Power (2005) reviewed three principal elements of supply chain integration: information systems, inventory management and supply chain relationships aiming at reducing costs and improving customer service levels. The emerging area of supply chain coordination is outsourcing practices in case of insufficient production capacity of suppliers (Sinha and Sarmah 2007).

A Review on Supply Chain Coordination Table 1 Various perspectives on supply chain coordination Author (year) Perspective Narus and Anderson Cooperation among independent but (1995) related firms to share resources and capabilities to meet their customers’ most extraordinary needs Lambert et al. (1999) A particular degree of relationship among chain members as a means to share risks and rewards that result in higher business performance than would be achieved by the firms individually Larsen (2000) Collaborative working for joint planning, joint product development, mutual exchange information and integrated information systems, cross coordination on several levels in the companies on the network, long term cooperation and fair sharing of risks and benefits Lee (2000) Supply chain coordination as vehicle for redesigning decision rights, workflow, and resources between chain members to leverage better performance Simatupang et al. Given the nature of the interdependencies (2002) between units, coordination is necessary prerequisite to integrate their operations to achieve the mutual goal of the supply chain as a whole as well as those of these units Larsen et al. (2003) Where two or more parties in the supply chain jointly plan a number of promotional activities and work out synchronized forecasts, on the basis of which the production and replenishment processes are determined Hill and Omar Coordination can be achieved when the (2006) supply chain members jointly minimize the operating costs and share the benefits after jointly planning the production and scheduling policies Arshinder (2008) Identifying interdependent supply chain activities between SC members and devise mechanisms for manage those interdependencies. It is the measure of extent of implementation of such aggregated coordination mechanisms, which helps in improving the performance of supply chain in the best interests of participating members

45

Context Resource sharing

Risk and reward sharing

Holistic view of coordination

Workflow/resource dependency

Mutuality

Joint promotional activities, forecasting

Joint decision-making, benefit sharing

Linking coordination mechanisms with SC performance

46

K. Arshinder et al.

Table 2 Different forms of coordination viewed in supply chain S. No. Coordination Author (year) perspectives 1 Coordination of Goyal and Deshmukh (1992), functions or process Thomas and Griffin across SC members (1996) Sarmiento and Nagi (1999) 2 Coordination by Hoyt and Huq (2000), Sahin information sharing and Robinson (2002), Huang et al. (2003), Simatupang et al. (2002) 3 Supply chain Power (2005) partnerships 4

5

Coordination mechanisms and performance Problems in coordinating SC

6

Coordination by IT

7

Implementation issues in coordination

Issues in coordination Integrated procurement, production, distribution and inventory systems Value of information sharing and sharing modes, incentive alignment

Communication, Inventory management and supply chain partnerships Lee et al. (1997) Channel coordination, operational efficiency and information sharing Fawcett and Magnan (2002), Lack of information transparency, incentive Simatupang and Sridharan misalignment (2002) Li et al. (2002), Mc Laren Internet based integration of et al. (2002) complex supply chain processes, cost and benefits of different information systems coordinating supply chain Barratt (2004) Cultural, strategic and implementation elements of supply chain coordination

The other pragmatic initiatives such as Collaborative Planning, Forecasting and Replenishment (CPFR) (Larsen et al. 2003) and Supply Chain Operations Reference (SCOR) (Huan et al. 2004) may have relevance from practitioner’s point of view. Even though coordination improves the performance of the supply chain, it may not always be beneficial to coordinate the supply chain members. The high adoption costs of joining inter-organizational information systems and information sharing under different operational conditions of organizations may hurt some supply chain members (Zhao and Wang 2002). Therefore, it is essential to investigate the conditions under which supply chain coordination is beneficial, so that it should not result in higher supply chain costs and imprecise information. Observations and Gaps Regarding Various Perspectives and Conceptual Models on SCC (a) There seems to be no standard definition of SCC. Various perspectives on SCC as reported in the literature are testimony to this. The differences in perceptions

A Review on Supply Chain Coordination

47

are there because of the different expectations of the various stakeholders and the respective problem domain. Some of these perspectives present the inherent capability or intangibles required to coordinate like responsibility, mutuality, cooperation and trust. The other perspectives can be visualized, based on the coordination effort required in achieving common goals in different activities of supply chain. Since the activities are different, the coordination requirements also vary with the complexity of the activity. The most challenging coordination perspective is to extend the concept of coordination from within an organization to coordination between organizations. (b) By looking at these different perspectives, the SCC can be viewed as a set of following steps: 1. Identify why supply chain members want to coordinate and for which activity/process they are interdependent? Different interdependencies among supply chain members can be: ordering, procurement, inventory management, production, design and development, replenishment, forecasting and distribution. 2. Identify which activity or a set of activities needs to be coordinated, complexities in the activity (activities) and degree of coordination required. 3. Identify the reason to coordinate. Is it the demand uncertainty and/or supply uncertainty, double marginalization or other external risk in the supply chain, which can be addressed by coordination? 4. Identify whether a single or a combination of coordination mechanism are required to tackle the complexities in managing the interdependencies like resource sharing, knowledge sharing, information sharing, joint working, joint decision making, joint design and development of product, joint promotions, implementing information systems, designing risk sharing contracts. (c) Though there are attempts to focus on coordinating the different processes of supply chain, most of the papers reviewed have discussed the work done on analytical models with joint decision making of different process. The literature seems to be lacking in developing empirical relationship between coordination means and mechanisms (Information sharing, trust and IT) and SCC. (d) There is a need to embrace a variety of perspectives on supply chain coordination, various coordination issues and the means and mechanisms to achieve coordination in a holistic manner. (e) Various coordination mechanisms suggested in these models help in improving the various performance measures of the supply chain. These mechanisms include: joint decision-making, information sharing, resource sharing, implementing information technology, joint promotional activities, etc. The other motivation seems to be the ability of supply chain members to share the risks and subsequently share the benefits. (f) There is a need to monitor coordination in supply chain because of the adverse effects of lack of coordination on supply chain performance. There seems to be no measure to quantify coordination. Some models can be proposed to

48

K. Arshinder et al.

quantify and assess the strength of coordination on the basis of coordination mechanisms. (g) More empirical studies are required regarding the proper implementation of coordination mechanisms, so that combinations of different feasible coordination mechanisms can capture the impact of coordination on various supply chain performance measures. The above conceptual models on supply chain coordination have been presented in a fragmented manner. It is important to understand various SC functions to be coordinated. The complexity in coordinating various SC members may also depend on the interface to which two supply chain members belong. The following section presents the importance of SC coordination in various SC functions as well as in different SC processes at various supply chain interfaces.

3 Joint Consideration of Functions or Processes by Supply Chain Members at Different Levels to Coordinate Supply Chain Coordination can be visualized in different functions such as logistics, inventory management, forecasting, transportation, etc. Similarly, various interface such as supplier-manufacturer; manufacturer-retailer, etc. can be effectively managed using coordination.

3.1

Coordinating Functions Across Supply Chain Members

The supply chain members perform different functions or activities like logistics, inventory management, ordering, forecasting and product design involved in management of flow of goods, information and money. In traditional supply chain individual members of supply chain have been performing these activities independently. The supply chain members may earn benefits by coordinating various activities as discussed in following subsections. Logistics has traditionally been defined as the process of planning, implementing, and controlling the efficient flow and storage of goods, services, and related information as they travel from point of origin to point of consumption. The uncertainty and complexity of decision making regarding logistics operations: diversified customers and their different requirements, different resources required, increasing rate of unanticipated change and level of goal difficulty among logistics provider and the customer (supplier, manufacturer, distributor and retailer), geographically dispersed networks of multiple manufacturing sites lead to the need of coordination in this process (Huiskonen and Pirttila 2002). The challenges lie in managing the network complexities to collectively create value to the end customer

A Review on Supply Chain Coordination

49

(Stank et al. 1999; Stock et al. 2000) and integrating the logistics with whole supply chain with the help of electronic communication. The major decisions regarding inventory management include: determination of the order quantity, the timing of order, reorder point and the replenishment of inventory. The factors which are considered while deciding the inventory policy are customer demand (deterministic and random), number of members in supply chain, replenishment lead time, number of different products stored, length of the planning horizon, service level requirements and costs comprised of cost of production, transportation, taxes and insurance, maintenance, obsolescence opportunity cost, stock out, etc. The changes even in one of the above factors affect the decisions regarding inventory policy. The factors related to inventory policy are highly dynamic because of changing market condition, supply uncertainty; different and conflicting inventory policies among supply chain members, and unavailability of inventory information of other members. To face the dynamic situation, the members of supply chain have realized the importance of coordination in inventory management. The supply chain members may coordinate by joint consideration of the system wide costs (Huq et al. 2006; Wu and Ouyang 2003; Gurnani 2001; Barron 2007), sharing cost and price information (Boyaci and Gallego 2002; Piplani and Fu 2005), synchronizing order processing time (Zou et al. 2004; Lu 1995; Yao and Chiou 2004; Barron 2007) and networked inventory management information systems (Verwijmeren et al. 1996). These policies may sometime hurt one of the supply chain members. To compensate losses, different mechanisms have been proposed as quantity discounts, revenue sharing contracts and incentive alignment policies (Li et al. 1996; Moses and Seshadri 2000; Chen and Chen 2005). The different models results in reduction in ordering cost, holding cost, purchasing cost, and supply chain system wide costs and improvement in customer service level and product availability and product variety. The organization has perceived the need of reviving the traditional purchasing function in view of degree of participation and expertise of suppliers to a new evolving function called “strategic sourcing”(Gottfredson et al. 2005). The suppliers can form strategic partnerships by having common goals and sharing forecast information to have updated single forecasting process, which results in substantial cost reduction in whole supply chain (Zsidisin and Ellram 2001; Aviv 2001). The increasing rate of changing technologies, innovation, customer expectations, competition, and risk involved with new product entry and at the same time keeping the product design process cost efficient, is a challenging job. Kim and Oh (2005) presented systems dynamics approach to coordinate supplier and manufacturer decisions regarding improvement in quality and the new product development. Petersen et al. (2005) presented the findings from an empirical survey about the capabilities of suppliers required in coordinating the product design process with supplier. The coordination at design stage may result in better design and improved financial performance if the supplier has sufficient knowledge required to design the product.

50

3.2

K. Arshinder et al.

Coordinating Different Processes of the Supply Chain

A supply chain process consists of a set of activities taken together. Various processes in supply chain are procurement, production and distribution. These processes can be accomplished when some activities are performed like procurement process comprised supplier management, ordering, acquisition, replenishment, inspection activities, etc. Integration of different processes into a single optimization model to simultaneously optimize decision variables of different processes that have traditionally been optimized sequentially helps in improving the performance of SC (Park 2005). These processes sometimes face conflicting issues which are presented in Table 3. Isolated decision making in functionally related supply chain processes might weaken the supply chain system wide competitiveness. The different supply chain processes can be coordinated by implementing joint production delivery policies, common cycle approach, identical replenishment cycle (Yang and Wee 2002) and joint lot scheduling models (Kim et al. 2006). The coordination problems and the related issues at the interfaces of supply chain are presented in Table 4.

3.2.1

Production and Distribution Coordination

Integration of production and distribution processes may lead to a substantial saving in global costs and to an improvement in relevant service by exploiting scale economies of production and transportation, balancing production lots and vehicle loads, and reducing total inventory and stockout. Chikan (2001) gave a theoretical background of integrated production/logistics systems on the basis Table 3 Conflicting issues in supply chain processes SC processes Conflicting issues in supply chain processes Production and distribution The difference in performance metrics such as improvement in coordination quality of production, reduction in cost and improvement in service levels for distribution may also give rise to conflict Production sub functions are usually concentrated in the organization, while distribution sub functions are spread over (Chikan 2001) Production function is obsessed with low cost production, with large batch sizes and efficient and smooth production schedules (Pyke and Cohen 1993) and the distribution function is concerned with customer service as first priority, small batch sizes and frequent changeovers (Pyke and Cohen 1993) Procurement and production Suppliers typically want manufacturers to commit themselves coordination to purchasing large quantities in stable volumes with flexible delivery dates Manufacturers require just-in-time (JIT) supply in small batches from their suppliers due to changing demand and their unwillingness to hold inventories

Information sharing and IT

Coordination mechanism

Logistics provider and clientsa

Structure of supply chain

Joint decision making and benefit sharing Quantity discounts

Manufacturer–retailer

Single-supplier–multibuyers

Joint system cost consideration, quantity discounts

Supply chain network

Single-supplier–multibuyer Seller–buyer

Joint decision making and quantity discounts

Need of system cooperation Independent management IT and mutual benefits of inventories

Different order intervals

Mismatch goals between Information sharing, aligning Logistics provider and shipper and goals, EDI, contracts clientsa transportation provider Lack of integration EDI Logistics provider and between logistics and clientsa supply chain Need of relation Information sharing, IT, Logistics provider and improvement integrating role manufacturer between logistics and client

Mismatch goals between shipper and transportation provider

Coordination problem

Moses and Seshadri (2000) Need of risk sharing, mismatch in stock level and review period Gurnani (2001) Mismatch in timing of order

Verwijmeren et al. (1996)

Inventory Lu (1995) and Yao and Chiou (2004) Li et al. (1996)

Huiskonen and Pirttila (2002)

Stock et al. (2000)

Stank and Goldsby (2000)

Logistics Stank et al. (1999)

Author

Table 4 Coordination in various activities and interfaces of supply chain

Analytical model

Optimization

Network solution

Game theoretic model

Analytical model

Conceptual survey

Empirical survey

Conceptual framework

Empirical survey

Methodology used

Minimize cost

(continued)

Improving customer service level, increasing product variety, and lower supply chain system wide costs Minimize cost

Minimize costs (ordering + holding + purchasing) Maximize profits

Good relationship

Operational performance and financial performance

Inventory level, transportation costs, warehousing costs, ordering costs, order cycle variance, on time deliveries and unacceptable deliveries Channel cycle time and inventory level

Performance measure

A Review on Supply Chain Coordination 51

Misaligned inventory decisions

Independent cost consideration Different cycle times

Piplani and Fu (2005)

Huq et al. (2006)

Inventory-distribution Haq and Kannan (2006)

Forecasting Aviv (2001)

Barron (2007)

Chen and Chen (2005)

Zou et al. (2004)

Wu and Ouyang (2003)

Potential lie in reducing costs by considering all costs jointly

Multi-supplier–singleassembler

Single-manufacturer– multi-retailer Vendor–buyer

Single-wholesaler– multi-retailer

Structure of supply chain

Joint decision making

Joint consideration of cost

Joint consideration of costs at Multi-echelon each level

Manufacturer–retailer

Multi-warehouse– multi-retailer Serial supply chain (multi-echelon)

Joint consideration of cost, Manufacturer–retailer savings sharing, quantity discounts Cost sharing and service level Multi-echelon contracts

Jointly plan pricing and inventory replenishment policies Order coordination, information sharing Joint cost consideration with shortages Information sharing and revenue sharing contracts

Coordination mechanism

Independent decision Joint decision making and making of forecasting demand information sharing

Lack of coordination in lot sizing decisions and pricing Mismatch in timing of order Independent cost calculation Different order processing times of suppliers and incentive conflicts Need of risk sharing

Boyaci and Gallego (2002)

Zhao et al. (2002)

Coordination problem

Author

Table 4 (continued) Performance measure

Pareto improvement

Maximize profits, minimize costs (holding + shortage)

Minimize cost and improve service level Minimize cost

Minimize cost

Fuzzy AHP and genetic Minimize costs (inventory algorithm carrying + production + transportation)

Analytical model

Multi agent technology Minimize inventory holding cost and genetic algorithm Mathematical model Minimize distribution cost and lead and simulation time Analytical model Minimize costs (ordering + holding)

Mathematical

Analytical model (extension to newsboy model)

Analytical model

Simulation

Analytical optimization Maximize channel profits (wholesale problem price-inventory related costs)

Methodology used

52 K. Arshinder et al.

a

Multi-echelon

Joint production distribution cost minimization Joint cost minimization with global BOM

Coordinate production scheduling and vehicle routing

Multi-echelon

Multi-echelon

Multi-echelon

Near optimal cost and service Single-manufacturer– level, plan jointly single-distributor– single-retailer

Joint production distribution cost minimization

Conflict in finding Joint decision making and Single-supplier–multinumber of deliveries quantity discounts buyer of an order by vendor and buyer Holding costs increases For different holding costs of Single-supplier–singleas goods move members, find order buyer downstream in supply quantity and share chain benefits Managing complexities Synchronizing production Multi-echelon (5 cycles and risk pooling levels) effects

Lack of integration in different processes of supply chain Conflict between large batch size (production) and small batch size (distribution) Costs of carrying inventory at multi location, results more inventory level in whole supply chain Lack of integration in different processes Need for coordinating production and distribution

Clients can be supplier, manufacturer, distributor and retailer

Hwarng et al. (2005)

Hill and Omar (2006)

Production-inventory Yang and Wee (2002)

Jang et al. (2002)

Ganeshan (1999)

Chandra and Fisher (1994)

Pyke and Cohen (1993)

Production-distribution Jayaraman and Pirkul (2001)

Simulation

Mathematical model

Mathematical model

Mathematical and simulation Lagrangian heuristics and genetic algorithm

Local improvement heuristics

Constrained optimization problem

Lagrangian relaxation scheme

Average stock level, average backlog and average total cost

Minimize costs (production + shipping + holding)

Minimize costs (holding + ordering)

Minimize costs (purchasing + production + distribution) Minimize costs (production + distribution)

Minimize costs (fixed cost of facilities + holding + distribution)

Production cost and service level

Minimize costs (purchasing + production + distribution)

A Review on Supply Chain Coordination 53

54

K. Arshinder et al.

of institutional economics, discussed business issues regarding integration of these two functions and how this connection is handled in education. Jayaraman and Pirkul (2001) developed an integrated production distribution model comprised fixed cost, purchasing cost, production cost and distribution cost, taken simultaneously. Pyke and Cohen (1993) presented an integrated production distribution model and examined its performance characteristics (production cost and service level). Hill (1997) determined the production and shipment schedule for an integrated system to minimize average total cost per unit time. Kim et al. (2006) developed a mathematical optimization problem in multiple plants in parallel and single retailer supply chain system. The joint optimization of costs was carried out to determine the production cycle length, ordering quantity and frequency, and production allocation ratios for multiple plants. Dotoli et al. (2005) proposed a three-level hierarchical methodology for a supply chain network design at the planning management level. The network is so designed where the members are selected based on the performance followed by optimizing the communication and transportation links of supply chain. The performance measures used were operating costs, cycle time, energy saving, product quality and environmental impact.

3.2.2

Procurement and Production Coordination

Goyal and Deshmukh (1992) reviewed the literature on Integrated ProcurementProduction (IPP) systems. The different models of IPP were classified into the categories based on number of products, planning horizon, solution method employed, joint replenishment orders, and algorithmic issues in their study. Munson and Rosenblatt (2001) presented a purchasing-production integrated model and compared the cases of centralized SC and decentralized SC. It was found that decentralized SC gives same results as that of centralized supply chain if quantity discounts are considered at both upstream and downstream interfaces.

3.2.3

Production and Inventory Coordination

Lu (1995) considered heuristics approach for single vendor multi-buyer problem based on equal sized shipments. With the coordination of the replenishments of different items, the vendor can reduce his total annual cost by 30%. The buyers also benefit from the multi-buyer model by reducing their costs. Hoque and Goyal (2000) developed an optimal solution procedure for optimal production quantity in single vendor single buyer production inventory system with unequal and equal sized shipments from the vendor to the buyer and under the capacity constraint of the transport equipment by using simple interval search approach. Arreola-Risa (1996) considered the situation of multi-item production–inventory system with stochastic demands and capacitated production under deterministic or exponentially distributed unit manufacturing times. The observed results are that variation

A Review on Supply Chain Coordination

55

in the production environment increases the optimal inventory levels. The impact of capacity utilization in optimal base stock level is non-linear function of demand rate. Grubbstrom and Wang (2003) developed a multi-level capacity constrained model with stochastic demand. The Laplace transform was used as tool to construct the model and dynamic programming was used to solve and to find out the net present value (NPV) as an objective function. It was observed that for higher levels of capacity, the stochastic solution continues to improve performance of the system, albeit at a very slow rate and then takes advantage of increasing availability of the capacity resources. Kim et al. (2006) considered common production cycle length, delivery frequency and quantity in three level supply chain in joint economic procurement, production and delivery policy.

3.2.4

Distribution and Inventory Coordination

Jayaraman (1998) developed an integrated mathematical programming mixedinteger model for minimization of the total distribution cost associated with all three decision components i.e. facility locations, inventory parameters and transportation alternative selection, all investigated jointly. The integrated model permits a more comprehensive evaluation of the different trade-off that exists among the three strategic issues. Yokoyama (2002) developed an integrated optimization model of inventory-distribution system in which any consumer point can be supplied by multiple distribution centers. The order-up-to-R, periodic review inventory policies and transportation problem are considered simultaneously. Simulation and linear programming was used to calculate the expected costs and a random local search method was developed to determine optimum target inventory, which was then compared with genetic algorithm. Haq et al. (1991) formulated a mixed integer programming for integrated production–inventory-distribution model. The objective of the model was to determine optimal production and distribution quantities through various channels, optimal levels of inventory at various production stages and at warehouses over 6-month planning periods considering set up time cost, lead time, production losses and recycling of losses with backlogging. Observations and Gaps in Different Activities and at the Interfaces of Supply Chain (a) In the literature, different problems in coordinating the activities with various approaches have been discussed. The main objective considered in coordinating different problems in some activity is either minimizing the costs or maximizing profits. The coordination of same activities at different levels of supply chain reduces the supply chain costs. (b) The common problems addressed in literature are the joint consideration of different costs in an activity. These costs are associated with the supply chain coordination problems of joint ordering by buyers to some supplier, jointly plan order quantity between supplier and buyer, jointly order delivery to the buyers and joint replenishment activities in terms of coordinated lead times.

56

K. Arshinder et al.

The coordination problems have also been extended for coordinating different processes to collectively consider the costs of different processes to minimize the overall cost of supply chain. (c) The methodologies adopted to tackle the problem include: analytical, mathematical and optimization tools. Most of the studies regarding SCC are conducted on a two-level dimension because of the simple supply chain structure (Ganeshan 1999; Hill 1997) and discussed production delivery policies and joint stocking with discounts (Weng and Parlar 1999) at two level supply chain. To effectively allocate the production requirement and capture supply chain dynamics, various models have been dealt with joint purchasing policies in multiple supplier environment (Zou et al. 2004) and considering total cost of logistics. The investigations are required in supply chain encompassing multiple levels that consider the complex interactions between the upstream and downstream sites and gives a more real picture of supply chain. (d) The following are some gaps, which if considered, may further enhance coordination and performance of supply chain: • The whole supply chain is required to coordinate, so models can be extended to consider more than one activity. • The only coordination mechanism used by most of the authors is joint consideration of costs. From the literature regarding coordination models it can be observed that a number of coordination mechanisms (information sharing, roles integration, information technology) are possible to solve the coordination problem. There can be situations where two mechanisms are required to reduce the supply chain costs for example information sharing and quantity discounts. • The consideration of one performance measure may not justify the value of coordination. So, a number of performance measures are required to capture the impact of coordination in a holistic manner. Along with the measures like costs and profits, the benefits of coordination may also be indicated with the help of performance measures like: improving responsiveness by timely information sharing in whole supply chain, reducing inventory delays and information lead time by implementing good information systems and evaluating risks and rewards due to coordination. • The analytical and mathematical approaches used to coordinate activities and processes of supply chain may not tackle the dynamics of supply chain. Hence, simulation approach may be a good choice to view the overall coordination scenario of the whole supply chain. • Most of the studies on coordination are done for two level supply chains. This assumption may restrict the usage of models, as these models may not handle the ever-changing variables of supply chain. • The assumption of integrated different functions and processes leads to cost reduction, but models are required to evaluate or measure the degree of coordination (which leads to improvement in the supply chain performance).

A Review on Supply Chain Coordination

57

• The recent trend of outsourcing the logistics operations to third party logistics provider (3PL) has reduced many discrepancies related to replenishment of goods (Jayaram and Tan 2010). The studies are required how to 3PLs can be an information source to coordinate suppliers and buyers. The knowledge and expertise of 3PLs on routes, fleet size and fleet type can be leveraged in optimizing the procurement-production-distribution problems and integrating with 3PLs. • To gain the advantage of common logistics provider and information systems, the supply chain members at same level may coordinate horizontally. Very few papers have discussed horizontal collaboration (Arshinder et al. 2006; Bahinipati et al. 2009) by using multi-criteria decision making models. Some quantitative models can be proposed to quantify such kind of coordination also. In this section we can observe that how supply chain coordination is required in each SC process. Various processes have been coordinated by adopting different means mechanisms of coordination. By looking at the need of coordination in SC, the researchers may like to know various existing coordination mechanisms, which can be adopted to coordinate supply chain across different industries. The next section presents various coordination mechanisms, which can be adopted as per the suitable supply chain environment.

4 Various Supply Chain Coordination Mechanisms Adopted in the Supply Chain The dependencies between supply chain members can be managed by some means and mechanisms of coordination. By utilizing coordination mechanisms, the performance of supply chain may improve. There are different types of coordination mechanisms as discussed in the following subsection.

4.1

Supply Chain Contracts

Supply chain members coordinate by using contracts for better management of supplier buyer relationship and risk management. The contracts specify the parameters (like quantity, price, time, and quality) within which a buyer places orders and a supplier fulfills them. The objectives of supply chain contracts are: to increase the total supply chain profit, to reduce overstock/understock costs and to share the risks among the supply chain partners (Tsay 1999). The contracts counter double marginalization that is by decreasing the costs of all supply chain members and total supply chain costs when they coordinate as against the costs incurred

58

K. Arshinder et al.

when the SC members act independently. The problem of double marginalization and risks like overstock and understock has been widely been observed single period inventory models with less shelf life of product. Most of the contracts have been proposed as single period models. Various contracts are defined in Table 5. Buyback contracts or returns policy has been widely used coordination contract in textile and fashion industry. In buyback contracts a manufacturer offers retailer either full credit for a partial return of goods a partial credit for all unsold goods. In case of retail competition the manufacturer will be benefited from the returns policy when the production costs are sufficiently low and demand uncertainty is not too great (Padmanabhan and Png 1997). Krishnan et al. (2004) have analyzed that Table 5 Definitions of supply chain contracts S. Supply chain Definition No. contract 1 Buy back The manufacturer (seller) agrees to buy back the unsold units from the retailer (buyer) for agreed upon prices at the end of the selling season 2 Revenue In a revenue sharing contract, the sharing buyer shares some of his revenues with the seller, in return for a discount on the whole sale price 3 Sales rebate The sales rebate provides a direct incentive to the retailer to increase sales by means of a rebate paid by the supplier for any item sold above a certain quantity 4 Quantity It couples the customer’s flexibility commitment to purchase no less than a certain percentage below the forecast with the supplier’s guarantee to deliver up to a certain percentage above 6 Trade policy This policy deals with how the total profit is shared among supply chain entities 7 Reservation This policy offers discounts to the policy products reserved and the products which are not reserved are sold at retail price After the selling season, the unsold 8 Markdown money units are sold at discounted price (price discount) 9 Quantity During the selling period, the seller discount offers discounts based on quantity of goods purchased

Author (year)

Remarks

Mantrala and Improves the coordination, Raman increases (1999), Hau and Li (2008) sales, risk sharing Yao et al. (2008), More flexible in terms in terms Zhou and of whole sale Wang (2009) price Wong et al. Provides direct (2009) incentives for retailers to increase sales Tsay (1999)

Gives more flexibility in order quantity

Ding and Chen (2008)

Offers better profit sharing

Chen and Chen (2009)

Reduces the uncertainty in demand

Lee (2001), Pan et al. (2009)

Improves profit of the channel

Weng (2004)

Improves the sales

A Review on Supply Chain Coordination

59

buyback contract coupled with promotional cost sharing agreements between manufacturer and retailer result in supply chain coordination. The other consideration in buyback contract is the case of information sharing and asymmetrical information between the supply chain members (Yao et al. 2005; Yue and Raghunathan 2007). Bose and Anand (2007) proposed that by assuming transfer price exogenous the buyback contract is Pareto efficient. Yao et al. (2008) proposed an analytical model to analyse the impact of stochastic and price dependent demand on returns policy between manufacturer and retailer. The other variants of buyback contracts discussed in literature are: stochastic salvage capacity in fashion industry (Lee and Rhee 2007); two period contract model in case of decentralized assembly system (Zou et al. 2008); in case of updating of information in supply chain (Chen et al. 2006) and by including the risk preferences of the SC members (He et al. 2006). In case of quantity flexibility contract, the buyer is allowed to modify the order within limits agreed to the supplier as demand visibility increases closer to the point of sale. The buyer modifies the order as he gains better idea of actual market demand over time. Tsay and Lovejoy (1999) proposed quantity flexibility contracts for two independent members of the supply chain model to design incentives for the two parties to determine system wide optimal outcome. The efficiency can be improved when buyer is ready to pay more to the supplier for increased flexibility. Tsay and Lovejoy (1999) proposed a framework for the design of quantity flexibility in three level supply chains, behavioural models in response to quantity flexibility contracts and the impact on the supply chain performance measures: inventory levels and order variability. More output flexibility comes at the expense of greater inventory cost, so inventory management has been viewed as the management of process flexibilities. It is observed that the quantity flexibility contracts can dampen the transmission of order variability throughout the supply chain. Milner and Rosenblatt (2002) analysed two period quantity flexibility contract in which the buyer is allowed to adjust second order paying a per unit order adjustment penalty. This contract can reduce the potentially negative effect of correlation of demand between two periods, but the order quantity flexibility reduces the profits of the buyer. Barnes-Schuster et al. (2002) proposed two period options contracts where buyer has flexibility to respond to market changes in second period and coordinate the supply chain channel. Sethi et al. (2004) developed a model to analyze a quantity flexibility contract involving multiple periods, rolling horizon demand and forecast updates including demand and price information updates. In revenue sharing contract, the supplier charges the buyer a low wholesale price and shares a fraction of the revenues generated by the buyer (Giannoccaro and Pontrandolfo 2004; Cachon and Lariviere 2005; Koulamas 2006). The SC members can design contracts based on discounts: lot size based or volume based. Yao et al. (2008) developed a revenue sharing model in the case of retail competition by considering price sensitivity. vander Rhee et al. (2010) has considered multi echelon (more than two) supply chain members and simultaneously installed revenue sharing contracts between all pairs of adjacent supply chain members to coordinate the supply chain.

60

K. Arshinder et al.

A discount is lot size based if the pricing schedule offers discounts based on the quantity ordered in a single lot. A discount is volume based if the discount is based on the total quantity purchased over a given period regardless of the number of lots purchased over that period (Rubin and Benton 2003; Weng 2004). Chauhan and Proth (2005) proposed a profit sharing model under price dependent demand proportional to their risks based on expected customer demand.

4.2

Role of Information Sharing and Information Technology

IT is used to improve inter-organizational coordination (McAfee 2002; Sanders 2008) and in turn, inter-organizational coordination has been shown to have a positive impact on select firm performance measures, such as customer service, lead-time, and production costs (Vickery et al. 2003). Information technology helps to link the point of production seamlessly with the point of delivery or purchase. It allows planning, tracking and estimating the lead times based on the real time data. Advances in Information Technology [e.g. internet, EDI (electronic data interchange), ERP (enterprise resource planning), e-business and many more] enable firms to rapidly exchange products, information, and funds and utilize collaborative methods to optimize supply chain operations. Internet and web can enhance effective communication, which helps members of supply chain review past performance, monitor current performance and predict when and how much of certain products need to be produced and to manage workflow system (Liu et al. 2005). Fin (2006) investigated the relation between EDI in apparel industry and three performance levels: operational, financial and strategic. This helped in reduction of lead time from several weeks to 3 days. According to Soliman and Youssef (2001), e-business strategy refers to the way internet tools are selected and used in relation to the needs of integration and coherent with other organizational and managerial tools: e-commerce (Swaminathan and Tayur 2003) can be used to support processes such as sales, distribution and customer service processes, support to sourcing, procurement, tendering, and order fulfillment processes, and e-manufacturing (Kehoe and Boughton 2001). Devaraj et al. (2007) analyzed the relationship between supplier integration and customer integration with supply chain performance when supported by e-business technologies. E-business capability supporting supply chain technologies such as customer orders, purchasing and collaboration between suppliers and customer enhances the production information integration intensity, which in turn improves the supply chain performance. Skipper et al. (2008) proposed a conceptual model to link level of interdependence among supply chain with supply chain performance moderated by different types of IT needed to achieve different levels of coordination. The framework is supported by interdependence theory and coordination theory. The coordination processes between globally dispersed and mobile supply chain members is becoming more and more information intensive. The recent trends in intelligent wireless

A Review on Supply Chain Coordination

61

web services have proved enhancement in the mobile real time supply chain coordination (Saroor et al. 2009). The various coordination problems handled by information systems are: little value to the supplier because of competitive bidding, forced implementation of IT, incompatible information system at different levels of supply chain, greater lead times, inefficient purchase order and misaligned e-business strategies and coordination mechanisms (Porter 2001). Stank et al. (1999) report that the food firms benefit from more accurate and timely information and IT or EDI improves inventory management and helps in comprehension of the order cycle. Yusuf et al. (2004) examined key dimensions of implementation of ERP system in Rolls Royce. The implementation of latest information system only may not be sufficient to integrate supply chain members, since at times; faulty implementation may result in the poor performance of supply chain. Li et al. (2009) carried out an empirical study to explore relationship between IT, supply chain integration and supply chain performance of Chinese manufacturing organization. Supply chain integration mediates the relationship between IT implementation and supply chain performance. Hence, IT can be a good enabler to integrate supply chain. But it is important to take into account the justification of IT in changing business environment. It must take into account the appropriate usage, investment justification and align with business environment to achieve competitive advantage (Gunasekaran et al. 2006). The supply chain members coordinate by sharing information regarding demand, orders, inventory, shipment quantity, POS data, etc. Timely demand information or advanced commitments from downstream customers helps in reducing the inventory costs by offering price discounts and this information can be a substitute for lead time and inventory (Reddy and Rajendran 2005). The value of information sharing increases as the service level at the supplier, supplier-holding costs, demand variability and offset time increase, and as the length of the order cycle decrease (Bourland et al. 1996; Chen et al. 2000). The higher the level of information sharing, the more important the effective supply chain practice is to achieve superior performance (Zhou and Benton 2007). Some comparative studies have done in which no information sharing policy is compared with full information sharing policy. Information sharing policy results in inventory reductions and cost savings (Yu et al. 2001). Cachon and Fisher (2000) presented a simulation-based comparative study, where the supply chain costs are 2.2% lower on average with full information sharing policy than with traditional information policy and the maximum difference is 12.1%. Also, this results in faster and cheaper order processing that leads to shorter lead times. The point of sales (POS) data helps the supplier to better anticipate future orders of the retailers and reduces the bullwhip effect (Dejonckheere et al. 2004). The supplier may take advantage of the retailers’ inventory information in allocating the stock to retailers optimally (Moinzadeh 2002). Ding et al. (2011) has investigated the mechanism of providing incentive to retailer by upstream partner for implementing demand information sharing in the context of three-echelon supply chain system. A cooperative game approach is

62

K. Arshinder et al.

proposed to address the problem of profit allotment between partners to effectively motivate the partners to be cooperative with each other.

4.3

Other Collaborative Initiatives

Joint consideration of replenishment (Yao and Chiou 2004; Chen and Chen 2005), inventory holding costs with dynamic demand (Boctor et al. 2004), collaborative planning (Aviv 2001), costs of different processes (Haq and Kannan 2006; Jayaraman and Pirkul 2001; Ganeshan 1999), frequency of orders (Yang and Wee 2002; Barron 2007), batch size (Pyke and Cohen 1993; Boyaci and Gallego 2002), product development (Kim and Oh 2005) to improve the performance of supply chain. A supply chain member may design a scheme to share profits at the end of period. The supply chain members share profit by determining optimal order quantity of single supplier and multi-buyer supply chain and achieve coordination (Jain et al. 2006). A coherent decision-making helps in resolving conflicts among supply chain members and in exceptions handling in case of any future uncertainty. There are many factors involved in achieving coordination like human, technology, strategies, relationship, rewards, sharing of knowledge, sharing benefits, aligning goals, scheduling of frequent meetings of stakeholders for conflict resolution, understanding of nature of intermediates and knowledge of supply chain concepts, status or power difference and resistance in following the instructions of other organizations (Lu 1995; Gittell and Weiss 2004). Simatupang et al. (2004) explored a fashion firm to see how coordination is driven by its responsibility interdependence, uncertainty, and inter-functional conflict. By properly identifying different points of coordination, the performance improvement was effected. Vendor Managed Inventory (VMI) is a supply chain initiative whereby a supplier assumes responsibility for maintaining inventory levels and determining order quantities for its customers. A number of benefits from VMI adoption have been reported in literature: reduction in inventories, shorter order intervals and more frequent deliveries. A VMI program typically involves the use of a software platform, the sharing of demand forecasts and/or cost information, timely communications, set liability levels, and risk-sharing parameters and common goal sharing between the buyer and the supplier. VMI can be particularly beneficial in the products with high demand variance and high outsourcing costs (Cheung and Lee 2002). Collaborative Planning, Forecasting and Replenishment (CPFR) is a collaboration initiative where two or more parties in the supply chain jointly plan a number of promotional activities and work out synchronized forecasts, on the basis of which the production and replenishment processes are determined (Larsen et al. 2003). Some of the benefits of CPFR are increased sales, higher service levels, faster order response time, lower product inventories, faster cycle times, reduced capacity requirements, reduced number of stocking points, improved forecast accuracy and lower system expenses. Danese et al. (2004) explored the relationship between the

A Review on Supply Chain Coordination

63

types of interdependencies (one way and two way communications) among the units involved in the CPFR processes and the activated coordination mechanisms (Liaison positions, meetings, task forces, standing committees and integrating managers) in three case studies for all the steps of implementation of CPFR. The case studies were considered from different industries: pharmaceutical, automotive and mechanical. This relationship may help managers in the decision making process to select the most appropriate action to perform to implement CPFR. Quick response (QR) is another inventory management initiative which can be undertaken to coordinate supply chain members by responding quickly to market changes with reduced lead time. The response time is reduced as a retailer sends POS data to its supplier. The supplier makes use of this information to improve the demand forecast and production/distribution schedules (Iyer and Bergen 1997; Simchi-Levi et al. 2007). Choi and Sethi (2010) have reviewed QR supply chains from both supply and demand perspectives and classified the literature as supply information management, demand information management and supporting technologies. It is concluded that there are challenges to implement QR in multiple decision points, which needs to be met by continuously innovating new technologies like Radio Frequency Identification Devices (RFID). The Supply Chain Operations Reference (SCOR) model helps in evaluating and improving enterprise wide supply chain performance and management. SCOR is structured on four levels: plan, source, make and deliver. It brings order to the diverse activities that make up the supply chain, and provides common terminology and standard process descriptions. The model allows companies to: evaluate their own processes effectively, compare their performance with other, companies both within and outside their industry segment, pursue specific competitive advantages, use benchmarking and best practice information to prioritize their activities, quantify the benefits of implementing change and identify software tools best suited to their specific process requirements (Huan et al. 2004). Observations and Gaps in Coordination Mechanisms (a) The supply chain contracts can be a useful mechanism to resolve the conflict and risk related problems. The use of information technology in handling transactions online between supply chain members reduces the response time. The members can plan their operational activities by sharing or retrieving the data from each other. It helps in streamlining the processes and reduces supply chain costs. (b) The members might have different technologies, skill and different type of knowledge about market. To handle any future exceptions or uncertainties, the members may jointly plan supply chain activities like ordering, replenishment, and forecasting and product design. (c) The following gaps regarding coordination mechanisms need attention to enhance coordination: • Since the role and utility of all coordination mechanisms is handling different phases of supply chain. To coordinate supply chain as a whole,

64

K. Arshinder et al.

the consideration of all coordination mechanisms may give very good performance. • Most of the models describing coordination mechanisms are dealt in two level supply chain, which can be extended to multi-level supply chain. The relation between different coordination mechanisms and the performance measures of supply chain need to be developed. The models handling the problems of coordination have emphasized on single performance measures. The supply chain dynamics may be captured by considering a number of performance measures of supply chain. • Supply chain contracts are designed to motivate the downstream member to order more than his/her optimal order quantity. The downstream member always faces uncertainty of overstock or under stock. The upstream member always faces uncertainty that whether the downstream member will send the order matching the upstream member’s capacity. The contracts like buyback and revenue sharing contracts can enhance expected sales and reduces stock outs. Quantity flexibility contracts can reduce the overstock problems of downstream members. These performance indicators are equally important, which needs more research attention. • The contract decision variables at different interfaces of the supply chain in multi echelon environment interact with each other. For example the contract adopted by supplier and manufacturer is sometimes dependent on the contract adopted by same manufacturer with his/her distributor in a same supply chain. There is a need to explore such relationship and to explore different combinations of contracts at different interfaces of supply chain. The major driver of SCC is the conflict or uncertainty, which needs to be addressed by selecting suitable coordination mechanism. But, it is important to understand at the same time, to what extent SCC can help in mitigating supply chain uncertainty (presented in the next section).

5 Supply Chain Coordination to Manage Uncertainty in the Supply Chain Supply chain uncertainty has been captured in various forms like supply uncertainty, production or operational uncertainty and demand uncertainty. In the supply chain coordination literature, various coordination mechanisms have been adopted to manage supply chain uncertainty like uncertainty in capacity, demand, lead time, quantity and production and supply disruptions (Tang and Musa 2011) as shown in Fig. 2. Many papers have emphasized on supply chain contracts and information exchange/sharing to manage supply chain uncertainties. Whereas, the other set of papers discussed the joint consideration of costs and profits of all supply chain members while taking decisions regarding ordering and replenishment. This joint

A Review on Supply Chain Coordination Supply side uncertainty

Supplier Production/ Operational uncertainty

65 Supply side uncertainty

Buyer Production/ Operational uncertainty

Buyer’s side uncertainty

Supplier

Coordination mechanisms

Demand uncertainty

Buyer

Coordinate as SC members are part of one system to manage uncertainty and to share risks and rewards

Supply Chain Performance Improvement Fig. 2 Managing supply chain uncertainty with supply chain coordination

consideration of costs or profits (centralized system) helps to improve the performance of supply chain over a decentralized case (independent decision making). Due to the increased technological innovations, the products’ lifecycle has largely shortened. Seasonal and perishable goods can be attributed to this kind. Such products have longer production and delivery lead time than their selling season (Mantrala and Raman 1999). So the orders should be placed before the selling season starts. Some of the important challenges in integrating the supply chain are tackling issues such as managing complex supply chain structures, demand uncertainty and leftover units after selling season. In a single period inventory model, better coordination can be achieved by inducing the retailer/ buyer to order more in order to avoid their risk of under stocking through some negotiations with the manufacturers/seller. The manufacturer offers integrated decision making policies like returns policy, sales rebate policy, price discount/ volume discount policy, etc. to raise the order quantity and improves sales (Yao et al. 2008). Past research has proved that introduction of various contracts improve the performance of the supply chain as well of each entity in supply chain.

66

K. Arshinder et al.

The contracts have been discussed for single period inventory models with either deterministic demand or uncertain demand or price dependent demand. Apart from contracts there can be some incentive function to achieve flexible cost allocation between supplier and buyer to coordinate the supply chain and to manage uncertainty in supply (Zimmer 2004). Hou et al. (2010) have proposed a model considering one manufacturer and two suppliers: the main supplier is cheaper but prone to disruption risk and backup supplier is more reliable but an expensive. The authors have developed a non-linear optimization model to determine the optimal values of buyer’s order quantity and optimal buyback price under both supply and demand uncertainty. Early supply involvement reduces the likelihood of supply disruptions and negative supply events (Zsidsin and Smith 2005). The contracts like advanced purchase commitments can help mitigating supply uncertainty, where unsatisfied demand can be backordered from risky supplier (Serel 2007). The other kind of uncertainty due to disruption can be observed as disruptions in the production process at manufacturer’s facility. Qi et al. (2004) proposed a model for short life cycle product with demand as decreasing function of retail price considering disruptions. The model considered the two periods wherein the second period demand change can lead to the disruption, which may affect the production plan of supplier. The wholesale quantity discounts may coordinate the supply chain in this scenario of disruption. The similar kind of disruption can be seen in terms of production costs. Xiao and Qi (2008) developed two-period model for onemanufacturer and two competing retailers supply chain under production costs disruption. The authors have analyzed two mechanisms; an all unit quantity discounts and incremental quantity discounts the under production disruptions for possible coordination scenarios. A risk sharing contract is proposed where at the end of period the retailer compensates manufacturer’s losses due to overproduction or manufacturer compensates retailer’s losses due to over stock in case of supply chain with two stage demand information updating (Chen et al. 2006). There can be several benefits of splitting the single period order into multiple ordering to update the demand information and revise the order in the subsequent orderings. It has impact on production costs of the manufacturer due to slow production and fast production as against the multiple different orders (Liu et al. 2004). The other effect of multiple ordering can be seen on holding cost, lead time, backorders, varying wholesale and retail price and consideration of demand for multiple periods. The methodology adopted for handling multiple ordering ranges from newsboy problem to analytical models with simulation to the dynamic programming. The decision variables have been the order quantities and/or the varying wholesale prices, retail prices and buyback prices in multi-period situation (Lee 2007; Zhou and Wang 2009; Pan et al. 2009). Other aspect of capturing demand uncertainty is by using fuzzy demand. The expected profits of coordinated supply chain outperform the expected profits in the case of no coordination under fuzzy demand (Xu and Zhai 2010). Barbarosoglu (2000) has proposed a decision support model for improving supplier–buyer coordination by using supply contracts where the buyer’s commitment is considered as a

A Review on Supply Chain Coordination

67

function of time at the contract renewal time to reduce the supply chain nervousness. A pricing model is formulated to address partnership expectations for a fair sharing of savings of the supply chain members. Observations and Gaps in Uncertainty and Supply Chain Coordination (a) Most of the studies are restricted to two level serial supply chains. In reality, supply chain can have divergent and convergent multi-echelon structures. The literature seems lacking to address the uncertainty concerns in such structures. (b) The literature has emphasized more on demand uncertainty, whereas, supply uncertainty can be of equal concern in the era of globalization and outsourcing. Moreover, the quantitative models can be proposed to explore the impact of supply uncertainty on supply chain performance. (c) There are very few studies on splitting the single period order into multiple orders. The supply chain members can take advantage of more accurate information over a period of selling season and hence resolve supply chain inefficiencies. (d) The buyback contract is the only contract which has been discussed in multi ordering models to manage the risk. There is a scope to explore combination of other contracts in multiple ordering over single season.

6 Discussion A number of difficulties in SCC are identified based on the literature. These difficulties have been identified from different activities, interfaces and the number of levels in the supply chain. It has been realized that the difficulties in SCC and independent working of supply chain members lead to poor performance. The coordination problems are solved by implementing some coordination mechanisms in supply chain activities, which may result in the improvement of some performance measures. The SC activities have been considered in isolation to solve their respective coordination problem. The coordination problems may not be same in all activities of supply chain. The requirements of coordinating whole SC may vary with SC activity, with some interface of SC, with number of echelons in SC and with process of SC. There are different activities and different coordination problems in whole supply chain. Coordinating one activity may not help to improve supply chain system wide performance.

6.1

Existing Models of Coordination and the Gaps in These Models

There are some initiatives and models (such as CPFR and SCOR) which may help in collaboration along the supply chain. These models consist of so many steps and

68

K. Arshinder et al.

the implementation of such processes takes time. Various guidelines are required to implement these models in practice. It is difficult to link the guidelines directly to the performance of supply chain. It may take a number of years to know the performance improvement by implementing these models, as there is no set measure to quantify coordination which can be linked with practice (or which may result due to implementation of these models) of these models. It is difficult to get a quantitative measure after implementing models like CPFR and SCOR, which may indicate about whether SC is coordinated or not. The coordination models discussed have different performance measures at single level and at interface of supply chain, which are not aligned with the whole supply chain. To monitor coordination in supply chain, same performance measures throughout will help in evaluating the value of coordination. There are different mechanisms, which when applied, result in different trade-offs of performance measures of coordinated supply chain because of different characteristics of performance measures. The complexity of considering whole supply chain and the performance trade-offs cannot be handled with the models discussed in the literature. These difficulties can be easily tackled by approaches like fuzzy logic (Ross 1997) and multi-objective genetic algorithms (Deb 2002). Fuzzy logic is applied in the situation where understanding is quite judgmental and the processes where human reasoning and human decision making is involved like the complexities in supply chain. The optimum values of decision variables in multi objective environment can be easily determined with the help of tools like Genetic Algorithm.

6.2

Proposed Framework to Quantify Coordination

The controlling parameter of achieving coordination is the impact of application of coordination mechanisms (CMs) on the performance measure. It can be observed from the Decision-coordination mechanism matrix given in Table 6 that how different coordination mechanisms can be used for various supply chain decisions. The proper implementation and usage of coordination mechanisms improve the performance of the supply chain (Arshinder 2008). It can be observed that the problems and conflicts in coordinating the supply chain members can be resolved through coordination mechanisms. The importance of coordination mechanism may help in determining the value of coordination in supply chain.

6.2.1

Framework Using Various Coordination Mechanisms

A framework has been proposed based on the usage of coordination mechanisms and their importance in managing uncertainty and resolving various kinds of conflicting problems in coordination. The coordination mechanisms can be classified as:

Coordination mechanism

Supply chain network Integrated procurementproduction

Integrated production-distribution Joint consideration of cost

X

X

Coordinated timing of the order

Coordinated timing of replenishment Inventory management in a network Forecasting

X

X

X

X

X

X

Supply Information chain technology contracts

Inventory Coordinated order quantity

Logistics Coordination issues in 3PL provider and customer Integrating the logistics activity geographically dispersed network/supply chain

Supply chain decision

Table 6 Decision-coordination mechanism matrix

X

X X

X

Jayaraman and Pirkul (2001), Pyke and Cohen (1993), Kim et al. (2006) Dotoli et al. (2005) Goyal and Deshmukh (1992), Munson and Rosenblatt (2001)

Aviv (2001)

X

X

Speed of delivery, status of order, accuracy of information, invoicing on delivery, cash-flow improvements, accurate invoicing, transportation costs, warehousing costs, inventory levels, ordering costs, stock-outs, order cycle time, order cycle variance, on time deliveries

Performance measures

Supply chain system wide cost

Supply chain system wide cost

Bullwhip effect, holding cost, system wide cost

Li et al. (1996), Boyaci and Gallego (2002), Inventory levels, ordering costs, customer Piplani and Fu (2005), Zou et al. (2004), service level, holding costs, product Wu and Ouyang (2003) variety, purchasing costs, product availability, unacceptable delivery, Lu (1995), Moses and Seshadri (2000), system wide costs Gurnani (2001), Zhao et al. (2002) Huq et al. (2006), Barron (2007)

Stock et al. (2000)

Huiskonen and Pirttila (2002)

Authors

Verwijmeren et al. (1996)

X

X

X

X

Joint decision making

X

X

X

X

Information sharing

A Review on Supply Chain Coordination 69

70

• • • •

K. Arshinder et al.

Supply chain contracts (M1) Information Technology (M2) Information sharing (M3) Joint decision making (M4)

This is not an exhaustive list of coordination mechanisms. These coordination mechanisms can be different in number as per the requirements of supply chain for example dependent on the type of industry and type of interdependencies between SC members. In the present framework four coordination mechanisms are considered because of their extensive discussion in literature. It can be assumed without any loss of generality, that if the coordination mechanism is applied properly, it will help in achieving SCC. Since, supply chain involves certain members who are human beings and the human system is the most complex system to be managed in organization study. There is bound to be conflicts and problems in the traditional supply chain, which call for an urgent need to implement coordination mechanisms in supply chain. The coordination mechanisms are from different domains, require different conditions and can operate in different situations. But, one thing common in all mechanisms is that all mechanisms are implemented to improve the performance of supply chain and to resolve confusion and uncertainty among SC members due to independent decision making. To know more about the importance of coordination mechanism, one way is to study all the activities in some process, identify the dependent activities in that process and select the coordination mechanism to coordinate all activities of a process (Arshinder et al. 2006). Since, whole supply chain needs to be coordinated; the usage of all four coordination mechanisms and performance improvement achieved by these mechanisms will help in evaluating SCC. A better way to find some quantitative index of supply chain coordination is by incorporating the strength of coordination mechanisms by following steps shown in Fig. 3. The quantitative index can be represented as Supply Chain Coordination Index (SCCI) can be viewed as a function of implementation of coordination mechanisms. SCCI for four coordination mechanisms can be represented as: SCCI ¼ f ðM1; M2; M3; M4Þ The above function is to be formulated in such a way that the combined impact of performance improvement by using all mechanisms is considered. This formulation poses two challenges: 1. It is required to represent all coordination mechanisms with a unique scale. 2. It is required to evaluate improvement in performance measures qualitatively or quantitatively by using coordination mechanisms. The methodologies like AHP and Fuzzy logic may help to represent coordination mechanisms with a unique scale. The performance improvement can be captured either empirically with the help of judgments given by managers or

A Review on Supply Chain Coordination

71

Define the structure of supply chain

Set performance measures for whole supply chain

Choose input variables

Run the simulation for The case when the members are working independently Observe the impact on performance measures and set as PM(w/c) (without coordination)

Select the coordination mechanisms and run the simulation

Supply chain contracts (M1) Buyback Revenue sharing Quantity flexibility Quantity discounts

Determine performance measure (PMM1C)

Information Technology (M2) Email Internet EDI ERP POS

Determine performance measure (PMM2C)

Information sharing (M3) Demand Inventory Lead time Production schedule Capacity Cost

Determine performance measure (PMM3C)

Joint decisionmaking (M4) Cost consideration Replenishment Forecasting Ordering

Determine performance measure (PMM4C)

Determine the percentage improvement in performance measures with respect to case of without coordination PMMi = (PMMiC – PMMi (w/c))/ PMMi (w/c) for i=1,2,3,4

Assign weights to different coordination mechanisms (WMi, for i= 1,2,3,4) based on relative improvement of percentage of all CMs by devising some scale (AHP).

WM1

WM2

WM3

WM4

SCCI = WM1PMM1+ WM2PMM2 + WM3 PMM3 + WM4 PMM4

Fig. 3 The proposed model to quantify supply chain coordination index (SCCI)

with the help of simulating the scenarios of using these coordination mechanisms to obtain same performance measures. The improvement in performance measures will motivate supply chain members to implement coordination mechanisms.

72

K. Arshinder et al.

One of the efforts has been proposed based on the implementation of all coordination mechanisms with the help of graph theoretic approach (Kaur et al. 2006). This methodology is based on allocation of relative importance of these coordination mechanisms given by the judgments of managers. These judgments are based on the implementation of mechanisms and the importance of mechanisms based on the performance improvement by these mechanisms.

6.2.2

Relation Between Coordination Mechanisms and Performance Measures with Simulation

A simulation approach can also be a useful tool in capturing the different scenarios of coordination mechanisms and their impact on selected performance measures. Certain assumptions can be considered regarding the levels of supply chain, one period or multiple period model and various operational variables like order quantity, holding and shortage costs, etc. The implementation of various coordination mechanisms can be simulated to analyze same performance measures with same assumptions. Some constraints can be included in the model which takes care of the fact that none of the supply chain member will face losses by implementing coordination mechanisms. The improvement in performance measures will give an idea about the capability of an organization to achieve coordination. The model proposed in Fig. 3, helps in evaluating SCCI. The first few steps can be used in simulation to determine the performance measures. Some input variables may be selected like different costs, price, inventory policies, lead time, capacity and type of coordination mechanisms at all levels of supply chain in a pre defined structure of supply chain. The assumptions for demand (uncertain and price dependent), lead time and time horizon can be set for the simulation and run the simulation to obtain certain performance measures. The performance measures are function of input variables. The problem may be multi objective based on the selected performance measures of supply chain. The results of simulation that is improvement in the performance measures by applying different coordination mechanisms can be combined using again some hybrid frameworks like: AHP, Fuzzy logic and/or Graph theoretic approach to determine SCCI.

6.2.3

Hybrid Framework Using Various Coordination Mechanisms and Simulation

The coordination mechanisms (M1, M2, M3 and M4) have different characteristics and their impact on the performance measures may also be different. The simulation can be carried out without implementing coordination mechanisms and then the results are compared with the situation with considering the coordination

A Review on Supply Chain Coordination

73

mechanisms as shown in Fig. 3. A framework is required which can capture and combine the values of performance improvements by coordination mechanisms and their relative importance. To make the results consistent, the performance improvements can be normalized in terms of percentages. This framework may have capability to find the relative importance of respective coordination mechanisms by using AHP and/or Fuzzy logic (Arshinder et al. 2007). A scale can be devised based on the difference in the percentage improvements by CMs. This scale may help in determining the relative importance or weights of CMs. The linear equation of SCCI can be derived from the proposed model to determine the value of SCCI.

6.3

Insights Gained from Proposed Framework

The proposed framework helps in defining and measuring SCC. Supply chain coordination can be used to enhance system wide performance enabled due to the implementation of coordination mechanisms selected based on the type of industry and the interdependencies between supply chain members keeping in view mutual interests of all SC members. Supply chains can be coordinated by identifying interdependent activities between supply chain members required to accomplish SC objectives. Once interdependencies are identified, some means of mechanism(s) are devised to manage the decision variables pertaining to interdependent activity. The independent evaluation of decision variables of interdependent activities by SC members represents the case of uncoordinated supply chains. Once, coordination mechanism is selected to manage interdependencies, SC members can simulate and compare the scenarios: one with using CM and other without coordination mechanisms. The expected values of improvement in certain performance measures may help to realize the value of coordination. Same steps can be used for all processes of supply chain. Various functions can be explored for SCCI depending on the number and implementation of CMs. Suitable techniques can be used such as Multi-CriteriaDecision-Making (MCDM) models to quantify SCCI as a function of various CMs.

6.4

Surrogate Measures of Supply Chain Coordination

To innovate continuously is the base line for all the organizations, which makes the supply chain more dynamic in nature. It is important to capture the performance of supply chain. The highly uncertain environment in supply chain brings in the challenges to have fix kind of performance measures. Gunasekaran et al. (2001) developed a framework for measuring supply chain performance for each activity

74

K. Arshinder et al.

of plan, source, make and deliver under strategic, tactical and operational decisions. The literature on supply chain performance measures is lacking in presenting standard performance metrics. The problem manifolds when the question comes to measure supply chain coordination. There is scarcity of studies to evaluate coordination in supply chain. The following performance measures can be good indicators of supply chain coordination. (a) Supply chain profitability. Joint consideration of order quantity, costs or profits may lead to improvement in supply chain performance. Regardless of the number of entities in supply chain, the joint consideration of order quantity in supply chain for single period model improves the profitability of whole supply chain (Arshinder et al. 2009a). Most of the contracts reported in literature have expected profits as a performance indicator too. (b) Supply chain flexibility. When the supply chain members coordinate with each other by using contracts, it gives more flexibility to supply chain members to change order quantity, price, cost and lead time. The lower and upper bound can be set for decision variables of contracts (coordination mechanisms) to ensure that the performance of each SC member in a centralized case (consideration of all SC members to be a part of one system) with appropriate coordination mechanism is better than decentralized case (individual supply chain member). Various supply chain contracts present different kinds of supply chain flexibility (Arshinder et al. 2008a). (c) Mitigating uncertainty or risk sharing. The recent issue in supply chain coordination is “How to allocate the total gain in the supply chain achieved due to coordination after mitigating risk?” Many studies have recently developed game theoretic models to fairly share the rewards among supply chain members. The risk mitigation in the form of gain in whole supply chain can be a surrogate measure of SCC. In similar way the extra share of profit allocated out of total gain in SC due to coordination can also reflect the coordinated supply chain. It has also been observed that as the demand variance is increased, the coordinated supply chain due to contracts outperform the independent case of supply chain (Arshinder et al. 2008b). The SC members can devise the contracts in which supplier gives assurance to the buyer to supply emergency orders in case of sudden surge in demand to share risk of losing a customer. Whereas, the buyer can share the extra cost incurred by the supplier in producing emergency orders in view of uncertainty in demand. How well such kind of contracts is designed can be a good indicator of coordination to share risks due to uncertainty in supply chain (Serel 2007). (d) Supply chain coordination index. As it has been discussed that various combination of coordination mechanisms can improve the performance of supply chain. Many situations in supply chain need more than one coordination mechanisms like VMI with quantity discounts, supply chain contracts with information sharing, supply chain contracts with joint decision making (joint consideration of costs). Such kind of index has been developed in Arshinder et al. (2009b) (also mentioned in the proposed framework).

A Review on Supply Chain Coordination

75

7 Major Challenges and Future Research Directions Coordinating the supply chain across organizational boundaries may be one of the most difficult aspects of supply chain management. Many firms simply are unaware of the fundamental dynamics of supply chains, but even those firms that are enlightened enough to understand these dynamics are often unable to realize inter-organizational coordination. Often the most effective supply chains have a dominating organization that sees the benefits of SCC and forces the rest of the supply chain to comply (i.e., global leader in retailing such as Wal-Mart). Many supply chains, however, either do not have a dominant organization, or the dominating organization is unenlightened. In these instances, coordinating the supply chain is most difficult. Typically, it is observed that the SCC problems could be due to the conflicting objectives that leads to a short time relationships with SC members, hence the environment and expectations changes frequently with dealing with new members. On this background, it is essential that the SC members need to appreciate the importance of coordination. This paper has attempted to deliberate on various theoretical perspectives on SCC. The objective to achieve coordination is limited only to the individual functions, to the single coordination mechanism at interfaces of supply chain and to achieve restricted performance measures. A holistic approach towards coordination in whole supply chain is a big challenge, which motivated to propose the issues of SCC in this paper. The mechanisms for coordination need to be studied in detail. The coordination mechanisms can further be of different sub types. To coordinate the whole supply chain, the aggregation of the impact of all coordination mechanisms on the performance of supply chain is required. Various combinations may be explored with the help of simulation. Supply chain contracts have proved to coordinate single period supply chains. The research is required to explore the utility of contracts in multi-period cases. In multi period model, the supply chain members are more expose to the uncertainty as they are dealing with supply chain members frequently. How various coordination mechanisms can be allied in multi period problems as well as can we evaluate coordination in such case? Very few studies have been reported to quantify risk or uncertainty in supply chain. The Bullwhip effect has extensively been discussed in the literature. Actually, there can be many variations seen in supply chain like supply uncertainty, delay in delivery having cascading effect as we go downwards in the supply chain, which is similar to the order variation in Bullwhip effect. How SCC can help in mitigating such uncertainties is one of the important research issues? Acknowledgements The authors are grateful to the reviewers and the whole editorial team of International Journal of Production Economics, Elsevier for constructive suggestions and for considering our paper for publication. The authors are also thankful to the referees of Springer’s Research Handbook Series on “Innovative Schemes for Supply Chain Coordination and Uncertainty” for the comments and suggestions to improve the quality of our paper.

76

K. Arshinder et al.

Appendix List of Journals Refereed in Review Paper 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16.

Computers & Industrial Engineering Computers & Operations research European Journal of Operational Research IIE Transactions International Journal of Logistics and System Management International Journal of Logistics Management International Journal of Operations and Production Management International Journal of Physical Distribution & Logistics Management International Journal of Production Economics International Journal or Production Research Journal of Operations Management Management Science Omega Supply Chain Management: An International Journal Transportation Research (Part E) Other Journals from Emerald, Inderscience and Sciencedirect portal

References Arreola-Risa A (1996) Integrated multi-item production-inventory systems. Eur J Oper Res 89(2):326–340 Arshinder K (2008) An integrative framework for supply chain coordination. Unpublished doctoral thesis, Indian Institute of Technology Delhi, New Delhi, Arshinder K, Kanda A, Deshmukh SG (2006) A coordination based perspective on the procurement process in supply chain. Int J Value Chain Manage 1(2):117–138 Arshinder K, Kanda A, Deshmukh SG (2007) Coordination in supply chains: an evaluation using fuzzy logic. Prod Plann Control 18(5):420–435 Arshinder K, Kanda A, Deshmukh SG (2008a) Supply chain coordination: perspectives, empirical studies and research directions. Int J Prod Econ 115(2):316–335 Arshinder K, Kanda A, Deshmukh SG (2008b) Development of a decision support tool for supply chain coordination using contracts. J Adv Manage Res 5(2):20–41 Arshinder K, Kanda A, Deshmukh SG (2009a) A coordination theoretic model for three level supply chains using contracts. Sadhana Acad Proc Eng Sci 35(5):767–798 Arshinder K, Kanda A, Deshmukh SG (2009b) A framework for evaluation of coordination by contracts: a case of two-level supply chains. Comput Ind Eng 56:1177–1191 Aviv Y (2001) The effect of collaborative forecasting on supply chain performance. Manage Sci 47(10):1326–1343 Bahinipati BK, Kanda A, Deshmukh SG (2009) Horizontal collaboration in semiconductor manufacturing industry supply chain: an evaluation of collaboration intensity index. Comput Ind Eng 57(3):880–895

A Review on Supply Chain Coordination

77

Barbarosoglu G (2000) An integrated supplier-buyer model for improving supply chain coordination. Prod Plann Control 11(8):732–741 Barnes-Schuster D, Bassok Y, Anupindi R (2002) Coordination and flexibility in supply contracts with options. Manuf Serv Oper Manage 4(3):171–207 Barratt M (2004) Understanding the meaning of collaboration in the supply chain. Supply Chain Manage Int J 9(1):30–42 Barron CLE (2007) Optimizing inventory decisions in a multi stage multi customer supply chain: a note. Transp Res E 43(5):647–654 Boctor FF, Laporte G, Renand J (2004) Models and algorithms for the dynamic joint replenishment problem. Int J Prod Res 42(13):2667–2678 Bose I, Anand P (2007) On returns policies with exogenous price. Eur J Oper Res 178(3): 782–788 Bourland KE, Powell SG, Pyke DF (1996) Exploiting timely demand information to reduce inventories. Eur J Oper Res 92(2):239–253 Boyaci T, Gallego G (2002) Coordinating pricing and inventory replenishment policies for one wholesaler and one or more geographically dispersed retailers. Int J Prod Econ 77(2):95–111 Burgess K, Singh PJ, Koroglu R (2006) Supply chain management: a structured review and implications for future research. Int J Oper Prod Manage 26(7):703–729 Cachon GP, Fisher M (2000) Supply chain Inventory management and the value of shared information. Manage Sci 46(8):1032–1048 Cachon GP, Lariviere MA (2005) Supply chain coordination with revenue sharing contracts: strengths and limitations. Manage Sci 51(1):30–44 Chandra P, Fisher ML (1994) Coordination of production and distribution planning. Eur J Oper Res 72(3):503–517 Chauhan SS, Proth JM (2005) Analysis of a supply chain partnership with revenue sharing. Int J Prod Econ 97(1):44–51 Chen TH, Chen JM (2005) Optimizing supply chain collaboration based on joint replenishment and channel coordination. Transp Res E 41(4):261–285 Chen LH, Chen YC (2009) A newsboy problem with simple reservation arrangement. J Comput Ind Eng 56(1):157–160 Chen F, Drezner Z, Ryan JK, Levi DS (2000) Quantifying the bullwhip effect in a simple supply chain: the impact of forecasting, lead times, and information. Manage Sci 46(3):436–443 Chen H, Chen J, Chen YF (2006) A coordination mechanism for a supply chain with demand information updating. Int J Prod Econ 103(1):347–361 Cheung KL, Lee HL (2002) The inventory benefit of shipment coordination and stock rebalancing in a supply chain. Manage Sci 48(2):300–306 Chikan A (2001) Integration of production and logistics – in principle, in practice and in education. Int J Prod Econ 69(2):129–140 Choi TM, Sethi S (2010) Innovative quick response programs: a review. Int J Prod Econ 114:456–475 Danese P, Romano P, Vinelli A (2004) Managing business processes across supply networks: the role of coordination mechanisms. J Purch Supply Manage 10(5):165–177 Deb K (2002) Multi-objective optimization using evolutionary algorithms. Wiley, Chichester Dejonckheere J, Disney SM, Lambrecht MR, Towill DR (2004) The impact of information enrichment on the Bullwhip effect in supply chains: a control engineering perspective. Eur J Oper Res 153(3):727–750 Devaraj S, Krajewski L, Wei JC (2007) Impact of eBusiness technologies on operational performance: the role of production information integration in the supply chain. J Oper Manage 25(6):1199–1216 Ding D, Chen J (2008) Coordinating three level supply chain with flexible returns policies. Omega 36(4):865–876 Ding H, Guo B, Liu Z (2011) Information sharing and profit allotment based on supply chain coordination. Int J Prod Econ 133(1):70–79

78

K. Arshinder et al.

Dotoli M, Fanti MP, Meloni C, Zhou MC (2005) A multi-level approach for network design of integrated supply chain. Int J Prod Res 43(20):4267–4287 Fawcett SF, Magnan GM (2002) The rhetoric and reality of supply chain integration. Int J Phys Distrib Logistics Manage 32(5):339–361 Fin B (2006) Performance implications of information technology implementation in an apparel supply chain. Supply Chain Manage Int J 11(4):309–316 Fisher ML, Raman A, McClelland AS (1994) Rocket science retailing is almost here: are you ready? Harv Bus Rev 72(3):83–93 Ganeshan R (1999) Managing supply chain inventories: a multiple retailer, one warehouse, multiple supplier model. Int J Prod Econ 59(1–3):341–354 Giannoccaro I, Pontrandolfo P (2004) Supply chain coordination by revenue sharing contracts. Int J Prod Econ 89(2):131–139 Gittell JH, Weiss L (2004) Coordination networks within and across organizations: a multi-level framework. J Manage Stud 41(1):127–153 Gottfredson M, Puryear R, Phillips S (2005) Strategic sourcing: from periphery to the core. Harv Bus Rev 83(2):132–139 Goyal SK, Deshmukh SG (1992) Integrated procurement-production systems: a review. Eur J Oper Res 62(1):1–10 Grubbstrom RW, Wang Z (2003) A stochastic model of multi-level/multi-stage capacityconstrained production-inventory systems. Int J Prod Econ 81–82(1):483–494 Gunasekaran A, Patel C, Tirtiroglu E (2001) Performance measures and metrics in a supply chain environment. Int J Oper Prod Manage 21(1–2):71–87 Gunasekaran A, Ngai EWT, McGaughey RE (2006) Information technology and systems justification: a review for research and applications. Eur J Oper Res 173(3):957–983 Gurnani H (2001) A study of quantity discount pricing models with different ordering structures: order coordination, order consolidation, and multi-tier ordering hierarchy. Int J Prod Econ 72(3):203–225 Haq AN, Kannan G (2006) Design of an integrated supplier selection and multi echelon distribution inventory model in a built-to-order supply chain environment. Int J Prod Res 44(10):1963–1985 Haq AN, Vrat PA, Kanda A (1991) An integrated production-inventory-distribution model for manufacture of urea: a case. Int J Prod Econ 25(1–3):39–49 Hau Z, Li S (2008) Impacts of demand uncertainty on retailer’s dominance and manufacturerretailer supply chain coordination. Omega 36(5):697–714 He J, Chin KS, Yang JB, Zhu DL (2006) Return policy model of supply chain management for single-period products. J Optim Theory Appl 129(2):293–308 Hill RM (1997) The single-manufacturer single retailer integrated production-inventory model with a generalized policy. Eur J Oper Res 97(3):493–499 Hill RM, Omar M (2006) Another look at the single-vendor single-buyer integrated productioninventory problem. Int J Prod Res 44(4):791–800 Hoque MA, Goyal SK (2000) An optimal policy for a single-vendor single-buyer integrated production-inventory system with capacity constraint of the transport equipment. Int J Prod Econ 65(3):305–315 Hou J, Zeng AZ, Zhao L (2010) Coordination with a backup supplier through buy-back contract under supply disruption. Transp Res E 46:881–895 Hoyt J, Huq F (2000) From arms-length to collaborative relationships in the supply chain: an evolutionary process. Int J Phys Distrib Logistics Manage 30(9):750–764 Huan SH, Sheoran SK, Wang G (2004) A review and analysis of supply chain operations reference (SCOR) model. Supply Chain Manage Int J 9(1):23–29 Huang G, Lau J, Mak K (2003) The impacts of sharing production information on supply chain dynamics: a review of the literature. Int J Prod Res 41(7):1483–1517 Huiskonen J, Pirttila T (2002) Lateral coordination in a logistics outsourcing relationship. Int J Prod Econ 78(2):177–185 Huq F, Cutright K, Jones V, Hensler DA (2006) Simulation study of a two-level warehouse inventory replenishment system. Int J Phys Distrib Logistics Manage 36(1):51–65

A Review on Supply Chain Coordination

79

Hwarng HB, Chong CSP, Xie N, Burgess TF (2005) Modelling a complex supply chain: understanding the effect of simplified assumptions. Int J Prod Res 43(13):2829–2872 Iyer AV, Bergen ME (1997) Quick response in manufacturer-retailer channels. Manage Sci 43:559–570 Jain K, Nagar L, Srivastava V (2006) Benefit sharing in inter-organizational coordination. Supply Chain Manage Int J 11(5):400–406 Jang YJ, Jang SY, Chang BM, Park J (2002) A combined model of network design and production/ distribution planning for a supply network. Comput Ind Eng 43(1–2):263–281 Jayaram J, Tan K (2010) Supply chain integration with third-party logistics providers. Int J Prod Econ 125(2):262–271 Jayaraman V (1998) Transportation facility location and inventory issues in distribution network design. Int J Oper Prod Manage 18(5):471–494 Jayaraman V, Pirkul H (2001) Planning and coordination of production and distribution facilities for multiple commodities. Eur J Oper Res 133(2):394–408 Kaur A, Kanda A, Deshmukh SG (2006) A graph theoretic approach to evaluate supply chain coordination. Int J Logistics Syst Manage 2(4):329–341 Kehoe DF, Boughton NJ (2001) New paradigms in planning and control across manufacturing supply chains: the utilization of internet technologies. Int J Oper Prod Manage 21(5–6):582–593 Kim T, Hong Y, Lee J (2006) Joint economic production allocation and ordering policies in a supply chain consisting of multiple plants and a single retailer. Int J Prod Res 43(17): 3619–3632 Kim B, Oh H (2005) The impact of decision making sharing between supplier and manufacturer on their collaboration performance, Supply Chain Management: An International Journal 10 (3-4):223–236 Kleindorfer PR, Saad GH (2005) Managing disruptions risks in supply chains. Prod Oper Manage 14(1):53–68 Koulamas C (2006) A newsvendor problem with revenue sharing and channel coordination, Decision Sciences 37(1):91–100 Krishnan H, Kapuscinski R, Butz DA (2004) Coordinating contracts for decentralized supply chains with retailer promotional effort. Manage Sci 50(1):48–63 Lambert DM, Cooper MC (2000) Issues in supply chain management. Ind Mark Manage 29(1): 65–83 Lambert DM, Emmelhainz MA, Gardner JT (1999) Building successful partnerships, Journal of Business Logistics 20(1):165–181 Larsen ST (2000) European logistics beyond 2000. Int J Phys Distrib Logistics Manage 30(6): 377–387 Larsen TS, Thernoe C, Anderson C (2003) Supply chain collaboration theoretical perspective and empirical evidence. Int J Phys Distrib Logistics Manage 33(6):531–549 Lee HL (2000) Creating value through supply chain integration. Supply Chain Manage Rev 4(4): 30–36 Lee HC (2001) Coordinated stocking, clearance sales, and return policies for a supply chain. Eur J Oper Res 131(3):491–513 Lee HC (2007) Coordination on stocking and progressive pricing policies for a supply chain. Int J Prod Econ 106(1):307–319 Lee CH, Rhee BD (2007) Channel coordination using product returns for a supply chain with stochastic salvage capacity. Eur J Oper Res 177(1):214–238 Lee HL, Padmanabhan V, Whang S (1997) Information distortion in supply chain: the bullwhip effect. Manage Sci 43(4):546–558 Li SX, Huang Z, Ashley A (1996) Improving buyer-seller system cooperation through inventory control. Int J Prod Econ 43(1):37–46 Li Z, Kumar A, Lim YG (2002) Supply chain modeling: a coordination approach. Integr Manuf Syst 13(8):551–561 Li G, Yang H, Sun L, Sohal AS (2009) The impact of IT implementation on supply chain integration and performance. Int J Prod Econ 120(1):125–138

80

K. Arshinder et al.

Liu K, Li JA, Lai KK (2004) Single period, single product news vendor model with random supply shock. Eur J Oper Res 158(3):609–625 Liu J, Zhang S, Hu J (2005) A case study of an inter-enterprise workflow-supported supply chain management system. Inf Manage 42(3):441–454 Lu L (1995) A one-vendor multi-buyer integrated model. Eur J Oper Res 81(2):312–323 Malone T, Crowston K (1994) The interdisciplinary study of coordination. ACM Comput Surv 26(1):87–119 Maloni MJ, Benton WC (1997) Supply chain partnerships: opportunities for operations research. Eur J Oper Res 101(3):419–429 Mantrala MK, Raman K (1999) Demand uncertainty and supplier’s returns policies for a multistore style-good retailer. Eur J Oper Res 115(2):270–284 McAfee A (2002) The impact of enterprise information technology adoption on operational performance: an empirical investigation. Prod Oper Manage 11(1):33–53 McCarthy S, Golocic S (2002) Implementing collaborative planning to improve supply chain performance. Int J Phys Distrib Logistics Manage 32(6):431–454 Mc Laren T, Head M, Yuan Y (2002) Supply chain collaboration alternatives: understanding the expected costs and benefits, Internet Research 12(4):348–364 Milner JM, Rosenblatt MJ (2002) Flexible supply contracts for short life cycle goods: the buyer’s perspective. Nav Res Logist 49(1):25–45 Moinzadeh K (2002) A multi-echelon inventory system with information exchange. Manage Sci 48(3):414–426 Moses M, Seshadri S (2000) Policy mechanisms for supply chain coordination. IIE Trans 32(3):254–262 Munson CL, Rosenblatt MJ (2001) Coordinating a three-level supply chain with quantity discounts. IIE Trans 33(5):371–384 Narus JA, Anderson JC (1995) Rethinking distribution: adaptive channels. Harv Bus Rev 74(4):112–120 Oke A, Gopalakrishnan M (2009) Managing disruptions in supply chains: a case study of a retail supply chain. Int J Prod Econ 118(1):168–174 Padmanabhan V, Png IPL (1997) Manufacturer’s returns policies and retailer’s competition. Mark Sci 16(1):81–94 Pan K, Lai KK, Liang L, Leung SCH (2009) Two period pricing and ordering policy for the dominant retailer in a two echelon supply chain with demand uncertainty. Omega 37(4):919–929 Park YB (2005) An integrated approach for production and distribution planning in supply chain management. Int J Prod Res 43(6):1205–1224 Petersen KJ, Handfield RB, Ragatz GL (2005) Supplier integration into new product development: coordinating product, process and supply chain design. J Oper Manage 23(3–4):371–388 Pfohl H, Kohler H, Thomas D (2010) State of the art in supply chain risk management research: empirical and conceptual findings and a roadmap for the implementation in practice. Logistics Res 2(1):33–44 Piplani R, Fu Y (2005) A coordination framework for supply chain inventory alignment. J Manuf Technol Manage 16(6):598–614 Porter ME (2001) Strategy and the internet. Harv Bus Rev 79(2):63–78 Power D (2005) Supply chain management integration and implementation: a literature review. Supply Chain Manage Int J 10(4):252–263 Pyke DF, Cohen MA (1993) Performance characteristics of stochastic integrated production distribution systems. Eur J Oper Res 68(1):23–48 Qi X, Bard JF, Yu G (2004) Supply chain coordination with demand disruptions. Omega 32:301–312 Reddy AM, Rajendran C (2005) A simulation study of dynamic order-up-to policies in a supply chain with non-stationary customer demand and information sharing. Int J Adv Manuf Technol 25(9–10):1029–1045 Ritchie B, Brindley C (2007) Supply chain risk management and performance: a guiding framework for future development. Int J Oper Prod Manage 27(3):303–322 Ross TJ (1997) Fuzzy logic with engineering applications. McGraw Hill, Singapore

A Review on Supply Chain Coordination

81

Rubin PA, Benton WC (2003) A generalized framework for quantity discount pricing schedules. Decis Sci 34(1):173–188 Sahin F, Robinson P (2002) Flow coordination and information sharing in supply chains: review, implications and directions for future research. Decis Sci 33(4):505–536 Sanders NR (2008) Pattern of information technology use: the impact on buyer–suppler coordination and performance. J Oper Manage 26(3):349–367 Sarmah SP, Acharya D, Goyal SK (2006) Buyer–vendor coordination models in supply chain management. Eur J Oper Res 175(1):1–15 Sarmah SP, Acharya D, Goyal SK (2007) Coordination and profit sharing between a manufacturer and a buyer with target profit under credit option. Eur J Oper Res 182(3):1469–1478 Sarmiento AM, Nagi R (1999) A review of integrated analysis of production–distribution systems. IIE Trans 31(11):1061–1074 Saroor J, Tarokh MJ, Shemshadi A (2009) Initiating a state of the art system for real-time supply chain coordination. Eur J Oper Res 196(2):635–650 Serel DA (2007) Capacity reservation under supply uncertainty. Comput Oper Res 34(4): 1192–1220 Sethi SP, Yan H, Zhang H (2004) Quantity flexibility contracts: optimal decisions with information updates. Decis Sci 35(4):691–712 Sharafali M, Co HC (2000) Some models for understanding the cooperation between supplier and the buyer. Int J Prod Res 38(15):3425–3449 Simatupang TM, Sridharan R (2002) The collaborative supply chain. Int J Logistics Manage 13(1):15–30 Simatupang TM, Wright AC, Sridharan R (2002) The knowledge of coordination for supply chain integration. Bus Process Manage J 8(3):289–308 Simatupang TM, Sandroto IV, Lubis SBH (2004) Supply chain coordination in a fashion firm. Supply Chain Manage Int J 9(3):256–268 Simchi-Levi D, Kaminsky P, Simchi-Levi E (2007) Designing and managing the supply chain, 3rd edn. McGraw Hill, New York Sinha S, Sarmah SP (2007) Supply chain coordination model with insufficient production capacity and option for outsourcing. Math Comput Modell 46:1442–1452 Skipper JB, Craighead CW, Byrd TA, Rainer RK (2008) Towards a theoretical foundation of supply network interdependence and technology-enabled coordination strategies. Int J Phys Distrib Logistics Manage 38(1):39–56 Soliman F, Youssef M (2001) The impact of some recent developments in e-business on the management of next generation manufacturing. Int J Oper Prod Manage 21(5–6):538–564 Stank TP, Crum MR, Arango M (1999) Benefits of interfirm coordination in food industry in supply chains. J Bus Logistics 20(2):21–41 Stank TP and Goldsby TJ (2000) A framework for transportation decision making in an integrated supply chain, Supply Chain Management: An International Journal 5(2):71–77 Stock GN, Greis NP, Kasarda JD (2000) Enterprise logistics and supply chain structure: the role of fit. J Oper Manage 18(5):531–547 Swaminathan JM, Tayur SR (2003) Models for supply chains in E-business. Manage Sci 49(10):1387–1406 Tang CS (2006) Perspectives in supply chain risk management. Int J Prod Econ 103(2):451–488 Tang O, Musa SN (2011) Identifying risk issues and research advancements in supply chain risk management, International Journal of Production Economics 133(1):25–34 Thomas DJ, Griffin PM (1996) Coordinated supply chain management. Eur J Oper Res 94(1): 1–15 Tsay A (1999) The quantity flexibility contract and supplier–customer incentives. Manage Sci 45 (10):1339–1358 Tsay AA, Lovejoy WS (1999) Quantity flexibility contracts and supply chain performance. Manuf Serv Oper Manage 1:89–111 vander Rhee B, vander Veen JAA, Venugopal V, Nall VN (2010) A new revenue sharing mechanism for coordinating multi-echelon supply chains. Oper Res Lett 38(4):296–301

82

K. Arshinder et al.

Verwijmeren M, Vlist PV, Donelaar KV (1996) Networked inventory management information systems: materializing supply chain management. Int J Phys Distrib Logistics Manage 26(6):16–31 Vickery SK, Jayaram J, Droge C, Calantone R (2003) The effects of an integrative supply chain strategy on customer service and financial performance: an analysis of direct versus indirect relationships. J Oper Manage 21(5):523–539 Weng ZK (2004) Coordinating order quantities between the manufacturer and the buyer: a generalized newsvendor model. Eur J Oper Res 156(1):148–161 Weng ZK, Parlar M (1999) Integrating early sales with production decisions: analysis and insights, IIE Transactions, 31(11): 1051–1060 Wong WK, Qi J, Leung SYS (2009) Coordinating supply chains with sales rebate contracts and vendor-managed inventory. Int J Prod Econ 120(1):151–161 Wu K, Ouyang L (2003) An integrated single-vendor single-buyer inventory system with shortage derived algebraically. Prod Plann Control 14(6):555–561 Xiao T, Qi X (2008) Price competition, cost and demand disruptions and coordination of a supply chain with one manufacturer and two competing retailers. Omega 36:741–753 Xu L and Beamon B (2006) Supply Chain Coordination and Cooperation Mechanisms: An Attribute-Based Approach, The Journal of Supply Chain Management 42(1):4–12 Xu R, Zhai X (2010) Analysis of supply chain coordination under fuzzy demand in a two-stage supply chain. Appl Math Model 34:129–139 Yang PC, Wee HM (2002) A single-vendor and multiple buyers production-inventory policy for deteriorating item. Eur J Oper Res 143(3):570–581 Yao MJ, Chiou CC (2004) On a replenishment coordination model in an integrated supply chain with one vendor and multiple buyers. Eur J Oper Res 159(2):406–419 Yao D, Yue X, Wang X, Liu JJ (2005) The impact of information sharing on a returns policy with the addition of a direct channel. Int J Prod Econ 97(2):196–209 Yao Z, Leung SCH, Lai KK (2008) Analysis of the impact of price sensitivity factors on the returns policy in coordinating supply chain. Eur J Oper Res 187(1):275–282 Yokoyama M (2002) Integrated optimisation of inventory-distribution systems by random local search and a genetic algorithm. Comput Ind Eng 42(2–4):175–188 Yu Z, Yan H and Cheng TCE (2001) Benefits of information sharing with supply chain partnerships, Industrial Management and Data Systems 101(3):114–119 Yue X, Raghunathan S (2007) The impacts of the full returns policy on a supply chain with information asymmetry. Eur J Oper Res 180(2):630–647 Yusuf Y, Gunasekaran A, Abthorpe MS (2004) Enterprise information systems project implementation: a case study of ERP in Rolls-Royce. Int J Prod Econ 87(3):251–266 Zhao W, Wang Y (2002) Coordination of joint pricing-production decisions in a supply chain. IIE Trans 34(8):701–715 Zhao X, Xie J, Zhang WJ (2002) The impact of information sharing and ordering coordination on supply chain performance. Supply Chain Manage Int J 7(1):24–40 Zhou H, Benton WC Jr (2007) Supply chain practice and information sharing. J Oper Manage 25(6):1348–1365 Zhou YW, Wang SD (2009) Manufacturer-buyer coordination for newsvendor products with two ordering opportunities and partial backorders. Eur J Oper Res 198(3):958–974 Zimmer K (2004) Supply chain coordination with uncertain just-in-time delivery. Int J Prod Econ 77:1–15 Zou X, Pokharel S, Piplani R (2004) Channel coordination in an assembly system facing uncertain demand with synchronized processing time and delivery quantity. Int J Prod Res 42(22):4673–4689 Zou X, Pokharel S, Piplani R (2008) A two-period supply contract model for a decentralized assembly system. Eur J Oper Res 187(1):257–274 Zsidisin GA, Ellram LM (2001) Activities related to purchasing and supply management involvement in supplier alliances. Int J Phys Distrib Logistics Manage 31(9):629–646 Zsidsin GA, Smith ME (2005) Managing supply risk with early supplier involvement: a case study and research propositions. J Supply Chain Manage 41(4):44–57

Control Policies for Multi-echelon Inventory Systems with Stochastic Demand Qinan Wang

Abstract Although supply chain management has become an important management paradigm, the optimal control of a stochastic multi-echelon supply chain inventory system is still largely an open issue. An inventory control policy for such a system has to consider at least three aspects: order coordination, information sharing, and stock or risk pooling. Each aspect can generate significant benefits. Nevertheless, an inventory control policy that can fully optimize system performance on all dimensions, even if it exists, would be very difficult to determine. In this paper, we provide a literature review of inventory control policies for multiechelon supply chain systems with uncertain demand. We first review generic policies developed by extending the basic inventory control policies at a single location. Subsequently, we discuss the themes of coordinated replenishment and information sharing for multi-echelon inventory systems and review the relevant literature. This framework highlights the key factors that drive the performance of multi-echelon inventory systems and shed lights on directions of future research in this area. Keywords Multi-echelon inventory control policy • Review • Supply chain management

1 Introduction A supply chain consists of all organizations involved directly or indirectly in the provision of a product and/or service required by end customers. Inventory management in a supply chain spans all movement and storage of raw materials, work-in-process inventory, and finished goods from point of origin to point of consumption. Traditionally, inventory decisions are made locally at each stocking Q. Wang (*) Nanyang Business School, Nanyang Technological University, Singapore, Singapore 639798 e-mail: [email protected] T.-M. Choi and T.C. Edwin Cheng (eds.), Supply Chain Coordination under Uncertainty, International Handbooks on Information Systems, DOI 10.1007/978-3-642-19257-9_4, # Springer-Verlag Berlin Heidelberg 2011

83

84

Q. Wang

point. However, companies have long realized that they can achieve tremendous benefits by integrating their inventory operations with their business partners. Contemporary supply chain management calls for the application of a total systems approach to supply chain inventory management (Jacobs et al. 2009). There has been a great deal written on the control and management of supply chain inventory systems. The literature on supply chain inventory systems has been reviewed from different perspectives (see, e.g., Clark 1972; Federgruen 1993; Axs€ater 1993a, 2003b; Tsay et al. 1998; Ganeshan et al. 1998; Tan 2001; Sahin and Robinson 2002; Li and Wang 2007). In particular, Federgruen (1993) discussed discrete-time centralized planning models, and Axs€ater (1993a, 2003b) discussed continuous-review centralized planning models for multi-level inventory systems with stochastic demand. Tan (2001) reviewed the evolution of the supply chain management philosophy. More recently, Li and Wang (2007) reviewed the literature on coordination mechanisms of supply chain systems. We review the recent developments on centralized control policies for multiechelon inventory systems with stochastic demand in this paper. Except for some special cases, the optimal control of a supply chain inventory system with stochastic demand is still largely an open problem. We attempt to review the literature to highlight the deficiencies of existing multi-echelon inventory control policies in their capacity to optimize system performance. For this purpose, we adopt a framework to review the literature based on the basic inventory control policies at a single facility. As multi-echelon inventory systems are comprised of individual facilities, their control has been treated as an extension of the inventory control problem at a single facility. Consequently, the multi-echelon inventory control literature has been built based on inventory control policies at a single facility. We first review generic inventory control models for multi-echelon inventory systems that are developed by applying inventory control policies at a single facility. This approach helps us understand the evolution, and highlights the fundamental issues and methodologies for the control of multi-echelon inventory systems. However, a multi-echelon inventory control system is not a simple and straightforward extension of inventory control at individual facilities. The control and management of a multi-echelon inventory system brings about completely new issues and challenges. We subsequently discuss these challenges and review the relevant literature. This discussion highlights the deficiencies of the current literature and enhances our understanding about the key factors that drive the performance of multi-echelon inventory systems. The review also points out new challenges and directions for future research on multi-echelon inventory control systems. The rest of the paper is organized as follows. In Sect. 2, we discuss three basic inventory control policies at a single location, and the structures and issues of multiechelon inventory control systems. Subsequently, in Sects. 3–5, we review multiechelon inventory control systems that are built respectively on the three basic installation inventory control policies. In Sect. 6, we discuss the key factors that drive the performance of multi-echelon inventory systems and review the relevant

Control Policies for Multi-echelon Inventory Systems with Stochastic Demand

85

literature. Finally, in Sect. 7, we conclude the study and summarize possible directions for future research.

2 Multi-echelon Inventory Systems and Control Policies There are various supply chain network structures. The simplest is a serial system that has a single facility at each stage. The next simplest structure is an assembly system in which multiple parts are assembled into a single component and, therefore, a single facility can have multiple suppliers. A distribution system supplies a product to many customers and, therefore, can have multiple customers for a supplier. A tree system combines assembly systems and distribution systems, and a general system can include any of the above as part of the system (Zipkin 2000). A prototype network structure for previous studies on multi-echelon inventory systems is a two-level distribution system whereby a central warehouse supplies a product to a group of retailers. This structure includes a serial system as a special case. An assembly system can also be considered as a special case as it can be decomposed into multiple serial systems under certain conditions (Rosling 1989). More importantly, this structure includes all the fundamental issues for a multiechelon inventory control system and yet does not have the complexity of a general system. Unless otherwise stated, we use the distribution system that is depicted in Fig. 1 for the discussion. Furthermore, we assume that the warehouse obtains supply from a perfectly reliable external source that can fill an order without delay. As illustrated in Fig. 1, an installation is a single facility, and an echelon comprises the facility at the current stage and all facilities at the downstream stages. The following assumptions are typical (1) demand generates randomly and independently at the retailers, (2) all shortages are backordered, (3) all lead times are constant and deterministic, and (4) inventory holding costs and backorder costs are linear. In addition, demand processes are assumed to have stationary and independent increments. This condition holds true for the commonly used compound Poisson demand process and the normal demand model (Rao 2003).

2.1

Basic Inventory Control Policies for a Single Facility

Assume that the central warehouse has unlimited capacity and is perfectly reliable in fulfilling customer orders. Under this condition, inventory control decisions at a retailer can be made independently of inventory control decisions at other facilities in the system. An inventory control policy for the retailer in this case is referred to as an installation inventory control policy. Inventory management at a single facility or installation consists of two fundamental decisions: how much to order and when to order (Zipkin 2000). The objective is to minimize long-run expected inventory related costs including

86

Q. Wang

Echelon Inventory System

External Supplier

Warehouse

Retailer 1

Customer

. . .

Retailer N

Customer

Fig. 1 A two-echelon distribution system

ordering costs, inventory holding costs, and stockout costs. Fundamental trade-offs exist for safety stock and order quantity. Namely, a higher safety stock lowers stockout costs but raises inventory levels or inventory holding costs, and a smaller order quantity (or more frequent ordering) lowers inventory levels or inventory holding costs but increases ordering costs. We desire an inventory control policy that can balance the cost components and minimize their total. There are three basic installation inventory control policies. If inventory positions can be monitored continuously, there are two approaches to decide when to order. We can adopt a stock-based approach to place an order when the inventory position (inventory on hand plus outstanding orders minus backorders) reaches a reorder point or a time-based approach to order periodically in a fixed replenishment interval. The stock-based approach leads to the well-known continuous-review batch-ordering (R, Q) policy, and the time-based approach leads to the fixed-interval order-up-to (S, T) policy. On the other hand, if inventory positions can be monitored only periodically, a base-stock policy is adopted.

2.1.1

The Continuous-Review Batch-Ordering (R, Q) Policy

Under this policy, a batch of Q units is ordered from the supplier when the inventory position drops to a pre-determined reorder point R. The structural properties and optimization procedures for this policy have been well discussed (Silver and Peterson 1998; Zipkin 2000). The two decision variables of this policy, i.e. the reorder point R and order quantity Q, cannot be determined separately. According to Zheng (1992), if the order quantity is determined by the deterministic economic order quantity model,

Control Policies for Multi-echelon Inventory Systems with Stochastic Demand

87

and the reorder point is determined optimally given the order quantity, the maximum cost deviation as compared to the optimal (R, Q) policy is 12.5%.

2.1.2

The Fixed-Interval Order-up-to (S, T) Policy

Under this policy, the inventory position is reviewed in a fixed time interval of T and inventory is ordered, if necessary, at a review point to raise the inventory position to S. Similar to R and Q for the (R, Q) policy, S and T are decision variables for this policy. This policy provides a time-based alternative to the stockbased (R, Q) policy. Although time-based control that uses fixed replenishment schedules to coordinate production and inventory activities has long been a common practice (Graves 1996), the (S, T) policy has not been widely applied in the literature. One reason is that it is dominated by the (R, Q) policy. The structural properties of this policy have been analyzed recently by Rao (2003). It is shown that the long-run inventory related cost under this policy can be much higher than that under the (R, Q) policy. The worst case scenario identified by Rao (2003) is a 43% cost increase. The superiority of the (R, Q) policy comes from the use of continuous review or real time stock information to optimize replenishment decisions.

2.1.3

The Periodic-Review Base-Stock (s, S) Policy

If inventory positions can be monitored only periodically, for example, due to restrictions of physical conditions such as 1 day, 1 week, etc., time is typically modeled as a sequence of discrete time points. Time periods are defined as intervals between time points. Assume that all significant events occur at the time points. The decision problem is then to determine the order quantity at each review point. The problem was first formulated as a dynamic programming model under the assumption that demands at different time points (or equivalently in different time periods) are independent (Arrow et al. 1951; Dvoretzky et al. 1952). Since then, a massive literature on this topic has accumulated. Porteus (1990) provided a thorough review. Assume that the planning horizon is finite or otherwise demand and costs are stationary. The optimal inventory control policy can be fully characterized for this problem. Namely, if there are no economies of scale in ordering, the optimal inventory control policy is to order at each time point to raise the inventory position to a fixed base-stock level S. This policy is referred to as the base-stock policy. If there are economies of scale in ordering, the optimal inventory control policy is a generalization of the single-parameter base-stock policy: order to raise the inventory position to S if the current inventory is less than s (and order nothing otherwise). This modified base-stock policy is referred to as a (s, S) policy. Various algorithms have been developed to compute optimal (s, S) policies (see, e.g., Veinott and Wagner 1965; Zheng and Federgruen 1991).

88

2.2

Q. Wang

Multi-echelon Inventory Control Policies

Regardless of the source of supply, the central warehouse is usually not perfectly reliable and, even if it is, this may not be economically desirable. As such, an order from a retailer will be filled depending on availability of stock at the warehouse. Consequently, the inventory decisions at the retailers and the warehouse cannot be made independently. In other words, inventory control decisions at the warehouse and retailers should be considered together and integratively to optimize system performance. This leads to a multi-echelon inventory control problem. Obviously, each facility in a multi-echelon inventory control system faces the same fundamental decisions and issues for an installation inventory control problem. Therefore, a multi-echelon inventory control policy can be developed by applying an installation inventory control policy at each facility. Indeed most multi-echelon inventory control policies have been developed in this way. These policies are generic multi-echelon inventory control policies. This development provides a framework to review and discuss the multi-echelon inventory control literature. We will review the multi-echelon inventory control models that are developed based on the three basic installation inventory control policies separately in the next three sections. A multi-echelon inventory control system, however, is not a simple and straightforward addition of separate inventory control problems at individual facilities. The control of a multi-echelon inventory system brings about completely new issues and challenges. First, inventory decisions and activities at the warehouse and the retailers must be carefully planned and coordinated. Control policies as a result of local optimization are usually not able to optimize system performance. This issue is particularly important for decentralized supply chains in which inventory control decisions are usually made locally (Li and Wang 2007). Second, information sharing becomes a key driving factor for system performance. Accurate, timely and easily accessible information on demand and stock or advanced commitment from downstream customers can significantly improve system performance. Cachon and Fisher (2000) demonstrated that information sharing can lower system costs by an average of 2.2%, with a maximum of 12.1%. The benefits of demand information may be even higher (Lee et al. 2000). Finally, pooling lead time demand at retailers can also lead to significant reductions in system inventories and/or stockouts. Because of these issues, conventional relationships between installation inventory control policies may no longer be applicable. For example, although the basic (R, Q) policy dominates the basic (S, T) policy at a single facility, a fixed-interval order-up-to policy may perform significantly better than a stock-based batch-ordering policy for a distribution system with multiple retailers. As shown by Wang and Axs€ater (2010) and Wang (2010), although a stock-based batch-ordering policy still has the advantage of using real time stock information, a time-based base-stock policy can provide a better mechanism to coordinate replenishments and pool lead time demand for retailers. When the benefits of order coordination and stock pooling are more significant than the value of stock

Control Policies for Multi-echelon Inventory Systems with Stochastic Demand

89

information, the latter outperforms the former. We discuss these issues and review the relevant literature in Sect. 6. This discussion will highlight the key factors that drive the performance of a multi-echelon inventory control system.

3 Stock-Based Batch-Ordering Policies A typical stock-based batch-ordering policy for a multi-echelon inventory system is to apply a continuous-review ðR; QÞ policy at each facility. This means that a batch of size Q is ordered when the inventory position drops to or below the reorder point R. The batch size and reorder point may vary for different facilities. In addition, the reorder point can be determined based on installation stock or echelon stock. For convenience of discussion, we define the following conditions and parameters (1) there are N retailers, (2) the demand at retailer i follows a simple N P Poisson process with mean li and l ¼ li , and (3) the lead time is Li for retailer i¼1

i and L0 for the warehouse.

3.1

One-for-One Replenishments

Early applications of the (R, Q) policy started with one-for-one replenishments at all facilities. Under this assumption, facility i sets up an order-up-to level Si and orders one unit from the supplier when its inventory position drops to or below Si , or equivalently facility i adopts a continuous-review (R, Q) policy with R ¼ Si 1 and Q ¼ 1. Consider the operations at the warehouse when the warehouse and retailers all adopt one-for-one replenishments. A retailer order at time t is filled immediately if there is stock at the warehouse or otherwise until stock is available. The maximum delay occurs when an order has to be met with a unit ordered from the external supplier at time t. Consequently, all retailer orders placed prior to time t have been filled by time t þ L0 . Therefore, the outstanding orders at time t þ L0 , i.e. units ordered from the external supplier but not yet delivered to the warehouse, are the retailer orders or demands that have occurred between t and t þ L0 . When demand is Poisson, the number of outstanding orders at the warehouse follows a Poisson distribution with mean l L0 . Backorders occur at the warehouse as soon as the number of outstanding orders at the warehouse exceeds the inventory position S0 . Let y denote the number of outstanding orders and EðB0 Þ denote the average number of backorders. We have EðB0 Þ ¼ E½ðy S0 Þþ , where xþ ¼ maxf0; xg. By Little’s law, the average delay at

90

Q. Wang

the warehouse is then equal to EðB0 Þ= l0 . This result holds even if the lead time at the warehouse is stochastic (Axs€ater 1993a). The actual lead time at a retailer is equal to the constant lead time (i.e. Li ) plus a random delay at the warehouse. According to the discussion above, the delay at the warehouse is a random variable with an expected value of EðB0 Þ= l0 . The average lead time at retailer i, denoted by Li , is then equal to Li ¼ Li þ EðB0 Þ= l0 . Sherbrooke (1968) first developed the METRIC approximation by assuming that the lead time at retailer i is constant at Li . With this assumption, expected inventory levels and backorders at the retailers can be easily evaluated. However, this approximation often leads to significant errors. Various subsequent studies have attempted to improve and extend the METRIC approach to other situations (Muckstadt 1973, 1979; Graves 1985; Sherbrooke 1986; Lee and Moinzadeh 1987a). Shortly after the METRIC approximation was introduced, Simon (1971) developed an exact solution for the problem. Exact solution procedures that are more efficient and/or under more general conditions followed (Shanker 1981; Svoronos and Zipkin 1991). In particular, Axs€ater (1990a, 1993c) provided a very efficient recursive procedure to evaluate one-for-one replenishment policies when retailers are identical. Forsberg (1995) provided a solution when demand follows a compound Poisson process. The exact solution has also been extended to the case in which the warehouse adopts a general batch-ordering policy. Let CðSw ; Sr Þ denote the total inventory holding and shortage costs per time unit when applying one-for-one replenishments with inventory positions Sw and Sr at the warehouse and a retailer, respectively. When retailers use one-for-one replenishments and the warehouse adopts a general batch-ordering policy ðRw ; Qw Þ, the total inventory holding and þ Qw RwP shortage costs per time unit is given by ð1= Qw Þ Cðj; Sr Þ (Axs€ater 1993a). j¼Rw þ1

3.2

General Batch-Ordering Policies

Apparently, one-for-one replenishments are applicable only when there are no setup costs for orders. When there are economies of scale in ordering, a general batchordering policy is more appropriate. However, a general batch-ordering policy is usually more difficult to evaluate. Let Qi denote the order quantity of retailer i. Then retailer i places orders to the warehouse according to an Erlang renewal process with Qi stages. As a result, the demand process at the warehouse is a superposition of N such processes. The problem is still tractable for serial systems. For a two-level distribution system with a single retailer or N ¼ 1, let the retailer use a batch-ordering policy ðRr ; Qr Þ and the warehouse adopt a batch-ordering policy ðRw ; Qw Þ (both as batches of Qr ). Using the results for one-for-one replenishments, the total inventory holding and shortage costs per time unit is given by þ Qw RrP þ Qr RwP ð1= Qw Qr Þ Cðj Qr ; kÞ (Axs€ater 1993c). j¼Rw þ1 k¼Rr þ1

Control Policies for Multi-echelon Inventory Systems with Stochastic Demand

91

Many studies have developed batch-ordering inventory control policies for general serial systems. The following are a few notable examples. De Bodt and Graves (1985) developed an approximate solution using a nested policy in which an order is delivered to the retailer upon the arrival of a shipment from the external supplier at the warehouse. Badinelli (1992) put forward a model of the steady-state values of on-hand inventory and backorders at each facility when each member adopts an installation-stock (R, Q) policy. More recently, Chen and Zheng (1994a) first evaluated echelon-stock (R, nQ) policies and developed a recursive procedure to compute the steady-state inventory levels of the system when demand follows a simple Poisson process. Subsequently, they extended the analysis to situations in which demand follows a compound Poisson process and developed near-optimal echelon-stock (R, nQ) policies (Chen and Zheng 1998) and to situations in which materials flow in fixed batch sizes (Chen 2000). In particular, Chen (1999a) proved that, for a two-stage serial system with zero lead time at the warehouse, a nested stock-based batch-ordering policy can achieve at least 94% system optimality. This is probably the only performance evaluation for a multi-echelon batch-ordering policy. When there are multiple retailers, decisions at the retailers must be coordinated and integrated in order to optimize system performance. However, members in a supply chain traditionally have little communication about their demand and inventory activities. As a result, early developments of stock-based batch-ordering policies considered installation policies in which each member makes its inventory decision separately using only local stock and demand information. Deuermeyer and Schwarz (1981) considered a distribution system consisting of one supplier and multiple identical retailers, and developed an installation-stock policy. They approximated each facility as a single-location inventory system and used decomposition as an adaptation of the METRIC technique. The problem has been examined by many others and more accurate approximate solutions have been developed since then (Moinzadeh and Lee 1986; Lee and Moinzadeh 1987a, b; Svoronos and Zipkin 1988). Exact evaluations of on-hand inventories and backorders for the system have been developed by Axs€ater (1993c) when retailers are identical and face independent simple Poisson demand processes. Forsberg (1996) and Axs€ater (2000) then generalized the results to the case of non-identical retailers under simple and compound Poisson demand. Cheung and Hausman (2000) provided exact performance evaluations for the supplier when retailers order in batches of a basic quantity. Other approximate optimizations have been developed by Axs€ater et al. (2002), Axs€ater (2003a) and Gallego et al. (2007). Instead of using local or installation inventory position, a stock-based backordering policy can be developed based on echelon inventory position, or the sum of the local inventory position and the inventory positions at all its downstream members. Since an echelon-stock policy incorporates stock information at downstream facilities for inventory control, it is superior to an installation-stock policy. As shown by Axs€ater and Rosling (1993) and Axs€ater and Juntti (1996), although the relative performance of the two policies seldom deviates by more than 5%, it is

92

Q. Wang

clear that the echelon-stock policy outperforms the installation policy for serial and assembly systems. However, there are two barriers for the wide application of the echelon-stock policy. First, its implementation requires a centralized information system that enables a supply chain member to have access to the demand and inventory information at all its downstream members in order to continuously monitor its echelon stock. Second, echelon-stock policies are usually more difficult to evaluate than installation-stock policies. Axs€ater (1997) and Chen and Zheng (1997) provided exact evaluations of echelon-stock batch-ordering policies for twolevel distribution systems. When there are multiple retailers, neither an installation-stock policy nor an echelon-stock policy fully utilizes the inventory state of a multi-echelon inventory system to optimize system performance. This is obvious for an installation policy since the inventory control at the supplier utilizes no inventory information from the retailers. For an echelon-stock policy, the inventory control decision at the supplier utilizes the echelon-stock (i.e. the total inventory position at all retailers) information rather than the inventory status at the individual retailers. In this sense, neither policy may be able to optimize system performance. Stock-based batch-ordering policies have also been widely used to analyze multi-echelon inventory systems with multiple supply modes (see, e.g., Lee 1987; Moinzadeh and Nahmias 1988; Axs€ater 1990b; Moinzadeh and Schmidt 1991; Johansen and Thorstenson 1998) and information sharing (see, e.g., Bourland et al. 1996; Chen 1998; Gavieneni 2002; Gurbuz et al. 2007). The issue of information sharing will be discussed further in Sect. 6.

4 Fixed-Interval Order-up-to Policies When inventory position can be monitored continuously, most previous research work on multi-echelon inventory systems has been confined to stock-based batchordering policies. Applications of the fixed-interval order-up-to (S, T) policy to build multi-echelon inventory control systems are relatively limited. One reason is that the (S, T) policy is dominated by the (R, Q) policy for a single location. This dominance, however, does not extend to multi-echelon inventory systems. Wang and Axs€ater (2010) and Wang (2010) demonstrated recently that time-based control can perform significantly better than stock-based control under certain conditions for distribution systems with multiple retailers. Previous studies have considered control policies for multi-echelon inventory systems with fixed replenishment intervals. These studies assume that replenishment intervals are determined exogenously but allow the flexibility of coordinating replenishments. A notable feature is the concept of nesting (Graves 1996; van Houtum et al. 2007). This condition is achieved under two constraints: the integer-ratio constraint requires that the replenishment interval at a stage be an integer multiple of the replenishment interval at the next downstream stage; and the synchronization constraint requires that a shipment from an upstream can be

Control Policies for Multi-echelon Inventory Systems with Stochastic Demand

93

forwarded immediately to the next downstream stage if so desired. These policies can be considered as applications of the fixed-interval order-up-to (S, T) with a given interval T. On the other hand, for a given interval T, the decision problem is similar to a periodic-review inventory control problem with constraints on the replenishments at certain time points. Therefore, base-stock policies are adopted at replenishment points. Eppen and Schrage (1981) first considered a fixed replenishment interval policy for retailers to synchronize their replenishments. The retailers are identical, face independent stochastic demand, and order together periodically from an outside supplier through a depot. The replenishment interval is given, and stock is allocated to the retailers entirely and immediately upon its arrival at the depot. In this way, the depot does not carry inventory and acts only as a cross docking point. By assuming large incoming orders that will always result in an equal probability of stockout at each retailer, they derived approximately optimal base- stock policies at the depot. Subsequently, Schwarz (1989) and Kumar et al. (1995) adopted fixed replenishment interval policies to study the benefits of stock pooling. McGavin et al. (1993, 1997) used this structure to analyze warehouse inventory allocation policies to minimize system lost sales. Atkins and Iyogun (1988), Viswanathan (1997), and Eynan and Kropp (1998), among others, studied joint replenishment policies, and Cetinkaya and Lee (2000) analyzed time-based consolidation policies for vendor-managed inventory systems. In addition, a number of studies have been devoted to optimal control policies for multi-echelon inventory systems when replenishment intervals at all facilities are given (Yano and Carson 1988; Jackson 1988; Graves 1996; Axs€ater 1993b; van Houtum et al. 2007). A few studies have considered full decision models to optimize system performance on both replenishment interval and inventory control policies recently. Feng and Rao (2007) applied the ðS; TÞ policy to a two-stage serial system. They considered the “fixed reorder interval, T, order-up-to base-stock level, R” policy and refer to this policy as the echelon-stock (R, nT) policy. The decision problem is formulated as a mixed-integer nonlinear programming model, in which the cost under each decision alternative is obtained by simulation. They compared the (R, nT) policy to the (R, nQ) policy (Chen and Zheng 1994a, b) numerically and showed that, although the latter dominates the former, the cost differences are often not significant. More recently, Wang and Axs€ater (2010) developed a fixed-interval order-up-to policy for a distribution system with multiple retailers. Let the warehouse set up a basic replenishment period T. Retailers are required to replenish through the warehouse in intervals that are integer multiples of the basic replenishment period. No inventory is carried at the warehouse. They compared this policy to the stockstock batch-ordering policy (Axs€ater 1993c) and showed that the dominance of the latter at a single location does not extend to distribution systems with multiple retailers. The time-based policy can perform significantly better than the stockbased batch-ordering policy in certain situations. However, since the warehouse is restricted to a cross-docking point that does not carry inventory, the efficiency of this policy could be low if the cost of carrying inventory at the warehouse is low.

94

Q. Wang

Subsequently, Wang (2010) generalized this model to allow the warehouse to carry inventory and developed a general integer-ratio policy for a distribution system with multiple and identical retailers. Another motivation for this development is the remarkable achievements of the integer-ratio policy in deterministic settings (Roundy 1985, 1986). Let the warehouse set up a base replenishment interval to replenish inventories from the external supplier. Retailers are required to review their inventory positions and order in fixed time intervals that are integer or integer-ratio multiples of the base replenishment interval at the warehouse. The warehouse and retailers each adopt an echelon-stock order-up-to policy, i.e. order the needed inventories to raise the echelon inventory position to a fixed order-up-to level at each review point. It is shown numerically that, although the stock-based batch-ordering policy generates a lower system cost in many cases, the integer-ratio policy can perform significantly better in certain settings.

5 Periodic-Review Base-Stock Policies When time is discrete, the problem is usually formulated as a dynamic programming or Markov decision model. However, except for some special cases, the large dimension of the inventory state for the exact formulation usually precludes an exact solution. The focus is then shifted to approximate solutions.

5.1

Serial Systems

Clark and Scarf (1960) showed that, for a serial system with M stages, if the planning horizon is finite and there is no setup cost except at the highest installation, the inventory problem for the system can be decomposed into exactly M separate single location problems, one for each echelon. These problems can be specified and solved recursively starting from the highest echelon. The optimal solution at the highest echelon is a ðsM ; SM Þ policy: order the necessary stock from the external supplier to raise the system inventory position to an order-up-to level SM whenever the starting inventory position is at or below a reorder point sM ; the optimal solution at a lower echelon j < M is a modified base stock policy: set up a base stock level Sj for echelon j < M, and ship the necessary stock from the upper level if there is sufficient stock and otherwise whatever available to raise the inventory position at the echelon to Sj . The model and solution were originally suggested by Clark (1958). The seminal work by Clark (1958) and Clark and Scarf (1960, 1962) initiated the research on the optimal control of multi-echelon inventory systems. The basic model and solution have been generalized to various settings since then. Federgruen

Control Policies for Multi-echelon Inventory Systems with Stochastic Demand

95

and Zipkin (1984a) extended the basic model to the infinite-horizon case for both discounted and average costs, and showed that the computations required are much easier than for the finite-horizon case. They also considered the case with multiple locations at the lower echelon and showed that the problem can be approximated by a problem with a single location at the lower echelon. Subsequently, Zipkin (1986a, b) considered the problem with uncertain lead times. Chen (2000) considered optimal control policies for multi-stage serial or assembly inventory systems with batch ordering, and generalized several existing results for the basic model. Recently, van Houtum et al. (2007) generalized the results to serial inventory systems with fixed replenishment intervals. They proved the optimality of the (s, S) policy for a serial system under the integer and synchronization constraints when replenishments intervals are given. Chen and Song (2001) considered a serial system in which demand generates at stage one in each period according to a distribution that is determined by the current state of an exogenous Markov Chain, and showed that the optimal policy for the system is an echelon-stock base-stock policy with state-dependent order-up-to levels. Shang and Song (2003, 2007) and Gallego and Ozer (2003, 2005) developed simple heuristics and bounds and approximations for the optimal policy parameters. Recently, Chao and Zhou (2009) extended the results to serial systems with batch ordering and fixed replenishment schedules. The results for a serial system have been generalized to assembly systems. Schmidt and Nahmias (1985) considered a simple assembly system in which two components are purchased from outside and assembled into a single end item, and characterized the optimal policy. Subsequently, building on Schmidt and Nahmias (1985), Rosling (1989) showed that, when all order and assembly cost functions are linear, an assembly system can be transformed into an equivalent serial system. All carrying, outside order and assembly costs remain linear in the equivalent system. Therefore, an optimal control policy is to apply modified base stock policies in each period for all nodes. The result holds for both finite planning horizon and infinite planning horizon models. Song and Yao (2002) considered the performance and optimization of assembly systems with random lead times. By modeling an assembly system under a given base-stock policy as a M=G=1 queuing system, they derived easy-to-compute performance bounds. When there are setup costs at all stages, it is well known that the optimal control of the inventory system is difficult. Only heuristic solutions and approximations have been developed. Chen and Zheng (1994b) developed lower bounds for multi-echelon stochastic inventory systems including serial, assembly and distribution systems with multiple retailers in which economies of scale in ordering exist at all locations. Recently, Shang and Zhou (2010) considered (r, nQ, T) policies. Under such a policy, each stage reviews its inventory in every T periods and orders according to an echelon-stock (r, nQ) policy. There are two types of fixed costs: one for each order batch Q and another for each inventory review. They developed a method for obtaining heuristic and optimal policy parameters.

96

5.2

Q. Wang

General Systems

Similar to serial systems with setup costs at all stages, an optimal control policy that can fully optimize system performance for a distribution system with multiple retailers is unknown. Previous studies have focused on heuristic solutions and approximations under various restrictions. The reader is referred to Federgruen (1993) for a systematic and detailed discussion on such restrictions and approximations. A number of studies have considered control policies under the restriction that the warehouse does not hold inventory. Under this condition, the warehouse acts as only a cross docking point and a delivery at the warehouse is allocated immediately and entirely to retailers. Eppen and Schrage (1981) started the use of this restriction. Subsequently, Federgruen and Zipkin (1984b) showed that a distribution system with multiple retailers can be approximated by a single-location problem under this condition. Recently, Wang and Axs€ater (2010) adopted this setting to develop a fixedinterval order-up-to policy for a two-level distribution system with multiple retailers. When multiple retailers order from the warehouse at a common point in time, stock allocation becomes an important issue. Federgruen and Zipkin (1984c) provided a systematic and detailed discussion on this problem. They showed that, when stock is allocated based on the inventory positions at individual retailers, the state space of the dynamic programming problem has a very large dimension, which usually precludes an exact solution of the model. They developed a computationally tractable approximate solution with myopic stock allocation. Under a myopic stock allocation policy, an incoming order at the warehouse is allocated to retailers to minimize their expected costs in the very first period in which the allocation has an impact, i.e. in the period where the shipments arrive at their destinations. Since then, different allocation approaches have been considered. A cycle allocation policy allocates an incoming order at the warehouse to retailers to minimize their expected costs in an ordering cycle, i.e. the period from the arrival of the current shipments to the arrival of the shipments from the next order (Federgruen 1993). A general reservation policy reserves a unit of supply at the time of a demand event at a retailer, and ships the reserved supply together to a retailer according to a fixed schedule or when inventory becomes available. This allocation method was also called “virtual allocation” by Graves (1996) and Axs€ater (1993c). To maximize the benefits of stock pooling, stock at the warehouse should be allocated in the lastminute or at the time of shipment (Marklund 2006). Several studies also assumed that inventory positions can be balanced at an allocation point if necessary (see, e.g., Eppen and Schrage 1981; Federgruen and Zipkin 1984c; McGavin et al. 1993; Axs€ater et al. 2002; Sosic 2006). This balance condition means simply that transshipments among retailers or allocation of negative quantities are allowed, although this is not always possible in practice. According to Wang and Axs€ater (2010), allocation policies can have a significant impact on system costs. They showed that system costs may increase by more than 30% under certain conditions when stock at the warehouse is allocated according to a complete reservation allocation policy as compared to a last-minute allocation policy.

Control Policies for Multi-echelon Inventory Systems with Stochastic Demand

97

Despite these progresses, exact evaluations and solutions are rare. Cachon (2001) provided exact evaluations for average inventory, backorders and fill rates for a two-echelon distribution system with multiple retailers and batch-ordering. Chen and Zheng (1994b) established lower bounds for serial, assembly and distribution systems with setup costs at all stages. Although many studies have provided full or partial characterizations of optimal polices for some special cases (see, e.g., Chen 2004a, b; Fox et al. 2006; Chao and Zipkin 2008; Sheopuri et al. 2010), a general optimal solution is hard, if not impossible. The reader may refer to Glasserman and Tayur (1994, 1995), Kapuscinski and Tayur (1998), Gallego and Toktay (2004), Ozer and Wei (2004), Chao and Zipkin (2008), and the references herein for capacitated multi-echelon inventory systems; and Tagaras and Vlachos (2001), Feng et al. (2006), Sheopuri et al. (2010), and the references herein for multi-echelon inventory systems with multiple supply modes.

6 Coordinated Replenishment and Information Sharing Despite the great progresses over the years, we seem to know little about the key factors that drive the performance of a multi-echelon inventory control system (Gallego et al. 2007). This is particularly the case for systems with multiple facilities at a stage (e.g. a distribution system with multiple retailers). Many control policies for multi-echelon inventory systems have been developed by extending installation control policies. However, a multi-echelon inventory control system is more than a simple addition of inventory control problems at individual facilities. It brings about completely new issues and challenges. We discuss two such issues and review the relevant literature in this section.

6.1

Coordinated Replenishment

In general, coordinated replenishment centralizes and synchronizes the ordering decisions for retailers. There are two distinct potential benefits. First, when retailers order together, they may achieve economies of scale in ordering such as quantity discounts from the outside supplier, savings in transportation costs, etc. Second, the system can choose to allocate the stock among the retailers at the time of shipment rather than at the time of ordering. Postponing the allocation allows the system to observe the demands at the retailers in the warehouse lead time, and thus to make a better informed allocation. This can result in significant system cost reductions. Eppen and Schrage (1981) coined the term “statistical economies of scale” for these benefits. This effect is also commonly called the benefits of stock pooling in the literature. Coordinated replenishment for multiple products is commonly referred to as the joint replenishment problem and has been studied extensively (Federgruen et al. 1984;

98

Q. Wang

Khouja and Goyal 2008). The objective is to achieve economies of scale in ordering. Apparently, similar benefits can be achieved by coordinating replenishments for multiple retailers in a supply chain. Axs€ater and Zhang (1999) attempted to develop a joint replenishment policy for a distribution system, but did not find an appropriate policy to improve system performance. A more efficient policy was developed subsequently by Cheung and Lee (2002). They considered a supplier serving multiple retailers and developed the following joint replenishment policy. Let each retailer set up a target inventory position. As demands arrive at the retailers, the inventory positions at the retailers drop below their respective targets. When the cumulative demands over all retailers reach a batch size Q, the supplier orders the quantity from a warehouse to restore the inventory positions at the retailers to their respective target levels. The orders from the warehouse are transported from the external supplier directly to the retailers. The supplier does not carry inventory and acts as only a cross-docking point. They showed that the joint replenishment policy dominates the installation-stock batch-ordering policy (Axs€ater 1993c). The benefits of coordinated replenishments for multiple retailers in a supply chain, however, are beyond the traditional economies of scale in ordering. Cheung and Lee (2002) showed that, if orders from the warehouse are first transported to the supplier and then allocated and shipped to the retailers upon their arrival at the supplier, system costs can be further reduced significantly. This is due to the statistical economies of scale. Eppen and Schrage (1981) first studied this effect by synchronizing ordering decisions for a group of identical retailers. Subsequently, Jackson (1988) extended the basic model to allow the warehouse to carry inventory and demonstrated empirically the benefits of centralizing at least a portion of the total system stock. Schwarz (1989) assessed the value of stock pooling by comparing two systems. In system one, inventories are ordered and shipped directly from an outside supplier to each retailer separately. In system two, this is done through a warehouse, thereby inventories are ordered together and allocated and shipped to the retailers upon their arrival. The effect of stock pooling is measured by the overall reduction in variance of the retailer end-of-cycle net inventory. Kumar et al. (1995) studied this effect along a fixed delivery route using a dynamic inventory control policy. McGavin et al. (1993, 1997) analyzed warehouse inventory allocation policies to minimize system lost sales. Other studies have considered stock pooling in various situations (see, e.g., Eynan and Fouque 2003; Benjaafar et al. 2005; Ve´ricourt et al. 2002; Kukreja et al. 2001; Wee and Dada 2005). In particular, Wang and Axs€ater (2010) and Wang (2010) showed that fixedinterval order-up-to polices are able to provide a more efficient mechanism for supply chain members to coordinate replenishments than stock-based batch-ordering policies. Because of the benefits of coordinated replenishment, particularly the statistical economies of scale, time-based control policies can perform significantly better than stock-based control policies for distribution systems with multiple retailers.

Control Policies for Multi-echelon Inventory Systems with Stochastic Demand

6.2

99

Demand and Stock Information

Iglehart (1964) initiated the research on the use of demand information in inventory control. He considered the inventory control problem for a single location facing a demand distribution with unknown parameters, and developed a Bayesian estimation scheme to update the demand distribution with new information for inventory control. This problem has been studied subsequently by many others (Azoury 1985; Lovejoy 1990; Milner and Kouvelis 2002). Furthermore, Zheng and Zipkin (1990) started to quantify the value of information sharing among multiple products and/or facilities. They considered the scheduling problem for two items competing for a single production facility, and showed that information about outstanding orders of the two products could improve system performance. Zipkin (1995) extended the analysis to a multi-item production facility subsequently. Supply chain management brings about a new issue of sharing information across supply chain members. A massive literature has accumulated on the use of stock and demand information (Gunasekaran and Ngai 2004). Bourland et al. (1996) considered the use of stock information to improve inventory control decisions. They considered a two-level serial system with a supplier and a single retailer, and demonstrated that the supplier could improve its replenishment decision by making use of stock information at the retailer. In the meantime, a number of studies analyzed the use of information sharing to reduce the bullwhip effect in supply chain systems (Lee et al. 1997; Lee and Whang 1999; Chen et al. 2000). The recent interest has focused on the use of advance demand information to improve the efficiency of multi-echelon inventory control systems (see, e.g., Hariharan and Zipkin 1995; Gallego and Ozer 2001; Iyer and Ye 2000; Ozer 2003; Marklund 2006; Axs€ater and Marklund 2008; Wang and Toktay 2008). Many studies have evaluated the value of stock and demand information in supply chain management. The evaluations, however, varied for different studies. Chen (1998) studied the benefits of stock information by comparing the costs for a multi-echelon serial system when using an echelon-stock batch-ordering policy and an installation-stock batch-ordering policy. He observed that echelon-stock information could reduce system cost by an average cost of 1.75%, with a maximum of 9%. Gavieneni et al. (1999) and Gavieneni (2002) considered a similar setting with a capacitated manufacturer and a single retailer, and observed higher values for stock information, ranging from 1 to 35% of system inventory related cost with an average of 14%. In the meantime, Cachon and Fisher (2000) considered a setting with one supplier and N identical retailers. They contrasted the value of information sharing with faster and cheaper order processing, which led to shorter lead times and smaller batch sizes, respectively, and found that implementing information technology to accelerate and smooth the physical flow of goods through a supply chain was significantly more valuable than to expand the flow of information. Furthermore, Gurbuz et al. (2007) extended the joint replenishment policy developed by Cheung and Lee (2002) to use retailer stock information to trigger a joint replenishment order. Following Moinzadeh (2002), they proposed a second-order

100

Q. Wang

trigger rule based on a minimum inventory position at each retailer so that an order is placed to the outside supplier when any retailer’s inventory position drops to a minimum requirement level or the demand at all retailers accumulates to Q units, whichever is earlier. They showed numerically that stock information could improve the joint replenishment policy. However, the improvements were only moderate. Evaluations for the value of demand information varied even greater. Lee et al. (2000) considered a two-level supply chain that consists of a manufacturer and a single retailer who faces a non-stationary auto-correlated demand process, and showed that the manufacturer would experience great savings when demand information was shared by the downstream member. The analysis was subsequently extended by Aviv (2002) to a setting in which companies could observe early market signals to improve their forecasting performance. However, Raghunathan (2001, 2003) pointed out later that the finding of Lee et al. (2000) depends on the critical assumptions that the manufacturer uses only the most recent order information from the retailer to forecast its future orders and that the parameters of the demand process are only accessed by the retailer. If the parameters of the autocorrelated demand process are known to both parties, sharing of demand information is actually of limited value. As the manufacturer can forecast the demand by using the retailer’s order history and the accuracy of forecast increases monotonically with each subsequent time period, the value of information decreases monotonically with each time period and converges to zero. Since the entire retailer order history is available to the manufacturer, the manufacturer is in a position to use the data in its forecasting process. Graves (1999) made similar observations in a slightly different setting.

7 Conclusions and Discussions To summarize the literature review, let us consider the distribution system that is described in Sect. 2. Suppose time is continuous and there are multiple retailers. Several alternative policies have been developed for the inventory control system in the literature. We list these policies in Table 1 below. Previous studies have shown that, if everything else holds constant, (1) Policy 2 dominates Policy 1 because echelon stock provides more accurate information about the system inventory state, (2) the joint replenishment (JR) policies, i.e. Policies 3, 4, and 5, dominate Policy 1, also because joint replenishment utilizes more accurate system stock information, (3) Policy 4 dominates Policy 3 due to the effect of stock pooling, and (4) Policy 5 dominates Policy 3 because more accurate information about the inventory state at retailers is used. Policy 4, Policy 5, and Policy 6 do not dominate each other. These relationships reflect the state of the art for the developments of multiechelon inventory control systems. Optimal control of a multi-echelon inventory system is still largely an open problem. First of all, little is known about the format

Control Policies for Multi-echelon Inventory Systems with Stochastic Demand

101

Table 1 Policies for a distribution system with multiple retailers Policy Policy description References Installation-stock Apply the stock-based (R, Q) policy Axs€ater (1993c, 2000), Forsberg batch-ordering at each facility based on (1997), Cheung and Hausman policy installation stock (2000) Echelon-stock batchApply the stock-based (R, Q) policy Axs€ater (1997), Axs€ater and ordering policy at each facility based on echelon Rosling (1993), Chen and stock Zheng (1997) JR policy without stock Order for all retailers when their Axs€ater and Zhang (1999), pooling demands accumulate to a fixed Cheung and Lee (2002) batch size and ship stock to each retailer directly from the outside supplier JR policy with stock Order for all retailers when their Cheung and Lee (2002) pooling demands accumulate to a fixed batch size and ship stock to a depot where the stock is optimally allocated JR policy using retailer Order for all retailers when their Gurbuz et al. (2007) stock information demands accumulate to a fixed but no stock pooling batch size and/or the inventory position at a retailer reaches a trigger level, and ship stock to each retailer directly from the outside supplier Fixed-interval orderApply the time-based (S, T) policy at Eppen and Schrage (1981), up-to policy each facility based on echelon Jackson (1988), Schwarz stock (1989), Wang and Axs€ater (2010)

of an optimal control policy. Stock-based batch-ordering policies and time-based order-up-to policies have been developed based respectively on the basic (R, Q) policy and (S, T) policy for a single facility. These formats have been adopted mostly because of convenience rather than optimality. For a single facility, the stock-based (R, Q) policy can be optimal and dominates the time-based (S, T) policy. This dominance can carry over to serial systems but not to distribution systems with multiple retailers. Second, little seems to be known about the factors that drive the performance for a multi-echelon inventory system. Previous studies have shown that an optimal policy has to consider at least three aspects (a) careful planning and coordination of inventory activities at the warehouse and retailers, (b) use of accurate and timely stock and demand information, and (c) pooling of stocks. However, the evaluations of these benefits varied significantly for different studies. For example, evaluations for the value of information varied from almost zero to an average system cost reduction of more than 10%. Apparently, the significance of these benefits depends on the setting and the inventory control policy adopted for the study, among other possible confounding factors. Trade-off often exists. For example, a setting or a format of inventory control policy that can fully utilize stock information may not

102

Q. Wang

be able to maximize the benefits of order coordination and stock pooling. Consequently, an inventory control policy that can fully optimize system performance, even if it exists, would be extremely difficult to structure and determine. As a result, the research on multi-echelon inventory control systems has focused on the identification of close-to-optimal policies that have a relatively simple structure and are reasonably easy to determine and implement. While this trend will continue, the question on the efficiency of an inventory control policy to optimize system performance will always be a challenge. Current inventory control policies have focused on some but not all driving factors for system performance. For example, stock-based policies focus more on using stock information to optimize replenishment decisions. In contrast, time-based policies focus more on coordinating system replenishments for both the traditional and statistical economies of scale. In view of these developments, while performance evaluations on multi-echelon inventory control policies to optimize system performance are very desirable, we may develop more efficient multi-echelon inventory control policies by combining the features of different policy formats. Finally, we have focused exclusively on multi-echelon inventory systems where decisions can be centralized by a unique decision maker to optimize system performance. Although many studies have considered decentralized multi-echelon inventory control systems (see, e.g., Cachon and Zipkin 1999; Chen 1999b; Axs€ater 2001, 2005; Caldentey and Wein 2003), the literature on decentralized systems is still limited as compared to the literature on centralized systems. This is inadequate, given the fact that most supply chains consist of members who are separate and independent economic entities. One reason for this inadequacy is that, although a centralized solution can optimize system performance, it is not always in the best interest of every individual member. Therefore, the implementation of a centralized solution or the coordination of a decentralized supply chain that aims to optimize system performance must have a mechanism to align the objectives of individual supply chain members. While a lot of research has been done on this subject for deterministic supply chains (Li and Wang 2007), relatively little seems to have been done on the topic for stochastic supply chain systems. Consequently, the coordination of decentralized multi-echelon inventory systems with stochastic demand and/or lead time represents great opportunities and challenges for future research.

References Arrow K, Harris T, Marschak J (1951) Optimal inventory policy. Econometrica 19:250–272 Atkins D, Iyogun P (1988) Periodic versus ‘can-order’ policies for coordinated multi-item inventory systems. Manage Sci 34:791–796 Aviv Y (2002) Gaining benefits from joint forecasting and replenishment processes: the case of auto-correlated demand. Manuf Serv Oper Manage 4(1):55–74 Axs€ater S (1990a) Simple solution procedures for a class of two-echelon inventory problems. Oper Res 38:64–69 Axs€ater S (1990b) Modeling emergency lateral transshipments in inventory systems. Manage Sci 36:1329–1338

Control Policies for Multi-echelon Inventory Systems with Stochastic Demand

103

Axs€ater S (1993a) Continuous review policies for multi-level inventory systems with stochastic demand. In: Graves S, Rinnooy Kan A, Zipkin P (eds) Handbooks in operations research and management science, vol 4, Logistics of production and inventory. North-Holland, Amsterdam, pp 175–197 Axs€ater S (1993b) Optimization of order-up-to-S policies in two-echelon inventory systems with periodic-review. Nav Res Logist 40:245–253 Axs€ater S (1993c) Exact and approximate evaluation of batch-ordering policies for two-level inventory systems. Oper Res 41(4):777–785 Axs€ater S (1997) Simple evaluation of echelon stock (R, Q) policies for two-level inventory systems. IIE Trans 29:661–669 Axs€ater S (2000) Exact analysis of continuous review (R, Q) policies in two-echelon inventory systems with compound Poisson demand. Oper Res 48(5):686–696 Axs€ater S (2001) A framework for decentralized multi-echelon inventory control. IIE Trans 33:91–97 Axs€ater S (2003a) Approximate optimization of a two-level distribution inventory system. Int J Prod Econ 81–82:545–553 Axs€ater S (2003b) Supply chain operations: serial and distribution inventory systems. In: Graves SC, de Kok T (eds) Supply chain management: design, coordination, and operation. Handbooks in operations research and management science, vol 11, Chap. 10. Elsevier, Amsterdam, pp 525–559 Axs€ater S (2005) A simple decision rule for decentralized two-echelon inventory control. Int J Prod Econ 93–94:53–59 Axs€ater S, Juntti L (1996) Comparison of echelon stock and installation stock policies for twolevel inventory systems. Int J Prod Econ 45:303–310 Axs€ater S, Marklund J (2008) Optimal position-based warehouse ordering in divergent twoechelon inventory systems. Oper Res 56(4):976–991 Axs€ater S, Rosling K (1993) Installation vs. echelon stock policies for multilevel inventory control. Manage Sci 39(10):1274–1280 Axs€ater S, Zhang W (1999) A joint replenishment policy for multi-echelon inventory control. Int J Prod Econ 59:243–250 Axs€ater S, Marklund J, Silver EA (2002) Heuristic methods for centralized control of onewarehouse, N-retailer inventory systems. Manuf Serv Oper Manage 4(1):75–97 Azoury K (1985) Bayes solutions to dynamic inventory models under unknown demand distributions. Manage Sci 31(9):1150–1160 Badinelli R (1992) A model for continuous-review pull policies in serial inventory systems. Oper Res 40(1):142–156 Benjaafar S, Cooper W, Kim J (2005) On the benefits of pooling in production-inventory systems. Manage Sci 51(4):548–565 Bourland K, Powell S, Pyke D (1996) Exploiting timely demand information to reduce inventories. Eur J Oper Res 92:239–253 Cachon G (2001) Exact evaluation of batch-ordering inventory policies in two-echelon supply chains with periodic review. Oper Res 49(1):79–98 Cachon G, Fisher M (2000) Supply chain inventory management and the value of shared information. Manage Sci 46(8):1032–1048 Cachon G, Zipkin P (1999) Competitive and cooperative inventory policies in a two-stage supply chain. Manage Sci 45(7):936–953 Caldentey R, Wein L (2003) Analysis of a decentralized production-inventory system. Manuf Serv Oper Manage 5(1):1–17 Cetinkaya S, Lee C (2000) Stock replenishment and shipment scheduling for vendor managed inventory. Manage Sci 46(2):217–232 Chao X, Zhou S (2009) Optimal policy for a multiechelon inventory system with batch ordering and fixed replenishment intervals. Oper Res 57(2):377–391

104

Q. Wang

Chao X, Zipkin P (2008) Optimal policy for a periodic-review system under a supply capacity contract. Oper Res 56(1):59–70 Chen F (1998) Echelon reorder points, installation reorder points, and the value of centralized demand information. Manage Sci 44(12):S221–S234 Chen F (1999a) 94%-effective policies for a two-stage serial inventory system with stochastic demand. Manage Sci 45(12):1679–1696 Chen F (1999b) Decentralized supply chains subject to information delays. Manage Sci 45 (8):1076–1090 Chen F (2000) Optimal policies for multi-echelon inventory problems with batch ordering. Oper Res 48(3):376–389 Chen S (2004a) The optimality of hedging point policies for stochastic two-product flexible manufacturing systems. Oper Res 52(2):321–323 Chen S (2004b) The infinite horizon periodic review problem with setup costs and capacity constraints: a partial characterization of the optimal policy. Oper Res 52(3):409–421 Chen F, Song J (2001) Optimal policies for multi-echelon inventory problems with Markovmodulated demand. Oper Res 49(2):226–234 Chen F, Zheng Y (1994a) Evaluating echelon stock (R, nQ) policies in serial production/inventory systems with stochastic demand. Manage Sci 40(10):1262–1275 Chen F, Zheng Y (1994b) Lower bounds for multi-echelon stochastic inventory systems. Manage Sci 40(11):1426–1443 Chen F, Zheng Y (1997) One-warehouse multi-retailer systems with centralized stock information. Oper Res 45(2):275–287 Chen F, Zheng Y (1998) Near-optimal echelon-stock (R, nQ) policies in multistage serial system. Oper Res 46(4):592–602 Chen F, Drezner Z, Ryan J, Simchi-Levi D (2000) Quantifying the bullwhip effect in a simple supply chain: the impact of forecasting lead time and information. Manage Sci 46(3):436–443 Cheung K, Hausman W (2000) An exact performance evaluation for the supplier in a two-echelon inventory system. Oper Res 48(4):646–653 Cheung K, Lee H (2002) The inventory benefit of shipment coordination and stock rebalancing in a supply chain. Manage Sci 48(2):300–306 Clark A (1958) A dynamic, single-item, multi-echelon inventory model. Report. Rand Corporation, Santa Monica, CA Clark A (1972) An informal survey of multi-echelon inventory theory. Nav Res Logistics Q 19:621–650 Clark A, Scarf H (1960) Optimal policies for a multi-echelon inventory problem. Manage Sci 6 (4):475–490 Clark A, Scarf H (1962) Approximate solutions to a simple multi-echelon inventory problem. In: Arrow KJ, Karlin S, Scarf H (eds) Studies in applied probability and management science. Stanford University Press, Stanford, CA De Bodt M, Graves S (1985) Continuous review policies for a multi-echelon inventory problem with stochastic demand. Manage Sci 31(10):1286–1299 Deuermeyer B, Schwarz L (1981) A model for the analysis of system service level in warehouse/ retailer distribution systems: the identical retailer case. In: Schwarz LB (ed) Studies in the management sciences, vol 16, Multi-level production/inventory control systems. North-Holland, Amsterdam, pp 163–193 Dvoretzky A, Kiefer J, Wolfowitz J (1952) The inventory problem. Econometrica 20:187–222 Eppen G, Schrage L (1981) Centralized ordering policies in a multiwarehouse system with leadtimes and random demand. In: Schwarz LB (ed) Studies in the management sciences, vol 16, Multi-level production/inventory control systems. North-Holland, Amsterdam, pp 88–110 Eynan A, Fouque T (2003) Capturing the risk-pooling effect through demand reshape. Manage Sci 49(6):704–717

Control Policies for Multi-echelon Inventory Systems with Stochastic Demand

105

Eynan A, Kropp D (1998) Periodic review and joint replenishment in stochastic demand environment. IIE Trans 30:1025–1033 Federgruen A (1993) Centralized planning models for multi-echelon inventory systems under uncertainty. In: Graves S, Rinnooy Kan A, Zipkin P (eds) Handbooks in operations research and management science, vol 4, Logistics of production and inventory. North-Holland, Amsterdam, pp 133–173 Federgruen A, Zipkin P (1984a) Computational issues in an infinite-horizon, multi-echelon inventory model. Oper Res 32(4):818–836 Federgruen A, Zipkin P (1984b) Approximations of dynamic, multilocation production and inventory systems. Manage Sci 30(1):69–84 Federgruen A, Zipkin P (1984c) Allocation policies and cost approximation for multilocation inventory systems. Nav Res Logistics Q 31:97–131 Federgruen A, Groenevelt H, Tijms H (1984) Coordinated replenishments in a multi-item inventory system with compound Poisson demands and constant lead times. Manage Sci 30:344–357 Feng K, Rao U (2007) Echelon-stock (R, nT) control in two-stage serial stochastic inventory systems. Oper Res Lett 35:95–104 Feng Q, Sethi S, Yan H, Zhang H (2006) Are base-stock policies optimal in inventory problems with multi delivery modes? Oper Res 54(4):801–807 Forsberg R (1995) Optimization of order-up-to-S policies for two-level inventory systems with compound Poisson demand. Eur J Oper Res 81:143–153 Forsberg R (1997) Exact evaluation of (R, Q)-policies for two level inventory systems with Poisson demand. Eur J Oper Res 96:130–138 Fox E, Metters R, Semple J (2006) Optimal inventory policy with two suppliers. Oper Res 56 (4):696–707 Gallego G, Ozer O (2001) Integrating replenishment decisions with advance demand information. Manage Sci 47(10):1344–1360 Gallego G, Ozer O (2003) Optimal replenishment policies for multi-echelon inventory systems under advance demand information. Manuf Serv Oper Manage 5:157–175 Gallego G, Ozer O (2005) A new algorithm and a new heuristic for serial supply systems. Oper Res Lett 33:349–362 Gallego G, Toktay L (2004) All-or-nothing ordering under capacity constrain. Oper Res 52 (6):1001–1002 Gallego G, Ozer O, Zipkin P (2007) Bounds, heuristics, and approximations for distribution systems. Oper Res 55(3):503–517 Ganeshan R, Jack E, Magazine M, Stephens P (1998) A taxonomic review of supply chain management research. In: Tayur S, Magazine M, Ganeshan R (eds) Quantitative models for supply chain management, vol 17, International series in operations research and management science. Kluwer, Norwell, MA, pp 839–879 Gavieneni S (2002) Information flows in capacitated supply chains with fixed ordering cost. Manage Sci 48(5):644–651 Gavieneni S, Kapuscinski R, Tayur S (1999) Value of information in capacitated supply chains. Manage Sci 45(1):16–24 Glasserman P, Tayur S (1994) The stability of a capacitated, multi-echelon production-inventory system under a base-stock policy. Oper Res 42(5):913–925 Glasserman P, Tayur S (1995) Sensitivity analysis for base-stock levels in multi-echelon production-inventory systems. Manage Sci 41(2):263–281 Graves S (1985) A multi-echelon inventory model for repairable item with one-for-one replenishment. Manage Sci 31(10):1247–1256 Graves S (1996) A multi-echelon inventory model with fixed replenishment intervals. Manage Sci 42(1):1–18 Graves S (1999) A single-item inventory model for a nonstationary demand process. Manuf Serv Oper Manage 1(1):50–61

106

Q. Wang

Gunasekaran A, Ngai E (2004) Information systems in supply chain integration and management. Eur J Oper Res 159:269–295 Gurbuz M, Moinzadeh K, Zhou Y (2007) Coordinated replenishment strategies in inventory/ distribution systems. Manage Sci 53(2):293–308 Hariharan R, Zipkin P (1995) Customer-order information, leadtimes, and inventories. Manage Sci 41(10):1599–1607 Iglehart D (1964) The dynamic inventory problem with unknown demand distributions. Manage Sci 10(3):429–440 Iyer A, Ye J (2000) Assessing the value of information sharing in a promotional retail environment. Manuf Serv Oper Manage 2(2):128–143 Jackson P (1988) Stock allocation in a two-echelon distribution system or “what to do until your ship comes in”. Manage Sci 34(7):880–895 Jacobs F, Chase R, Aquilano N (2009) Operations and supply management, 12th edn. McGrawHill, New York Johansen S, Thorstenson A (1998) An inventory model with Poisson demands and emergency orders. Int J Prod Econ 56–57:275–289 Kapuscinski R, Tayur S (1998) A capacitated production-inventory model with period demand. Oper Res 46(6):899–911 Khouja M, Goyal S (2008) A review of the joint replenishment problem literature: 1989–2005. Eur J Oper Res 186:1–16 Kukreja A, Schmidt C, Miller D (2001) Stocking decisions for low-usage items in a multilocation inventory system. Manage Sci 47(10):1371–1383 Kumar A, Schwarz L, Ward J (1995) Risk-pooling along a fixed delivery route. Manage Sci 41:344–362 Lee H (1987) A multi-echelon inventory model for repairable items with emergency lateral transshipments. Manage Sci 33:1302–1316 Lee H, Moinzadeh K (1987a) Two-parameter approximations for multi-echelon repairable inventory models with batch ordering policy. IIE Trans 19:140–149 Lee H, Moinzadeh K (1987b) Operating characteristics of a two-echelon inventory system for repairable and consumable items under batch ordering and shipment policy. Nav Res Logistics Q 34:365–380 Lee H, Whang S (1999) Decentralized multi-echelon supply chains: incentives and information. Manage Sci 45(5):633–640 Lee H, Padmanabhan V, Whang S (1997) Information distortion in a supply chain: the bullwhip effect. Manage Sci 43(4):546–558 Lee H, So K, Tang C (2000) The value of information sharing in a two-level supply chain. Manage Sci 46(5):626–643 Li X, Wang Q (2007) Coordination mechanisms of supply chain systems. Eur J Oper Res 179:1–16 Lovejoy W (1990) Myopic policies for some inventory models with uncertain demand distributions. Manage Sci 36(6):724–738 Marklund J (2006) Controlling inventories in divergent supply chains with advance order information. Oper Res 54(5):988–1010 McGavin E, Schwarz L, Ward J (1993) Two-interval inventory-allocation policies in a onewarehouse N-identical retailer distribution system. Manage Sci 39(9):1092–1107 McGavin E, Ward J, Schwarz L (1997) Balancing retailer inventories. Oper Res 45(6):820–830 Milner J, Kouvelis P (2002) On the complementary value of accurate demand information and production and supplier flexibility. Manuf Serv Oper Manage 4(2):99–113 Moinzadeh K (2002) A multi-echelon inventory system with information exchange. Manage Sci 48(3):414–426 Moinzadeh K, Lee H (1986) Batch size and stocking levels in multi-echelon repairable systems. Manage Sci 32:1567–1581 Moinzadeh K, Nahmias S (1988) A continuous review model for an inventory system with two supply modes. Manage Sci 34:761–773

Control Policies for Multi-echelon Inventory Systems with Stochastic Demand

107

Moinzadeh K, Schmidt C (1991) An (S-1, S) inventory system with emergency orders. Oper Res 39:308–321 Muckstadt J (1973) A model for a multi-item, multi-echelon, multi-indenture inventory system. Manage Sci 20:472–481 Muckstadt J (1979) A three-person, multi-item model for recoverable items. Nav Res Logistics Q 26:199–221 Ozer O (2003) Replenishment strategies for distribution systems under advance demand information. Manage Sci 49(3):255–272 Ozer O, Wei W (2004) Inventory control with limited capacity and advance demand information. Oper Res 52(6):988–1001 Porteus E (1990) Stochastic inventory theory. In: Heyman D, Sobel M (eds) Stochastic models, handbooks in operations research and management science, vol 12. Elsevier (North-Holland), Amsterdam Raghunathan S (2001) Information sharing in a supply chain: a note on its value when demand is nonstationary. Manage Sci 47(4):605–610 Raghunathan S (2003) Impact of demand correlation on the value of and incentives for information sharing in a supply chain. Eur J Oper Res 146:634–649 Rao U (2003) Properties of the periodic-review (R, T) inventory control policy for stationary, stochastic demand. Manuf Serv Oper Manage 5(1):37–53 Rosling K (1989) Optimal inventory policies for assembly systems under random demands. Oper Res 37(4):565–579 Roundy R (1985) 98%-effective integer-ratio lot-sizing for one-warehouse multi-retailer systems. Manage Sci 31(11):1416–1429 Roundy R (1986) A 98%-effective lot sizing rule for a multi-product, multi-stage production/ distribution system. Math Oper Res 11(4):699–727 Sahin F, Robinson E (2002) Flow coordination and information sharing in supply chains: review, implications, and directions for future research. Decis Sci 33(4):505–535 Schmidt C, Nahmias S (1985) Optimal policy for a two-stage assembly system under random demand. Oper Res 33:1130–1145 Schwarz L (1989) A model for assessing the value of warehouse risk-pooling: risk-pooling over outside-supplier leadtimes. Manage Sci 35(7):828–842 Shang K, Song J (2003) Newsvendor bounds and heuristic for optimal policies in serial supply chains. Manage Sci 49(5):618–638 Shang K, Song J (2007) Serial supply chains with economies of scale: bounds and approximations. Oper Res 55(5):843–853 Shang K, Zhou S (2010) Optimal and heuristic echelon (r, nQ, T) policies in serial inventory systems with fixed costs. Oper Res 58(2):414–429 Shanker K (1981) Exact analysis of a two-echelon inventory system for recoverable items under batch inspection policy. Nav Res Logistics Q 28:579–601 Sheopuri A, Janakiraman G, Seshdri S (2010) New policies for the stochastic inventory control problem with two supply sources. Oper Res 58(3):734–747 Sherbrooke C (1968) METRIC: a multi-echelon technique for recoverable item control. Oper Res 16:122–141 Sherbrooke C (1986) VARI-METRIC: improved approximation for multi-indenture, multi-echelon availability models. Oper Res 34:311–319 Silver E, Peterson R (1998) Decision systems for inventory management and production. Wiley, New York Simon R (1971) Stationary properties of a two-echelon inventory model for low demand items. Oper Res 19:761–777 Song J, Yao D (2002) Performance analysis and optimization of assemble-to-order systems with random lead times. Oper Res 50(5):889–903 Sosic G (2006) Transshipment of inventories among retailers: myopic vs. farsighted stability. Manage Sci 52(10):1493–1508

108

Q. Wang

Svoronos A, Zipkin P (1988) Estimating the performance of multi-level inventory systems. Oper Res 36(1):57–72 Svoronos A, Zipkin P (1991) Evaluation of one-for-one replenishment policies for multiechelon inventory systems. Manage Sci 37:68–83 Tagaras G, Vlachos D (2001) A periodic review inventory system with emergency replenishments. Manage Sci 47(3):415–429 Tan K (2001) A framework of supply chain management literature. Eur J Purch Supply Manage 7:39–48 Tsay A, Nahmias S, Agrawa N (1998) Modeling supply chain contracts: a review. In: Tayur S, Magazine M, Ganeshan R (eds) Quantitative models for supply chain management, vol 17, International series in operations research and management science. Kluwer, Norwell, MA, pp 299–336 van Houtum G, Scheller-Wolf A, Ji X (2007) Optimal control of serial inventory systems with fixed replenishment intervals. Oper Res 55(4):674–687 Veinott A, Wagner H (1965) Computing optimal (s, S) inventory problems. Manage Sci 11:525–552 Ve´ricourt F, Karaesmen F, Dallery Y (2002) Optimal stock allocation for a capacitated supply system. Manage Sci 48(11):1486–1501 Viswanathan S (1997) Note. Periodic review (s, S) policies for joint replenishment inventory systems. Manage Sci 43:1447–1454 Wang Q (2010) Integer-ratio policies for distribution systems with multiple retailers and stochastic demand. Working paper, Nanyang Business School, Nanyang Technological University, Singapore Wang Q, Axs€ater S (2010) Fixed-interval order-up-to policies for distribution systems with multiple retailers and stochastic demand. Working paper, Nanyang Business School, Nanyang Technological University, Singapore Wang T, Toktay B (2008) Inventory management with advance demand information and flexible delivery. Manage Sci 54(4):716–732 Wee K, Dada M (2005) Optimal policies for transshipping inventory in a retailer network. Manage Sci 51(10):1519–1533 Yano C, Carson C (1988) Safety stocks for assembly systems with fixed production schedules. J Manuf Oper Manage 1:182–201 Zheng Y (1992) On properties of stochastic inventory systems. Manage Sci 38(1):87–103 Zheng Y, Federgruen A (1991) Finding optimal (s, S) policies is about as simple as evaluating a single policy. Oper Res 39(4):654–665 Zheng Y, Zipkin P (1990) A queuing model to analyze the value of centralized inventory information. Oper Res 38(2):296–307 Zipkin P (1986a) Stochastic leadtimes in continuous-time inventory models. Nav Res Logistics Q 33:763–774 Zipkin P (1986b) Inventory service level measures: convexity and approximation. Manage Sci 32 (8):975–981 Zipkin P (1995) Performance analysis of a multi-item production-inventory system under alternative policies. Manage Sci 41(4):690–703 Zipkin P (2000) Foundations of inventory management. McGraw-Hill, New York

Supply Chain Models with Active Acquisition and Remanufacturing Xiang Li and Yongjian Li

Abstract Nowadays economic, marketing, and environment legislation are increasingly driving firms to consider product reuse, thus active acquisition of used products is prevailing in industry. This chapter focuses on the problem regarding the active acquisition and remanufacturing within supply chain scope. We introduce some recent and important research developments, including the centralized system problem with price-sensitive acquisition and demand, the decentralized supply chain problem with vertical channels, and the decentralized supply chain problem with horizontal competition. For each problem, analytical models are presented and main results are elucidated. Finally, further research directions are also pointed out. Keywords Acquisition management • Closed-loop supply chain • Coordination • Remanufacturing

1 Introduction Supply chain management considering return flows has received substantial interests from both industrial and academic worlds. Product returns represent a growing financial concern for firms, with an estimation of $35 billion annually for the USA alone (Meyer 1999). From a traditional perspective, these returns fall into two categories: consumer returns to the retailer during the return period, and product overstocks returned to the upstream manufacturer, both incurring considerable reprocessing cost and revenue loss. This return flow is regarded as “a bad X. Li (*) Research Center of Logistics, College of Economic and Social Development, Nankai University, Tianjin 300071, P.R. China e-mail: [email protected] Y. Li Business School, Nankai University, Tianjin 300071, P.R. China e-mail: [email protected] T.-M. Choi and T.C. Edwin Cheng (eds.), Supply Chain Coordination under Uncertainty, International Handbooks on Information Systems, DOI 10.1007/978-3-642-19257-9_5, # Springer-Verlag Berlin Heidelberg 2011

109

110

X. Li and Y. Li

thing”, and the associated costs are reduced by developing various return policies and incentive strategies (Ferguson et al. 2006). On the other hand, nowadays legislations usually encourage the reuse of product returns. Remanufacturing technology has been developed as an environmentally and economically sound way to deal with the returns after customer usage. As a result, a trend of active acquisition of used product is replacing the traditional passive return, and the remanufacturing industry is being boosted. The relevant activities, however, require effective coordination at multiple locations to retrieve the used product and recover its value, which is usually referred to as reverse supply chain management (Prahinski and Kocabasoglu 2006). The acquisition and remanufacturing management turns out to be a complicated problem, considering the uncertainty of the product returns, and the different, usually conflicting objectives of the supply chain members. This chapter focuses on the supply chain problem with active acquisition and remanufacturing, hoping to provide some recent research models and results. More specifically, there are three streams of researches drawing our attention: the centralized system problem with price-sensitive acquisition and demand, the decentralized supply chain problem with vertical channel, and the decentralized supply chain problem with horizontal competition. First of all, the used product can be actively collected in a market-driven channel by paying a price called acquisition price to end users or core dealers. The related problem is first proposed by Guide and Jayaraman (2000), which delineates the necessity of a careful coordination to balance the returns with the demand. Guide et al. (2003) builds up a quantitative model of a remanufacturing system in which the returns and demand could be controlled by the acquisition price and selling price, respectively, and Bakal and Akcali (2006) extends it into a random remanufacturing yield case. Ray et al. (2005) studies the optimal pricing/trade-in strategies for a durable, remanufacturable product, by characterizing some key factors of such product. Other related papers include Robotis et al. (2005), Qu and Williams (2008), Liang et al. (2009), etc. In sum, all the above papers focus on the centralized control of integrated acquisition and remanufacturing system. As for the decentralized supply chain, the acquisition and remanufacturing problems are usually linked up with game theory and contracting. One stream of the research is on the vertical channel, which is studied under Stackelberg leaderfollower game framework. Savaskan et al. (2004) considers the problem of choosing an appropriate reverse channel for the acquisition of used product. Three channel structures are analyzed and compared, involving the interaction among supply chain members such as the OEM, retailer, and 3PL collector. In the succeeding work, Savaskan and Van Wassenhove (2006) further explores the reverse channel choice problem with one upstream manufacturer and two competing retailers. The reverse supply chain model is also studied in Karakayali et al. (2006) without considering the forward distribution channel. As most papers focus on the policy of commercial returns, e.g., Pasternack(1985), Padmanabhan and Png (1997), Mukhopadhyay and Setoputro (2005), Yue and Raghunathan (2007) and so on, the research on vertical supply chain management with the active acquisition is rather limited.

Supply Chain Models with Active Acquisition and Remanufacturing

111

Furthermore, the study on the acquisition and remanufacturing competition at a horizontal supply chain level is being a hot topic recently. Majumder and Groenevelt (2001a) describes a two-period model under the assumption of indistinguishable quality between remanufactured and new product, but distinguishable brand between the competing OEM and remanufacturer. Majumder and Groenevelt (2001b) extends the model by allowing for competition in the acquisition of used product. Ferrer and Swaminathan (2006) extends Groenevelt and Majumder (2001a) into a multi-period setting where the brand competition is carried on in the second and subsequent periods. Ferguson and Toktay (2006) adopts a similar two-period model focusing on the strategic role of OEM remanufacturing as an entry deterrent to the local remanufacturer. The remanufacturing cost has a quadratic form and the competition is between the sales of new and remanufactured products. Atasu et al. (2008) also analyzes a two-period model to study the effect of timing of the remanufactured product introduction and the use of remanufacturing as a marketing strategy to compete with low-cost OEM competitors. Other related papers include Debo et al. (2005), Heese et al. (2005), Webster and Mitra (2007, 2008), etc. Most of the above models are under Nash game framework and two-period setting. In this chapter, we choose to present six models from the above three research streams, and provide an in-depth discussion on their managerial insights. The purpose is not to give an all-around review or survey on the numerous literature related to product returns, but to offer some connected and evolving works, hoping to reflect new research trends. Also, one of our main focuses is on the collection effort (and price) paid by supply chain members during the active acquisition of the used product, therefore some important researches outside this scope are not involved in the chapter. For example, some papers study the acquisition decision of the used product under a traditional newsvendor framework, where the used product acquisition can be simply regarded as a normal product order, e.g., Ferrer (2003), Vlachos and Dekker (2003), Galbreth and Blackburn (2006), Zikopoulos and Tagaras (2007), Kaya (2010), etc. Some other papers consider the acquisition problem with the location decision of acquisition centers, e.g., Wojanowski et al. (2007), Aras and Aksen (2008), Aras et al. (2008), etc. For a more comprehensive review on these subjects, we refer the reader to Dekker et al. (2004), Guide and Van Wassenhove (2009), Pokharel and Mutha (2009), and Ilgin and Gupta (2010).

2 Centralized System with Price-Sensitive Acquisition and Demand As stated in the last section, in the market-driven channel an acquisition price is paid to end users or core suppliers for the returned used product. This approach is widely adopted as it grants the firm a partial control on the used product returns. Guide and Van Wassenhove (2001) provides a detailed case study of the telephone remanufacturer, ReCellular Inc., buying used phones from a variety of sources and

112

X. Li and Y. Li

and selling remanufactured ones. Similar examples can also be found in the automotive industry, where automotive parts from the end-of-life vehicles are remanufactured and reused (Bakal and Akcali 2006). A common feature of the above cases is that the remanufacturing firm has full pricing power in both used product acquisition and remanufactured product demand, the quantities of which are sensitive to the acquisition price and the selling price, respectively. In this section, we investigate this pricing problem faced by the remanufacturing firm, with the objective of maximizing the profit from remanufacturing. The pricing decision is regarded as a lever to match the demand and supply, and both the deterministic and random yield models can be established.

2.1

Deterministic Model

A remanufacturing firm, ReCellular, collects used phones from the providers grading the phones and selling them in different quality classes. The used phones with quality classification differ in their condition, like the appearance, damage, and age, and the remanufacturing costs for different quality classes are different. On the other hand, all the remanufactured phones have the same quality and the same selling price. The acquisition quantities and demand are deterministic and dependent on the acquisition prices and selling price, respectively. This quantitative model is studied in Guide et al. (2003), and introduced in the following. Suppose that there are N quality classes, 1,2,. . .,N. The remanufacturing cost of class i is ci. The acquisition price of the used product of class i is denoted as fi, and the corresponding return quantity is ri( fi). It is assumed that ri( fi) is a continuous, increasing and twice differentiable function defined on [bi, g ci], where bi is the minimal acquisition price and g is the maximum price at which a remanufactured product can be sold. For convenience, the classes are ordered in such a way that b1 þ c1 < b2 þ c2 < < bN þ cN . On the demand side, let d(p) be the demand of the remanufactured product when the selling price is p, which is a continuous, decreasing and twice differentiable function defined on [b1 + c1, g]. The inverse function of d( p) is denoted by P(d), i.e., P(d(p)) ¼ p. And the inverse function of ri( fi) is denoted by Ai(ri), i.e., Ai(ri( fi)) ¼ fi, for all i. The objective is to determine the acquisition prices fi ; i ¼ 1; . . . ; N, and the selling price p to maximize the profit. The optimal acquisition price is denoted by fi and the resulting acquisition quantity is ri ; i ¼ 1; . . . ; N. As it is a deterministic model, the demand of the remanufactured product should be equal to the quantity of the used product returns to attain the profit maximization, i.e., the prices should satisfy dðpÞ ¼

N X i¼1

ri ðfi Þ:

(1)

Supply Chain Models with Active Acquisition and Remanufacturing

113

Thus, the optimization problem can be formulated as ðP1Þ

max P

f1 ;;fN

N X

! ri ðfi Þ

i¼1

N X

ri ðfi Þ

i¼1

N X

½ri ðfi Þðfi þ ci Þ;

(2)

i¼1

or equivalently, ðP2Þ

N X

max P

r1 ;...;rN

! ri

i¼1

N X i¼1

ri

N X

½ri ðAðri Þ þ ci Þ:

(3)

i¼1

It is natural to ask if the objective function (P1) or (P2) has joint concavity. Examples have shown that the function is not always concave and even not unimodal in some cases. Then the following two questions turn out to be essential: 1. On what conditions the objective function is concave (or unimodal) and in that case, is there any analytical property helpful to solve the problem? 2. Is there any algorithm/heuristic useful for the problem? Guide et al. (2003) provides the following answers. First, the objective (P2) is jointly concave if dP(d) is concave and riA(ri) is convex for any i. Second, under these conditions (called Condition C1 and C2, respectively), there is an M such that fi >bi for all i ¼ 1; . . . ; M and fi ¼ bi for all i ¼ M þ 1; ; N. This property indicates it is optimal to only acquire the used product in class i ¼ 1; ; M for some 1 M N. For the second question, define functions GðpÞ :¼ ddðpÞ and 0 ðpÞ þ p Fi ðfi Þ :¼ rr0iiðfðfiiÞÞ þ ðfi þ ci Þ. Then, the first-order optimality conditions for problem (P1) can be expressed by Fi ðfi Þ ¼ G P

N X

!! ri ðfi Þ

for i ¼ 1; ; M;

(4)

for i ¼ M þ 1; ; N:

(5)

i¼1

and Fi ðfi Þ G P

N X

!! ri ðfi Þ

i¼1

The above properties yield the following algorithm to calculate the optimal acquisition prices under Conditions C1 and C2. Step 1. M: ¼ 1. M ðaM Þ :¼ aM . If M 2, use the equation Fi(ai) ¼ FM(aM) to express ai as Step 2. HM a function of aM, denoted by HiM ðaM Þ, for i ¼ 1; 2; ; M 1.

114

X. Li and Y. Li

P M Step 3. Use the equation dðpÞ ¼ M i¼1 ri Hi ðaM ÞÞ to express p as a function of aM, denoted by KM(aM). Step 4. Obtain the value aM M for aM by equalizing FM(aM) to G(KM(aM)). Step 5. If either M ¼ N or FM ðaM M Þ FMþ1 ðbMþ1 Þ, output the optimal prices: fi ¼ M M M M Hi ðaM Þ for i ¼ 1; ; M, and fi ¼ bi for i ¼ M þ 1; ; N; And p ¼ K ðaM Þ. Otherwise, M :¼ M þ 1 and then go to Step 1. This algorithm shows the procedure to compute the exact optimal prices. However, Steps 4 and 5 have to be carried out numerically if the demand and return functions have complex shapes. Moreover, the algorithm can be only applied when both the conditions C1 and C2 are satisfied. In this regard, a heuristic is derived in Guide et al. (2003) based on the idea of making the prices such that the profit per product is fixed. It is set that fi ¼ max(bi, fnew ci), where fnew is the theoretical acquisition price that the firm is willing to pay for an as-good-as-new returned item, which doesn’t need the remanufacturing. The associated total return P quantity r 0 ðfnew Þ :¼ i;bi þci

2.2

Random Remanufacturing Yield Model

The uncertainty on product returns is regarded as an important characteristic of reverse logistics. In a market-driven collection channel, the quantity uncertainty of the returns is considerably controlled by acquisition price, while the quality uncertainty still exists. This quality uncertainty can be reflected by remanufacturing yield randomness and is considered in this subsection. The problem is motivated by the automotive remanufacturing industry. Typically, the automotive dismantling/remanufacturing firm buys end-of-life vehicles (ELVs) and sells remanufacturable parts, facing both the acquisition pricing and selling pricing problems. However, due to the uncertainty of the returned product quality, there is a random yield in the part remanufacturing process. That is to say, the quality requirement for remanufacturing is not satisfied by all returned parts, and the percentage of remanufacturable ones is uncertain. The random yield is a major concern in this remanufacturing system. Specifically, suppose that the supply of ELVs r(f) is a deterministic and linear function of the acquisition price f, i.e., r(f) ¼ a + bf, where a,b > 0. The demand for the remanufactured part d(p) is a deterministic and linear function of the selling price p, i.e., d(p) ¼ a bp. The random yield is also dependent on the acquisition price and modeled as the product of R and t(f). t(f) is a deterministic, concave and

Supply Chain Models with Active Acquisition and Remanufacturing

115

nondecreasing function of f which converges to 1 as f increases, and R is a random variable denoting the maximum attainable yield rate. The part not remanufacturable or not remanufactured has a salvage value s. It is also supposed that each ELV has a hulk that can be salvaged with a unit price h. The testing cost for each part is c and the remanufacturing cost for remanufacturable part is cr. The objective is to maximize the total expected profit from the remanufactured part sale and the hulk sale. Bakal and Akcali (2006) studies three different models to explore the acquisition pricing and selling pricing decisions of this problem. The first model is the deterministic yield case serving as a benchmark for the random yield cases. The pricing problem is formulated as follows: max f ; p Pðf ; pÞ ¼ ða bpÞðp s cr Þ þ ða þ bf Þðh þ s f cÞ s:t: ða bpÞ Rtðf Þða þ bf Þ

(6)

where the yield R now is a constant. Let K ¼ h + s c and r0 ¼ [a b(s + cr)]/ [t(K/s a/2b)(a + bK)]. It turns out that r0 is a threshold yield rate. If r0 1, then the profit is increasing in R until R ¼ r0 and remains constant thereafter. Otherwise, the profit is strictly increasing in R. The second model is called Postponed Pricing Model (PPM), in which the yield is random and the remanufacturer has the opportunity to set the selling price of the remanufactured part after the realization of random yield. Therefore, a two-step dynamic programme can be formulated for this model as follows. ðP3Þ

max p P1 ðpjr; f Þ ¼ ða bpÞðp s cr Þ þ ða þ bf Þðh þ s f cÞ s:t:ða bpÞ rtðf Þða þ bf Þ (7)

and ðP4Þ

max f Pðf Þ ¼ ER ½P1 ðp ðR; f ÞjR; f Þ

(8)

where p ðr; f Þ denotes the optimal selling price of Problem (P3) given the yield is realized as r and the acquisition price is f. It is generally difficult to explore the property of P(f) in (P3) for arbitrary t(f). Even in the simple case t(f) ¼ (f + m)/ (f + n), the unimodularity of P(f) is hard to prove analytically, though it could be validated by a thorough numerical experiment. In the third model, called Simultaneous Pricing Model (SPM), the acquisition price and selling price are determined simultaneously before the realization of random yield. Then the problem is equal to max f ; p Pðf ; pÞ ¼ ER fðp cr sÞ min½a bp; Rtðf Þða þ bf Þg ðc sÞða þ bf Þ þ hða þ bf Þ;

(9)

116

X. Li and Y. Li

in which v is the unit shortage penalty. The problem is also hard to analyze theoretically and it is conjectured that the objective function is pseudo-concave. The effect of the random yield on this active acquisition and remanufacturing system is explored by a comprehensive computational study. The threshold yield rate r0 turns out to be a critical indicator. The system is referred to as “high margin”/ “low margin” if r0 1/r0 1, which indicates the profit of used product acquisition h + s c f is high/low. In the high margin case, the expected supply always exceeds the demand in both the PPM and SPM scenarios, which indicates the firm creates a buffer inventory. This effect is especially prominent in SPM since the firm has no opportunity to adjust the selling price according to the realization of the yield and the buffer inventory helps to deal with this uncertainty more effectively. Moreover, when the deviation of the maximum attainable yield rate is moderate, both the models generate profits close to the benchmark DPM. In the low margin case, the performance of PPM is also close to DPM, whereas SPM generates lower profit and the firm does not hold buffer inventory any more since all the remanufacturable parts are used to satisfy the demand due to lower profit margin. It is shown that PPM always outperforms SPM by postponing the selling pricing decision after the realization of random yield. However, in the high margin case the difference is not prominent since the high hulk salvage value offsets the loss of delayed yield information to some extent. In fact, the benefit of postponing the selling pricing becomes more significant with lower profit margin, lower yield rate and higher variation. The value of perfect yield information is also studied by the comparison of deterministic maximum yield R ¼ 0.5 and uniformly distributed yield R ~ U(0,1). It can be concluded that the yield information is crucial for the operations of the firm, especially for the lower margin case.

3 Decentralized Supply Chain with Vertical Channels Outsourcing allows a firm to concentrate on its own core competency, reduce operational cost, and lower the financial risk. For these reasons, an original equipment manufacturer (OEM) usually outsources part of the reverse logistic activities, like used product acquisition or remanufacturing. In this section, we study the decentralized supply chain with vertical structure, in which the reverse logistics activities could be conducted by different supply chain members. The section starts from exploring a typical reverse supply chain consisting of a collector and a remanufacturer, engaged in the acquisition and remanufacturing of used products, respectively. The OEM can choose to outsource the collection or remanufacturing activity to a specific collector or remanufacturer, namely collectordriven or remanufacturer-driven channel, respectively. The decentralized decision problems within these channels are studied in Karakayali et al. (2006) and will be introduced in Sect. 3.1.

Supply Chain Models with Active Acquisition and Remanufacturing

117

Next we discuss the problem on closed-loop supply chain structure, which is regarded as the integration of the forward and reverse channels. The involved entities might be OEM, retailer, and 3PL collector, while the product returns could be through OEM-collection channel, retailer-collection channel or 3PLcollection channel. The related models are investigated in Savaskan et al. (2004) and will be presented in Sect. 3.2.

3.1

Reverse Supply Chain Channels

The reverse supply chain consists of a collector and a remanufacturer. The acquisition of used product and the demand of the remanufactured product are both deterministic and price-sensitive, denoted as r(f) ¼ a + bf and d(p) ¼ a bp, respectively. The collector and the remanufacturer incur unit transaction cost cl and cm, respectively. The salvage value per part is s and per hulk is h. The unit remanufacturing cost is cr. The above assumptions and notations are very similar to Sect. 2, with the difference that the collection and remanufacturing processes are operated by the collector and remanufacturer independently. However, the remanufacturing yield is not considered here. Karakayali et al. (2006) studies the problem on this reverse supply chain channel. Although the main focus is the decentralized interaction, the centralized problem serves as a benchmark for the decentralized one, which is formulated as follows: max f ; p Pðf ; pÞ ¼ ða bpÞðp s cr Þ þ ða þ bf Þðh þ s f cl cm Þ s:t: a bp a þ bf

(10)

It is easy to see this problem is a special case of Sect. 2.2 for R ¼ t(f) 1. For the decentralized scenario, two different channel settings are remanufacturerdriven channel and collector-driven channel. In the remanufacturer-driven/collector-driven channel, the OEM outsources the remanufacturing/collection activity, giving the remanufacturer/collector the leadership role in the channel. The analysis is based on the wholesale price contract under Stackelberg game framework. In the remanufacturer-driven channel, the remanufacturer acts as a leader and the other party responds as a follower. As a result, the collector’s problem is to choose the optimal acquisition price f*(w) for a given wholesale price w, maximizing the following objective: maxPc ¼ ðw f cl Þða þ bf Þ: f

(11)

The remanufacturer’s problem is max w ; p Pr ¼ ða bpÞðp s cr Þ þ ða þ bf ðwÞÞðh þ s w cm Þ s:t: a bp a þ bf ðwÞ:

(12)

118

X. Li and Y. Li

The remanufacturer’s problem can be solved by substituting the simple solution of the collector’s problem and then obtaining the optimal decision. In the collection-driven channel, the collector proposes the wholesale price w and the acquisition price f as the leader, and the remanufacturer correspondingly chooses the selling price of remanufactured product p*(w, f) and the quantity of used products that are not remanufactured t*(w, f). The remanufacturer’s problem is formulated as follows: max p ; t Pr ¼ ða bpÞðp cm cr w þ hÞ þ tðh þ s w cm Þ s:t:

t þ a bp a þ bf

(13)

t 0: The collector sets the wholesale price w and the collection price f accordingly, to maximize her profit function: max w ; f Pc ¼ ðw cl f Þða þ bf Þ s:t: tðw; f Þ þ a bpðw; f Þ a þ bf :

(14)

The solving procedure is similar to the remanufacturer-driven problem. The only delicate notion is that the collector should ensure the remanufacturer’s profit margin to be non-negative, or he is not willing to purchase any unit from the collector. As expected, the centralized channel is shown as the most effective strategy from both environmental and consumer’s perspectives. The acquisition price is highest paid and the selling price lowest charged, which yields the largest quantities of used products collected and remanufactured parts sold. It is also shown that a two-part tariff contract, characterized by a wholesale price and a fixed payment, can be used to coordinate the decentralized supply chain to achieve the collection efficiency attained in the centralized one. For the decentralized channels, a natural question is when the OEM would prefer the collection-driven or remanufacturer-driven channel. Numerical experiment shows that the parameters on both the supply and demand sides can influence such channel choice decision. As a decreases, and/or b increases, and a and/or b increases, it becomes more favorable for an OEM to outsource the collection activity. Otherwise the remanufacturer-driven is more preferable. On the other hand, the OEM can also consider altering the outsourcing decision due to a change in the cost and revenue parameters cr and h. For example, an OEM originally preferring a remanufacturer-driven channel for lower cr may choose a collectiondriven channel as cr increases and exceeds a threshold. Similar behavior could be observed for increasing h. Note that an implicit assumption in the above cases is the homogeneousness of the used product quality. If the used products are heterogeneous, the quality condition of the used product influences the acquisition and remanufacturing costs, and the hulk and part salvage values. Any remanufactured part, however, is

Supply Chain Models with Active Acquisition and Remanufacturing

119

of the same quality level and the same selling price. This setting is the same as in Sect. 2.1. For m quality classes, Karakayali et al. (2006) derives O(m2) iterative algorithms for the centralized and remanufacturer-driven channel problems, and an O(m22m) algorithm for the collector-driven channel problem. The supply chain can also be coordinated by two-part tariff contracts with a proper set of wholesale prices and fixed payments.

3.2

Closed-Loop Supply Chain Channels

We now consider a product in which there is no distinction between a remanufactured item and newly manufactured one, thus OEM could use a hybrid remanufacturing/manufacturing strategy to satisfy the market demand. The goal of this subsection is to present the model on a manufacturer’s reverse channel choice and to discuss its impact on the whole supply chain performance. Specifically, the manufacturer has three options to collect used products for remanufacturing (1) collect directly from customers, called Manufacturer Collection Channel; (2) delegate the downstream retailer to collect, called Retailer Collection Channel; and (3) subcontract the collection to a 3PL collector, like in Sect. 3.1, called 3PL Collection Channel. Suppose that remanufacturing is less costly than manufacturing new product for the manufacturer, i.e., cr < cm. The demand is linear decreasing in the selling price, i.e., d(p) ¼ a bp. The acquisition cost structure, on the other hand, is independent of the collection agent, i.e., the cost of collecting a certain amount of used units is the same for the manufacturer, retailer or 3PL. Let 0 t 1 denote the collection rate, which is the ratio of acquisition quantity of used units to the demand quantity. The total cost of collection C(t) is dependent on the collection rate of used products and given by C(t) ¼ CLt2 + Atd(p), where CL is a scaling parameter and A is the variable collection cost of each unit returned item. The problem is studied in a single-period setting. In the following we present three decentralized closed-loop supply chain channels, and the centralized scenario as well, which serves as a benchmark for the decentralized ones. The objective for the centralized system is to choose the selling price p and collection rate t to maximize max PC ¼ ða bpÞðp cm þ tDÞ CL t2 Atða bpÞ; p;t

(15)

where D ¼ cm cr denotes the economic attractiveness of remanufacturing compared to manufacturing. The analytical solution pC ; tC can be obtained by solving the first-order optimality condition of (15). In the decentralized systems, the manufacturer is assumed to have sufficient channel power over the other entities. She acts as a Stackelberg leader and uses her foresight about the retailer’s and 3PL’s reactions to maximize her own profit. In the

120

X. Li and Y. Li

Manufacturer Collection Channel, for a given wholesale price w, the retailer’s problem is maxp PM R ¼ ðp wÞða bpÞ. Substituting the best response pM ¼ ða þ bwÞ=ð2bÞ, we have the manufacturer’s problem as follows, maxPM M ¼ w;t

a bw a bw ðp cm þ tDÞ CL t2 At : 2 2

(16)

The optimal solutions wM and tM can be obtained by solving the first-order optimality condition. In the Retailer Collection Channel, the manufacturer delegates the retailer to collect used products and pays a transfer price b per unit acquired. Thus the retailer’s problem is maxp;t PRR ¼ ðabpÞðpwÞþbtðabpÞCL t2 AtðabpÞ, and the optimal pR and tR can be solved since the profit function is concave. Given the best response of the retailer, the manufacturer’s problem is R R R R max PM M ¼ ða bp Þðw cm þ t DÞ bt ða bp Þ; w;b

(17)

and the corresponding solution wR and bR can be obtained. An interesting observation is that the manufacturer’s profit is increasing in b, thus it is optimal to set bR ¼ D by the manufacturer. In the 3PL Collection Channel, the manufacturer outsources the collection to an independent 3PL collector and pays a transfer price b per unit collected. In this case, the retailer only engages in the product distribution and chooses the optimal selling price p , for a given wholesale price set by the manufacturer. The 3PL’s problem is R 2 R 3P maxt P3P 3P ¼ btða bp Þ CL t Atða bp Þ, and the optimal solution t can be obtained. As a Stackelberg leader, the manufacturer determines the optimal w3P and b3P by solving R 3P Þ: max P3P M ¼ ða bp Þðw cm þ ðD bÞt w;b

(18)

We have b3P ¼ ðD þ AÞ=2, indicating that b is a direct cost for the manufacturer and the closed-loop supply chain. Savaskan et al. (2004) compares the above cases and obtains the following results on the different closed-loop supply chain channels: 1. tC > tR > tM > t3P : 2. pC < pR < pM < p3P , and consequently dC > dR > d M > d3P . M 3P R M 3P C R M 3P 3. pR M > pM > pM , pR > pR > pR , and consequently p > pT > pT > pT . It shows that the collection rate increases with the 3PL, manufacturer, retailer as collecting agent, respectively, and reaches the highest value in the centralized system. Consequently, the acquisition of used product and sales of remanufactured product are highest in quantity in the centralized system. This result, which has been proved in Sect. 3.1, is validated again in the closed-loop supply chain setting.

Supply Chain Models with Active Acquisition and Remanufacturing

121

On the other hand, the selling price decreases in the same sequence. Moreover, both the profits of the retailer and manufacturer reach the highest values in the Retailer Collection Channel and lowest in the 3PL Collection Channel. In summary, the preferred collecting agent is the retailer, followed by the manufacturer, and then the third-party (3P), no matter from the perspective of the manufacturer, retailer, supply chain or the consumer welfare. Finally, Savaskan et al. (2004) indicates that a simple coordination mechanism can be designed for the Retailer-Collection Channel and thus the maximum of the supply chain profit can be attained.

4 Decentralized Supply Chain of Horizontal Competition Horizontal competition among supply chain members is a common phenomenon. Generally, retailers compete in sales performance, and manufacturers compete for market share. In the closed-loop supply chain area, the horizontal competition is more fierce between the OEM who manufactures the product originally and the small agents who only involve in the used product acquisition and remanufacturing. Such instances can be found in the industry of toner cartridge (Majumder and Groenevelt 2001a) or single-used camera (Ferrer and Swaminathan 2006). In this section, we introduce two models seizing the essential feature of horizontal competition between OEM and local remanufacturer. Both models capture the dynamic nature of the problem by a two-period setting, in which the OEM manufactures in the first period and faces competition with the remanufacturer in the second period. The main differences are that the first model assumes a linear remanufacturing cost and distinguishable product brands between the OEM and remanufacturer, while the second model assumes convex collection and remanufacturing costs and distinguishable product quality between the new product and remanufactured one. Nearly all papers in this field fall into the category of brand competition or quality competition, and this section incorporates typical problems of these two types.

4.1

Distinguishable Brand and Linear Remanufacturing Cost

We first consider a two-period model with two players, an OEM and a local remanufacturer. In the first period, the OEM manufactures and sells new product. A fraction of sold items after use is available for the OEM and remanufacturer to be acquired and remanufactured in the second period, and the OEM can also manufacture new units in addition to remanufacturing used ones. Therefore, the players compete in selling their products to consumers in the second period.

122

X. Li and Y. Li

Since the competition for used product acquisition is not the focus of this problem, the model is analyzed under given allocation mechanisms for returned units. Specifically, the OEM has access to a fraction of the products sold in the first period, namely g, and the remanufacturer has 1 g. Four allocation mechanisms are considered (1) OEM can acquire any used product left over by the remanufacturer, however the remanufacturer cannot do likewise; (2) the remanufacturer can acquire any used product left over by OEM, but OEM cannot do likewise; (3) neither can acquire used product left over by the other; (4) both can acquire used product left over by the other. An important feature is that the two players face a brand competition in selling their products. In other words, there is a perfect substitution between the OEM’s new and remanufactured items like in Sect. 3.2, but customers can distinguish between the remanufacturer and OEM’s products. The demand functions in the second period are of Bertrand-type: Do ðpo ; pl Þ ¼ ao bo po þ co pl ; Dl ðpo ; pl Þ ¼ al bl pl þ cl po ;

(19)

where the subscript o means the variables and parameters of the OEM, and the subscript l means those of the local remanufacturer. We suppose that the quantity of total used products available in the second period is equal to a fraction a of the production quantity in the first period. The problem is analyzed in two stages, starting with Nash game analyze in the second period. Given this equilibrium prediction, the OEM chooses the manufacturing quantity and selling price in the first period. Specifically, the remanufacturer’s problem in the second period is max ql ; pl Pl ¼ ql ðpl rl Þ s:t: ql r; ql Dl ql 0; pl 0

(20)

where ql and pl are the remanufacturer’s decision variables, denoting the remanufacturing quantity and selling price, respectively. rl is unit remanufacturing cost and r is the quantity of attainable used products according to specific allocation rule. Similarly, the OEM’s problem in the second period is max z ; qor ; po Pl ¼ zðpo cÞ þ ðc ro Þqor s:t: qor r; qor z Do qor 0; p0 0

(21)

where qor, z and po are the OEM’s decision variables, denoting the remanufacturing quantity, total selling quantity and the selling price, respectively. The unit

Supply Chain Models with Active Acquisition and Remanufacturing

123

remanufacturing cost is ro and manufacturing cost is c. Here, a crucial assumption is that the variable costs for remanufacturing each unit are constants (though they can be different for the OEM and local remanufacturer, i.e., rO 6¼ rl). Different remanufacturing cost structure will be discussed in the next subsection. Groenevelt and Majumder (2001a) proves that a unique pure strategy Nash Equilibrium exists in the second period game. The total produced quantity (of the newly manufactured and remanufactured units) and the newly manufactured quantity are both nondecreasing in the quantity of returned items R. In fact, as R increases, the remanufacturer no longer uses up all available used products to remanufacture, while the OEM first stops manufacturing new units and then no longer uses up all the used units available for him. In contrast, the first period problem for OEM is difficult to analyze theoretically, as the profit function doesn’t have a convenient structural property. Nevertheless, some important insights can be obtained through numerical experiments. For example, we can consider that OEM is a monopoly without or with remanufacturing option. These cases, called B1 (without remanufacturing) and B2 (with remanufacturing) serve as benchmarks to examine the competitive case. It is shown that a monopoly OEM in B2 produces most in the first period and earns the highest total profit among three cases. However, the comparison of B1 with the competitive case is inconclusive. Under some circumstances the OEM does have higher profit in the competitive case than in B1, indicating that the OEM might prefer to facilitate the remanufacturing even if it would incorporate the reverse channel competition. The effects of parameter changes are also examined. An interesting observation is that although competing with the OEM in remanufacturing, the remanufacturer is better off when the OEM’s remanufacturing cost r0 is reduced, which would induce the OEM to produce more in the first period and thus benefit the remanufacturer. The fraction a is another crucial parameter, the increase of which would enlarge the remanufacturing quantity. However, such an increase does not necessarily raise the OEM’s profit due to the competition effect. Note that the above observations are valid for all allocation mechanisms, indicating the allocation rule is not an essential factor as long as the quantity fraction between players is exogenously imposed.

4.2

Distinguishable Quality and Convex Remanufacturing Cost

Ferguson and Toktay (2006) studies the horizontal competition problem following Majumder and Groenevelt (2001a) in somewhat similar settings. The OEM manufactures new product in the first period, and in the second period faces the entry threat of a local remanufacturer that competes with the OEM by selling the remanufactured product. The quantity of available units to be remanufactured is a fraction g of the production quantity of the first period. Some other key assumptions, which are distinct from the last model, are stated as follows.

124

X. Li and Y. Li

First, the qualities of the new product and remanufactured one are differential for customers. More specifically, each consumer’s willingness-to-pay for a remanufactured product is only a fraction d (∈ (0, 1)) of that for the new one. It is further assumed that the consumer’s willing-to-pay is heterogeneous and uniformly distributed in interval [0,1]. This leads to the demand functions entirely different from Sect. 4.1. Second, the collection/remanufacturing cost is increasing and convex with respect to the collection/remanufacturing quantity, respectively. More specifically, the remanufacturing cost is a quadratic function hq2, for a given remanufacturing quantity q. Finally, instead of an exogenous allocation mechanism in Sect. 4.1, this model incorporates a costly collection and remanufacturing option. More specifically, the firm choosing to remanufacture incurs a fixed cost F ¼ Fc + Fr where Fc and Fr are the fixed costs for building up the collection operation and remanufacturing operation, respectively. Under the above assumptions, it is investigated that whether a monopoly OEM should process the collection and remanufacturing. For a monopoly OEM without remanufacturing, it turns out to be a simple optimization problem maxq(p c)q for each period. For the OEM with remanufacturing, a two-step dynamic programming model is formulated as follows: max q2n ;q2r P2 ðq2n ; q2r jq1 Þ ¼ ðp2n cÞq2n þ ðp2r hq2r Þq2r s:t:

q2r gq1

(22)

and max pðq1 Þ ¼ ðp1 cÞq1 þ p2 ðq1 Þ:

(23)

q1 0

Note that in the above formulation we suppose the fixed cost of remanufacturing F ¼ 0. In this case, remanufacturing is always more profitable than not remanufacturing. However, if F > 0, then there exists a threshold level on the remanufacturing cost factor h, above which the OEM would not remanufacture any unit in the second period. The closed-form solutions of the pricing and quantity decisions can be obtained. Another key issue of this problem is to explore the strategic role of the OEM remanufacturing as an entry deterrent to the local remanufacturer. Ferguson and Toktay (2006) studies the motivation for an OEM without remanufacturing to deter the remanufacturer’s entry into the reverse channel. In fact, an OEM might not remanufacture due to a too high remanufacturing cost factor h or fixed investment F. In this case, however, it is shown that remanufacturing could still be profitable for an external remanufacturer thus the OEM may suffer from it. Specifically, Nash Equilibrium can be obtained by simultaneously solving the following problems faced by the OEM and remanufacturer: q2r Þ ¼ ðp2n ðq2n ; qc max q2n P2 ðq2n jc 2r Þ cÞq2n ;

(24)

Supply Chain Models with Active Acquisition and Remanufacturing

125

and q2r jq2n ; q1 Þ ¼ ðc p2r ðq2n ; qc q2r Þc q2r F max qb P2 ðc 2r Þ hc 2r

s:t:

qc 2r gq1 :

(25)

In the first period, the OEM determines the production quantity q1 expecting the equilibrium result of the second period. By analyzing the above problem, Ferguson and Toktay (2006) characterizes the potential profit loss for the OEM due to the sales competition with the remanufacturer. As a result, the OEM could adopt two entry-deterrent strategies: remanufacturing, and preemptive collection. By the first strategy, the OEM chooses to remanufacture for the sole purpose of discouraging an external remanufacturer to do so, even if the remanufacturing would not be preferable in a monopoly setting. By the second strategy, the OEM would choose to collect returned units with the quantity large enough to deter the remanufacturer from collecting and remanufacturing. The collected used product, however, is never remanufactured by the OEM. They further investigate the impact of collection cost, unit manufacturing cost and consumer willing-to-pay on the choice of which deterrent strategy to use for the OEM. As the collection cost/unit manufacturing cost/willingness-to-pay increases, the profitability of the remanufacturing becomes more prominent. Moreover, if the collection cost increases linearly in the quantity, the collection strategy would also become less attractive, which indicates the significance of an accurate modeling for the collection cost curve.

5 Conclusion The importance of remanufacturing has been widely recognized in terms of environmental sustainability and economical benefit. The topic of this chapter relates to the supply chain models on active acquisition and remanufacturing, with a discussion of the following three classes of problems. The first class is on the centralized system with acquisition price and sale price decisions, faced by remanufacturing firms utilizing a market-driven acquisition channel. The used product supply can be controlled actively by the acquisition price, and the acquisition management is shown as a significant driver of remanufacturing profitability. In the deterministic model, the efficient strategies are developed to balance the supply and demand through the pricing level. In the random yield model, the effect of remanufacturing random yield is explored, and the benefit of delaying pricing decisions to mitigate the yield randomness is analyzed. Both models belong to the class of centralized optimization. It is also common for a dominant party in the supply chain, usually the manufacturer, to dictate terms to other supply chain members to process some activities,

126

X. Li and Y. Li

such as acquisition or remanufacturing. It is the objective of Sect. 3 to investigate this vertical channel design problem for the decentralized supply chain. In the pure reverse logistic setting, it is shown that the channel preference is conditional on some parameters, although an OEM might prefer to outsource the collection activity rather than the remanufacturing activity from a practical perspective. On the other hand, in a closed-loop supply chain setting, the OEM is most likely to assign the collection activity to the retailer and most reluctantly to the 3PL collector. The supply chain coordination issue is also considered in both the cases. Moreover, conflict between entities at the same supply chain level is also prevailing, which is the central issue of Sect. 4. Different settings of product distinction and cost structure lead to different results, while the effects of horizontal competition are explored by comparing with the monopoly OEM case. We believe there is still great potential on the study of supply chain with active acquisition and remanufacturing. Possible further directions include the problem with more random factors, e.g., the problem with random acquisition, random yield, and random demand, etc. In this case, both the centralized and decentralized issues are more complicated, yet more interesting. Another promising research opportunity might be information asymmetry. The information issue in the supply chain field has been a hot topic, but very scarce research has been done related to the product acquisition and remanufacturing. Finally, it is to be studied on negotiation problem between the manufacturer and other supply chain members, within the framework of bargaining theory or other cooperative game theories. Acknowledgment This work is partly supported by National Natural Science Foundation of China (NSFC) Nos. 70971069, 71002077 and 71002106, the Fok Ying-Tong Education Foundation of China (Grant No. 121078), and 2009 Humanities and Social Science Youth Foundation of Nankai University (Grant No. NKQ09027).

References Aras N, Aksen D (2008) Locating collection centers for distance and incentive dependent returns. Int J Prod Econ 111(2):316–333 Aras N, Aksen D, Tanugur AG (2008) Locating collection centers for incentive dependent returns under a pick-up policy with capacitated vehicles. Eur J Oper Res 191(3):1223–1240 Atasu A, Sarvary M, Van Wassenhove LN (2008) Remanufacturing as a marketing strategy. Manage Sci 54(10):1731–1746 Bakal IS, Akcali E (2006) Effects of random yield in reverse supply chains with price-sensitive supply and demand. Prod Oper Manage 15(3):407–420 Debo L, Toktay B, Van Wassenhove LN (2005) Market segmentation and product technology selection for remanufacturable products. Manage Sci 51(8):1193–1205 Dekker R, Fleischmann M, Inderfurth K, Van Wassenhove LN (2004) Reverse logistics: quantitative models for closed-loop supply chains. Springer, New York Ferguson M, Toktay B (2006) The effect of competition on recovery strategies. Prod Oper Manage 15(3):351–368 Ferguson M, Guide VDR Jr, Souza G (2006) Supply chain coordination for false failure returns. Manuf Serv Oper Manage 8(4):376–393

Supply Chain Models with Active Acquisition and Remanufacturing

127

Ferrer G (2003) Yield information and supplier responsiveness in remanufacturing operations. Eur J Oper Res 149:540–556 Ferrer G, Swaminathan J (2006) Managing new and remanufactured products. Manage Sci 52 (1):15–26 Galbreth MR, Blackburn JD (2006) Optimal acquisition and sorting policies for remanufacturing. Prod Oper Manage 15:384–392 Guide VDR Jr, Jayaraman V (2000) Product acquisition management: current industry practice and a proposed framework. Int J Prod Res 38(16):3779–3800 Guide VDR Jr, Van Wassenhove LN (2001) Managing product returns for remanufacturing. Prod Oper Manage 10(2):142–155 Guide VDR Jr, Van Wassenhove LN (2009) The evolution of closed-loop supply chain research. Oper Res 57(1):10–18 Guide VDR Jr, Teunter R, Van Wassenhove LN (2003) Matching supply and demand to maximize profits from remanufacturing. Manuf Serv Oper Manage 5:303–316 Heese HS, Cattani K, Ferrer G, Gilland W, Roth AV (2005) Competitive advantage through takeback of used products. Eur J Oper Res 164:143–157 Ilgin M, Gupta S (2010) Environmentally conscious manufacturing and product recovery (ECMPRO): a review of the state of the art. J Environ Manage 91(3):563–591 Karakayali I, Emir-Farinas H, Akcali E (2006) An analysis of decentralized collection and processing of end-of-life products. J Oper Manage 25(6):1161–1183 Kaya O (2010) Incentive and production decisions for remanufacturing operations. Eur J Oper Res 201:442–453 Liang Y, Pokharel P, Lim GH (2009) Pricing used products for remanufacturing. Eur J Oper Res 193(2):390–395 Majumder P, Groenevelt H (2001a) Competition in remanufacturing. Prod Oper Manage 10 (2):125–141 Majumder P, Groenevelt H (2001b) Procurement competition in remanufacturing. Working paper, Duke University School of Business, Durham, NC Meyer H (1999) Many happy returns. J Bus Strategy 80(7):27–31 Mukhopadhyay S, Setoputro R (2005) Optimal return policy and modular design for build-to-order products. J Oper Manage 23:496–506 Padmanabhan V, Png IPL (1997) Manufacturer’s returns policies and retail competition. Mark Sci 16(4):81–94 Pasternack BA (1985) Optimal pricing and return policies for perishable commodities. Mark Sci 4 (2):166–176 Pokharel S, Mutha A (2009) Perspectives in reverse logistics: a review. Resour Conserv Recycl 53 (4):175–182 Prahinski C, Kocabasoglu C (2006) Empirical research opportunities in reverse supply chains. Omega 34(6):519–532 Qu X, Williams JA (2008) An analytical model for reverse automotive production planning and pricing. Eur J Oper Res 190(3):756–767 Ray S, Boyaci T, Aras N (2005) Optimal prices and trade-in rebates for durable, remanufacturable products. Manuf Serv Oper Manage 7:208–228 Robotis A, Bhattacharya S, van Wassenhove LN (2005) The effect of remanufacturing on procurement decisions for resellers in secondary markets. Eur J Oper Res 16(3):688–705 Savaskan C, Van Wassenhove LN (2006) Reverse channel design: the case of competing retailers. Manage Sci 52(1):1–14 Savaskan C, Bhattacharya S, Van Wassenhove LN (2004) Closed-loop supply chain models with product remanufacturing. Manage Sci 50(2):239–252 Vlachos D, Dekker R (2003) Return handling options and order quantities for single period products. Eur J Oper Res 151:38–52 Webster S, Mitra S (2007) Competitive strategy in remanufacturing and the impact of take-back laws. J Oper Manage 25(6):1123–1140

128

X. Li and Y. Li

Webster S, Mitra S (2008) Competition in remanufacturing and the effects of government subsidies. Int J Prod Econ 111(2):287–298 Wojanowski R, Verter V, Boyaci T (2007) Retail-collection network design under deposit-refund. Comput Oper Res 34:324–345 Yue X, Raghunathan S (2007) The impacts of the full returns policy on a supply chain with information asymmetry. Eur J Oper Res 180:630–647 Zikopoulos C, Tagaras G (2007) Impact of uncertainty in the quality of returns on the profitability of a single-period refurbishing operation. Eur J Oper Res 182:205–225

Part II

Analytical Models for Innovative Coordination under Uncertainty

.

Facilitating Demand Risk-Sharing with the Percent Deviation Contract Matthew J. Drake and Julie L. Swann

Abstract Suppliers do not have much incentive to build capacity for supply chains with stochastic demand in which the buyer bears little or no inventory risk. This hinders the supply chain from satisfying the optimal amount of customer demand from a channel perspective. We describe and analyze the percent deviation contract as an innovative mechanism to improve the overall performance of this type of supply chain. This contract induces a dynamic game of perfect information, and we characterize the subgame-perfect Nash Equilibria under various contract scenarios. We establish ways to set the contract parameters to coordinate the supply chain under uncertainty and show that the percent deviation contract is able to achieve channel coordination in some cases where the quantity flexibility contract fails. In order to aid the implementation of the percent deviation contract in practice, we develop ways to set the parameters to satisfy the buyer’s individual-rationality constraint. Keywords Supply chain coordination • Contracting • Newsvendor • Model • Risksharing

1 Introduction The proliferation of computerized information systems in the 1990s facilitated the establishment of supply chain partnerships in which demand information is shared between firms. The upstream firms can use this information to reduce the traditional M.J. Drake (*) Palumbo-Donahue Schools of Business, Duquesne University, Pittsburgh, PA 15282, USA e-mail: [email protected] J.L. Swann H. Milton Stewart School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, GA 30332-0205, USA e-mail: [email protected] T.-M. Choi and T.C. Edwin Cheng (eds.), Supply Chain Coordination under Uncertainty, International Handbooks on Information Systems, DOI 10.1007/978-3-642-19257-9_6, # Springer-Verlag Berlin Heidelberg 2011

131

132

M.J. Drake and J.L. Swann

demand distortion due to the bullwhip effect. Some firms have also incorporated this information into contracts that induce their supply chain partners to share demand risk, thereby improving supply chain efficiency. Many researchers and practitioners (e.g., Lee 2004; Finley and Srikanth 2005) have advocated demand risk-sharing as a necessary condition for supply chain collaboration efforts to be successful in practice. In this paper we analyze one such contracting mechanism, an innovative structure which we denote as the percent deviation contract. The percent deviation contract is most applicable to supply chains in which the buyer does not traditionally bear any inventory obsolescence risk in satisfying stochastic consumer demand. In these channels the buyer places orders with the supplier only when the consumer demand is known with certainty; that is, the buyer does not carry any excess inventory. This environment occurs in many service operations. One industry that stands to benefit from application of the percent deviation contract is truckload transportation. In fact, a large truckload carrier originally proposed the idea for this particular contract structure but did not know how to set the parameters or whether or not the contract would be beneficial. While these carriers generally have standing weekly orders for loads with their bigger customers, many shippers call dispatch requesting a pickup in a few hours. This limits the carrier’s ability to utilize its equipment effectively by allocating trailers in advance or coordinating backhauls from prior shipments. The percent deviation mechanism is applicable in many traditional manufacturer–retailer channels as well, such as home construction, equipment integrators, window replacement, or door-to-door sales. In all of these industries, the supplier bears most, if not all, of the consumer demand risk in many arrangements. We analyze the strategic properties of the percent deviation contract in which the buyer gives an initial order estimate and the supplier pre-acquires inventory at a low cost. Once the buyer’s consumer demand is realized, the buyer places the actual order, and the supplier fulfills all or a portion of the order, possibly by expediting, at a higher cost. The buyer pays a penalty if the final order is outside of an allowable range established around the initial order estimate. We characterize the subgameperfect Nash Equilibria (SPNE) decisions when the supplier has a fixed expediting capacity and discuss methods of channel coordination to optimize the performance of the entire system. Since the buyer assumes some consumer demand risk under the percent deviation contract, its expected profit may be less than that under a traditional contracting structure; therefore, we develop a method that the supplier can use to satisfy the buyer’s individual-rationality constraint. Our contribution to the existing literature on supply chain collaboration includes analysis of a risk-sharing contract where decisions made by the buyer and supplier explicitly depend on each other and are solvable in the framework of a dynamic, extensive form game. This necessarily results in a more complex contract, but we also show that this contract can be strictly Pareto-improving for both parties. Our contract has a structure similar to the quantity flexibility contract, but ours does not enforce limits on the buyer’s final behavior; thus, this contract can coordinate the supply chain in some cases where quantity flexibility cannot. Many models consider a supply chain with infinite capacity; whereas, the total capacity in our model is a function of the supplier’s decisions as well as an external constraint.

Facilitating Demand Risk-Sharing with the Percent Deviation Contract

133

2 Literature Review The breadth of supply chain contracting literature has grown significantly over the last two decades as researchers and practitioners have examined strategic relationships between supply chain partners. One stream of supply chain contracting literature has proposed and analyzed methods of coordinating decentralized decisions to attain the optimal supply chain profit. See Tsay et al. (1999) and Fugate et al. (2006) for extensive reviews of the supply chain contracting literature. We discuss below the most relevant contracting references, which model a system with multiple, sequential decisions. Tsay (1999) analyzes a quantity flexibility contract in which the retailer commits to purchasing no less than a certain percentage of the initial forecast while the supplier agrees to fulfill up to a certain percentage above the forecast. He also evaluates the sharing of demand risk that produces the coordinated channel. Tsay and Lovejoy (1999) extend these results to a rolling horizon decision environment. More recently, Lian and Deshmukh (2009) develop a finite-horizon dynamic programming model for a quantity flexibility contract where the buyer can adjust previously-determined order quantities in response to updated demand information for a higher price. Bassok and Anupindi (2008) discuss a similar model where the buyer must establish quantity commitments for multiple periods into the future and then can adjust them within a certain tolerance in a rolling horizon fashion. In contrast to quantity flexibility, the percent deviation contract places no limits on the buyer’s final order, although it adds complexity to the decision environment by including additional contract parameters. We show in Sect. 4.3 that this added complexity can be justified because the percent deviation contract succeeds in coordinating the supply chain in several cases where the quantity flexibility contract is known to be unable to coordinate the channel. Donohue (2000) and Cachon (2004) analyze contracts with two-tier pricing structure that induce early commitment from buyers. In both of these contracts the buyer is bound to its order in both periods, whereas in our contract the first order is only an estimate of demand and can be freely adjusted once demand is known. These two papers only consider the full compliance contract regime where the supplier must fulfill the entire order; whereas, we model the supplier’s compliance decision explicitly. Several contracts employ an options framework where the buyer makes a firm order commitment and purchases options for additional goods to be exercised if demand is high. Cachon and Lariviere (2001) consider a single period model with options and forecast sharing. Since the buyer has an incentive to provide a biased forecast, they develop conditions that facilitate the credible sharing of forecasts under both full and voluntary compliance. Barnes-Schuster et al. (2002) extend the options framework using a two-period model with correlated demand between periods. Wang and Liu (2007) develop some structural properties of coordinating options contracts in channels with powerful retailers, and Zhao et al. (2010) show that an options contract can coordinate the supply chain and can be

134

M.J. Drake and J.L. Swann

Pareto-improving compared with a traditional wholesale-price contract. In our model we have no firm commitment and no upper bound on the final order amount, so our contract cannot be reduced to an option-based model. An additional series of studies (see, for example, Erkoc and Wu 2005; Serel 2007) have analyzed reservation fee supply contracts in which the buyer pays a (usually) deductible fee to reserve capacity along with an exercise fee for the final order quantity. The manufacturer builds capacity based on the reservations made, but it can also build excess capacity to offer at a higher spot rate once demand is realized. The aforementioned studies only consider linear reservation fee contracts–where each unit ordered is charged the same prices. The percent deviation contract is a special case of a piecewise-linear reservation fee contract in which the reservation and exercise prices differ for various portions of the order. In addition to contracting, several papers (e.g., Lee et al. 2000; Cachon and Fisher 2000; Balakrishnan et al. 2004) have examined various ways of reducing the bullwhip effect through information sharing in decentralized supply chains. Kulp et al. (2004) study the benefits the manufacturer gains under different degrees of information sharing and collaboration. They find that most of the manufacturers’ benefit from information integration comes from collaborative activities such as vendor-managed inventory and collaborative forecasting instead of simply sharing information. On the contrary, our results suggest that the risk sharing induced by the percent deviation contract enables the supplier and the entire channel to attain higher profit. We develop a model in the next section for the general case where the supplier has an expediting capacity constraint as well as the special case of infinite capacity. In Sect. 4 we identify conditions on the contract parameters that satisfy each party’s participation constraints, detail ways to coordinate the channel in each decentralized scenario, and compare the percent deviation contract with the wellknown quantity flexibility contract. We demonstrate the use of the percent deviation contract on several numerical examples in Sect. 5, and we discuss the study’s conclusions and suggestions for future research in the final section of the chapter.

3 Models and Scenarios The percent deviation contract accommodates the following sequence of decisions. The buyer provides an initial estimate of its final-order demand that will be placed at a later date. The seller can then use this information to acquire goods in advance (e.g., a truckload carrier can preposition trucks or coordinate backhauls to optimize its transportation network) at a low cost in anticipation of this demand. When the buyer’s demand is known with certainty, the buyer places actual order with the supplier. Depending on the contract parameters, the seller can choose to satisfy additional demand by expediting or subcontracting at a high cost or can choose to fulfill only the demand equal to the number of previously-acquired goods. The percent deviation penalty is the mechanism that punishes the buyer for unrealistic

Facilitating Demand Risk-Sharing with the Percent Deviation Contract

135

estimates. If the buyer’s final order is within a certain percentage above or below the initial estimate, no penalty is charged. If the order exceeds the limits, the supplier charges a penalty on all goods ordered outside of the tolerable range.

3.1

Notation and Assumptions

We employ notation adapted from Donohue (2000). The buyer receives r dollars in revenue for each unit and pays a wholesale price, w, to the supplier. We assume that the buyer earns a positive gross margin from these transactions (i.e., r > w). Consumer demand for a period is given by the random variable X, which has a continuous, differentiable probability distribution function, f(x). If the buyer cannot satisfy the customers’ demand (due to lack of product availability), it incurs a customer penalty of b per unit.1 The seller faces a cost of c1 dollars to acquire a unit of inventory in anticipation of demand and must pay c2 dollars to satisfy demand after the firm order has been placed. We assume that c2 > c1, so the c2 can be thought of as an expediting or subcontracting cost. If the supplier has excess inventory at the end of the period, it receives a unit salvage value of v. It is natural to assume that w > v and c1 > v, which ensure that the supplier does not receive too much of a benefit from selling goods for salvage. Since the seller may choose not to satisfy the buyer’s entire order, it must pay the buyer a for each unit ordered but not delivered. We assume that a < b, which signifies that lost customers are more costly for the buyer. The per-unit penalty that the buyer must pay the supplier for orders outside of the allowable deviation range is denoted by p, while d ∈ [0,1] is the percentage that defines the range. For orders above the upper limit of the range, the buyer only pays the percent deviation penalty for the units that the supplier fulfills. The buyer’s initial forecast is given by q1, and the actual order is q2. The number of units the supplier acquires in advance of demand is t1, and the additional goods expedited or subcontracted are denoted by t2. The supplier has a maximum expediting capacity of M units. This particular way of modeling the supplier’s capacity bears further consideration. It is important to note that the capacity for the supplier’s pre-acquisition decision is infinite. By setting the t1 value, the supplier is de facto determining the capacity of the system as a whole, which is equal to t1 + M. This structure is appropriate for buyer–supplier transactions in which the supplier has a lot of capacity in its system but must make the allocation decisions across many customers before production occurs. Therefore, if the supplier knows in advance that demand will be high, it can allocate sufficient capacity to satisfy the large order; closer to the purchase date, however, it can only provide a limited amount of 1 This b could also be viewed as the higher cost from using an alternative supplier not under longterm contract.

136

M.J. Drake and J.L. Swann

excess capacity because the rest of the system is dedicated to fulfilling orders from other customers. We make the following assumptions that improve tractability but are not likely to impede the application of the results. The first assumption is that all costs are constant for each unit of demand for a single product line because we are interested in examining the structure of the incentives. Another assumption is the existence of complete, symmetric cost, capacity, and demand information between the two parties. When the buyer places the final order, it knows the exact demand as is usual in the other relevant service and manufacturer–retailer channels discussed in Sect. 1. If the actual demand exceeds the upper limit of the deviation range, the cost and penalty parameters determine whether or not the buyer’s order equals the full demand. In order for the buyer to order above the deviation threshold, the net cash flow must exceed the shortage penalty owed to the end consumer. r w p> b

(1)

If the inequality holds, then q2 ¼ X, or the actual customer demand. If this inequality is not satisfied (for instance, if the penalty for ordering outside of the deviation range is too high), q2 ¼ min{X,(1 þ d)q1}. The additional assumption w > p assures that if the actual demand is below the lower limit of the deviation range, (1 d)q1, the buyer orders the actual demand.

3.2

General Model with Finite Expediting Capacity

We begin our analysis with the decentralized structure in which each party makes decisions to optimize its individual expected profit. Even though the supplier has an expediting capacity of M units, the cost of expediting these units, c2, might be too high for the supplier to choose to do so. In order for the supplier to use any of this expediting capacity, the cash flow from expediting must be higher than the cost of failing to expedite. These flows are dependent on whether or not the supplier will receive the deviation penalty on some or all of these units. While the two buyer scenarios discussed above, which are dependent on whether or not the buyer is willing to place orders above the upper limit of the deviation range, generate different supplier responses, the derivation and form of these optimal decisions are the same in both cases. Thus, in this chapter we only examine the case in which the buyer is willing to order the entire demand even if it must pay the deviation penalty. (See Drake (2006) for the case in which the buyer will not order above the upper limit of the deviation range.) When the buyer is willing to order the actual demand, the supplier’s expediting decision can be determined a priori, without knowledge of how many units for which the buyer will pay the deviation penalty. If w c2 > a, then the supplier finds it beneficial to expedite whether or not the buyer will pay the deviation penalty

Facilitating Demand Risk-Sharing with the Percent Deviation Contract

137

on any units; consequently, t2 ¼ ðminfq2 t1 ; MgÞþ . We will denote this as Case A. Similarly, if w c2 þ p < a, the supplier would not choose to expedite any units even if the buyer paid the deviation penalty on all of the units; and thus, t2 ¼ 0. This will be Case B. We will use backward induction to solve for the subgame-perfect Nash Equilibria in each scenario. We now formulate the expected profit functions for the buyer and supplier in Case A, where q2 ¼ X and t2 ¼ ðminfq2 t1 ; 0gÞþ . The supplier chooses t1 to maximize the following expected profit function: Z t1 þM Z 1 S PA ¼ w xf ðxÞdx þ ðt1 þ MÞð1 Fðt1 þ MÞÞ a ðx t1 MÞf ðxÞdx Z

t1 þM

0

minft1 þM;ð1dÞq1 g

þp "Z0 þp

t1 þM

ð1þdÞq1

Z

t1

þv

ðminft1 þ M; ð1 dÞq1 g xÞf ðxÞdx # þ

ðx ð1 þ dÞq1 Þf ðxÞdx þ ðt1 þ M ð1 þ dÞq1 Þ ð1 Fðt1 þ MÞÞ Z

ðt1 xÞf ðxÞdx c1 t1 c2

0

t1 þM

ðx t1 Þf ðxÞdx þ Mð1 Fðt1 þ MÞÞ :

t1

(2) There are three separate functions that represent the realizations of the expected profit function in (2) based on the relationship between the total system capacity, t1 þ M, and the boundaries of the deviation range, (1 d)q1 and (1 þ d)q1. The expected profit regions are depicted in Fig. 1. In region S1 the supplier sets capacity so that it cannot even satisfy the lower limit of the deviation range. Region S2 prescribes that the total system capacity lies somewhere in the deviation range. In these first two regions, the buyer will never pay the deviation penalty for orders above the upper limit of the range because these units will never be fulfilled. Region S3 specifies that the system capacity exceeds the upper limit of the deviation range. In each region only one of the three separate expected profit realizations is feasible, regardless of which expected profit is higher in the region. The following observation establishes that the overall expected profit function is continuous. Observation 1 The individual expected profit function realizations that are active in two adjacent feasible regions are equal at the boundary (i.e., and PSA:II: ðð1 þ dÞq1 MÞ PSA:I: ðð1 dÞq1 MÞ ¼ PSA:II: ðð1 dÞq1 MÞ 2 S ¼ PA:III: ðð1 þ dÞq1 MÞ). Fig. 1 Regions of capacity defining the form of the supplier’s expected profit function

S1

S2 (1-d)q1

S3 (1+d)q1

t1+M

The three separate functions (PSA:I: , PSA:II: , and PSA:III: ) are defined in (24)–(26) in the Appendix.

2

138

M.J. Drake and J.L. Swann

Lemma 1. If w þ a > p þ c2, the supplier’s expected profit function in (2) is piecewise-concave, a continuous, piecewise function whose separate segments are individually concave. Each of the individual realizations of (2) has a corresponding maximizing value that is derived from the solution to the following equations, w þ a c1 ðc2 vÞFðtÞ (3) t1A:I: 2 t : Fðt þ MÞ ¼ w þ a c2 p w þ a c1 ðc2 vÞFðtÞ A:II: (4) t1 2 t : Fðt þ MÞ ¼ w þ a c2 t1A:III: 2

t : Fðt þ MÞ ¼

w þ a c1 þ p ðc2 vÞFðtÞ ; w þ a c2 þ p

(5)

which are all dependent on the supplier’s available expediting capacity. It is interesting to note that all of these values are independent of the buyer’s order quantity; they can easily be computed from the exogenous parameters. We can always solve these equations by applying the Intermediate Value Theorem since the left-hand sides are all bounded between 0 and 1. To understand the supplier’s best response to q1, we can consider the individual functional maximizers in (3)–(5) and their relationship to each other and the boundaries of the feasible regions. The following theorem characterizes the supplier’s best response, which is dependent on the specific value of the buyer’s decision, q1, via the feasible region boundary conditions. Theorem 1. The supplier’s best w þ a > p þ c2 is 8 A:I: t1 ; > > > > A:II: > ; t > > >1 > > > > > > > > tA:III: ; > >1 > > < ð1 dÞq1 M; t1 ðq1 Þ ¼ > > > > S > arg maxtA:I > ;t1A:III: PA ; > 1 > > > > arg maxð1dÞq1 M;t1A:III: PSA ; > > > > > > > > > : arg max A:II: A:III: PS ; t1 ;t1 A

response to a given value of q1 when if t1A:I: ð1 dÞq1 M & tA:III: ð1 þ dÞq1 M; 1 if ð1 dÞq1 M tA:II: ð1 þ dÞq1 M & 1 A:III: t1 ð1 þ dÞq1 M; if tA:II: ð1 þ dÞq1 M & tA:III: ð1 þ dÞq1 M; 1 1 if tA:II: ð1 dÞq1 M t1A:I: & 1 ð1 þ dÞq1 M; tA:III: 1 if t1A:I: ð1 dÞq1 M & tA:III: ð1 þ dÞq1 M; 1 if tA:II: ð1 dÞq1 M t1A:I: & 1 ð1 þ dÞq1 M; tA:III: 1 if ð1 dÞq1 M tA:II: ð1 þ dÞq1 M tA:III: : 1 1 (6)

The supplier’s best response function in (6) is admittedly complicated and difficult to interpret. To aid the reader’s understanding of this function, we provide

Facilitating Demand Risk-Sharing with the Percent Deviation Contract

Supplier I

t1 t1III

{S3}

{S1} t1I

{S1,S2} t1III

t1

argmax t1I, t1III

t1III

{S1,S2}

(1-d)q1-M

t1

{S2}

{S3}

t1II I

t1I

t1III t1II

{S1}

II

{S1}

{S1,S2}

argmax (1-d)q1-M, t1III

t1

{S2}

{S1,S2}

{S3}

{S1,S2} t1II

t1III

t1III

{S3}

argmax (1-d)q1-M, t1III t1III

III

II

{S3} t1III

{S 3}

{S3} t1

t1III

{S2} (1-d)q1-M

t1III

t1II

139

t1II

argmax t1II, t1III

argmax t1II, t1III

Fig. 2 Supplier’s best response depiction for Scenario A

a decision tree-type depiction of the supplier’s optimal decision for various problem parameters in Fig. 2. The buyer must choose the q1 that maximizes its expected profit while anticipating the supplier’s response to the chosen value. The buyer’s expected profit function is given by "Z # t1 ðq1 ÞþM B xf ðxÞdx þ t1 ðq1 Þ þ M 1 F t1 ðq1 Þ þ M PA ¼ ðr wÞ Z p "Z0 p

0

minfð1dÞq1 ;t1 ðq1 ÞþMg

t1 ðq1 ÞþM ð1þdÞq1

minfð1 dÞq1 ; t1 ðq1 Þ þ Mg x f ðxÞdx

ðx ð1 þ dÞq1 Þf ðxÞdx

(7)

i þ þ t1 ðq1 Þ þ M ð1 þ dÞq1 1 F t1 ðq1 Þ þ M Z 1 þ ða bÞ x t1 ðq1 Þ M f ðxÞdx: t1 ðq1 ÞþM

The supplier’s best response function in (6) is comprised of four explicit values as well as three situations where the supplier chooses the profit-maximizing quantity from a set of two of the explicit values. We first determine how the buyer should set q1 when the supplier will respond with each of the four possible t1 values. Each of these cases results in a different realization of the buyer’s expected profit function in (7). We apply the Karush–Kuhn–Tucker (KKT) conditions (c.f. Bazaraa et al. 1993: 151–155) over each realization’s feasible range of decisions to determine the constrained optimal values of q1.

140

M.J. Drake and J.L. Swann

Table 1 Possible SPNE pairs and feasibility conditions for Case A’s explicit t1(q1) decisions ðq1 ; t1 ðq1 ÞÞ Feasibility conditions n n A:I: oo A:III: þM t þM t ; t1A:I: Always feasible q1A:I: q : q max 11d ; 1 1þd

A:III: þM 1 maxftA:II: g 1 ;t1 ; t1A:II: Always feasible qA:II: max F1dð0Þ ; 1 1þd ðqA:III: fq : ð1 dÞFðð1 dÞqÞ ¼ 1

qA:III: 1

Þ dÞqÞÞg; tA:III: 1

ð1 þ dÞð1 Fðð1 þ q : Fðð1 dÞqÞ ¼

ðq1:A:IV:

rwaþb ; rwaþbþp

ð1 dÞq1A:IV: MÞ

A:III: þM minftA:II: g 1 ;t1 1þd

n A:II: o t þM tA:III: þM q1A:IV: max 1 1d ; 1 1þd qA:IV: 1

tA:I: 1 þM 1d

Theorem 2. The possible subgame-perfect Nash Equilibrium (SPNE) decision pairs for explicit t1 ðq1 Þ decisions are given in Table 1. We must also determine the buyer’s optimal decision over the regions where it knows that the supplier will be selecting the maximizing argument of a set of two values. From Theorem 1, we establish the following ranges of q1 under which each situation is possible. arg maxtA:I PSA : ;tA:III: 1 1 arg maxð1dÞq1 M;tA:III: PSA: : 1

t1A:I: þ M tA:III: þ M q1 1 (8) 1d 1þd A:I: tA:II: þM t1 þ M t1A:III: þ M 1 (9) q1 min ; 1d 1d 1þd

PSA : q1 arg maxtA:II: ;tA:III: 1 1

tA:II: þ M tA:II: þM tA:III: þ M 1 & 1 q1 1 1d 1þd 1þd

(10)

Since all three situations involve the possible decision t1A:III: , we define the difference function, DðtÞ PSA ðt1A:III: Þ PSA ðtÞ, where t is any other possible supplier pre-acquisition amount. While it is difficult to determine the exact feasible region for q1 that induces each of the possible t1(q1) values, we can use these difference functions to explain how a buyer would determine its optimal decision for a given set of problem parameters. Proposition 1. The structure of the three difference functions for the decisions in (8)–(10) enables us to determine the specific ranges of q1 that induce each of the two possible supplier values for t1. For each instance there are at most seven decision pairs from Table 1 and obtained from the procedure in Proposition 1, but some of these decisions may not be mutually feasible given a set of problem parameters. Since this set contains a maximum of seven elements, the buyer can evaluate its expected profit in (7) with respect to each of the feasible pairs and select the value of q1 that yields the highest

Facilitating Demand Risk-Sharing with the Percent Deviation Contract

141

expected profit to obtain the overall subgame-perfect Nash Equilibrium for this sequential supply chain game. Each of the formulas used in computing the potential decision pairs has an economic interpretation. The buyer always sets the initial order estimate with a goal of minimizing the expected deviation penalty that must be paid under each scenario. In the case where the supplier has system capacity larger than the upper limit of the deviation range, the optimal quantity balances the expected lower and upper deviation penalties. The supplier also seeks to balance the expected revenue from pre-acquiring inventory with the cost of doing so as well as the expected expediting and shortage costs. Even though the resulting formulas are more complicated, the supplier follows the same rationale as in a traditional newsvendor contract. We now consider Case B, where the supplier chooses not to expedite any units after the buyer places the final order because of a high expediting cost. This case is analogous to Case A when M ¼ 0 since the supplier can be viewed as having an effective expediting capacity of zero units if it chooses not to expedite. The maximizing values below correspond with the equations in (3)–(5) with M ¼ 0. t1B:I: 2

w þ a c1 t : FðtÞ ¼ wþavp

tB:II: 1

2

w þ a c1 t : FðtÞ ¼ wþav

t1B:III: 2

t : FðtÞ ¼

(11)

w þ a c1 þ p wþavþp

(12) (13)

Using the relationships in Lemma 4 (stated in the Appendix) to simplify the feasibility conditions, the supplier’s best response in this scenario is characterized by the following theorem. While it may not seem like it at first glance, the feasibility conditions for each of the decisions in (14) correspond to those in (6). Theorem 3. The supplier’s best response to a given value of q1 when w + a > p + v is 8 B:I: t1 ; > > > > tB:II: > 1 ; > > B:III: > t > 1 ; > > < ð1 dÞq1 ; t1 ðq1 Þ ¼ arg max B:I: B:III: PS ; t1 ;t1 B > > > S > B:III: arg max > ð1dÞq1 ;t1 PB ; > > > > > > : S B:III: P ; arg maxtB:II: B 1 ;t1

if t1B:I: ð1 dÞq1 & t1B:III: ð1 þ dÞq1 ; if ð1 dÞq1 tB:II: & t1B:III: ð1 þ dÞq1 ; 1 if tB:II: ð1 þ dÞq ; 1 1 if tB:II: ð1 dÞq1 t1B:I: & t1B:III: ð1 þ dÞq1 ; 1 if t1B:I: ð1 dÞq1 & t1B:III: ð1 þ dÞq1 ; if tB:II: ð1 dÞq1 t1B:I: & 1 t1B:III: ð1 þ dÞq1 ; if ð1 dÞq1 tB:II: ð1 þ dÞq1 t1B:III: : 1 (14)

142

M.J. Drake and J.L. Swann

The possible subgame-perfect Nash Equilibrium decision pairs in Case I.B. are the same as those for Case A (given in Table 1) except with the B supplier decision values replacing the A decisions. The buyer’s decisions are exactly the same as those in Case A since the actual q1 values are independent of M. The buyer can again apply the methods described in the proof of Proposition 1 to determine the optimal value of q1 in the cases in which the supplier’s best response is the value among a set of maximizing arguments for two of the expected profit realizations. Once the feasible set of possible decision pairs is determined, the buyer can again substitute each of them into its expected profit function to find the maximizing decision pair, which is the SPNE. If the contract’s parameters are such that neither of the above scenarios (A or B) apply, then the supplier’s expediting decision is dependent on the magnitude of the final order. The most interesting case is when the supplier will receive the deviation penalty for some of the expedited units and not for others, so we consider the case where q2 > (1 + d)q1 and t1 + M > (1 + d)q1. For it to be Pareto optimal for the supplier to fulfill the entire order, the cash flow from satisfying must be greater than the cash flow from not satisfying. When q2 t1 + M, which means that the supplier has enough capacity to satisfy the full order if it wants to, we must have (w c2) (q2 t1) + p(q2 (1 + d)q1) a(q2 t1). Solving for q2, the supplier satisfies the extra demand if q2

ðw c2 þ aÞt1 þ pð1 þ dÞq1 L1 : w c2 þ a þ p

Formally, the supplier’s expediting decision is ðq2 t1 Þþ ; if q2 L1 t2 ¼ 0; if q2 < L1 : When the actual order exceeds the supplier’s total capacity (i.e., q2 > t1 + M), the supplier can only satisfy an additional M units beyond t1. If the supplier chooses to supply the additional M units, it will have to pay the buyer the a penalty on each of the q2 t1 M that were ordered and not fulfilled. Thus, we must have (w c2)M + p(t1 + M (1 + d)q1) a(q2 t1 M) a(q2 t1) for the supplier to want to supply the extra M units. This condition simplifies to M

pðð1 þ dÞq1 t1 Þ L2 : w c2 þ a þ p

This means that the supplier’s decision to expedite the M additional units is M; if M L2 t2 ¼ 0; if 0

Facilitating Demand Risk-Sharing with the Percent Deviation Contract

143

added supply uncertainty. Knowing that the expediting decision rests with the magnitude of the order might induce the buyer to inflate the final order so that the supplier will fulfill the entire amount. The buyer’s strategic behavior in this case is detrimental for the supplier because it could be induced to expedite when it would not otherwise. To alleviate these difficulties, we recommend that the parties set the negotiated parameters–p and a–such that the contract assumes another case. This could be accomplished by letting a > c2 w, shifting the contract to the A case. Of course, shifts to other scenarios are possible through negotiations, depending on the relative market power of the parties. Since both parties have an incentive to set the contract parameters to move the contract to other cases, we omit this intermediate situation from further analysis and subsequently only consider Cases A and B.

3.3

Infinite Expediting Capacity

We conclude our presentation of the general models by considering a special case in which the supplier’s expediting capacity is infinite (or especially large for practical purposes). These uncapacitated models have an especially simple structure that enables us to develop (quasi) closed-form optimal decisions. Since this extension is based on the expediting capacity, we must only develop models for case analogous to A above, in which the supplier chooses to expedite. It does not matter how much extra capacity the supplier has if it chooses not to use it. Again we only consider the case where the buyer orders the actual demand in all cases (i.e., q2 ¼ X and t2 ¼ ðminfq2 t1 ; MgÞþ ) since (1) holds and w c2 > a. (It is straightforward to extend these models to the case in which the buyer does not order above the deviation range; see Drake (2006) for details.) We denote this scenario as A.1. The supplier’s expected profit function is again the same as in (2) with M ¼ 1, but this substitution results in the simpler function Z PSA:1

1

¼w

Z

0

þp þv

Z

ð1dÞq1

xf ðxÞdx þ p 1

ð1þdÞq1 Z t1

ðð1 dÞq1 xÞf ðxÞdx

0

ðx ð1 þ dÞq1 Þf ðxÞdx

(15) Z

1

ðt1 xÞf ðxÞdx c1 t1 c2

0

ðx t1 Þf ðxÞdx:

t1

The expected profit function in (15) is concave by Lemma 1, so we can solve for the optimal pre-acquisition amount using first order conditions. This yields c 2 c1 tA:1 ; (16) 2 t : FðtÞ ¼ 1 c2 v which is independent of the buyer’s initial order estimate because of the symmetric information assumption and the infinite total system capacity.

144

M.J. Drake and J.L. Swann

Similarly, the buyer’s expected profit function is the same as in (7), but infinite supplier capacity yields the following simplified form. Z PBA:1

¼ ðr wÞ Z 1 p

1

0

ð1þdÞq1

Z

ð1dÞq1

xf ðxÞdx p

ðð1 dÞq1 xÞf ðxÞdx

0

(17)

ðx ð1 þ dÞq1 Þf ðxÞdx

Because the supplier is willing to expedite to satisfy the buyer’s order regardless of its size, the buyer’s expected profit function is no longer dependent on the supplier’s t1 decision. In this case, the t1 decision only affects the supplier’s profitability and not its ability to fulfill the buyer’s order. The buyer’s optimal initial order estimate is given by qA:1 fq : ð1 dÞ 1 Fðð1 dÞqÞ ¼ ð1 þ dÞð1 Fðð1 þ dÞqÞÞg, which corresponds with the optimal decision in Case A.III in which the supplier has the capacity to satisfy orders is the value of q1 above the upper limit of the deviation range. The decision qA:1 1 that equates the marginal expected deviation penalty for demands below the lower limit of the range, p(1 d)F((1 d)q1), to the marginal expected penalty for orders above the upper limit, p(1 + d)(1 F((1 + d)q1)). Since the nominal deviation penalty, p, is the same regardless of whether the deviation was a lower deviation or an upper deviation, it is irrelevant to the buyer’s decision. Of course, if there were two deviation penalties, pl and pu, they would affect the buyer’s decision. A:1 Thus, the SPNE for the A.1 case is ðq1 ; t1 ðq1 ÞÞ ¼ ðqA:1 1 ; t1 Þ for all parameter sets such that w c2 > a. It applies in situations where the supplier always has enough extra capacity in its network to satisfy the buyer’s order. It would be most reasonable when the buyer’s requirements are small compared with the supplier’s capabilities. Consequently, the supplier would only need to utilize the more complicated capacitated contracts for customers who require a large portion of its capacity. Since these buyers are larger, they are presumably more important to the supplier, so it would have more incentive to utilize a more complicated contract for these customers.

4 Economic Analysis and Model Extensions 4.1

Individual Rationality Constraints

The practical implementation of the percent deviation contract is necessarily impacted by the competitive power of the parties. If the buyer has a powerful market presence, it will likely be able to negotiate favorable contract terms by threatening to use another supplier who offers a more traditional agreement. (We assume that the contract is used in a competitive industry, so the buyer can find

Facilitating Demand Risk-Sharing with the Percent Deviation Contract

145

another supplier with comparable service performance and quality.) The terms of the contract, therefore, must satisfy the buyer’s individual-rationality constraint, which says that under the percent deviation contract the buyer must be able to attain an expected profit at least as great as it could under a traditional mechanism. See Tirole (1988) for a detailed discussion of individual-rationality constraints. If this constraint is not satisfied, the buyer will switch to another supplier. In this section we compare our percent deviation contract to the status quo of a traditional wholesale-price contract. In the cases where the supplier’s expediting capacity is limited, the percent deviation mechanism can induce the supplier to preacquire significantly more inventory than it would under the traditional wholesaleprice contract. This additional ability to meet demand is beneficial for both parties, resulting in higher expected profits without further contract modifications. This is the situation demonstrated in the numerical example in Sect. 5.1. In situations where the supplier does not increase its pre-acquisition quantity significantly (i.e., the deviation penalty is not high enough to induce the supplier to pre-acquire much more inventory than under the traditional wholesale-price contract), it is clear that the buyer will earn less expected profit under the percent deviation contract because it now shares additional demand risk by paying the deviation penalty for orders outside of the allowable range. There are several ways in which the parties can adjust the terms of the percent deviation contract to satisfy the buyer’s individual-rationality constraint. The supplier can offer the buyer a fixed transfer payment to share some of its gain. In some cases the supplier can offer a discounted wholesale price, w0 , that gives the buyer the same expected profit as it would attain under the traditional wholesaleprice contract. The remainder of this section illustrates the methodology required to find the requisite discounted wholesale price. The A.1 infinite capacity model is comparable to the traditional newsvendor, wholesale-price (NV) contract, since the supplier chooses and has the capacity to satisfy the entire order. The Rbuyer’s expected profit function under the NV contract 1 is given by PBNV ¼ ðr wÞ 0 xf ðxÞdx. Notice that this function is not dependent on any decision by the supplier, because under a traditional contract in this setting the buyer places orders for exactly the number of units needed with no demand risk. Comparing this expected profit to that under the percent deviation contract in (15) with the qA:1 decision, the contracting parties wish to find w0 such that PBNV ðwÞ 1 PBA:1 ðw0 Þ, to ensure that the buyer earns at least as much expected profit under the percent deviation contract as it does under the original wholesale-price contract. We find that the discounted wholesale price given by 0

2R ð1dÞqA:1

w0 @wp4

1

0

31þ R1 ðð1dÞqA:1 xÞf ðxÞdxþ ð1þdÞqA:1 ðxð1þdÞqA:1 Þf ðxÞdx 1 1 1 5A R1 0 xf ðxÞdx (18)

satisfies the buyer’s participation constraint. The term in brackets represents the percentage of order periods in which the deviation penalty will be paid.

146

M.J. Drake and J.L. Swann

Consequently, the supplier must provide an allowance for this expected penalty if the buyer is to realize the same expected profit as in the newsvendor contract. If the right side of (18) assumes the value of zero, there is no discounted wholesale price mechanism that can satisfy the buyer’s rationality constraint with the given contract parameters. The supplier also has an individual-rationality constraint that should be considered. To induce the buyer to participate in the percent deviation contract in this case, the supplier must offer the discounted wholesale price discussed above. This reduces its expected profit from the high theoretical profit that could be earned under the percent deviation contract with the original newsvendor wholesale price. This is not a problem when we compare the supplier’s expected profit to that of the wholesale-price contract. The percent deviation contract induces the supplier to establish a higher system capacity (t1 + M) than the wholesale-price contract does. This increases the total expected profit of the entire supply chain because the supply chain is able to satisfy more of the consumer demand. If the supplier offers w0 equal to the right-hand side of (18), the buyer’s expected profit under percent deviation will be equal to its expected profit under the wholesale-price contract, and the supplier captures all of the additional supply chain profit. This makes the supplier better off than it was under the wholesale-price contract. Even if the supplier decides to make the buyer strictly better off by offering a slightly smaller value of w0 than the right-hand side of (18) requires, there is a range of values where both parties can improve their position by splitting the increased supply chain expected profit. This is the situation demonstrated in the numerical example in Sect. 5.2.

4.2

Channel Coordination

Supply chain research has shown that the total supply chain profit is maximized by a centralized firm making decisions that are best for the system as a whole. One main objective of supply chain contracts is to align each entity’s own incentives to induce decentralized decisions that attain the maximal centralized supply chain profit. This achievement is commonly referred to as “channel coordination.” We first examine the performance of the centralized channel and then develop mechanisms to coordinate the channel.

4.2.1

Centralized Channel Benchmark

In terms of a centralized channel, the buyer and the seller are viewed as a single entity trying to maximize its own expected profit. Hence, there is no wholesale price (w) paid from the sales department (the buyer) to the manufacturing department (the seller), and the penalties levied under the percent deviation contract (p and a) are not valid. The buyer’s decisions are not relevant either since the single company does not order from itself; the combined firm must only determine the number of units to acquire early and the number to expedite.

Facilitating Demand Risk-Sharing with the Percent Deviation Contract

147

If the cost structure for the centralized channel is such that r c2 > b, the firm will satisfy additional demand beyond the number of pre-acquired goods up to capacity M. In this case the number of units to be expedited is given by t2 ¼ (min {X t1, M})+. The channel’s expected profit function is Z

Z t1 xf ðxÞdx þ ðt1 þ MÞð1 Fðt1 þ MÞÞ þ v ðt1 xÞf ðxÞdx c1 t1 0 0 Z t1 þM c2 ðx t1 Þf ðxÞdx þ Mð1 Fðt1 þ MÞÞ t Z 11 ðx t1 MÞf ðxÞdx: (19) b

PC:I: ¼ r

t1 þM

t1 þM

This newsvendor-type profit function is concave, so first order optimality n C:I: conditions show that the optimal solution for t is t 2 t : Fðt þ MÞ ¼ 1 1 o rþbc1 ðc2 vÞFðtÞ . rþbc2 If r c2 < b the loss from expediting or subcontracting to meet the marginal demand is larger than the cash outlay from the penalty paid to the customer for not satisfying its demand. Accordingly, the centralized channel will not expedite at the higher cost c2 ; formally, we have t2 ¼ 0. The channel expected profit function now becomes Z

t1

PC:II: ¼ r 0

Z

Z xf ðxÞdx þ t1 ð1 Fðt1 ÞÞ þ v 1

b

t1

ðt1 xÞf ðxÞdx c1 t1

0

(20)

ðx t1 Þf ðxÞdx:

t1

The profit function in (20) is very similar to (19), except the expected revenue has been adjusted to reflect the fact that the centralized channel will not satisfy any demandnmore than t1. The o optimal number of units to acquire early is given by

1 tC:II: 2 t : Fðt1 Þ ¼ rþbc 1 rþbv .

4.2.2

Finite Expediting Capacity Channel Coordination

The subgame-perfect Nash Equilibria for the scenarios in which the supplier has finite expediting capacity have a complicated form. As a result, different mechanisms are required for each possible decision pair. Consequently, we consider one possible decision pair to show how the system can be coordinated given that particular decision. The procedure described below is applicable to all other possible decision pairs and case scenarios. We consider Case B, in which the buyer orders the entire demand but the supplier chooses not to expedite, where the corresponding decision pair is q1B:III: ; t1B:III: . The

148

M.J. Drake and J.L. Swann

following theorem contains the channel coordinating condition for this case, which also applies in Case A when the same decision pair is optimal. Theorem 4. The decentralized channel in scenario B (and A) in which the SPNE decision pair is q1B:III: ; t1B:III: will be coordinated if the contract parameters are set such that a þ p þ w ¼ r þ b:

(21)

The left-hand side of (21) comprises parameters that represent payments between the buyer and the supplier. These are set during contract negotiations as opposed to the right-hand side, which only contains parameters that we assumed were exogenous to the contract because they involve an outside party to the contract (the buyer’s customer). The parties can coordinate the channel by setting a, p, and w according to (21).

4.3

Comparison to Quantity Flexibility Contracts

Since the percent deviation contract provides the buyer with order flexibility around an initial order estimate, it is constructive to compare its channel performance with the quantity flexibility contract, which affords the buyer similar flexibility. Tsay (1999) establishes that the quantity flexibility contract cannot coordinate the supply chain when the buyer is not bound by a minimum purchase commitment. The percent deviation contract, on the other hand, does coordinate the channel without establishing a floor on the buyer’s order. Let us consider analysis for a particular case, e.g., the B scenario in which the SPNE is q1B:III: ; t1B:III: . Recall that this scenario can be coordinated by setting a + p + w ¼ r + b. To compare the quantity flexibility and percent deviation contracts, we need to analyze them in a similar framework. We apply the basic quantity flexibility contract structure but modify as follows to correspond to the percent deviation decision environment. We assume that the buyer’s actual order in the quantity flexibility contract is made after the customer demand has been realized, as in the percent deviation scenario. Consequently, the supplier commits to fulfilling a maximum of t1 units. The buyer establishes a minimum purchase commitment of (1 d)q1 units when it provides the initial order estimate, q1. If the buyer ends up being forced to order more units than are ultimately required to satisfy the realized demand (as a result of the minimum purchase quantity), these units generate u dollars per unit as a salvage value. We assume that leftover units of inventory are no more valuable to the buyer than they are to the supplier (i.e., u v). This is practical for several reasons. While it is true that goods generally appreciate in value as they move downstream in a supply chain, the buyer is not physically performing additional functions to add value to the product; consequently, the actual sale price of the salvaged product

Facilitating Demand Risk-Sharing with the Percent Deviation Contract

149

should be no higher than that which the supplier could receive if the good were sold it in the secondary market. Leftover product should be more valuable to the supplier in terms of expected revenue since the supplier could likely use the product to fulfill demand from another buyer while the buyer may have limited outlets to offload the extra product. This is especially true in the market for truckload transportation, which was an inspiration for the percent deviation contract. Carriers would obviously place more value on an unassigned truck than any one particular shipper might. Following the same backward-induction methodology we used in identifying the other equilibria, Theorem 5 provides the equilibrium decision for the quantity flexibility contract. Theorem 5. The SPNE decisions for the quantity flexibility contract are 80

1 rwþba > 1 >

F > > B rwþba C > rvþba > B C; if ðc1 vÞðrþbaÞ > ;F1 > 1d @ > rvþba A < > QF ¼ qQF 1 ;t1 þwvc1 u > > > > ðwþaÞðwuÞ > >

> > wþac > 1 > ; otherwise: : fqjFðð1dÞqÞ¼0g;F1 wþav (22) Suppose the parameter values are such that the quantity flexibility equilibrium decisions are the first pair in (22). We can write the expected total supply chain profit as the sum of the agents’ individual expected profit functions, which reduces to "Z QF # Z tQF t1 1 QF QF QF SC PQF ¼ r xf ðxÞdx þ t1 1 F t1 tQF þu 1 x f ðxÞdx c1 t1 0

Z

b

1 tQF 1

f ðxÞdx: x tQF 1

0

(23) SC Note that if u ¼ v, for any value of t1 we have PSC QF ðt1 Þ ¼ PC:II: ðt1 Þ, where is the centralized supply chain profit in (20).

PSC C:II: ðt1 Þ

Theorem 6. The percent deviation contract coordinates the supply chain in the following cases where the quantity flexibility contract fails to coordinate: 1. When the salvage value is higher at the supplier (u < v), there are cases in which the centralized supply chain profit under the percent deviation contract always exceeds that attainable from the quantity flexibility contract. 2. When the salvage values are equal for both parties (u ¼ v), channel coordination efforts for quantity flexibility require either setting a < 0 or w < c1 , both of which violate the underlying assumptions of the model.

150

M.J. Drake and J.L. Swann

In other supply chain contracting structures such as revenue-sharing agreements, it is possible for suppliers to benefit by selling goods for a wholesale price below their marginal cost of production as the second part of Theorem 6 requires. This strategy is successful because the supplier is receiving part of the buyer’s revenue in addition to the wholesale price. Looking at the supplier’s expected profit function under the quantity flexibility contract in (28), the supplier can either obtain w or v for each of the t1 pre-acquired units. If each of these values are less than c1, the supplier cannot earn positive expected profit by selling below the marginal cost. We have thus shown that there are cases in which the quantity flexibility contract cannot coordinate the supply chain, while the percent deviation contract is able to achieve coordinated performance. The main difficulty the quantity flexibility contract has in this decision environment is that it establishes a minimum purchase commitment for the buyer. The percent deviation contract provides buyers more flexibility by allowing them to choose to pay the penalties associated with ordering outside of the deviation range. Of course, in order to gain this flexibility, the contract must be more complex; therefore, the percent deviation contract would likely be more costly to manage in practice.

5 Numerical Analysis In this section we provide several numerical examples that illustrate the behavior of the percent deviation contract in various decision environments discussed above as well as how parameters can be set to satisfy individual-rationality constraints and to coordinate the channel. We estimated the demand distributions used below from weekly shipping data provided by a major US manufacturer. The demand random variable represents the number of shipments per week required from the supplier to a retailer on a particular origin-destination lane; we consider two such lanes. For one of the lanes, the exponential distribution gave the best fit, and for the other the uniform distribution was appropriate. We failed to reject chi-squared goodness of fit statistics at the 10% significance level for each of the two distributions. For the cost and contract parameters, we constructed values that make relative sense in this manufacturer’s business setting.

5.1

Exponential Demand Example (Case A.)

Consider weekly demand that follows an exponential distribution with l ¼ 0.17297 and the cost parameters listed in Table 2. These parameters define a contract in Case A., since r w + b > p and w + a > c2; all of the supplier’s expected profit function realizations are concave because w + a > c2 + p. Thus, the buyer orders the exact demand, and the supplier chooses to expedite units (up to the capacity of 5). Under a traditional wholesale-price contract with inventory pre-acquisition and

Facilitating Demand Risk-Sharing with the Percent Deviation Contract Table 2 Parameter declarations for numerical examples

Parameter r w c1 c2 a

Exp (.17297) 60 17 7 13 3

a

Unif (0,18) 30 18 6 22 1

151

Parameter b v d p M

Exp (.17297) 12 1 0.2 5 5

Unif (0,18) 4 1 0.2 13 5

b

225

50 25

200

0

175

−25

150

−50

125

−75 30 20 t

10

00

10

20 q

30

30

30

20

20 t

10 00

10 q

Fig. 3 Expected profit functions of (a) the supplier and (b) the buyer for the Exp(0.17297) example

n o 2 vÞFðtÞ , or expediting, the supplier pre-acquires tNV 2 t : Fðt þ MÞ ¼ wc1 ðc wc2 tNV ¼ 4.7667 units. This results in an expected profit of 189.89 for the buyer and 29.21 for the supplier; thus, the total supply chain profit for the wholesale-price contract is 219.10. The centralized-channel pre-acquisition quantity is 10.4930, yielding a maximal channel expected profit of 243.45. The main problem with the wholesale-price contract is that the supplier does not have an incentive to pre-acquire enough inventory because the buyer is not sharing any of the demand risk. This low pre-acquisition amount restricts the total system’s ability to satisfy realized customer demand, which dampens the system’s profit potential. For the same parameter set, the percent deviation contract with (p,d) ¼ (5,0.2) is Pareto-improving for both parties as compared with the wholesale-price contract. Figure 3a, b depict the expected profit functions for the supplier and the buyer, respectively, as a function of the two main decision variables, q1 and t1. Note the piecewise form of these expected profit functions, which reflects the different profit function realizations with their distinct optimal solutions. Applying solu the ; t ðq Þ ¼ tion procedure detailed in Sect. 3.2, the SPNE decision pair is q 1 1 1 A:III: A:III: ¼ ð5:1810; 6:0391Þ. These decisions yield an expected profit of q1 ; t1 192.55 for the buyer and 37.37 for the supplier and a total expected supply chain

152

M.J. Drake and J.L. Swann

profit of 229.92. While both parties are better off in relation to the wholesale-price contract, there are still more gains to be realized because there is a 6% efficiency loss with this solution compared with the centralized solution. We can design the percent deviation contract parameters to ensure that the channel coordination condition in (21) is met. Namely, we need a + w + p ¼ r + b, so we 3 can by setting satisfy this inequality p ¼ 52. This induces an equilibrium pair of A:III: A:III: C:I: ¼ t1 ¼ ð5:1810; 10:4930Þ, which gives channel-optiq1 ; t1 ðq1 Þ ¼ q1 ; t1 mal expected profits of 84.35 and 159.10 for the buyer and supplier, respectively, and a total expected supply chain profit of 243.45, as designed. To induce the supplier to pre-acquire the channel-optimal inventory amount, the buyer has had to relinquish a substantial amount of profit to the supplier. The buyer’s coordinated expected profit does not satisfy its individual-rationality constraint, which requires expected profit of at least 189.89, the buyer’s expected profit from the wholesale-price contract. If the buyer received a fixed transfer payment, then it would be willing to accept the percent deviation contract. In this case, the fixed transfer payment must be larger than 105.54.4 This payment, denoted F, should not be too high, though; or else the supplier would be better off under the original wholesale-price contract as well. Thus, for any fixed supplier-to-buyer transfer payment in the range F ∈ (105.54,129.89), the percent deviation contract is coordinated and strictly Pareto-improving for both parties as compared to the wholesale-price contract.

5.2

Uniform Demand Example (Case B.)

We now consider an example with uniform demand and parameters as defined in Table 2. Since r w + b > p and w + a < c2, a percent deviation contract in this case would fall in scenario B., where the buyer orders the full demand and the supplier chooses not to expedite because it is too expensive.5 Under a traditional wholesale-price contract, the supplier pre-acquires 12.7059 units of inventory. The buyer and supplier expected profits are 95.54 and 76.24, respectively, resulting in a total supply chain expected profit of 171.78. If the firms acted as a centralized

3 Since the deviation penalty is so large, the condition for concavity on the supplier’s first profit function realization is no longer satisfied. This does not matter, though, because the buyer would never choose an equilibrium in this realization, which requires that it pay the deviation penalty for every unit of demand satisfied. 4 Note that, in this case, the channel coordinating condition is a function of the wholesale price, so we do not attempt to satisfy the buyer’s individual-rationality constraint using a wholesale price discount as discussed in Sect. 4.1. 5 We see in Table 2 that the supplier has M ¼ 5 units of available expediting capacity. This number is irrelevant here because regardless of the amount of extra capacity available, the supplier will not use any of it because expediting is too costly.

Facilitating Demand Risk-Sharing with the Percent Deviation Contract

a

153

b

150

50

100

25

50

0

0

15 15

10 t

5 5

0

0

10 q

−25 15

10 t

5

00

5

10 q

15

Fig. 4 Expected profit functions of (a) the supplier and (b) the buyer for the Unif(0,18) example

channel, the pre-acquisition amount would be 15.2727 with a total expected profit of 177.82. Figures 4a, b depict the expected profit functions for the supplier and the buyer under a percent deviation contract in this example. We can apply the solution procedure B:III: B:III:for Case B. to determine the SPNE decision pair of q1 ; t1 ðq1 Þ ¼ q 1 ; t1 ¼ ð10:3846; 15:0968Þ, which results in expected profits of 71.53 and 106.26 for the buyer and the supplier, respectively, and a total supply chain expected profit of 177.79. Note that this decentralized percent deviation contract produces a supply chain profit very close to that of the centralized channel; this is due to the fact that the supplier’s t1 decision value is approximately equal to that of the centralized channel. While the above percent deviation contract is close to coordinated as currently constructed, it does not satisfy the buyer’s individual-rationality constraint when compared with the wholesale-price contract. Consequently, the percent deviation contract must be modified to give the buyer an incentive to accept it over the status quo. If the supplier offers a discounted wholesale price (as discussed in Sect. 4.1) of 15.2346, which represents an approximate discount of 15% off the original price of 18, the equilibrium decision pair becomes q1 ; t1 ðq1 Þ ¼ q1B:III: ; t1B:III: ¼ ð10:3846; 14:1812Þ. This contract results in expected profits of 95.54 and 82.08 for the buyer and the supplier, respectively, and a total supply chain expected profit of 177.62, which is still close to the centralized optimum of 177.82. This percent deviation contract with a discounted wholesale price satisfies the buyer’s participation constraint and provides a higher profit for the supplier in relation to the traditional wholesale-price contract. Thus, for this example the individual-rationality constraint and the Pareto-improving condition are more important than channel coordination since the decentralized percent deviation contracts are close to being coordinated without any additional consideration.

154

M.J. Drake and J.L. Swann

6 Conclusions and Further Research In this chapter we have characterized the subgame-perfect Nash Equilibria of a dynamic supply chain game induced by the percent deviation contract, an innovative mechanism that was motivated by our discussions with a major firm in the transportation industry. Due to the sequential extensive form of this supply chain game, many of the decisions are functions of those decisions made in earlier stages of the game. The main result we have shown is that the percent deviation contract is a viable, albeit somewhat complicated, mechanism whereby the supplier can transfer some of its demand risk to the buyer. The prospect of receiving a deviation penalty for large or small buyer orders induces the supplier to pre-acquire more inventory than it ordinarily would, which increases the total capacity of the system. This extra ability to satisfy end-user demand benefits the entire system, enabling Pareto improvements. Several trajectories exist for future research in this area. The first direction includes relaxing some of the assumptions that we made in these models. A natural extension would be adding some information asymmetry by including one party’s proprietary information on costs or capacity. One could also include nonlinear costs or another pricing policy such as quantity discounts. For completion, it would also be interesting to examine supply chain coordination mechanisms for the other possible decision pairs. More generally, future work incorporating dynamic decision environments could be useful, especially in multi-echelon supply chains. Comparison studies of various contracting mechanisms applied to the same scenario could lead to Pareto-improvements similar to the ones we found. Further analysis is also needed to incorporate the advanced demand information into operational production and transportation network models. Only then will the true value of the percent deviation contract be estimated for the entire system. Acknowledgements This research was funded, in part, by The Logistics Institute Leaders in Logistics Grant from Lucent Technologies and NSF Grants DMI-0223364 and DMI-0348532.

Appendix Proof of Lemma 1. We will define the three realizations of (2) as follows: Z

PSA:I:

Z t1 þM ¼w xf ðxÞdx þ ðt1 þ MÞð1 Fðt1 þ MÞÞ þ p ðt1 þ M xÞf ðxÞdx 0 0 Z t1 þM Z t1 ðt1 xÞf ðxÞdx c1 t1 c2 ðx t1 Þf ðxÞdx þ Mð1 Fðt1 þ MÞÞ þv 0 t1 Z 1 a ðx t1 MÞf ðxÞdx (24Þ t1 þM

t1 þM

Facilitating Demand Risk-Sharing with the Percent Deviation Contract

Z PSA:II: ¼ w

t1 þM

155

xf ðxÞdx þ ðt1 þ MÞð1 Fðt1 þ MÞÞ

0

Z

ð1dÞq1

þp

ðð1 dÞq1 xÞf ðxÞdx Z t1 þM Z t1 ðt1 xÞf ðxÞdx c1 t1 c2 ðx t1 Þf ðxÞdx þ Mð1 Fðt1 þ MÞÞ þv t1 0 Z 1 ðx t1 MÞf ðxÞdx (25Þ a 0

t1 þM

Z PSA:III: ¼ w Z

t1 þM

0 ð1dÞq1

þp "Z0 þp

t1 þM ð1þdÞq1

Z

t1

þv 0

Z a

xf ðxÞdx þ ðt1 þ MÞð1 Fðt1 þ MÞÞ ðð1 dÞq1 xÞf ðxÞdx # ðx ð1 þ dÞq1 Þf ðxÞdx þ ðt1 þ M ð1 þ dÞq1 Þð1 Fðt1 þ MÞÞ Z

t1 þM

ðt1 xÞf ðxÞdx c1 t1 c2

1

t1 þM

ðx t1 Þf ðxÞdx þ Mð1 Fðt1 þ MÞÞ

t1

ðx t1 MÞf ðxÞdx:

(26Þ

The second derivative of (24) taken with respect to t1 is (p + c2 w a)f(t1 + M) (c2 v)f(t1), which is negative for all values of t1 if w + a > p + c2 based on the parameter conditions of scenario A. The second derivative of (25) is (c2 w a)f(t1 + M) (c2 v)f(t1), and the second derivative of (26) is (c2 w a p)f(t1 + M) (c2 v)f(t1). Both of these expressions are negative for all values of t1 without the extra condition. □ Proof of Theorem 1. There are five possible values for the supplier’s best response. Each of the three realizations of the supplier’s expected profit function has an individual maximizer, shown in (3)–(5). In addition, the two points where the pieces of the profit function converge (t1 ¼ (1 d)q1 M and t1 ¼ (1 + d) q1 M) are possible solutions. These solutions would occur when the maximizing t1 values do not lie in their corresponding feasible regions. In order to establish the result in Theorem 1, we first make some observations about the expected profit function that will help us with the main proof. Observation 2 For all values of t1 less than the lower boundary ðð1 dÞq1 MÞ, PSA:II: ðt1 Þ>PSA:I: ðt1 Þ because the term representing the expected value of the lower deviation penalty paid is larger in PSA:II: . For values of t1 greater than the lower boundary, PSA:II: ðt1 Þ

156

M.J. Drake and J.L. Swann

Observation 3 For all values of t1 less than the upper boundary ðð1 þ dÞq1 MÞ, PSA:II: ðt1 Þ > PSA:III: ðt1 Þ because the term representing the expected value of the upper deviation penalty paid in PSA:III: is negative. For values of t1 greater than the upper boundary, PSA:II: ðt1 Þ < PSA:III: ðt1 Þ. The supplier’s best response function depends on the values of the three maximizers relative to the feasible boundaries. There are 27 possible cases because each of the three decisions can potentially lie in three regions; however, the following results show that several of these cases are not possible. Lemma 2. The following results hold in Case A. It is not possible to have t1A:I: < A:III: ð1 dÞq1 M < tA:II: <ð1 þ dÞq1 M < tA:II: 1 , and it is also not possible to have t1 1 . After using Lemma 2 to reduce the number of possible cases, we can determine the overall best response for each given the values of the individual maximizers by applying Observations 1–3. Summarizing all of the scenarios, we obtain the solutiongiven in (6). □ Proof of Proposition 1. We use the difference function to determine when the supplier chooses each t1 for given values of q1 in a range where its best response is known to be the maximizer of its expected profit from a set of two values. We then use this information to characterize explicitly the ranges of q1 that induce each value of t1. If the difference function is positive for a value of q1, then the supplier will choose t1A:III: ; it will select the other possible decision if the function is negative. The number of ranges for q1 is determined by the number of changes of sign in the difference function. The difference function related to the decision in (8) is given by D t1A:I: "Z A:III: # t1 þM A:III: A:III: A:I: A:I: xf ðxÞdxþ t1 þM 1F t1 þM t1 þM 1F t1 þM ¼w þM tA:I: 1

"Z

ð1dÞq1

þp "Z þp

t1A:III: 0

"Z a

t1A:III: þM

c2 Z

t1A:I: þM t1A:I:

Z f ðxÞdx

tA:I: 1

0

xt1A:III: M f ðxÞdx

tA:III: þM 1

tA:III: 1

# f ðxÞdx

ðxð1þdÞq1 Þf ðxÞdxþ t1A:III: þMð1þdÞq1 1F t1A:III: þM

t1A:III: x

1

"Z

t1A:I: þMx

0 t1A:III: þM

þv

t1A:I: þM

ðð1dÞq1 xÞf ðxÞdx

0

ð1þdÞq1

"Z

Z

t1A:I: x

Z

1

t1A:I: þM

#

f ðxÞdx c1 t1A:III: t1A:I:

xt1A:I: M

# f ðxÞdx

xt1A:III: f ðxÞdx

xt1A:I:

# A:I: A:III: f ðxÞdxM F t1 þM þF t1 þM :

#

Facilitating Demand Risk-Sharing with the Percent Deviation Contract

157

This difference function is convex in q1 since @@qD2 ¼ pð1 dÞ2 f ðð1 dÞq1 Þþ 1 difference function at the pð1 þ dÞ2 f ðð1 þ dÞq1 Þ>0. We can begin by evaluating the tA:I: þM tA:III: þM two endpoints of the region defined in (8); that is, q1 ¼ 11d and q1 ¼ 1 1þd . If the difference function is positive for one value and negative for the other, then convexity implies that there exists a single threshold value of q1 in the interval where the difference function changes sign. The buyer can use these supplier decision values to evaluate its best selection of q1 in this region with respect to its expected profit function. If the difference function is positive for both endpoint values of q1, then it is possible that there are zero, one, or two points where the function switches sign. If there are zero or one switching points, then the supplier will choose t1 ¼ t1A:III: for all values of q1 in the region. If there are two switching points, then for values of q1 between these two values, the supplier will choose t1 ¼ t1A:I: . It will choose t1 ¼ t1A:III: for all other values of q1. If the difference function is negative for both endpoint values, then convexity implies that it will be negative for all values of q1; thus, the supplier will always choose t1 ¼ t1A:I: . The difference function related to the decision in (10) is given by A:II: D t1 "Z A:III: t1 þM ¼w xf ðxÞdxþ t1A:III: þ M 1 F t1A:III: þ M 2

t1A:II: þM

þ M 1 F tA:II: þM tA:II: 1 1 "Z A:III: # t1 þM A:III: A:III: þp ðx ð1þ dÞq1 Þf ðxÞdxþ t1 þ M ð1þ dÞq1 1 F t1 þ M ð1þdÞq1

"Z

t1A:III:

þv 0

"Z a c2 Z

1

tA:III: þM 1

"Z

tA:III: 1

Z f ðxÞdx

tA:II: 1

0

x t1A:III: M f ðxÞdx

tA:III: þM 1

t1A:II: þM t1A:II:

t1A:III: x

tA:II: x 1

Z

1 tA:II: 1 þM

#

f ðxÞdx c1 t1A:III: tA:II: 1

x tA:II: M f ðxÞdx 1

#

x t1A:III: f ðxÞdx

# x tA:II: þ M þ F t1A:III: þ M : f ðxÞdx M F tA:II: 1 1

This difference function is convex in q1 since @@qD2 ¼ pð1 þ dÞ2 f ðð1 þ dÞq1 Þ > 0, 1 @D ¼ pð1 þ dÞð1 Fðð1 þ dÞq1 ÞÞ < 0. and it is also decreasing in q1 because @q 1 tA:II: þM This means that if the difference function is negative when q1 ¼ 1 1þd , which is the lower limit of the range defined in (10), then the supplier will always choose t1 ¼ tA:II: if the difference function is positive at the upper endpoint of 1 . Likewise, n A:II: o t þM tA:III: þM , then the supplier will always select the range q1 ¼ min 1 1d ; 1 1þd 2

158

M.J. Drake and J.L. Swann

t1 ¼ t1A:III: . If the difference function is positive for the lower endpoint and negative for the upper endpoint, then there exists exactly one point where the difference function changes sign, and we have two distinct ranges of q1 values where the two t1 decisions are chosen. The difference function related to the decision in (9) is given by Dðð1dÞq1 MÞ "Z A:III: t1 þM xf ðxÞdxþ t1A:III: þM 1F t1A:III: þM ¼w ð1dÞq1

ð1dÞq1 ð1Fðð1dÞq1 ÞÞ "Z A:III: # t1 þM A:III: A:III: þp ðxð1þdÞq1 Þf ðxÞdxþ t1 þM ð1þdÞq1 1F t1 þM "Z

ð1þdÞq1 tA:III: 1

þv

t1A:III: x

0

f ðxÞdx

Z

ð1dÞq1 M

tA:III: þM 1

c2 Z

t1A:III: þM t1A:III:

ðð1dÞq1 M xÞf ðxÞdx

0

c1 t1A:III: ð1dÞq1 þM "Z Z 1 A:III: a xt1 M f ðxÞdx "Z

#

#

1 ð1dÞq1

ðxð1dÞq1 Þf ðxÞdx

xt1A:III: f ðxÞdx

# A:III: ðxð1dÞq1 þMÞf ðxÞdxM Fðð1dÞq1 ÞþF t1 þM :

ð1dÞq1 ð1dÞq1 M

Here one of the potential supplier decisions is an explicit function of the buyer’s q1 decision, so the difference function is more complex. Specifically, the function is not necessarily convex or concave. For a given set of parameters, then the exact switching points can be determined by simple numerical search methods. In many realizations the difference function will be well-behaved; thus, a similar analysis to that performed for the previous two cases above would suffice for these situations. □ Proof of Theorem 4. This case can easily be compared with the centralized Case C.II. in which the centralized firm also does not expedite. The total expected supply chain profit for the voluntary compliance case is Z PSC B: ¼ r

t1

Z xf ðxÞdx þ t1 ð1 Fðt1 ÞÞ þ v

0

t1

ðt1 xÞf ðxÞdx c1 t1

0

Z

1

b t1

ðx t1 Þf ðxÞdx:

(27)

Facilitating Demand Risk-Sharing with the Percent Deviation Contract

159

Comparing (27) with the centralized supply chain profit in (20), it is easily seen that the two profits will be equal if the t1 decisions are equal, which is accomplished rþbc1 1 þp if wþac wþavþp ¼ rþbv . Simplifying this equality yields the channel coordinating condition. □ Proof of Theorem 5. We will solve for the subgame-perfect Nash equilibrium decisions under a quantity flexibility contract via backward induction. The parameters in the B. scenario are such that the buyer orders q2 ¼ maxfX; ð1 dÞq1 g, where X denotes the realized customer demand. The supplier’s expected profit can thus be written as "Z PSQF ¼ w

ð1dÞq1

Z ð1 dÞq1 f ðxÞdx þ

0

"Z

ð1dÞq1

þv Z

#

t1 ð1dÞq1

xf ðxÞdx þ t1 ð1 Fðt1 ÞÞ Z

ðt1 ð1 dÞq1 Þf ðxÞdx þ

0 1

a

#

t1 ð1dÞq1

ðt1 xÞf ðxÞdx c1 t1

ðx t1 Þf ðxÞdx:

(28)

t1

Since the supplier’s expected profit function is concave, first-order optimality wþac conditions imply that the supplier’s optimal decision is t1 ¼ F1 wþav1 . There is one additional consideration, though, since the buyer is guaranteed to order at least (1 d)q1. The supplier should pre-acquire at least the minimum purchase quantity because these n sales are guaranteed. o Thus, the supplier’s optimal decision is 1 tQF 1 ¼ max ð1 dÞq1 ; F

wþac1 wþav

.

The buyer’s expected profit function is given by "Z PBQF

tQF 1

¼r

# xf ðxÞdx þ

0

"Z

ð1dÞq1

w 0

Z

ð1dÞq1

þu 0

tQF 1 ð1

FðtQF 1 ÞÞ Z

ð1 dÞq1 f ðxÞdx þ

tQF 1 ð1dÞq1

# xf ðxÞdx þ

ðð1 dÞq1 xÞf ðxÞdx þ ða bÞ

Z

1 tQF 1

tQF 1 ð1

FðtQF 1 ÞÞ

ðx tQF 1 Þf ðxÞdx: (29)

We can solve for the buyer’s optimal decision, as before, by assuming that the supplier’s decision takes on each of the two possible values and then optimizing the buyer’s profit subject to the constraint the supplier’s decision valid. that makes F1 rwþba rvþba QF 1 rwþba ; t ; F is optimal if ¼ The decision pair qQF 1 1 1d rvþba

160

F1

M.J. Drake and J.L. Swann

F1

, which reduces to (c1 v)(r + b a) + wv c1u QF ¼ (w+ a)(w u). If this inequality is reversed, qQF 1 ; t1

F1 wþac1 wþav 1 . In this case the supplier’s decision is fixed regard; F1 wþac 1d wþav rwþba rvþba

wþac1 wþav

QF less of the value of qQF 1 , so the buyer can reduce its demand risk by offering q1 2 fqjFðð1 dÞqÞ ¼ 0g such that there is no probability of customer demand below the minimum quantity. □ SC Proof of Theorem 6. If u < v, then clearly PSC QF PC:II: for every value of t1, and there exist some values of t1 where the inequality is strict. Consequently, coordination is not possible in these cases because leftover goods are less valuable in the buyer’s possession, which is where they reside under quantity flexibility. SC SC C:II Now let u ¼ v. If tQF 1 ¼ t1 , then PQF ¼ PC:II: , and we would have a coordirþbc1 nated supply chain. Thus, we want to have rwþba rvþba ¼ rþbv . Since a(0), the penalty the supplier pays the buyer for not satisfying units ordered, is the one parameter over which the parties are assumed to have control under quantity flexibility, we solve for the coordinating condition

a¼

ðr þ b vÞðc1 wÞ : c1 v

(30)

Examining the components of (30) individually, we see that the first term in the numerator is greater than zero because r > v and b 0, as is the denominator. So if w > c1 by our initial assumption, then this would require a negative a. We could have a positive coordinating a if we allowed the supplier to sell the goods below cost. □ Proof of Lemma 2. The first result follows directly from Lemma 3. To establish the second result by contradiction, assume that this relationship is true. Since the piecewise functions are concave from Lemma 1, t1A:III: is the single maximum of PSA:III: , and PSA:III: ðt1 Þ is decreasing for values of t1 >t1A:III: . Consequently, S PSA:III: ðt1A:III: Þ>PSA:III: ðð1 þ dÞq1 MÞ>PSA:III: ðtA:II: 1 Þ. Since PA:II: ðð1 þ dÞq1 MÞ ¼ S PA:III: ðð1 þ dÞq1 MÞ from Observation 1, we have PSA:III: ðt1A:III: Þ> S A:II: PSA:II: ðð1 þ dÞq1 MÞ>PSA:III: ðtA:II: 1 Þ. Observation 3 states that PA:III: ðt1 Þ> S S S S A:II: A:III: PA:II: ðt1 Þ, which implies PA:III: ðt1 Þ>PA:II: ðð1 þ dÞq1 MÞ> PA:III: ðtA:II: 1 Þ> S S A:II: PSA:II: ðtA:II: Þ. The statement P ðð1 þ dÞq MÞ>P ðt Þ contradicts the 1 A:II: A:II: 1 1 S maximizes P . □ result that tA:II: A:II: 1 Lemma 3. If w + a > p + c2, then t1A:I: tA:II: 1 . A:I: Proof. Suppose, on the contrary, t1A:I: < tA:II: 1 , which implies that Fðt1 þ MÞb FðtA:II: þ MÞ. Substituting the values given in (3) and (4) and simplifying, we have 1

A:I: A:II: ðc2 vÞðw þ a c2 Þ FðtA:II: 1 Þ Fðt1 Þ p w þ a c1 ðc2 vÞFðt1 Þ : (31)

Facilitating Demand Risk-Sharing with the Percent Deviation Contract

161

The left side of (31) is positive, and the right side is negative since the numerator in (4) must be positive at tA:II: 1 . (The denominator is positive due to the parameter relationship defining Case A.) This leads to a contradiction. □ B:I: B:III: B:II: Lemma 4. If w + a > p + v, then t1 min t1 ; t1 . Proof. The proof follows the same contradiction procedure as in that of Lemma 3. □ Proof of Lemma 5. We will define the four functions resulting from (7) as follows: "Z PBA:I: ¼ ðr wÞ Z

# xf ðxÞdx þ t1 ðq1 Þ þ M 1 F t1 ðq1 Þ

t1 ðq1 ÞþM 0

t1 ðq1 Þ

p

t1 ðq1 Þ þ M x

0

Z f ðxÞdx þ ða bÞ

1

t1 ðq1 Þ

x t1 ðq1 Þ M f ðxÞdx (32)

"Z PBA:II: ¼ ðr wÞ Z

t1 ðq1 ÞþM 0

Z

ð1dÞq1

p

# xf ðxÞdx þ t1 ðq1 Þ þ M 1 F t1 ðq1 Þ

ðð1 dÞq1 xÞf ðxÞdx þ ða bÞ

0

1

t1 ðq1 Þ

x t1 ðq1 Þ M f ðxÞdx (33)

"Z PBA:III:

t1 ðq1 ÞþM

¼ ðr wÞ 0

Z

ð1dÞq1

p

# xf ðxÞdx þ t1 ðq1 Þ þ M 1 F t1 ðq1 Þ Z

ðð1 dÞq1 xÞf ðxÞdx þ ða bÞ

0

"Z p

t1 ðq1 ÞþM ð1þdÞq1

1

t1 ðq1 Þ

x t1 ðq1 Þ M f ðxÞdx

ðx ð1 þ dÞq1 Þf ðxÞdx

þ t1 ðq1 Þ þ M ð1 þ dÞq1 1 F t1 ðq1 Þ þ M : (34) "Z PBA:IV:

ð1dÞq1

¼ ðr wÞ Z p 0

# xf ðxÞdx þ ð1 dÞq1 ð1 Fðð1 dÞq1 ÞÞ

0 ð1dÞq1

Z ðð1 dÞq1 xÞf ðxÞdx þ ða bÞ

1 ð1dÞq1

ðx ð1 dÞq1 Þf ðxÞdx (35)

The concavity result follows the same logic as that used in Lemma 1.

□

162

M.J. Drake and J.L. Swann

Proof of Theorem 2. The following lemma establishes the piecewise-concavity of (7). □ Lemma 5. The buyer’s expected profit function realizations resulting from (7) are concave. Since the buyer’s four individual profit function realizations are concave from Lemma 5, we can use the KKT conditions to solve for the optimal q1 for each function over the region of q1 values where the function is valid, as defined in Theorem 1. t1A:I: þM We first maximize (32) n over theA:I:region o q1 1d . Since (32) is not dependent t þM on q1, any value q1A:I: q : q 11d is optimal. tA:II: þM T tA:II: þM T tA:III: þM We consider (33) over the region q1 1 1d q1 1 1þd q1 1 1þd . Taking the partial derivative and setting it equal to zero yields p(1 d)F((1 d) q1) ¼ 0. Since only the lower deviation penalty exists in this profit function realization, the buyer wants to make the initial order estimate as small as possible to avoid A:II: A:III: F1 ð0Þ maxft1 ;t1 gþM A:II: paying the penalty. Consequently, q1 ¼ q1 max 1d ; . 1þd A:II: A:III: T t þM t þM q1 1 1þd . First order We want to maximize (34) over the region q1 1 1þd A:III: fq : ð1 þ dÞ ¼ ð1 dÞFðð1 dÞqÞþ optimality conditions yield q1 ¼ q1 A:III: þM minftA:II: g 1 ;t1 ð1 þ dÞFðð1 þ dÞqÞg, which is feasible if it is smaller than 1þd A:II: . T t þM Finally, we maximize (35) over the region q1 1 1d q1 A:I: A:III: T t1 þM t1 þM A:IV: q1 1þd . The first order conditions give us q1 ¼ q1 n1d o n A:II: o t þM tA:III: þM rwaþb q : Fðð1 dÞq1 Þ ¼ rwaþbþp , which is feasible if max 1 1d ; 1 1þd qb1

t1A:I: þM 1d .

□

Proof of Theorem 3. The proof of this result follows the same logic as that of Theorem 1, utilizing the results from Lemma 4. □

References Balakrishnan A, Geunes J, Pangburn MF (2004) Coordinating supply chains by controlling upstream variability propagation. Manuf Serv Oper Manage 6(2):163–183 Barnes-Schuster D, Bassok Y, Anupindi R (2002) Coordination and flexibility in supply contracts with options. Manuf Serv Oper Manage 4(3):171–207 Bassok Y, Anupindi R (2008) Analysis of supply contracts with commitments and flexibility. Nav Res Logist 55:459–477 Bazaraa MS, Sherali HD, Shetty CM (1993) Nonlinear programming: theory and algorithms, 2nd edn. Wiley, New York Cachon GP (2004) The allocation of inventory risk in a supply chain: push, pull, and advancepurchase discount contracts. Manage Sci 50(2):222–238 Cachon GP, Fisher M (2000) Supply chain inventory management and the value of shared information. Manage Sci 46(8):1032–1048

Facilitating Demand Risk-Sharing with the Percent Deviation Contract

163

Cachon GP, Lariviere MA (2001) Contracting to assure supply: how to share demand forecasts in a supply chain. Manage Sci 47(5):629–646 Donohue KL (2000) Efficient supply contracts for fashion goods with forecast updating and two production modes. Manage Sci 46(11):1397–1411 Drake MJ (2006) The design of incentives for the management of supply and demand. PhD thesis, Georgia Institute of Technology, Atlanta, GA Erkoc M, Wu SD (2005) Managing high-tech capacity expansion via capacity reservation. Prod Oper Manage 14(2):232–251 Finley F, Srikanth S (2005) Seven imperatives for successful collaboration. Supply Chain Manage Rev 9(1):30–37 Fugate BS, Sahin F, Mentzer JT (2006) Supply chain management coordination mechanisms. J Bus Logistics 27(2):129–161 Kulp SC, Lee HL, Ofek E (2004) Manufacturer benefits from information integration with retail customers. Manage Sci 50(4):431–444 Lee HL (2004) The triple-A supply chain. Harv Bus Rev 82(10):102–112 Lee HL, So KC, Tang CS (2000) The value of information sharing in a two-level supply chain. Manage Sci 46(5):626–643 Lian Z, Deshmukh A (2009) Analysis of supply contracts with quantity flexibility. Eur J Oper Res 196:526–533 Serel DA (2007) Capacity reservation under supply uncertainty. Comput Oper Res 34:1192–1220 Tirole J (1988) Theory of industrial organization. MIT, Cambridge, MA Tsay AA (1999) The quantity flexibility contract and supplier–customer incentives. Manage Sci 45 (10):1339–1358 Tsay AA, Lovejoy WS (1999) Quantity flexibility contracts and supply chain performance. Manuf Serv Oper Manage 1(2):89–111 Tsay AA, Nahmias S, Agrawal N (1999) Modeling supply chain contracts: a review. In: Tayur S, Ganeshan R, Magazine M (eds) Quantitative methods for supply chain management. Kluwer, Norwell, MA Wang X, Liu L (2007) Coordination in a retailer-led supply chain through option contract. Int J Prod Econ 110:115–127 Zhao Y, Wang S, Cheng TCE, Yang X, Huang Z (2010) Coordination of supply chains by option contracts: a cooperative game theory approach. Eur J Oper Res 207:668–675

.

Value-Added Retailer in a Mixed Channel: Asymmetric Information and Contract Design Samar K. Mukhopadhyay, Xiaowei Zhu, and Xiaohang Yue

Abstract With increasing regularity, manufacturers are opening a direct selling channel using internet, keeping their traditional retail channel in place. This mixed channel is attractive to the manufacturers because they retain the advantage of the retailer’s traditional services while increasing their sales base to customers purchasing online. One disadvantage of this model is the potential for channel conflict because they are in direct competition with their own retailers. In this chapter, we propose an innovative way to mitigate this channel conflict, where the manufacturer allows the retailer to add value to the base product so that it is differentiated from their own offering through the direct channel. We model this supply chain where the retailer is also given the authority to price the value added product. Design of an optimal contract from the manufacturer’s point of view is complicated due to the fact that the manufacturer does not know the retailer’s cost of adding value. This chapter develops the closed form solution of the optimal contracts under this information asymmetry. Comparison with channel coordinating contracts is provided. This chapter develops a number of new managerial guidelines and identifies future research topics.

S.K. Mukhopadhyay Graduate School of Business, Sungkyunkwan University, Jongno-Gu, Seoul 110-745, South Korea e-mail: [email protected] X. Zhu (*) College of Business and Public Affairs, West Chester University of Pennsylvania, West Chester, PA 19383, USA e-mail: [email protected] X. Yue Sheldon B. Lubar School of Business, University of Wisconsin-Milwaukee, P.O. Box 742, Milwaukee, WI 53201, USA e-mail: [email protected] T.-M. Choi and T.C. Edwin Cheng (eds.), Supply Chain Coordination under Uncertainty, International Handbooks on Information Systems, DOI 10.1007/978-3-642-19257-9_7, # Springer-Verlag Berlin Heidelberg 2011

165

166

S.K. Mukhopadhyay et al.

Keywords Channel conflict • Information asymmetry • Mixed channel • Valueadding retailer

1 Introduction In addition to their traditional retailer channel, firms are opening direct channels in increasing numbers. This is a new business model facilitated by emerging internet technology. The motivation is the increased control over product distribution and pricing, order capture and customer information. The traditional retail channel is also kept in place because it has important roles to play. These include creating and satisfying demand for the product, engaging in activities to build brand awareness, gathering market information, and providing customer support. The example of this type of mixed channel strategy includes Compaq starting it in 1998. Other firms that have adopted similar strategies include IBM (Nasiretti 1998), HP (Janah 1999), Mattel (Bannon 2000), Nike (Collinger 1998). Balasubramanian (1998) and Levary and Mathieu (2000) suggest that such a strategy could work well. The disadvantage of this model is that the manufacturer is now in direct competition with its channel partners. As Frazier (1999) showed, mixed channel would increase revenue, but would lead to decreased support from the channel partners. In fact, this led to some retailers actually taking action against the manufacturers who opened a direct channel in competition with them. Channel conflict is the biggest deterrent for the manufacturer to go ahead with the mixed channel business model. Because channel conflict is detrimental for the supply chain relationship, there needs to be ways to mitigate this conflict. Some of the ways are separating the brands sold directly and sold through retailers, taking orders over the Internet and then fulfilling the order through the retailers, and sharing a part of the profit from each direct sale with their retailers. They can also maintain the price at par with the retailer so as not to undercut them. Hann (1999) gives an example of Zurich, an insurance company. Another way would be to sell a basic version of the product direct, and let the retailer add further value to the product before selling to the final customer (Fay 1999). In this chapter, we address the mixed channel strategy where the channel conflict is eliminated by the use of a value-adding retailer. We study a business model where the retailer-manufacturer conflict is alleviated by a contract. We explore a number of cases in this scenario (1) a base case, for benchmarking purpose, where the channel is integrated and a joint profit function is maximized; (2) a case where the channel partners are separate but they share full information with each other; and (3) a more general case where there is information asymmetry in the channel. Under the information asymmetry, one partner offers a lump sum side payment to the other to alleviate channel conflict. In all cases we find the optimum price in each channel, the optimum value added by the retailer, and the optimum side payment. This book chapter is based on authors’ original work of Mukhopadhyay et al. (2008a).

Value-Added Retailer in a Mixed Channel

167

2 Literature Survey Supply chain coordination can be accomplished through appropriate contract design. Cachon and Lariviere (2005) study revenue-sharing contracts in a general supply chain model with revenues determined by each retailer’s purchase quantity and price. They compare revenue sharing to a number of other supply chain contracts, like buy-back contracts, price-discount contracts, quantity-flexibility contracts, sales-rebate contracts, franchise contracts, and quantity discounts. Plambeck and Taylor (2008) study how the potential for renegotiation influences the optimal structure of supply contracts. They show that renegotiation can greatly increase the firms’ investments and profits, provided that the contracts are designed correctly. Tsay et al. (1999) and Frazier (1999) survey channel structure and incentive design for performance enhancement. Cohen et al. (1995) study an intermediary who perform specific value-adding functions, and get compensated for this service by the manufacturer or distributors by a side payment. A mixed channel strategy in products that do not provide large value are studied by Chiang et al. (2003) who show that adding a direct channel can mitigate the profit loss. Yao and Liu (2003) study diffusion of customer between two channels and find that, under certain conditions, both channels would enjoy stable demand. Viswanathan (2000) study the mixed channel issue from the product differentiation point of view and conclude that the more different the product is in the two channels, the more the benefit for the channels. Khouja et al. (2010) indicates that the most critical factor in channel selection is the variable cost per unit of product sold using the direct versus the retail channels. There is an increased competition between the manufacturer and retailer (Agatz et al. 2008) as the manufacturer expands his channels to the customers. Though channel structures have been extensively researched in literature (i.e., Hua et al. 2010; Su et al. 2010; Chiang 2010) relatively few have studied mixed channel with value-adding retailer to decreasing the competition between channels. Mukhopadhyay et al. (2008b) find that the retailer would be willing to share information with the manufacturer if her cost of adding value is lower than a threshold value. One of contribution for our research is that we study the mixed channel under asymmetric information, not under full information like most of existing literature. Asymmetric information and supply chain coordination have been the subject of a number of recent studies. Desiraju and Moorthy (1997) study the case of information asymmetry about a price and service-sensitive demand curve. They show that coordination can be achieved by requirement of service performance. Cakanyildirim et al. (2010) find that information asymmetry about manufacturer’s production cost does not necessarily cause inefficiency in the supply chain. Value of information in a capacitated supply chain is derived by Gavirneni et al. (1999). Lee et al. (2000) show that, with a demand process correlated over time, it could be worthwhile to share information about the demand. Corbett and de Groote (2000) derive optimal discount policy for both full and incomplete information cases.

168

S.K. Mukhopadhyay et al.

Corbett et al. (2004) study different types of contracts to coordinate the supply chain for both complete information and asymmetric information. Ha (2001) finds that in case of private information, optimal order quantity is smaller and optimal selling price is higher than for the case with complete information. Mukhopadhyay et al. (2009) designed a contract for the manufacturer to motivate the retail’s marketing effort under asymmetric information of retailer’s sale effort. Section 3 of this chapter presents our mixed channel model. The optimal contracts for the complete information case and asymmetric information are shown in Sect. 4. We will also compare the two cases, and derive a number of managerial insights. In Sect. 5, we report the results of an extensive numerical experimentation to see how the changes in the parameters affect the contracts. Section 6 will conclude the chapter with some further research ideas.

3 The Model Our supply chain consists of a traditional manufacturer and a retailer. There is also a direct channel selling to the same customer pool (see Fig. 1). The retailer augments the basic product by adding value for the customer. Let p1 be the price of the basic product charged by the manufacturer in the direct channel. Let v be the value added to the basic product by the retailer who prices it as p2. The cost to the retailer for adding a value v is cv per unit. It can be assumed that p2 > p1. The effective price to the customer of the augmented product is p2 v because of the additional value compared to the basic product sold in the direct channel. Let the wholesale price charged by the manufacturer to the retailer be w. Customers evaluate both the products and compare their value with the respective prices. Let the equilibrium demands be d1 for the direct channel and d2 for the retailer channel. The decision variables in our model are p1 and w for the manufacturer and p2 and v for the retailer, each maximizing their own profit functions.

MANUFACTURER

(pM) w RETAILER

p1, d1

(pR) p2, cv, d2

Fig. 1 Mixed retail and direct channel distribution system

POTENTIAL CUSTOMERS

Value-Added Retailer in a Mixed Channel

3.1

169

Characterizing the Demands

The retail channel demand, in case of no direct channel is written as: d2 ¼ ða2 bp2 Þ þ bv where a2 is the base demand b is price sensitiveness, and b is the sensitivity of demand with respect to the value added, i.e., it is the increase in demand per unit value added. Similarly, in the absence of the retail channel, the demand from the direct channel is d1 ¼ ða1 bp1 Þ where a1 is the base demand for this channel. Literature in this area uses similar linear demand function (Cotterill and Putsis 2001) and we follow their lead here. We now consider the modified demand when both the channels are operating at the same time. Now, customers would make a purchase decision by considering the two prices p1 and p2, and also the value added v by the retailer. d1 and d2, therefore, would be functions of p1, p2 and v. As a result, there would be migration of customers from one channel to another. We assume that this migration is proportional to the price difference and the additional value. Then the demand of the two channels would be: The direct channel: d1 ¼ ða1 bp1 Þ rðp1 ðp2 vÞÞ ¼ a1 ðb þ rÞp1 þ rðp2 vÞ The retailer channel: d2 ¼ ða2 bp2 Þ þ bv rðp2 v p1 Þ ¼ a2 ðb þ rÞp2 þ ðb þ rÞv þ rp1 where r is the migration effectiveness. To maintain analytical tractability, we assume that a1 ¼ a2 ¼ a, b ¼ b and normalize (b + r) and (b + r) to 1. The demand function is thus simplified as follows. Direct channel: d1 ¼ a p1 þ rðp2 vÞ

(1)

d2 ¼ a ðp2 vÞ þ rp1

(2)

The retailer channel:

We assume that r < 1, so that own channel effects are greater than cross channel effects. a and r are assumed to be common knowledge.

3.2

Value-Adding Cost for the Retailer

When the retailer is allowed to add value to the product, there is a cost denoted by cv per unit. We assume a quadratic cost function for the retailer value-adding process. Specifically, we use the functional form: cv ¼

v2 2

(3)

170

S.K. Mukhopadhyay et al.

where , an efficiency parameter for the retailer’s value added cost, is retailer’s private information. Note that we have defined cv as per unit quadratic cost to capture the phenomenon that adding a large quantum of value is proportionately more costly than adding minimal amount of value. In some cases, there would be a fixed cost (due to infrastructure creation, e.g.) which could be applied to the total sales volume. We are not including this cost here. In this chapter we consider contracts under two information structures. In full information scenario, the retailer shares its private information to the manufacturer. In asymmetric information scenario, the manufacturer does not know . We assume that the manufacturer holds a prior cumulative distribution F() with density function f(), defined on ½ 0 ; 3 , where 0 0 3 1.

3.3

Retailer’s and Manufacturer’s Profit

The practice of a side payment, L, from the manufacturer to the retailer to alleviate the channel conflict is used in some cases. Incorporating this side payment would give the following profit functions: The retailer’s profit function: pR ¼ ðp2 w cv Þd2 þ L

(4)

And the manufacturer’s profit function: pM ¼ p1 d1 þ wd2 L

(5)

Where d1 and d2 are given by (1) and (2), w is the wholesale price charged by the manufacturer to the retailer. We include L as the manufacturer’s decision variable to make the contract more flexible and to achieve the supply chain coordination (Corbett et al. 2004). To maintain analytical tractability, we further assume there are no marginal costs incurred by the manufacturer for selling through direct channel and through the retailer. In reality, both the retailer and the manufacturer have a reservation profit level which they intend to achieve in order for a trade to take place. Let reservation profit levels for the retailer and manufacturer be pR and pM respectively.

4 Two Types of Contracts One type of contract we consider is full information (F) contract. The other type is the asymmetric information (A) contract. An integrated channel (I) provides the base case. Under (I), the contract is designed by maximizing the total profit of

Value-Added Retailer in a Mixed Channel

171

manufacturer and retailer and taking as common knowledge. Under (F), the manufacturer knows the retailer’s and designs the contract taking as common knowledge to maximize its own profit. Under (A), the manufacturer does not know the retailer’s and designs the contract using prior density function f() and cumulative distribution F() defined on ½0 ; 3 .

4.1

Integration Channel Contract (I)

In this case, the channel is integrated and thus would provide the first best solution. It is expected that the profits for the channel would be highest under this scenario and thus, can be used for comparison with other cases. In this case, the two channels are integrated and they together will behave as a single firm and therefore will optimize the joint channel profit. pI ¼ p1 d1 þ ½p2 cv d2

(6)

The optimal prices p1 and p2 and value added v can be obtained by taking first order condition and solve them simultaneously. We get, p1 ¼

a a 3 1 ; p ¼ þ ; v ¼ 2ð1 rÞ 2 2ð1 rÞ 4

The optimal joint profit for an integrated channel is given by a a2 1 þ 2ð1rÞ þ 16 p ¼ 4 2 Even though the two channel partners are integrated, they still need to decide how this total profit, derived above, be divided between the two. Suppose that the retailer has her own reservation profit as pR and the retailer, therefore, would participate in the contract only if the profit pR pR . One possible way of dividing the total profit is that the retailer is given pR to ensure her participation. The manufacturer, therefore, receives the remainder pI pR . A contract like this is proposed by Corbett et al. (2004). It can be shown that pM is a decreasing function of . Therefore, it is possible that when is high enough (approaching 3), pIM could be so low that it would be lower than his reservation profit pM . In that situation, the contract would be unattractive to the manufacturer and there would be no trade. Thus, the contract, to be viable, should be such that the manufacturer is guaranteed at least pM . Let N be that value of above which pIM is lower than pM . N, therefore, is the cut-off point above which there would be no trade, and the manufacturer is said to be following a cutoff policy (Ha 2001). Under this scenario, we also need to find the value of N. This is done in Proposition 1 which gives the complete channel contract under the base case, and the optimum channel profit and its division to the two partners. Proofs of all results are shown in the Appendix, unless stated otherwise. I

172

S.K. Mukhopadhyay et al.

Proposition 1. (a) The optimal contract under channel integration is given by: pI1 ¼

a 2ð1 rÞ

NI ¼

1 qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2a þ 2 4pM þ4pR a2 1þr 1r

pI2 ¼

a 3 þ 2ð1 rÞ 4

1 2 a ð1 þ rÞ where pM þpR 4ð1 rÞ vI ¼

(b) The optimal profits for the retailer and the manufacturer under channel integration are given by: pIR ¼ pR ( pIM ¼ pI ¼

a 4

2

1 a þ 16 2 þ 2ð1rÞ pR pM

N

)

N

2

a a 1 þ þ 4 2ð1 rÞ 162

It is interesting to note that the direct channel price does not depend on the retailer’s cost. Also, when r increases, the manufacturer will increase its direct channel price. Recall that r is the migration factor, and an increasing r will enable the manufacturer to attract more customers away from the retailer. This will enable manufacturer to increase his price, and thus his revenue. This result gives a managerial insight that the manufacturer should try operational and marketing means to increase r. This can be done, for example, by advertising, or by offering incentives like easy return policy for the internet purchase.

4.2

Full Information Contract (F)

The private information held by the retailer about her cost structure (about ) is shared with the manufacturer. The moves of manufacturer and retailer follow a Stackelberg type game: the manufacturer acts as the leader, announcing the p1 and w first; the retailer acts as the follower, announcing the p2 and v after that. The solution of this game follows. The manufacturer decides about his decision variables basing on the retailer’s best response function. This best response function is in terms of the manufacturer’s parameters. This function is obtained by maximizing the retailer’s profit pR with respect to her decision variables, namely p2 and v. Equation (7) gives the retailer’s best response function, as functions of p1 and w. pr2 ¼

3 w a p1 r þ þ þ 4 2 2 2

vr ¼

1

(7)

Value-Added Retailer in a Mixed Channel

173

Next, in stage 1 of the game, the manufacturer derives the optimal p1 and w by maximizing its own profit pM , given in (5), and substituting the optimum values of p2 and v thus making it a function of p1 and w alone. Using the first order conditions, we obtain the manufacturer’s optimal policies as: pF1 ¼

a 2ð1 rÞ

wF ¼

a 1 þ 2ð1 rÞ 4

(8)

In Stage 2 of the game, the retailer uses the manufacturer’s policy announcement given in (8), and maximizes her own profit function to obtain her own optimal policies as: pF2 ¼

ð3 rÞa 7 þ 4ð1 rÞ 8

vF ¼

1

(9)

From (8) and (9), we observe that w p1 . So, the manufacturer sees that selling one unit to the retailer at the wholesale price brings in more revenue than selling in the direct channel, he will have no incentive to open a direct channel, under the full information scenario, unless he wants to do it for reasons other than maximizing profits. These reasons could be to make customers aware of the product, provide product information, not to lose ground to competitors who have web presence, and so on. In that case, the cost penalty for the sub-optimal operation can be thought of as the cost of the above mentioned benefits.

4.3

Asymmetric Information Contract (A)

This is the most realistic case where the manufacturer does not know . As noted earlier, he knows the prior density function f() and cumulative distribution F() defined on ½0 ; 3 . The manufacturer offers the retailer a contract, which is a menu of {p1, w, L} meaning that it offers a number of alternative values for this tupple. The retailer has a choice of not accepting the contracts if none of the alternatives are attractive enough to her. Or she may select one alternative from the menu and decides to accept that. We include a side payment L in this case to formulate a two part nonlinear contract which gives the most flexible contract type (Corbett et al. 2004). Thus the profit for the manufacturer is pAM ¼ wd2 þ p1 d1 L and for the retailer is pAR ¼ ðp2 w cv Þd2 þ L pR . L > 0 is defined as a per-period payment from the manufacturer to the retailer. As noted earlier, this side payment is designed to alleviate the channel conflict in case the retailer is aggravated with the prospect of having competition with the manufacturer. This is also necessary if the retailer is more powerful than the manufacturer. For example, companies like Wal-Mart and Home depot can stop the manufacturer from opening a direct channel. For example, in 1999, Home Depot

174

S.K. Mukhopadhyay et al.

sent mail to more than 1,000 suppliers to ask them to stop online sale (Brooker 1999). We also include the possibility that L can be negative. L < 0 can be interpreted as a payment by the retailer to the manufacturer for getting the opportunity to do the business, as in the case of airline ticket. This can also be applied to the case of franchise, where the retailer has to pay the manufacturer. In most of these cases, there is little gain for the retailer to add value to the product, and the retailer has little power to impede the manufacturer’s sales in the direct channel strategy. The manufacturer offers a menu of contracts which is a function of because is unknown to the manufacturer. Thus the manufacturer offers fp1 ðÞ; wðÞ; LðÞg, and the retailer chooses a ^ to announce. Once the retailer has announced ^, direct Þ, wholesale price wð^ Þ and side payment Lð^Þ are fixed, so the channel price p1 ð^ Þ; wð^ ÞÞ using the best response function retailer will set retail channel price p2 ðp1 ð^ given in (7). This follows from the Revelation Principle of Fudenberg and Tirole (1991). The mechanism by which the retailer should chose which ^ to announce is as follows. First, she uses her own profit function pR and applies a first order condition and a local second order condition given by: @pR ð^ ; Þ ¼ 0 @^

and

@ 2 pR ð^; Þ 0 @^ @

Noting that, by substituting from Proposition 1(a) and earlier deductions, pR ð; ^Þ : ¼ ðpr2 ðp1 ð^ Þ; wð^ ÞÞ wð^ Þ crv Þd2 ðp1 ð^ Þ; wð^ Þ; pr2 Þ þ Lðwð^ ÞÞ 2 a 1 wð^ Þ rp1 ð^ Þ þ þ ¼ þ Lð^ Þ 2 4 2 2 Taking first order condition of pR w.r.t. ^ and solving at ^ ¼ we get: _ LðÞ ¼

1 a þ rp1 w _ _ ðwðÞ r pðÞÞ þ 4 2

This is what is called as the IC or Incentive Compatibility constraint. It can be also shown that _ ÞÞ @ 2 pR ðr p_ ð^ Þ wð^ 0 ð^ ; Þ ¼ 1 @^ @ 42 This is true under the common assumption that F() has decreasing reverse hazard rate, i.e., F()/f() is increasing in . Given that the IC constraint is derived as functions of the manufacturer’s variables, the next step for the manufacturer is to devise his optimal menu of contracts. This is done when the manufacturer maximizes his own profit function over the range of subject to the IC constraint and the individual-rationality (IR) constraint that the retailer will at least recover her own reservation profit. This is given in the following formulation.

Value-Added Retailer in a Mixed Channel

ðN Max

p1 ;w;L;N

0

175

ðwd2 þ p1 d1 LÞf ðÞdþ F

(10)

1 a þ rp1 w _ ðw_ r pÞ þ 4 2

(11)

subject to IC : L_ ¼

IR : pR ¼ ðp2 w cv Þd2 þ L pR

(12)

The first term in (10) gives the expected value of the manufacturer’s profit over the range of from the lowest possible value in the range, viz. 0 and N, the cutoff value explained earlier. For the range of between ½N; 3 , the manufacturer will gain his reservation profit giving him a total amount of F. Equation (11) is the IC constraint, forcing the retailer to truly announce ^ , as derived above. Equation (12) is the IR constraint. The structure of the above formulation fits the standard optimal control formulation with a salvage value. Solution. The above problem is complex as it is, but in this case its intractability is increased even further due to the fact the behavior of F would change the way the problem is solved. The reason is that depending upon the value of F, the “transversality condition” (see Kamien and Schwartz 1981, p. 148 for details) would be free. These points will be elaborated later. Now we enumerate below all possible cases that would arise for the transversality condition. There will be three possible cases depending on how the retailer’s and the manufacturer’s profits are behaving with respect to each other. Case 1. In this case, the manufacturer’s profit decreases in and the retailer’s profit increases in . Then at the cutoff point N, the manufacturer’s profit will hit his reservation profit pM and then remain constant at that value. Therefore F ¼ pM ð1 FÞ and transversality condition (used at cut-off point N) is free (as in case v of Kamien and Schwartz 1981, p. 148). Case 2. This is the case where both the manufacturer’s and the retailer’s profits decrease in . Thus again we have F ¼ pM ð1 FÞ and the transversality condition using when the K(IR) 0 will be required at cut-off point N (as in case vii of Kamien and Schwartz 1981, p. 148). Case 3. The manufacturer’s profit increases in Z and the retailer’s profit decreases profit in . Here the manufacturer’s profit is no longer fixed at pM like in the two earlier cases. Now the salvage value is given by F ¼ ½p1 ðNÞd1 þ wðNÞd2 LðNÞð1 FÞ. The transversality condition using when the K(IR) 0 will be required at cut-off point N (as in case vii of Kamien and Schwartz 1981, p. 148). It is not possible to have a case where both the manufacturer’s and retailer’s profits increase with . Because as increases, the total profit from these two channels is decreasing.

176

S.K. Mukhopadhyay et al.

Next, we will take Case 1, and solve the optimal control problem for this case. This is given in Proposition 2(a) which shows the menu of optimal pricing policies of the manufacturer and his side payment given that the retailer’s IR constraint is satisfied. In Part (b) of Proposition 2, we get the optimal profits for both channels. Proposition 2. (a) The manufacturer’s optimal contract under Asymmetric information for Case 1 is given by: a F ra ; wA ¼ ; 2ð1 rÞ 2f 2 2ð1 rÞ @LA a F 1 2 @wA ¼ ; þ @ @ 2 4f 2 4 pA1 ¼

a 3 F þ þ 2ð1 rÞ 4 4f 2 The optimal profits of the retailer and the manufacturer under asymmetric information are given by: (b) The retailer’s optimal price is given by p2 A ¼

pAR ¼ ðpA2 wA cv A Þd2A þ LA ¼ pAM ¼ pA1 d1A þ wA1 d2A LA ¼

a 1 F þ 2 4 4f 2

2 þ LA

F2 aF F a2 ð1 þ rÞ þ þ þ LA 84 f 2 42 f 83 f 4ð1 rÞ

During the course of the proof (found in Appendix), we obtain: @pAR ¼ 0, 0 satisfies ðpA2 wA cv A Þd2 þ LA ¼ pR , 1 ¼ N A ; 1 satisfies 2 @ F aF F a2 ð1 þ rÞ LðNÞA ¼ pM L(N)A satisfies 4 2 þ 2 þ 3 þ 8N f 4N f 8N f 4ð1 rÞ We derived above the analytical solution to the MEP problem giving the optimal policies of both parties when information asymmetry exists in the supply chain. We have done this for one of the cases, namely, Case 1. Given the complexity of a “twopoint boundary value” problem, this analytical exposition is a significant contribution. For the other cases, we will show our results of numerical solution. We will generate a number of insights into those cases and develop significant guidelines for decision making. These will be reported in Sect. 5. Next, we will study how the information asymmetry affects the optimal policies and the nature of profit from using these strategies by comparing them with those of the first best case, i.e., the case of channel integration.

4.4

Comparison of the Types of Contracts

The profits under the channel integration (I) case and the asymmetric information (A) case are compared here. Note that, intuitively, the channel integration is an

Value-Added Retailer in a Mixed Channel

177

ideal case in a supply chain, where both parties work for the benefit of the supply chain as a whole. But given the reality that the channel partners more often than not are separate entity, how does the more real case of asymmetric channel information compare with the ideal case? This question is answered in the following proposition. Proposition 3. If a supply chain moves from the information asymmetry case to complete channel integration, then (i) The manufacturer’s profit will increase, i.e., pIM pAM (ii) The retailer’s profit will decrease, i.e., pIR pAR (iii) The supply chain profit will increase, i.e., pI pAR þ pAM As we saw, under I, the manufacturer’s profit is decreasing in . The insight we gain here is that the manufacturer would prefer lower or higher v*. The less is the information asymmetry or the more the product is different, the more the manufacturer would benefit. Also, by channel integration, the manufacturer would always gain. He can, therefore, offer incentive to the retailer for being willing to share information. It is interesting to see that when supply chain integration is achieved, the retailer tends to lose some of its profit, even though the total supply chain profit increases. Thus the manufacturer, who was the aggrieved party under the information asymmetry, will benefit more than the total channel benefit, at the cost of the retailer. It intuitively follows that, if the manufacturer wishes to motivate channel coordination, he must offer some incentive to the retailer to make up for her lost profit. The difference of profit of the retailer between the cases of A and I can be thought of as the value of this information to the retailer. The profit realized under supply chain integration is always higher than the sum of the retailer’s and the manufacturer’s profits without such integration. This is generally the same result found in most supply chain research into other aspects of supply chain decision making. Basically, information asymmetry is inefficient for the supply chain as a whole. Assuming some known distribution function for , e.g., uniform, we can show that the more the products are different through the retailer’s value added process, the more the supply chain benefits. It is true for both scenarios, I and A. Next, we analyze the retailer’s optimal policies about its value added process and pricing under the two cases where information is shared and where it is not. We do the same for the manufacturer’s pricing policies. These results are given in the next two propositions. The proofs are straightforward using earlier results and are omitted. Proposition 4. (i) The retailer’s optimal value added amount remains same under both cases of A and I. (ii) The retailer can set higher retail price under A, i.e., pA2 pI2 . Proposition 5. The manufacturer set the same direct channel price p1 under I and A, i.e. pI1 ¼ pA1 . Also, this price is independent of .

178

S.K. Mukhopadhyay et al.

Again, it is interesting to see that the value added process by the retailer becomes unaffected by the level of coordination in the supply chain. It is also seen that the optimum value of the value added depends only on one parameter, namely, . Therefore, if the retailer can use operational means to reduce her cost of adding value, the optimal value-added will be higher. This action will start a ripple effect by allowing the retailer to charge a higher price and increase her profitability. Of course, this needs to be weighed against the cost of such operational means to reduce cost. Our study gives a useful managerial insight as to the retailer’s action about her own cost structure. The manufacturer, on the other hand, does not change his price with the supply chain structure, because his price is not dependent on , and thus on the amount of information sharing. But when this policy is coupled with the retailer’s policy of pA2 pI2 (from Proposition 4), we can see that double marginalization occurs due to the asymmetric information. This, in turn, reduces the whole supply chain efficiency. It will be also interesting to study the behavior of demand for both channels under the two scenarios. We do this in the next proposition. Proposition 6. The retail channel will experience increased demand if the supply chain structure moves from being asymmetric to integrated. At the same time the direct channel demand will decrease. This is a rather surprising finding. We can explain this by using the example of uniform distribution, as shown in the proof of this proposition. With increasing, the retailer channel demand d2 is decreasing and direct channel demand d1 is increasing for both the cases of I and A. The explanation of this can be found in the fact that when increases, the amount of value added decreases. This makes buying from the retailer channel less attractive to the customer. Therefore, more and more customers choose to buy the product from the direct channel. We also find that the cut-off point is higher under I than under A, i.e., N I N A . It means that the manufacturer and retailer can trade longer under I, before it becomes unattractive to either of them to trade (by possibly hitting one of the reservation profit levels). It can be explained as follows. In case of A, the manufacturer does not know . The manufacturer, therefore, would feel safer to trade with retailer only within a small range of . The manufacturer and retailer would both lose some trading opportunities to earn higher profit.

5 Results of Numerical Analysis To validate our analytical results and to gain more insights into the optimal policies, we carried out some numerical analysis. The results are briefly reported here. The numerical values used in this experiment are: a 40

r 0.500

pM 1,100

pR 600

0 0.03

3 0.07

Value-Added Retailer in a Mixed Channel

5.1

179

Variation of Profits Under Different Scenarios

We will start by investigating the three cases detailed in Sect. 4.3. We used Case 1 for our analytical exposition, but here we study the other two cases to see when they occur and what kind of guidelines we can achieve from these cases. We assume a Uniform distribution for with f ðÞ ¼

1 3 0

FðÞ ¼

0 3 0

and

over the interval ½0 ; 3 . The manufacturer’s profit as function of is: pAM ¼

0 0 2 a a0 a2 ð1 þ rÞ þ þ þ LA ðÞ 83 84 4 42 4ð1 rÞ

and the retailer’s profit is pAR ¼ pA1 d1A þ wA1 d2A þ LA ðÞ ¼

a2 0 a 2 þ 2 þ 0 4 þ LA ðÞ: 4 4 16

We plot these two profit expressions as functions of as shown in Fig. 2. This shows that both the manufacturer’s and the retailer’s profit are non-monotone function with . There are three very obvious regions of in the Profit

Region 1 (n0, n1)

Region 2 (n1, n2)

Region 3 (n2, n3)

1250.000 1150.000 1050.000 950.000 850.000 750.000 650.000 n 550.000 0.047 0.050 0.053 0.056 0.059 0.062 0.065 0.068

Fig. 2 The manufacturer and retailer’s profit for various

M (A) R (A)

180

S.K. Mukhopadhyay et al.

graph. We break the range ½0 ; 3 into three regions. In any one of these three regions, the profits show monotone property. Region 1. In this region, in the interval ð0 ; 1 Þ, the manufacturer’s profit decreases and the retailer’s profit increases with . This corresponds to Case 1 in Sect. 4.3. We start with Z0 ¼ 0.03. To obtain 1, we find that 1 satisfies 2 @pAR 0 1 0 ð161 4 þ 41 3 Þ0 2 a þ a ¼ @ 41 3 8 1 2 41 5 321 7 3 ð81 2 þ 1 Þ0 2 þ 0 7 ¼0 161 321 7 This gives 1 ¼ 1:910 ¼ 0:0573. Region 2. The interval ð1 ; 2 Þ gives us Region 2 where both the manufacturer’s and the retailer’s profits decrease with . This corresponds to Case 2 in Sect. 4.3. 1 ¼ 0:0573 as obtained above. We obtain Z2 from: @pAM 0 1 0 2 ð162 4 þ 42 3 Þ0 1 2 ¼ þ þ a þ a @ 42 3 82 2 42 5 322 7 42 2 3 ð162 2 þ 2 Þ0 2 30 ¼0 0 7þ 162 322 7 82 4 Giving

@pAR þ @

a 0 2 30 þ 42 45 84

¼ 0:

We get 2 ¼ 2:010 ¼ 0:0603 Substituting 1 ¼ 0:0573. Region 3. This is the remaining region and the range is given by ð2 ; 3 Þis (0.0603, 0.07). In this region, the manufacturer’s profit increases in and the retailer’s profit decreases. This corresponds to Case 3 in Sect. 4.3. In Region 1, is relatively small, giving the value added (¼1/) as relatively high. The manufacturer’s profit decreases and the retailer’s profit increases in . This can be seen in certain industries like electronics or computer industry. Here the manufacturers are generally locked into working with their retailers. In these industries, the retailers are “unlikely to get dis-intermediated”, says AMR Research’s Bob Parker (Gilbert and Bacheldor 2000). In fact, “Manufacturers are looking to strengthen the channel rather than circumvent it” (Gilbert and Bacheldor 2000). The higher the value the retailer adds to the base product, the higher profit the manufacturer earns. So the manufacturer should cooperate with the retailer when opening a new direct online channel and push the retailer to add more value to the base product on the retail channel. We see this kind of practice from IBM and HP. We also find that the retailer prefers to add small amount of value (with a large ) to avoid heavy cost burden. Another extreme case is shown in Region 3. In the region, is relatively high or value added is relatively low, the manufacturer’s profit increases in and the

Value-Added Retailer in a Mixed Channel

181

retailer’s profit decreases in . This result can explain the competition between manufacturer and retailer in industries such as an airline company and its airline ticket agent. It is hard for the agent (or the retailer) to add any value to the base product. Our study suggests that the airline company should open direct sales channel and compete with the agent. We also see that the less the value added, the more the manufacturer benefit from competition. Region 2 is the middle of the other two extreme cases. We see that both the manufacturer’s and the retailer’s profits decrease in and increase in value added. The high degree of value added would differentiate the product on the two channels and reduce channel conflict. Both the manufacturer and retailer would prefer higher value added and earn higher profits. From the manufacturer’s point of view, he would want to limit within Region 1. There are several reasons. First, when comparing Region 1 with the Region 2, we find that the manufacturer’s profit in Region 1 is dominant to that of Region 2. It implies that the manufacturer should always push the retailer add more value and reduce to bring it back within Region 1. Next, consider Region 3. From the analysis given above, the manufacturer should open a direct sales channel to compete with retail channel. For example, the manufacturer could set wholesale price w equal to or very close to direct sale price p1 and squeeze the retailer out of the market if she could not play any role in the value-adding process. In the airline industry, the ticket agents are facing this fate.

5.2

Behavior of Profits with Varying h

We analyze the behavior of the profits when the parameter is varied. This is shown in Region 1 of Fig. 3. From the figure, we have several findings besides those shown in Propositions 4–6. We can see that pAR pIR and pIM pAM are increasing with . Profit

η0 1250.000

η1

1150.000 1050.000

M (I)

950.000

M (A)

850.000

R (A) R (I)

750.000 650.000 05 7 0.

05 5 0.

05 1

04 9

05 3 0.

0.

0.

04 7 0.

04 5

n

0.

0.

04 3

550.000

Fig. 3 Variation of the manufacturer and retailer’s profit for various

182

S.K. Mukhopadhyay et al.

It means that the value of information is increasing in . With increasing, the retailer earns higher profit because she holds private information, but the manufacturer’s profit decreases. We also see that the cut-off point N I N A . We already know that under A, the manufacturer’s profit drops down to his reservation profit at the cut-off point, i.e., ¼ NA ¼ 1 . While at this point the manufacturer’s profit under I is still higher than his reservation profit, so the manufacturer could continue to trade with the retailer until his profit drops down to pM . At cut-off point NA, the manufacturer and the retailer both earn their reservation profits pM and pR , respectively. At this point, they both have exhausted all trading opportunity. We also carried out sensitivity analyses on the effect on profit of other parameters like base demand a and the migration parameter r. Because of length consideration, we do not report the entire findings here. In short, we find that, with the base demand a increasing, the value of information (given by the difference between the profits in A and I cases) increases. The managerial guideline here is that the retailer should try to increase the base demand by means of, say, advertising or offering better return policy etc. We also found that when the migration parameter r increases, the value of information increases. Again, the guideline for the retailer is that she should use marketing means to influence r.

6 Conclusions An important aspect of supply chain management in the Internet era has been studied in this chapter. More and more firms are introducing a direct channel in addition to the traditional retailer channel. In this new business model, the traditional channel is differentiated by an augmented role for the retailer namely modifying the basic product by adding value for the customer. We have presented a game theoretic formulation for this new business model. We studied two cases: the case with complete information and the case with asymmetric information. We obtained closed form contracts for both the channel partners in terms of market parameters and contrasted the optimum policies with those when the channels are completely integrated. Our study found that the manufacturer – the partner suffering from the asymmetry of the information – would always benefit (increased profit) with more information. We also found that, with information asymmetry, the direct channel price does not change, while the retailer enjoys higher price. One interesting finding is that the quantum of value-added does not change under any scenario and is only dependent on the retailer’s cost structure. Information asymmetry imposes inefficiency to the manufacturer and to the supply chain as a whole. The managerial insight gained from all these results would enable the manufacturer to decide about an information sharing contract which would include suitable incentive for the retailer. Our study showed that the actual values of the decisions variables in the optimum policy depend on the various market parameters like the base demand, and the migration parameter. Our results can be used as a guideline to set decisions

Value-Added Retailer in a Mixed Channel

183

about other variables, like product quality and return policy in order to influence these market parameters to move in the direction which would be beneficial to the channel partners. We also showed the benefit of the complete channel integration. Our model can be extended in many different directions. We can allow the manufacturer to provide a value added service, instead of the retailer. Customer’s special order requirement could be easily delivered to the manufacturer through online direct channel. So some companies let the manufacturers handle the customized order through the direct channel. For example, Disney only takes personalized orders (like putting customer’s name on the product) online and sell standardized product at stores. Another interesting extension could allow both the retailer and manufacturer to offer value added service on both channels, the traditional retailer channel and direct online channel. The value added service through the manufacturer on the direct channel could be modeled by a make-toorder process and the value added service on the traditional retailer channel through the retailer could be modeled by a make-to-stock process and/or a mass customization process. These two approaches of offering variety to the customers, namely, make-to-order and mass customization, could be analyzed and compared. We can expect further research in the mixed channel field with more and more companies adding online stores to their traditional brick-and-mortar stores. We hope that the research methodology and topic presented in this chapter are helpful for future research project in this field.

Appendix Proof of Proposition 1(a) v2 ða p2 þ v þ p1 rÞ pI ¼ p1 d1 þ ðp2 cv Þd2 ¼ p1 ða p1 þ rðp2 vÞÞ þ p2 2 Then we take first order condition with respect to p1, p2 and v, and set them equal to zero, respectively. After that, solving these three equations simultaneously, we can get the desired result. Proof of Proposition 1(b) pFR ¼ ðp2 w Cv Þd2 þ LF ¼ pR 2 2a þ 1 F ) L ¼ pR 4 ( pM ¼

a 1 a2 pR þ þ 4 162 2ð1 rÞ pM

N N

ðaÞ ðbÞ

)

184

S.K. Mukhopadhyay et al.

Setting (a) ¼ (b), we get N¼

1 rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1þr 2a þ 2 4pM þ4pR a2 1r

where

pM þpR

a2 ð1 þ rÞ 4ð1 rÞ

Due to N > 0, we only keep the one with positive value. Proof of Proposition 2(a) The (10)–(12) can be written as: ðN max

mðÞd þ FðNÞ

s:t: _ _ LðÞ ¼ g1 ðÞ; wðÞ ¼ g2 ðÞ;

p_ 1 ðÞ ¼ g3 ðÞ

This is obtained by making the following variable substitution: m : ¼ ðp1 d1 ðpr2 Þ þ wd2 ðpr2 Þ LÞf 2 r r r a 1 w 1 p1 aþ þw þ rwp1 L f ; þ ¼ p1 1 þ 2 2 4 2 4 2 1 a þ rp1 w ðu1 ru2 Þ; g2 ¼ u1 ; g3 ¼ u2 ; þ g1 ¼ 4 2 FðNÞ ¼ pM ð1 FÞ: Using the multiplier equations gives following results: l_ 1 ¼ f

and l1 ¼ f

(13)

_l2 ¼ w þ a þ 1 þ rp1 f þ l1 ðu1 ru2 Þ 2 2 4

(14)

2

_l3 ¼ a 1 þ r þ 2p1 r 1 r þ rw f l1 rðu1 ru2 Þ 2 2 4 2

(15)

Using the optimality conditions gives following results: l1

1 a þ rp1 w þ l2 ¼ 0 þ 4 2

rl1

1 a þ rp1 w þ l3 ¼ 0 þ 4 2

(16)

(17)

Value-Added Retailer in a Mixed Channel

185

Taking derivative on both sides of (16) and using (13), we get 1 a þ rp1 w 1 r u1 f F 2 þ u2 þ l_ 2 ¼ 2 4 2 4 2

(18)

Solving (18) with (14), we get F ¼ fw frp1 22

(19)

Taking derivative on both sides of (17) and using (13), we get l_ 3 ¼ rf

1 a þ rp1 w 1 r u1 þ rF 2 þ u2 þ 2 4 2 4 2

(20)

Solving (20) with (15), we get 2 rF 3r rw ¼ f a þ ar þ þ 2 p 1 2 42 2

(21)

Solving (19) and (21) together, we get desired result pA1 ¼

a F ra ; wA ¼ 2ð1 rÞ 2f 2 2ð1 rÞ

and _ LðÞ ¼ g1 ðÞ ¼

1 a þ rp1 w 1 a þ rp1 w _ ðu1 ru2 Þ ¼ w: þ þ 4 2 4 2

Using the transversality conditions if N is free mðNÞ þ l1 ðNÞg1 ðNÞ þ l2 ðNÞg2 ðNÞ þ l3 ðNÞg3 ðNÞ þ FN ¼ 0 at N we get the following results: ðp1 d1 þ wd2 L pM Þf ¼ 0. Because f 6¼ 0, p1 d1 þ wd2 L pM must equals to 0. The manufacturer can make p1 d1 þ wd2 L pM 0 binding at 1 , 1 ¼ N. Then substitute p1 and w with pA1 (N) and wA (N), we get that L(N)A satisfies

F2 aF F a2 ð1 þ rÞ LðNÞA ¼ pM : þ 2 þ 3 þ 4 2 8N f 4N f 8N f 4ð1 rÞ

0 can be solved by let ðpA2 wA cv A Þd2 þ LA pR binding at 0 .

186

S.K. Mukhopadhyay et al.

Proof of Proposition 3 (i) Manufacturer: Adding pIR þ pIM pAR þ pAM from Proposition 3(ii) and pAR I pR from Proposition 3(iii), we can get pIM pAM . (ii) Retailer: Under (I), the retailer earns her reservation profit through the whole range of . Under (A), it is always higher or equal to her reservation profit. Therefore, pIR pAR for all . As an example, suppose follows a Uniform distribution where F¼

0 1 ; f ¼ : 3 0 3 0

Then, pAR ¼

a 0 þ 2 42

which is decreasing with to a value of pR at ¼ NA. At the same time, profit for the case I is constant at pR for all , so we have pR I pR A . (iii) The supply chain: The supply chain profit under (I) pI ¼

a 1 a2 þ þ 2 2ð1 rÞ 4 16

and the supply chain profit under (A) is pA ¼ pAR þ pAM ¼

a 1 a2 F2 þ þ 4 162 2ð1 rÞ 164 f 2

Obviously, pI > pA. Proof of Proposition 6 Under uniform distribution: d2I ¼

a 1 þ 2 4

and

d2A ¼

a 1 F : So d2I d2A : þ 2 4 4f 2

d1I ¼

a r 2 4

and

d1A ¼

a r rF : So d1I d1A : þ 2 4 4f 2

References Agatz NAH, Fleischmann M, van Nunen JAEE (2008) E-fulfillment and multi-channel distribution – a review. Eur J Oper Res 187(2):339–356 Balasubramanian S (1998) Mail versus mall: a strategic analysis of competition between direct marketers and conventional retailers. Mark Sci 17:181–195

Value-Added Retailer in a Mixed Channel

187

Bannon L (2000) Selling Barbie online may pit Mattel vs. stores. Wall Street J, 17 Nov Brooker K (1999) E-rivals seem to have home depot awfully nervous. Fortune 140(4):28–29 Cachon GP, Lariviere MA (2005) Supply chain coordination with revenue-sharing contracts: strengths and limitations. Manage Sci 51(1):30–44 Cakanyildirim M, Feng Q, Gan X, Sethi S (2010) Contracting and coordination under asymmetric production cost information. http://ssrn.com/abstract¼1084584 Chiang W, Chhajed D, Hess JD (2003) Direct marketing, indirect profits: a strategic analysis of dual-channel supply chain design. Manage Sci 49(1):1–20 Chiang W (2010) Product availability in competitive and cooperative dual-channel distribution with stock-out based substitution. Eur J Oper Res 200(1):111–126 Cohen M, Agrawal N, Agrawal V, Raman A (1995) Analysis of distribution strategies in the industrial paper and plastic industries. Oper Res 43(1):6–18 Collinger T (1998) Lines separating sales channels blur: manufacturers, direct sellers, retailers invade each other’s turf. Advert Age 34, 30 March Corbett C, de Groote X (2000) A supplier’s optimal quantity discount policy under asymmetric information. Manage Sci 46(3):444–450 Corbett C, Zhou D, Tang CS (2004) Designing supply contracts: contract type and information asymmetry. Manage Sci 50(4):550–559 Cotterill RW, Putsis WP (2001) Do models of vertical strategic interaction for national and store brands meet the market test? J Retailing 77(1):83–109 Desiraju R, Moorthy S (1997) Managing a distribution channel under asymmetric information with performance requirement. Manage Sci 43(12):1628–1644 Fay CJ (1999) The direct approach. Best’s Rev 99(11):91–93 Frazier GL (1999) Organizing and managing channels of distribution. J Acad Mark Sci 27 (2):226–241 Fudenberg D, Tirole J (1991) Game theory. MIT, Cambridge, MA Gavirneni S, Kapuscinski R, Tayur S (1999) Value of information in capacitated supply chains. Manage Sci 45(1):16–24 Gilbert A, Bacheldor B (2000) The big squeeze-in suppliers’ rush to sell directly to consumers over the Web, sales agents, distributors, and other channel partners worry that they’ll be pushed out of the picture. Inform Week, 27 March. http://www.informationweek.com/779/channel. htm Ha AY (2001) Supplier-buyer contracting: asymmetric cost information and cutoff level policy for buyer participation. Nav Res Logist 48:41–64 Hann LW (1999) Progressive, zurich show how it’s done. Best’s Rev 99(9):75 Hua G, Wang S, Cheng TCE (2010) Price and lead time decisions in dual-channel supply chains. Eur J Oper Res 205(1):113–126 Janah M (1999) Tech firms slow to use direct web sales. San Jose Mercury News, 24 Feb, C1. http://www.wsj.com Kamien MI, Schwartz N (1981) Dynamic optimization: the calculus of variations and optimal control in economics and management. North Holland, New York Khouja M, Park S, Cai G (2010) Channel selection and pricing in the presence of retail-captive consumers. Int J Prod Econ 125(1):84–95 Lee H, So KC, Tang C (2000) The value of information sharing in a two-level supply chain. Manage Sci 46:626–643 Levary R, Mathieu RG (2000) Hybrid retail: integrating e-commerce and physical stores. Ind Manage 42(5):6–13 Mukhopadhyay SK, Su X, Ghose S (2009) Motivating retail marketing effort: optimal contract design. Prod Oper Manage 18(2):197–211 Mukhopadhyay SK, Zhu X, Yue X (2008a) Optimal contract design for mixed channels under information asymmetry. Prod Oper Manage 17(6):641–650 Mukhopadhyay SK, Yao D, Yue X (2008b) Information sharing of value-adding retailer in a mixed channel hi-tech supply chain. J Bus Res 61:950–958

188

S.K. Mukhopadhyay et al.

Nasiretti R (1998) IBM plans to sell some gear directly to fight its rivals. Wall Street J, 5 June Plambeck EL, Taylor TA (2008) Implications of renegotiation for optimal contract flexibility and investment. Manage Sci 54(12):1997–2011 Su X, Wu L, Yue X (2010) Impact of introducing a direct channel on supply chain performance. Int J Electron Bus 8(2):101–125 Tsay AA, Nahmias S, Agrawal N (1999) Modeling supply chain contracts: a review. In: Tayur S, Ganeshan R, Magazine M (eds) Quantitative models for supply chain management, International series in operations research and management science. Kluwer, Norwell, MA, pp 299–336 Viswanathan S (2000) Competition across channels: do electronic markets complement or cannibalize traditional retailers. International Conference on Information Systems (ICIS) 2000 Proceedings, Brisbane, pp 513–519. http://aisel.aisnet.org/icis2000/51 Yao D, Liu J (2003) Channel redistribution with direct selling. Eur J Oper Res 144:646–658

Capacity Management and Price Discrimination under Demand Uncertainty Using Option Contracts Fang Fang and Andrew Whinston

Abstract This chapter considers the use of option contracts as a price discrimination tool under demand uncertainty to improve supplier profit and supply chain efficiency. Option contracts have long been used to manage demand or supply uncertainty, and the cost of the option is simply considered as the cost to avoid uncertainties. We give an example in a supply chain setting where a supplier has more than one downstream customer with private information. Under such a scenario, our game theoretical model shows that the option price shall be set taking into account the fact that only the customers who are more concerned about the demand uncertainty will purchase. Therefore, the supplier should be able to charge more for a unit of option contract compared to the traditional pricing method where simple expectations are taken. The supplier’s profit improves in three ways. First, the high type customers pay higher marginal prices on average. Second, the high type customers’ demand is satisfied as a first priority, guaranteeing allocation efficiency. Third, the supplier can observe the number of options being purchased and so determine customer types, improving capacity decision efficiency. We compare our results to those of classical second degree price discrimination literature. We show that the use of option contracts guarantee the same level of supplier profit as the level of second degree price discrimination. The overall supply chain efficiency improves to the full information benchmark. Keywords Capacity management • Demand uncertainty • Monopoly revenue management • Option contracts • Price discrimination F. Fang (*) Department of ISOM, College of Business Administration, California State University at San Marcos, 333 S. Twin Oaks Valley Road, San Marcos, CA 92096, USA e-mail: [email protected] A. Whinston Department of IROM, McCombs School of Business, The University of Texas at Austin, 1 University Station B6000, Austin, TX 78712, USA e-mail: [email protected] T.-M. Choi and T.C. Edwin Cheng (eds.), Supply Chain Coordination under Uncertainty, International Handbooks on Information Systems, DOI 10.1007/978-3-642-19257-9_8, # Springer-Verlag Berlin Heidelberg 2011

189

190

F. Fang and A. Whinston

1 Introduction Nowadays, the trend of Globalization has significantly intensified competition among companies. The requirement of effectively managing uncertainties has also been raised to an unprecedent level. Companies are striving to find novel ways to manage any kind of uncertainties. Demand uncertainty, as one of the outstanding example, has attracted a lot of attention in the recent supply chain management literature. A lot of attention has been focused on how to improve forecasting accuracies by sharing information among supply chain partners to reduce the uncertainty (e.g. Cachon and Fisher 2000; Chen 2003; Guo et al. 2006; Li 2002; etc.). In this chapter, we will discuss an innovative method of using option contracts for demand forecast and profit management. Real options contracts has been studied in supply chain management literature as a tool to protect risk-averse partners from potential uncertainties, such as demand and material cost changes (see e.g., Huchzermeier and Cohen 1996). In an incomplete contract setup, option contracts can also improve contracting efficiency by solving the famous hold-up problem (see e.g. Noldeke and Schmidt 1995). In this chapter, we explore using option contracts as a price discrimination tool under demand uncertainty. In the classic economics literature, price discrimination relies on the supplier’s ability to determine customers’ different levels of willingness to pay and hence to charge different prices. When customer types are not observable, the supplier can offer a variety of products with different prices for the customers to choose from. This practice is known as second degree price discrimination (e.g. Tirole 1988). In a classical second degree price discrimination model, the monopoly knows that the customers’ have different valuations over the quality (in some other applications, the bundled quantity) of the product, hence produces a product with two different qualities (or two bundles with different quantities) and charges different prices. The customers with different types will then have to select the product quality/quantity bundles that meet their needs the best. We extend the classical model in a setting where the customers’ demand quantity is uncertain and the supplier’s capacity is tight. In addition, the customers do not obtain higher valuation from the quantity satisfied that exceeds their own need. Example of such scenario could be when customers may want to purchase a certain number of tickets to a game and will not be able to use the extra tickets they do not need. Other cases can be found in the network/telephone service, when a customer may need to send out a certain size of email or text message. Additional network traffic would be of no use. In both cases, demand might be uncertain ex ante when customers cannot decide whether they need to go to the game or to send the email. Under such scenarios, quantity bundling strategy used in traditional second degree discrimination is not feasible. This is because the customers’ desired quantities vary according to their realized demand and would not benefit if obtain a bundle with superior quantity.

Capacity Management and Price Discrimination under Demand Uncertainty

191

In this chapter, we propose using a new form of option contracts to solve such an issue. The main reason that the option contracts would be a good device to use is because of the capacity constrain. In situations when capacity may be tight, customers have an incentive to hedge the risk when their demand cannot be satisfied. The hedging incentive is higher for those customers with higher willingness to pay. Under such circumstance, we suggest the supplier sells a form of option contracts to the customers. Executing one unit of the option contracts guarantees a customer the availability of one unit of demand and meets customers’ hedging incentive. To distinguish customers with different types, we propose that the supplier should price the option contracts in such a way that only those customers with higher willingness to pay will buy the options. The supplier can take into account the fact that those customers who have purchased the options have a higher willingness to pay to the product/service and can charge a higher option price to those customers. Under such a pricing scheme, the high type customers (who will purchase the options) will self-select to pay more than their low type counterparts (who will not purchase the options) to ensure execution of their demand. The supplier would then be able to identify the customers’ types from the purchase of the option contracts. To demonstrate the effectiveness of the option contracts as a price discrimination tool, we present a game theoretic model where one monopolistic supplier or service provider (he) faces two customers each with uncertain future demand. Each customer (she) has private information about their willingness to pay for one unit of the future demand. The supplier has to build capacity before the customers’ demand is realized. Since capacity may be insufficient for the highest level of demand realization, customers suffer from potential demand losses. When demand exceeds capacity, the supplier can only serve the demand randomly. Option contracts can be adopted in the following manner. The supplier opens the option market to the customers before the capacity investment and uncertain demand realization. At that time, customers can purchase options with unit price po. After observing customers’ option purchase decisions, the supplier invests in capacity to prepare for future uncertain demand. Afterwards, the customers find out their actual demand, observe the supplier’s capacity and decide whether to exercise their options. Alternatively, the customers can submit regular demand. Each customer pays a strike price, pe, for each unit of option executed. The amount of demand protected by the options (referred to as the “option demand” in the context) will be satisfied as the first priority. The remaining demand will be satisfied at a unit price p if there is leftover capacity. Our proposed framework improves the supplier’s revenue in three ways. First, customers with a higher willingness to pay will pay a higher unit price when the capacity is tight, increasing the supplier’s overall revenue. Second, the customers’ option demands are executed as a first priority. The remaining demand will be executed only when there is extra capacity. Third, customers’ options purchases reveal their types. This knowledge allows the supplier to more efficiently adjust the capacity levels, better accommodating the potential demand. The last two effects

192

F. Fang and A. Whinston

also improve the supplier’s decision efficiency, leading to an enhanced overall social welfare. In order to successfully induce high type customers into purchasing options, the supplier needs to convince the customers that the capacity could be insufficient. If a supplier is able to change capacity after the options are purchased, however, it may undermine the customers’ initial incentives to purchase them. This is because customers know that the supplier will want to guarantee enough capacity to meet the demand, so as to maximize the revenue by serving as much customer demand as possible. Knowing their own types and the supplier’s capacity cost, a rational customer can always conjecture the expected capacity level that the supplier will invest in contingent on their counterpart’s type. They can therefore calculate the overall benefit of purchasing the options based on these “rationally expected” capacity levels. The supplier cannot mislead the customers. However, the capacity investment levels could be different if the supplier adopts different option contracts (e.g. different po and pe). Hence, the supplier’s pricing strategy of those option contract is critical in our framework. The supplier can decide not to build sufficient capacity to guarantee the execution of the entire option demand. The decision depends on what commitment the supplier makes in the option contracts when he fails to meet the option demand. If the supplier does not compensate unsatisfied customers, the customers will have less incentive to purchase the options. This disappoints the high type customers and reduces their valuation of the options. If the supplier promises too high a compensation, he has to always ensure to overinvest in capacity. This may be inefficient especially when the capacity cost is high. In this paper, we suggest the supplier offer an option buy-back price as a compensation mechanism which leaves the high type customers indifferent between whether to execute their options or to sell them back to the supplier. This buy-back price scheme reduces the high type customers’ strategic decision of exercising the options and induces the supplier’s efficient capacity investment which maximizes ex ante social welfare. Our option framework has the potential to improve revenue management in many industries wherever demand uncertainty and information asymmetry exist. One potential application is in network traffic management. Since business communication relies heavily upon emails and video conferences, network congestion can cause severe economic losses. Companies are willing to pay a premium more than what the regular users pay to ensure important business emails being delivered promptly. Another example application to use the options framework is in the ticket sale business. Many people want to go to a concert or a game but face the risk of not being able to attend ex post. They do not want to pay the full price for the tickets in advance because they do not want to waste money if they are unable to attend. However, if they wait too long, the tickets could be sold out. The option contracts are a good choice for these people. Similar applications can be implemented in airline ticket management, hospital facility management and the hotel reservations business. In hotel reservation business, customers who care more about getting the hotel room can make reservations

Capacity Management and Price Discrimination under Demand Uncertainty

193

before arrival. It bears resemblance to the option framework. However, in practice, reservation is often free (i.e. the option price po is zero). This chapter provides an analytical framework to determine the option price. Supply chain partners can also consider using the option contracts when coordinating with each other. Although some research in supply chain management has proposed using options contracts for risk hedging and improve downstream partners’ ordering efficiency, there has been no study when suppliers have incomplete information about downstream partners’ types. Our proposed framework filled the blank by suggesting that the supplier offer different pricing schemes and allow the downstream partners (e.g. retailers) to self select the type of contracts they prefer. Retailers who are in the regions where the product is more popular would be willing to pay more than other retailers. The supplier should take such fact into consideration when determining the prices of the option contracts, rather than applying traditional options evaluation equations widely used in finance literature. In the options framework, the supplier employs price discrimination by extending the customers’ decision problem into an intertemporal one. That is, the customers have to decide whether and how many options they should purchase to hedge the future risk of demand loss before their demand realization and the supplier’s capacity investment. To make the decision, they have to figure out the capacity level the supplier may invest in and the possibility when they will use the options. When they evaluate the options, rational customers also take into account the fact that their option purchase reveals their types to the supplier and affects supplier’s the capacity level. To illustrate the implementation of option contracts, we present a two-period game-theoretic model. The model has one monopolistic supplier and two potential customers with private valuation of the service. Figure 1 shows the time line of events.

Fig. 1 The time line of the base model

194

F. Fang and A. Whinston

In the first period – the contracting period – the monopolistic supplier announces the option purchase price, po, and the strike price, pe, to maximize his expected overall profit. Then each customer decides the number of options to purchase, Oi, according to their types. At the beginning of the second period – consumption period – the supplier observes the customers’ options purchases and decides on an optimal capacity investment, K. Afterwards, the customers demands, Di, are realized. Each customer decides how many options they are going to exercise (Doi ) based on the observation of the aggregate demand and the capacity level. The supplier satisfies the option demand as a first priority. If Do > K, some of the options cannot be executed and the supplier will compensate the customers. If Do < K, the extra capacity will be used to satisfy the remaining regular demand. The rest of this chapter is organized as follows. A brief literature review is provided in Sect. 2. Section 3 presents the game model and Sect. 4 analyzes the equilibrium strategies of the supplier and two customers. We compare the model outcome to two benchmark cases. In one of the benchmarks, the supplier cannot distinguish the customer types at all. In the other, the supplier can determine each customer’s type and can charge different prices to each of them. In Sect. 5, we discuss the implications and extensions of our model. Section 6 concludes the paper.

2 Literature Review This chapter discusses the use of the options contract for price discrimination under demand uncertainty. The idea of using options contract in supply chain coordination is not a new idea. Cachon (2002) and Lariviere (1999) have surveyed a variety of supply chain contracts, particularly the options contract that induce the supply chain partners to take the right actions under different circumstances. Sethi et al. (2004) formulate a multi-period model to study procurement strategy under an option-like quantity flexible contract with spot market purchasing opportunity. Kleindorfer and Wu (2003) integrated the use of options with operational decisions, such as capacity, technology choice, and production to improve the profitability in supply chain coordination and risk management. In addition, research has also shown that the quantity flexible contracts, buy-back contracts, and pay-to-delay contracts are special cases of a combination of a priceonly contract and a call-option contract (e.g. Cachon and Lariviere 2001; BarnesSchuster et al. 2002; Martı´nez-de-Albe´niz and Simchi-Levi 2005; etc.). Lin and Kulatilaka (2007) introduce an real options approach to evaluate companies’ high-risk investments, e.g. IT outsourcing decisions, IT procurements, etc. Our framework moves one step further to discuss the possible use of option contracts in price discrimination. There has been a rich economics literature on price discrimination. (e.g. Varian 1985; Armstrong and Vickers 2001; Corsetti and Dedola 2005; Mortimer 2007) An

Capacity Management and Price Discrimination under Demand Uncertainty

195

extensive discussion on price discrimination can be found in Tirole (1988) and Mas-Colell et al. (1995). Maskin and Riley (1984) discussed optimal mechanism design that can induce different agents to report truthfully their types. Their model primarily concerns a principle who is trying to allocate limited resources to a number of agents. The assumption was that the resource is rare and demand uncertainty is not an issue. The outstanding result shows that the lowest type of agent will be left no surplus and the higher type agents enjoy certain level of surplus, which is necessary for them to report their true information. Such surplus is called “information rent” in the literature of second degree price discrimination. Deshpande and Schwartz (2002) extend the mechanism into a constrained capacity setup. In this mechanism, non-linear pricing rules are adopted to guarantee incentive compatibility. Boyaci and Ray (2006) discussed the impact of capacity costs on product differentiation in a product delivery model. In our framework, the demand is uncertain so that the allocation for the two types of customers cannot be predeterimined. In this sense, Maskin and Riley’s mechanism cannot be directly applied to solve our problem. An alternative pricing scheme is the spot pricing scheme, which suggests that the price should be dynamically adjusted according to the congestion levels. Gupta et al. (1996, 1999) suggests using priority classes with different spot prices to efficiently allocate network resource. The idea is that the customers with higher delay costs will choose to pay more and send their demand to the network with lower expected delay time. Afeche (2006) suggests to add strategic delay in the queue to further discriminate customers and maximize the revenue of the supplier. However, under the spot pricing mechanism, customers decide whether to execute the demand when observing the spot price. The supplier cannot discover the customers’ type distribution before adjusting the capacity. In addition, consumers will face additional uncertain future spot price due to the different realizations of future. In practice, the spot price is not preferred by both individual consumers and companies due to the management difficulties.

3 Model Development A monopolistic supplier sells to two risk neutral customers (i ¼ 1,2). Each customer can be one of two unknown types. A “high” type customer enjoys higher marginal utility from the satisfied demand than a “low” type customer. Specifically, customer i receives total utility ui ¼ vti Dei mi if she is of type ti ∈ {l, h}. vti represents the customer’s marginal value of satisfied for type ti and vh > vl. Dei is customer i’s demand being satisfied by the supplier, and mi is the total monetary transfer from customer i to the supplier. Each customer knows her own type ti but does not know for certain the other’s type. The supplier can observe neither of the customers’ types. The common belief is that ti ¼ h with probability l ∈ (0,1) and ti ¼ l with probability 1 l. The realizations of t1 and t2 are independent.

196

F. Fang and A. Whinston

Each customer’s demand, Di, is uncertain and could be either DH (with probability a) or DL (with probability 1 a). The realization of (D1, D2) is independent of customers’ type realization (t1, t2). We denote D ¼ ðD1 ; D2 Þ as the demand vector and D ¼ D1 + D2 as the aggregate realized demand. To avoid the trivial case when the supplier will only concern the high demand situation, we assume that DH < 2DL. After observing the demand realization Di, each customer decides how much of the demand should be submitted to the supplier, which we denote as Dsi . In this model, we restrict Dsi bDi , implying that the customer cannot submit a demand higher than the realization of their actual demand. This restriction makes sense when the demand can be verified ex post. Taking the example of network traffic management, customer sends out files with certain sizes. A customer can increase the demand by extending the size of the file. However, she cannot benefit from doing so. Failure to impose the above restriction introduces customers’ strategic behavior when submitting their demand. Cachon and Lariviere (1999) analyzed this kind of strategic behavior under different allocation rules. They show that customers’ order inflation could be an equilibrium strategy and the supplier is worse-off due to the concern of such strategic behavior. However, this is not the focus of our paper. To serve the customer demand, the supplier has to invest in a certain level of capacity K before the demand, D is realized. The marginal cost of capacity investment is c0. Observing the customers’ submitted demand Ds ¼ ðDs1 ; Ds2 Þ, the supplier decides how much demand to execute for each customer, De ¼ ðDe1 ; De2 Þ, ðDei bDsi for i ¼ 1; 2Þ to maximize his expected revenue. The total amount of executed demand, De ¼ De1 þ De2 is constrained by the supplier’s capacity K. To avoid trivial results, we will impose the following three assumptions: Assumption 1. c0 > minfð2a a2 Þvl ; a2 vh g: This condition guarantees that the supplier’s capacity cost is non-trivial so that the supplier cannot simply decide to invest in the highest capacity level possible to avoid congestion. Violation of this condition will result in the trivial case where the customers do not expect any congestion and hence will not purchase any option. Assumption 2. c0 < ð2a a2 Þvh : This condition guarantees that the capacity cost is not prohibitively high so that the supplier can make a profit. Otherwise, the supplier will always choose to invest qﬃﬃﬃﬃﬃﬃﬃﬃﬃ in zero capacity level. h 0 Assumption 3. l < 1 v vc h . This condition reduces our focus to the case where the chance of having a high type customer is small enough so that the supplier will not ignore the possible existence of low type customers. Assuming the supplier is risk neutral and there is no cost for executing the customers’ demand, the potential supplier’s profit P is calculated as P ¼ m1 þ m2 c0 K: We propose that the supplier can use option contracts to manage congestion (i.e. in the event that D > K). When capacity is insufficient to meet the aggregate

Capacity Management and Price Discrimination under Demand Uncertainty

197

demand D, customers can choose to execute their options, guaranteeing that their demand be satisfied. One unit of the option contract guarantees the customer one unit of satisfied demand regardless of congestion. To implement price discrimination, the options are priced such that only high type customers will buy the options because they will suffer more from demand loss than low type customers. By using the options, high type customers can avoid demand loss but may pay a higher unit price on average. In addition, low type customers’ demand will be executed at a lower priority, increasing their chance of facing a demand loss. The supplier benefits from using the option contracts since he can, in effect, charge the high types a higher fee and is able to adjust capacity after observing customers’ actual types. From a social optimal perspective, the allocation efficiency is always guaranteed since those demands with higher marginal value always receive first priority execution.

4 Analysis In order to demonstrate the supplier’s revenue improvement and the overall supply chain efficiency increase, we first introduce two benchmark models to compare the result of the proposed options framework. The first benchmark outlines the problem when the supplier does not have the ability to distinguish customers’ types. He also have to invest in a certain capacity level before the demand uncertainty is resolved and can only charge a linear price, p, for each unit of executed demand. Our proposed options framework should be able to improve situations outlined in the first benchmark. For one, options contracts can help supplier identify the high type customers, if any. The high type customers, with a higher average unit price paid to the supplier, can be guaranteed demand satisfaction when capacity is tight. The second benchmark model is an ideal situation where the supplier has full information. That is, the supplier can observe the customers’ types before the capacity investment. The supplier can also charge different prices ph and pl to the high types and low types, respectively. In this case, allocation efficiency is guaranteed since the supplier always satisfies demand from high type customers first and uses the remaining capacity (if there is any) to serve the low type customers. In addition, the supplier can set the prices that leave no surplus to both types of customers. The capacity investment is efficient in this case since the supplier maximizes his own profit as well as the social welfare. Our options framework, with revenue and efficiency improvement from benchmark 1, would hope to be able to achieve efficiency level demonstrated in the second benchmark. Section 4.3 analyzes the option framework and provide results. In all three frameworks, we examine both the supplier’s expected profit EP and the overall supply chain efficiency W, defined as the sum of the supplier’s expected profit and the expected utilities of both customers. We then compare the outcomes to demonstrate the effectiveness of the proposed options framework.

198

4.1

F. Fang and A. Whinston

Benchmark I: No Discrimination Case

In this section, we examine the case where the supplier can only charge a single linear price p to the customers regardless of their types. In this case, the customers will only be able to submit their demand as regular demand, rather than the option demand. In addition, there is no reason why they will submit their demand fewer than the realized demand. Therefore, we have Dsi ¼ Di for both i ¼ 1,2. However, since the capacity might be constrained, the total satisfied demand would be the minimum between the supplier’s capacity level and the total submitted demand. That is, De ¼ min{K, Ds}. Each customer is charged mi ¼ pDei and the supplier’s profit is P(p,K) ¼ pDe c0K. In this case, we have Dsi ðpÞ

¼

Di 0

if pbvti otherwise:

To decide an optimal price level, the supplier have several alternatives. By charging a price higher than vl, the supplier only serves the high type customers. The supplier can enjoy a higher marginal profit from each unit of demand, but loses business if a customer is of low type. If the price is lower than or equal to vl, both types will be served. The following Proposition 1 concludes that the supplier should serve both types of customers by charging a price p ¼ vl under the conditions we discussed in Sect. 3. The proposition also derives the optimal capacity ND , the supplier’s expected profit EPND, and the overall efficiency decision K ND . The superscript ND represents the no discrimination W ND ¼ E PND þ uND 1 þ u2 case. Proposition 1. Assume that the supplier can only charge a unit price to both customers. The optimal unit price pND ¼ vl and the capacity KND ¼ 2DL. Both types of customers will submit their demand and the supplier’s profit is EPND ¼ ðvl c0 Þ2DL : The overall efficiency is W ND ¼ lvh þ ð1 lÞvl c0 2DL : Note. The proofs of all the lemmas and propositions are provided in Appendix 2. Proposition 1 provides a benchmark outcome when the supplier is least capable of identifying customer types and charge different pricing schemes. Under such conditions, the supplier is conservative in his capacity investment decision and the capacity level is only enough for the least possible level of aggregate demand, K* ¼ 2DL ¼ inf{D}. The low type customers are left no surplus and the high type customers can make strictly positive surplus which equals the product of the difference in the marginal utility of these two types, vh vl, and the capacity 2DL.

Capacity Management and Price Discrimination under Demand Uncertainty

199

In this case, the supplier’s profit (vl c0)2DL equals the surplus it can extract as if the customers are all low type and the demand is always DL. Therefore, the first benchmark is an inefficient case where the supplier cannot take advantage of the fact that there is possible demand increase and higher willingness to pay. In order to improve this situation, the supplier needs to seek effective ways for revenue management and price discrimination.

4.2

Benchmark II: Full Information Case

In the full information benchmark case, we assume that the supplier can distinguish the different types of the customers before the capacity is invested and charge different prices according to the customer types. We use superscript FI to indicate the “full information” case. It is straightforward to show that the optimal prices will be pFI(t ¼ h) ¼ vh and FI p (t ¼ l) ¼ vl. In this case, since the customers only have one venue to submit their demand, the submitted demand Dsi equals to their realized demand Di, for i ¼ 1,2. Under such prices, the supplier leaves no surplus to both types of customers and extracts the maximum profit he can get under capacity constraint KFI. When the capacity is insufficient to meet the total submitted demand Ds ¼ Ds1 þ Ds2 , the supplier will satisfy the high type customers’ demand first to obtain a higher margin. The other decision the supplier has to make is on the capacity level. Since the supplier already knows the types of the customers, the capacity level KFI will then be different according to different customers’ types (t1, t2). Due to symmetry, we have KFI(h,l) ¼ KFI(l,h). Lemma 1 summarizes this optimal capacity level. Lemma 1. In the full information benchmark case, the capacity investment decision is made contingent on the actual types of the two customers. KFI ðt1 ; t2 Þ ¼

DH þ DL 2DL

if t1 ¼ t2 ¼ h otherwise:

The result of Lemma 1 shows that the supplier is able to set a higher capacity level when both customers are of high type. However, when at least one customer is of low type, the supplier will maintain a relatively low capacity level. This is because the supplier is able to identify and charge different prices to customers and to allocate the capacity to serve high type customers first when capacity is tight. When there is at least one low type customer, only the low types who pays a lower unit price suffer. The supplier is expecting less revenue loss which cannot motivate him to increase capacity investment. Proposition 2. When the supplier can observe customer types before making a capacity investment and charges different prices accordingly, the expected supplier

200

F. Fang and A. Whinston

profit equals the overall efficiency. That is, EPFI ¼ WFI. Comparing with EPND and WND yields

EPFI ¼ EPND þ D1 þ D2 þ D3 W FI ¼ W ND þ D1 þ D3

where: 1. D1 ¼ 2lð1 lÞaðDH DL Þðvh vl Þ; 2. D2 ¼ ðvh vl Þl2DL and 3. D3 ¼ l2 ðð2a a2 Þvh c0 ÞðDH DL Þ: It can be shown that D1, D2, and D3 are all strictly positive. Therefore, the results that EPFI > EPND and WFI > WND always hold. Proposition 2 identifies the three sources where the supplier gains higher profit: 1. The profit gain from the supplier’s ability to prioritize the customers’ demand when capacity is tight, represented by D1. 2. The profit gain from the supplier’s ability to charge different prices according to different customer types, represented by D2. 3. The profit gain from the supplier’s ability to invest in capacity levels according to the actual realization of the customer types, represented by D3. The efficiency gain comes in only two parts and does not include D2. This is because the supplier’s ability to charge the high type customers a higher price only affects the monetary transfer among the three supply chain parties. It does not change overall efficiency when adding the profits of all three parties.

4.3

Option-Capacity Game

In this section, we discuss the framework in which the supplier offers the customers a form of option contracts to hedge their demand risk. The customers choose the number of options to purchase at a unit price po before the supplier builds his capacity. The supplier will then observe the number of options purchased by each customer, conjecture the customers types, and decide how much capacity he should build to meet the future demand. After the capacity investment and demand realization, customers observe the supplier’s capacity and the aggregate demand and decide how many options to execute, if they have purchased any. The number of options a customer chooses to execute is called the option demand and is denoted as Doi . For each unit of option demand executed, the customers pay a unit execution price pe. We assume that the customer cannot execute more options than the actual demand, that is Doi bDi . If Doi < Di , the customer’s remaining demand Di Doi will be satisfied randomly. A unit price p is charged if the remaining demand is satisfied.

Capacity Management and Price Discrimination under Demand Uncertainty

201

To successfully discriminate the customers, the supplier must set the option contract parameters (po, pe) in such a way that only high type customers will buy them. The supplier also sets a unit price p ¼ vl for the regular demand. If the aggregate option demand, Do ¼ Do1 þ Do2 , exceeds the capacity K, the supplier needs to buy back some of the options. The option buy back price has to be high enough so that the high type customers are willing to sell it back. Meanwhile, it cannot be too high to make the customers want to sell all the options back instead of executing them. Thus, the buy-back price, pb, should equal vh pe, the marginal benefit the high type customers get from the demand being executed. According to the time line in Fig. 1, the strategic interactions among the supplier and the two customers can be described in a three-stage game. In the first stage, the supplier announces the option contract parameters, po and pe. The customers simultaneously decide the number of options, Oi, to buy based on their own types. We are interested in the equilibrium cases where Oi ðt ¼ lÞ ¼ 0 and Oi ðt ¼ hÞ > 0. With our assumptions on the customer demand, the high type customers have actually two choices: whether to buy Oi ¼ DL for a minimal hedge or to buy Oi ¼ DH for a maximal hedge. The minimum hedge strategy, buying DL units of options, guarantees the customers to execute at least the minimal demand level, when capacity is tight. In our setup, the customers’ demand equals to DL with probability 1 a. When a is small, the customer may consider minimum hedging because the chance of its demand exceeds DL is small. However, the customers also have to consider the real chance when the option is executed. For example, when a is small, the other customers’ demand is also unlikely to be high, so the possibility that the capacity is tight is also very low. Therefore, the customers may not have a strong incentive to choose minimum hedge strategy. Maximum hedge strategy guarantees the customers’ ability to execute all the future realized demand. However, the customers may not need to use all of the options purchased. The option price must be properly set by the supplier to induce the high type customers purchase options with an appropriate hedge strategy. The supplier also need to calculate which customer hedge strategy is the most profitable to him. If the option price is set properly and only the high type customers will purchase them, then the supplier can observe (t1, t2) through the sale of the options and decide the capacity level, K(t1, t2), to maximize his expected profit in the following period, Ep(K). After the demand is realized, the customers simultaneously decide how much demand to submit to the supplier, denoted as Dsi , and how much of Dsi is submitted as option demand, Doi . When the regular price p ¼ vl, Dsi ¼ Di hold for both types of customers. The supplier gathers the total demand (Do, Ds) and decides how to allocate the constrained capacity with the priority of the option demand. Assume that the supplier executes Dei amount of demand from the customers, then if Dei > Doi , the additional demand Dei Doi will be executed as the regular demand at a unit price p ¼ vl. If Dei < Doi , the supplier did not satisfy all the option demand,

202

F. Fang and A. Whinston

and therefore has to buy back the additional option demand Doi Dei at the option buy back price pb ¼ vh pe. The total charge to a customer will be mi ¼ po Oi þ pe Doi þ vl ðDei Doi Þþ vh ðDoi Dei Þþ ; where the last term is the customer’s expected compensation if the option demand is not executed. For the low type customers, since they won’t buy any option in equilibrium, we can simplify the total charge by substituting condition Oi ¼ Doi ¼ 0 and obtain mi ðt ¼ lÞ ¼ vl Dei . The customer’s overall utility: ui ¼

ðvh pe ÞDoi þ ðvh vl ÞðDei Doi Þþ po Oi 0

if t ¼ h if t ¼ l

In the following Sects. 4.3.1–4.3.3, we use backward induction to solve the three-stage game.

4.3.1

The Consumption Period

In period 3, each customer observes her own demand, which could be either DH or DL . After observing the aggregate demand, D, and the capacity, K, the two customers simultaneously decide how many options to execute. Denote O ¼ ðO1 ; O2 Þ. There are six configurations that need to be discussed: O ¼ ðDH ; DH Þ; ðDH ; 0Þ; ðDL ; 0Þ; ðDL ; DL Þ; (0,0) and (DH ; DL Þ. Due to symmetry, we do not need to analyze the cases of (DL, DH), (0, DH) and (0, DL). On the equilibrium path, only the high type customers will buy options. Therefore, supplier will infer that both customers are of high type when O equals (DH, DH), (DL, DL), or (DH, DL). Configurations (DH, 0) and (DL, 0) indicate that only one customer is high type, and configuration (0, 0) indicates both customers are of low type. For the low type customers, it is straightforward that Doi ¼ 0 since they don’t have any options. For the high types, Doi is decided based on the following optimization problem: max

Doi b minfOi ;Di g

ðvh pe ÞDoi þ ðvh vl ÞðDi Doi Þ f

ðK Doi Doi Þþ s:t: f ¼ min 1; D Doi Doi

where Doi is the option demand submitted by the other customer. Doi ¼ 0 if that other customer is of low type. f indicates the probability that regular demand is satisfied. When D < K, there is no congestion and all the demand will be satisfied, f ¼ 1. When Do > K, the option demand along will exceeds the capacity. The supplier does not have additional capacity to serve the regular demand and hence f ¼ 0.

Capacity Management and Price Discrimination under Demand Uncertainty

203

Lemma 2. Denoting D o as the solution set of the above optimization problem, we have Do f0; minfDi ; Oi gg. Lemma 2 suggests that a high type customer will either execute all the options she has purchased up to her realized demand or not execute the options at all, depending on the option execute price pe. As pe increases, the customer pays more on executing the options and hence is increasingly reluctant to do so. Tables 1–6 show the solutions of Doi as functions of the six different realizations of O, respectively. In each table, Doi also varies according to different values of the realized demand D, the option exercise price pe and the available capacity K. From the results summarized in Tables 1–6, we can observe that the amount of options a high type customer will execute decreases as the option execution price pe increases. When pe rvh ðvh vl Þ2DKH , no option contracts will be executed in any possible configuration of O and the demand realization D. This is because that the options are too expensive to execute. Therefore, the options have no value. We can then conclude that the supplier will never charge such an option execution price. In the following discussion, we focus on the cases where the execution price pe < vh ðvh vl ÞDKH when the option contracts will possibly be executed.

Table 1 Option demand when O ¼ ðDH ; DH Þ O ¼ ðDH ; DH Þ

vh ðvh vl Þ minf2DKH ; 1g < pe bvh vh ðvh vl Þ minfDH KþDL ; 1g < pe bvh ðvh vl Þ minf2DKH ; 1g h K v ðvh vl Þ minf2D L ; 1g < h h l pe bv ðv v Þ minfDH KþDL ; 1g K pe bvh ðvh vl Þ minf2D L ; 1g

Table 2 Option demand when O ¼ ðDH ; DL Þ O ¼ ðDH ; DL Þ vh ðvh vl Þ minf2DKH ; 1g < pe bvh vh ðvh vl Þ minfDH KþDL ; 1g < pe bvh ðvh vl Þ minf2DKH ; 1g K vh ðvh vl Þ minf2D L ; 1g < pe bvh ðvh vl Þ minfDH KþDL ; 1g K vl < pe bvh ðvh vl Þ minf2D L ; 1g

p e b vl

D ¼ ðDH ; DH Þ Do1 ¼ 0 Do2 ¼ 0 Do1 ¼ DH Do2 ¼ DH Do1 ¼ DH Do2 ¼ DH Do1 ¼ DH Do2 ¼ DH

(DH,DL)

D ¼ ðDH ; DH Þ Do1 ¼ 0 Do2 ¼ 0 Do1 ¼ DH Do2 ¼ 0

(DH,DL)

Do1 ¼ DH Do2 ¼ 0 Do1 Do2 Do1 Do2

¼ DH ¼0 ¼ DH ¼ DL

Do1 Do2 Do1 Do2 Do1 Do2 Do1 Do2

¼0 ¼0 ¼0 ¼0 ¼ DH ¼ DL ¼ DH ¼ DL

(DL,DH) Do1 Do2 Do1 Do2 Do1 Do2 Do1 Do2

¼0 ¼0 ¼0 ¼0

¼0 ¼0 ¼0 ¼0

Do1 Do2 Do1 Do2

Do1 Do2 Do1 Do2 Do1 Do2

¼ DH ¼ DL ¼ DH ¼ DL ¼ DH ¼ DL

Do1 Do2 Do1 Do2 Do1 Do2

¼0 ¼0 ¼0 ¼0 ¼0 ¼0

¼ DL ¼ DH ¼ DL Do1 ¼ DL ¼ DH Do2 ¼ DL

(DL,DH)

Do1 Do2 Do1 Do2

(DL,DL)

Do1 ¼ DL Do2 ¼ 0

Do1 Do2 Do1 Do2 Do1 Do2

¼0 ¼0 ¼0 ¼0 ¼0 ¼0

¼ DL ¼0 ¼ DL ¼ DL

Do1 Do2 Do1 Do2

¼ DL ¼0 ¼ DL ¼ DL

Do1 Do2 Do1 Do2

¼0 ¼0 ¼0 ¼0

(DL,DL)

204

F. Fang and A. Whinston

Table 3 Option demand when O ¼ ðDH ; 0Þ O ¼ ðDH ; 0Þ v ðv v h

h

l

Þ minf2DKH ; 1g < pe

bv

D ¼ ðDH ; DH Þ Do1 ¼ 0 Do2 ¼ 0 Do1 ¼ DH Do2 ¼ 0

h

vh ðvh vl Þ minfDH KþDL ; 1g < pe bvh ðvh vl Þ minf2DKH ; 1g l v < pe bvh ðvh vl Þ minfDH KþDL ; 1g

Do1 Do2 Do1 Do2

p e b vl

Table 4 Option demand when O ¼ ðDL ; DL Þ O ¼ ðDL ; DL Þ D Þ h vh ðvh vl Þ minfDKþ2ðD H þ2ðD H DL Þ; 1g < pe bv H

L

vh ðvh vl Þ minfDH KþDL ; 1g <

D Þ pe bvh ðvh vl Þ minfDKþ2ðD H þ2ðDH DL Þ; 1g H

L

vl < pe bvh ðvh vl Þ minfDH KþDL ; 1g

Table 5 Option demand when O ¼ ðDL ; 0Þ O ¼ ðDL ; 0Þ 1

vh ðvh vl Þ minf2

KDL D2L þDH DL DL ð2DH DL Þ ; 1g < pe

bvh

vh ðvh vl Þ minfDH KþDL ; 1g <

1 KDL D2 þDH DL

pe bvh ðvh vl Þ minf2 DL ð2DHL DL Þ ; 1g þD D D vh ðvh vl Þ minfKD ðD H þDL ÞDH L

b b

H

H

L

DL

; 1g <

pe v ðv v Þ minfDH KþDL ; 1g L þDH DH DL DL ; 1g vl < pe vh ðvh vl Þ minfKD ðD H þDL ÞDH h

h

l

p e b vl

Table 6 Optimal demand when O ¼ ð0; 0Þ O ¼ ð0; 0Þ D ¼ ðDH ; DH Þ For all pe bvl

Do1 ¼ 0 Do2 ¼ 0

D ¼ ðDH ; DH Þ Do1 ¼ 0 Do2 ¼ 0 Do1 ¼ DL Do2 ¼ DL Do1 Do2 Do1 Do2

p e b vl

¼ DH ¼0 ¼ DH ¼0

¼ DL ¼ DL ¼ DL ¼ DL

(DH,DL) Do1 Do2 Do1 Do2 Do1 Do2 Do1 Do2

¼0 ¼0 ¼0 ¼0 ¼ DH ¼0 ¼ DH ¼0

(DH,DL) Do1 Do2 Do1 Do2 Do1 Do2 Do1 Do2

¼ DL ¼0 ¼ DL ¼0

(DL,DH)

Do1 Do2 Do1 Do2 Do1 Do2 Do1 Do2

¼0 ¼0 ¼0 ¼0 ¼0 ¼0 ¼ DL ¼0

(DL,DL)

¼0 ¼0 ¼0 ¼0

Do1 Do2 Do1 Do2

¼0 ¼0 ¼0 ¼0

¼ DL ¼ DL ¼ DL ¼ DL

Do1 Do2 Do1 Do2

¼ DL ¼ DL ¼ DL ¼ DL

Do1 Do2 Do1 Do2

¼0 ¼0 ¼ DL ¼ DL

(DH,DL)

Do1 ¼ DL Do2 ¼ 0

Do1 ¼ 0 Do2 ¼ 0 Do1 Do2 Do1 Do2

¼0 ¼0 ¼0 ¼0

(DL,DL)

¼0 ¼0 ¼0 ¼0

Do1 Do2 Do1 Do2

¼ DL ¼0 ¼ DL ¼0

Do1 Do2 Do1 Do2 Do1 Do2 Do1 Do2

Do1 Do2 Do1 Do2

D ¼ ðDH ; DH Þ Do1 ¼ 0 Do2 ¼ 0 Do1 ¼ DL Do2 ¼ 0

Do1 Do2 Do1 Do2

(DL,DH)

¼0 ¼0 ¼0 ¼0

¼ DL ¼0 ¼ DL ¼0

(DL,DH) Do1 Do2 Do1 Do2

¼0 ¼0 ¼0 ¼0

(DL,DL) Do1 Do2 Do1 Do2

¼0 ¼0 ¼0 ¼0

Do1 ¼ DL Do1 ¼ 0 Do2 ¼ 0 Do2 ¼ 0 Do1 Do2 Do1 Do2

¼ DL ¼0 ¼ DL ¼0

Do1 Do2 Do1 Do2

¼ DL ¼0 ¼ DL ¼0

(DH,DL)

(DL,DH)

(DL,DL)

Do1 ¼ 0 Do2 ¼ 0

Do1 ¼ 0 Do2 ¼ 0

Do1 ¼ 0 Do2 ¼ 0

Capacity Management and Price Discrimination under Demand Uncertainty

4.3.2

205

Capacity Investment Game

In the above subsection, we have determined both customers’ equilibrium decisions on Doi . This decision is contingent on the number of options both customers have purchased O, the realized demand D, the option strike price pe, and the capacity K. This subsection will analyze the supplier’s optimal capacity decision K*, which is made before the actual demand is realized. As stated previously, the optimal capacity decision K* depends on the supplier’s observation of option purchase by the customers O and the option execution price pe. Based on such information, the supplier shall expect the possible future demand realization D and the customers’ reaction of Doi , i ¼ 1,2. Once the customers have purchased the options, the supplier will only look at the future revenue excluding the revenue from option purchase, po(O1 + O2), to decide the optimal capacity. When K* increases, the chance of congestion decreases. The supplier’s revenue comes more from serving the regular demand. When pe < vl, the supplier has an incentive to prepare enough capacity to avoid the possible customers option demand. However, if pe > vl, the supplier tends to induce the customers into exercising their options by restricting the capacity K. However, restricting capacity may create problems when K < Do because the supplier fails to execute the option demand as he has promised in the option contract. He must compensate those customers whose option demands cannot be executed. The supplier’s capacity decision is to figure out an optimal level of capacity to trade off among the revenue, capacity investment cost, and the possible compensation. Customers are more willing to execute their options as pe decreases. Foreseeing this, the supplier tends to increase the capacity level to guarantee that all the option demand be satisfied. This will minimize the expected compensation. Hence, we predict capacity K* decreases as pe increases. Proposition 3. In a subgame perfect equilibrium, the supplier’s optimal capacity decision is as follows: 1. When O ¼ ðDH ; DH Þ, the optimal capacity 8 L L > if vh ðvh vl ÞDDH < pe bvh < 2D h H L h h l DL e K ¼ vvhp 2DH if vh ðvh vl ÞD2DþD H < pe bv ðv v ÞDH vl > H L : H D þ DL if pe bvh ðvh vl ÞD2DþD H 2. When O ¼ ðDH ; DL Þ, the optimal capacity

K ¼

8 L > < 2D

v pe H þ DL Þ h l ðD > : v v DH þ DL h

if vh ðvh vl ÞDH2DþDL < pe bvh L

if vl < pe bvh ðvh vl ÞDH2DþDL if pe bvl L

3. When O 2 fðDL ; DL Þ; ðDH ; 0Þ; ðDL ; 0Þ; ð0; 0Þg, the optimal capacity K* ¼ 2DL for all pe < vh.

206

F. Fang and A. Whinston K*

DH+DL

K*(D H,D H) K*(DH,D L)

2DL

K*(DL,DL),K*(DH,0),K*(DL,0), and K*(0,0)

h

h

l

L

H

L

v −(v −v )2D /(D +D )

vl

h

h

l

L

H

v −(v −v )D /D h

h

l

H

L

v −(v −v )(D +D )/2D

H

pe

Fig. 2 Optimal capacity levels

Figure 2 summarizes the optimal capacity level K* as a function of pe in all six configurations of O. The capacity is at least the minimal level of the aggregate demand 2DL. It is greater than 2DL only when both customers have purchased the options and at least one of them bought DH units of options. In all other cases, we have O1 + O2 < 2DL and hence the total number of options demand, which is smaller than the total number of options purchased, must be smaller than 2DL. As a result, the supplier will remain unconcerned about the compensation even when he sets the capacity K ¼ 2DL. However, when both customers have purchased options and at least one bought DH units, the supplier has to worry about the possibility that both customers execute all their options and the capacity may be insufficient for satisfying the aggregate option demand. If he stays with the capacity level K ¼ 2DL, the probability of providing compensation will be as high as 2a a2 when pe is low enough. So our assumption that (2a a2)vh > c0 suggests that the supplier should increase the capacity from 2DL to avoid the situation. The assumption that a2vh < c0 suggests that the supplier cannot be better off by increasing the capacity from DH + DL to avoid the possible compensation when D ¼ ðDH ; DH Þ. The optimal capacity is between 2DL and DH + DL.

Capacity Management and Price Discrimination under Demand Uncertainty

4.3.3

207

Optimal Option Prices

Given the optimal decisions of K* and ðDo1 ; Do2 Þ, we can now analyze the first stage of the option-capacity game to derive the optimal option prices (po, pe) and the supplier’s expected profit by using the option contracts. All the decisions analyzed in Sects. 4.3.1 and 4.3.2 are contingent on both the option execution price pe and the customer’s option purchase O. In addition, the customers decide O based on their own types (t1, t2) and how the supplier prices the options. The fundamental question to be addressed is: what is the optimal po and pe that induces the customer to purchase the right number of options and maximizes the supplier’s profit? In this period, the supplier announces the option prices (po, pe). Then the customers submit their option purchase demand Oi to the supplier simultaneously. In order to detect the customer types, the supplier will price the options so that the low type customers will not purchase but the high type customers will. The customers’ valuation of the option contracts is different when the other customer’s option purchase demand, Oi, varies. If we denote the expected value of a unit of option contract as fo(Oi, Oi, pe), then the value of fo can be calculated as follows: fo ðOi ; Oi ; pe Þ ¼

1 h Eui ðOi ; Oi ; pe Þ Euhi ð0; Oi ; pe Þ Oi

where Euhi ðOi ; Oi ; pe Þ represents the expected utility a high type customer gets after purchasing Oi units of options while the other customer has purchased Oi units of options. Lemma 3. fo ðOi ; DH ; pe Þrfo ðOi ; DL ; pe Þrfo ðOi ; 0; pe Þ: Lemma 3 demonstrates an important result that is useful in proving the following Propositions. The lemma mathematically proves the fact that the options contract is more valuable to a high type customer if the other customer also buys more. This is so because the two customers will compete for the limited capacity resource in the consumption period. Buying more options protects them in the competition. Consequently, if the supplier can induce one customer to buy DH units of options, the other customer would want to pay more for the options if she is also a high type. This implies that the supplier prefers to induce the subgame equilibrium where the high type customers choose the maximal hedge strategy. Based on this conclusion, the following Proposition 4 gives the optimal option price and execution price (po, pe). Proposition 4. The supplier maximizes his expected profit when setting pe 2 vl ; H L H H H vh ðvh vl ÞD2DþD H Þ and po ¼ lfo ðD ; D Þ þ ð1 lÞfo ðD ; 0Þ. In such an equilibrium, a high type customer will choose a maximal hedging strategy by purchasing Oi ¼ DH units of options. The expected supplier profit will be EP ¼ ðvl c0 Þ2DL þ 2alðvh vl ÞðDH DL Þ þ l2 ð2a a2 Þvh c0 ðDH DL Þ

208

F. Fang and A. Whinston

and the equilibrium capacity investment K ¼

D H þ DL 2DL

if O1 ¼ O2 ¼ DH otherwise:

The supplier’s capacity investment K* is the same as the one in the full information benchmark case (see Sect. 4.2) given the belief that Oi ðt ¼ hÞ ¼ DH and Oi ðt ¼ lÞ ¼ 0. By using option contracts, the supplier can make a contingent capacity investment based on the customer’s option purchase decision O, which reveals the customer types (t1, t2) in equilibrium. This flexibility improves the capacity investment decision. The option compensation pb ¼ vh pe is important in achieving the efficient capacity level K* ¼ KFI. Under the options framework, the supplier’s incentive to increase the capacity when both customers are of high type is to satisfy as many options as possible while reducing the compensation. That is, the marginal benefit the supplier gets is pe + pb ¼ vh when K < Do. This incentive is well-aligned with the full information benchmark case where the incentive of increasing capacity is to increase the ability of serving high type demand, which yields a marginal profit pFI(t ¼ h) ¼ vh. We can rewrite the supplier’s expected profit as: EP ¼ EPND þ D1 þ D3 þ D4 where EPND is the supplier’s expected profit in the first benchmark. D1 and D3 are defined in Proposition 2. D1 represents the gain from prioritizing the high type customer’s demand over the low type one’s. D3 refers to the gain achieved when the supplier increases capacity after observing two high type customers. D4 ¼ 2al2 (vh vl)(DH DL) > 0. a(vh vl)(DH DL) is the expected loss a high type customer suffers if she doesn’t buy options but competing for capacity with another high type customer who has options. D4 is 2l2 times this expected loss representing the supplier’s expected gain from the competition between two high type customers. Comparing the supplier’s expected profit using the option contracts, EP*, to the supplier’s expected profit in the full information benchmark, EPFI, we can characterize the supplier’s profit loss when he cannot distinguish the high type customer and extract full surplus from them as follows: EPFI EP ¼ D2 D4 ¼ 2l lEuhi ð0; DH Þ þ ð1 lÞEuhi ð0; 0Þ

lEuhi ð0; DH Þ þ ð1 lÞEuhi ð0; 0Þ is the reserve utility a high type customer has when she does not buy options. This reserve utility is also known as the “Information Rent” in the price discrimination literature, (Mas-Colell et al. 1995) representing the cost the supplier pays to induce a high type customer to reveal her type. The expected information rent the supplier pays is exactly D2 D4, which increases as the probability that a customer being high type, l, increases.

Capacity Management and Price Discrimination under Demand Uncertainty

209

Proposition 5. W* ¼ WFI. The capacity investment is efficient in equilibrium and high type customers will always be served as the first priority. Hence, it is not surprising that the option contracts can achieve the same efficiency level as the full information benchmark case. Proposition 5 justifies the optimality of our proposed options contract framework.

5 Discussion and Extensions 5.1

Using Options for Price Discrimination

In the literature of price discrimination with unobservable types, the supplier’s ability to employ a non-linear pricing scheme (e.g. quantity discount and bundling) is critical to his profit. In this paper, we show that with properly priced options, the supplier can achieve the same profit level with a simple linear pricing function using an option framework. Particularly, our framework works when the customer’s demand is uncertain. In our framework, a customer faces two purchase decisions. She chooses the number of options to purchase before the demand realization and the amount of demand (including the numbers of regular demand and option demand) to request afterwards. In both decisions, she has full flexibility to choose the quantities. A linear pricing scheme is applied in all the purchases. The customers prefer such flexibility when they suffer from demand uncertainty. To discriminate the customers, we pay our attention to the case where only high type customers will buy the options. A high type customer’s total charge in equilibrium can be divided into two parts. A fixed payment poDH is charged ex ante regardless of her actual demand realization D to guarantee the priority of their demand execution. In addition, she pays a contingent payment based on her actual demand realization. Adjusting the option price po and pe, the supplier essentially changes the ratio between the ex ante and ex post payments to affect the high type customers hedging incentive and exploit their willingness to pay for the demand. The option framework helps the supplier to conduct price discrimination when the customer’s demand is uncertain. When there is no demand uncertainty (i.e., DH ¼ DL), the capacity will always be enough for the aggregate demand under our assumption. Hence, the option has no value and the supplier cannot discriminate among the customers. Our assumption of the supplier’s marginal capacity cost is critical to derive the result. The discrimination framework is built based on the high type customer’s concern of potential demand loss. If the capacity cost is small, the customers can infer that the supplier will always build enough capacity for the demand and won’t pay money ex ante to hedge the potential demand loss. The discrimination is not implementable in this case. If the capacity cost is large, the supplier will find it

210

F. Fang and A. Whinston

profitable to charge a high regular price (e.g. vh) to exploit the high type customers’ surplus only.

5.2

Multiple Agents and Multiple Types

In this paper, we use a parsimonious model with one supplier and two customers to illustrate price discrimination. The model can be extended to a case where multiple customers with two possible types. In such a model, we could still design the option contracts so that only those high type customers will buy them. The major difference would be a more complicated aggregate demand pattern. In a symmetric equilibrium where all high type customers adopt the same strategy, the supplier can separate the customers into two groups and treat them as two representative agents. The same analysis can then be applied to figure out the optimal option contract. The efficiency level of the second degree price discrimination still holds. The model gets further complicated when customers are of multiple types. In that case, the supplier could design multiple option contracts with different combinations of strike prices pe and compensation pb for each type. The customers could then self-select the options and decide when to exercise them. The demand should then be prioritized according to the supplier’s marginal punishment of not fulfilling the demand (i.e. pb pe). As a result, different demand associated with different option contracts are categorized into different priority levels. Allocation efficiency can be achieved if the customers with higher willingness to pay will always buy the option contracts with higher priority (characterized by pb pe). The challenge is how to price the options to induce the customers to purchase the right options to reveal their types.

5.3

Spot Exchange Options Market

In our setup, customers purchase options to hedge their demand risk. After their demand is realized, they can decide how many of their options to exercise. In this setup, a customer cannot buy additional options from other customer ex post. This raises a question: what if they can exchange their options after observing their demand? On the one hand, an option exchange leads to more efficient option utilization. If a customer is able to sell her extra options after the demand is realized, she is more willing to buy the options ex ante. Thus, the ex post exchange encourages the option purchase ex ante. On the other hand, the customers’ incentive for a maximal hedge decreases with the possibility of option exchange. This is because she may find someone who will sell options to her if her demand is high. Between this conflicting incentives, it is not clear which incentive is stronger in general. However, if we assume that customer types may change ex post, then the existence of an option exchange market helps in some cases. A detailed analysis

Capacity Management and Price Discrimination under Demand Uncertainty

211

of when a spot exchange market helps increase a monopolist’s profit can be found in Geng et al. (2007). Moreover, ex post exchange reduces the customer’s ex post risk of purchasing options. Hence, the existence of an ex post exchange market always improves the option purchase incentive if the customers are risk-averse.

6 Conclusion In this paper, we propose using a form of options contract framework that allows a monopolistic supplier to conduct price discrimination among customers, thereby maximizing his expected revenue under demand uncertainty and a capacity constraint. Our analysis shows that option contracts can benefit the supplier because the high type customers will pay more for hedging potential demand loss. The supplier gains the additional benefit of being able to adjust capacity according to his observation of customer types. We also analyze the strategic interactions among the supplier and customers. We show that, in equilibrium, the efficient capacity level can be induced by setting a compensation price which leaves the high type customers indifferent about whether to exercise their options or to ask for compensation. Overall efficiency is guaranteed and the supplier and the high type customers share the efficiency gain from the efficient capacity investment. Our proposed structure replicates the classical price discrimination outcome where the low type customers do not gain surplus and the high type customers enjoy an information rent. Our proposed structure can easily be adopted in situations where the supplier is not allowed to sell a bundled product with fixed quantity and situations where the actual demand and capacity is not contractible. Our framework has significant revenue management implications for various industrial applications such as network capacity management, airline ticket reservation, and telephone and electricity providers.

Appendix 1: Notation Table i Di D D Dsi Ds Dei De De

Customer index The realized demand of customer i. Di ∈ {DH, DL} D ¼ D1 + D2 the aggregate demand D ¼ ðD1 ; D2 Þ is the demand vector of both customers The demand customer i submits to the supplier Ds ¼ ðDs1 ; Ds2 Þ is the vector of customers’ submitted demand The demand of customer i, being satisfied by the supplier De ¼ De1 þ De2 is the aggregated demand satisfied by the supplier De ¼ ðDe1 ; De2 Þ is the vector of customers’ satisfied demand

212

F. Fang and A. Whinston

Doi Doi Do fo K mi Oi O p po pe pb ti ui vt a l P p

The amount of options executed by customer i when capacity is tight The amount of options executed by the customer other than i Do ¼ Do1 þ Do2 the aggregated option demand submitted by both customers The high type customer’s valuation of one unit of option contract The supplier’s capacity level The amount of monetary transfer made from customer i to the supplier The amount of option contracts customer i will purchase O ¼ ðO1 ; O2 Þ the vector of customers’ options purchase The unit price for the regular demand The option price The option execution price The option buy back price Type of customer i. ti ∈ {l,h} The utility customer i receives The marginal value of demand satisfaction for each type t The probability that a customer’s realized demand is high (Di ¼ DH) The probability that a customer is a high type one The supplier’s profit The supplier’s profit gained after period 1, excluding the sale from the option contracts The probability that regular demand is satisfied under the option framework

f

Appendix 2: Proof of Lemmas and Propositions Proof of Proposition 1. If p ⩽ vl, both customers submit all their demand to the supplier, Dsi

¼ Di ¼

DH DL

with prob ¼ a with prob ¼ 1 a:

The supplier’s expected profit: h i EP ¼ p a2 minfK;2DH gþ2að1aÞminfK;DH þDL Þþð1aÞ2 minfK;2DL g c0 K: Maximizing the profit under the condition p ⩽ vl, we have p* ¼ vl and K* ¼ 2DL. The supplier’s expected profit is EP(p ¼ vl) ¼ (vl c0)2DL. Customer i’s expected utility is ui ¼ (vi vl)DL. If p ∈ (vl, vh), only high type customers will submit the demand. Therefore, Dsi ¼ Di when ti ¼ h and Dsi ¼ 0 for ti ¼ l. When vl < p ⩽ vh, the supplier’s expected profit

Capacity Management and Price Discrimination under Demand Uncertainty

213

h i EPðpÞ ¼ l2 p a2 minfK;2DH gþ2að1agminfK;DH þDL gþð1aÞ2 minfK;2DL g

þ2lð1lÞp aminfK;DH gþð1aÞminfK;DL g c0 K qﬃﬃﬃﬃﬃﬃﬃﬃﬃ h 0 Maximizing the expected profit and applying the assumption l < 1 v vc h , we * h * * h have p ¼ v and K ¼ 0. Thereby, EP (p ¼ v ) ¼ 0 and it is not worthwhile to build capacity and serve high type customers only, due to the low probability of a high type customer’s existence. Compare the two cases, we conclude that the supplier’s best strategy is to set pND ¼ vl and serve both types of customers. The optimal capacity will be KND ¼ 2DL. The expected profit EPND ¼ (vl c0)2DL and the overall efficiency: ND W ND ¼ EPND þ lEuND i ðti ¼ hÞ þ ð1 lÞEui ðti ¼ lÞ

¼ lvh þ ð1 lÞvl c0 2DL :

□ FI

Proof of Lemma 1. The optimal capacity K is made contingent on the customer types T ¼ ðt1 ; t2 Þ. When T ¼ ðh; hÞ, the supplier’s expected profit h EPFI ðK;TÞ¼vh a2 minfK;2DH gþ2að1aÞminfK;DH þDL g i þð1aÞ2 minfK;2DL g c0 K which is maximized when K(h,h) ¼ DH + DL due to the assumption avh < c0 < (2a a2)vh. When T ¼ ðl; lÞ, similarly, we can have KFI(l,l) ¼ 2DL, since (2a a2) l v < c 0 < v l. When T ¼ ðh; lÞ or (l,h) and K > DH, the supplier’s expected profit EPFI ðK; ðh; lÞÞ ¼ a2 vh DH þ vl minfK DH ; DH g þ að1 aÞ vh DH þ vl minfK DH ; DL gÞ þ að1 aÞ vh DL þ vl minfK DL ; DH g þ ð1 aÞ2 vh DL þ vl minfK DL ; DL g c0 K which is maximized when K* ¼ 2DL. It can also be shown that K ⩽ DH cannot be optimal. Therefore, KFI(h,l) ¼ KFI(l,h) ¼ 2DL. □ Proof of Proposition 2. For the supplier, the probability that both customers are high types is l2. The expected profit EPðh; hÞ ¼ vl c0 2DL þ vh vl 2DL þ ð2a a2 Þvh c0 DH DL :

214

F. Fang and A. Whinston

With probability 2l(1 l), one customer is of high type and the other is of low type. The expected profit EP(h, l) ¼ (vl c0)2DL + (vh vl) (aDH + (1 a)DL). With probability (1 l)2, both customers are of low type. The expected profit EP (l,l) ¼ (vl c0)2DL. Since ui(t ¼ h) ¼ ui(t ¼ l) ¼ 0, W FI ¼ EPFI ¼ l2 EPðh; hÞ þ 2lð1 lÞEPðh; lÞ þ ð1 lÞ2 EPðl; lÞ ¼ ðvl c0 Þ2DL þ vh vl 2alð1 lÞ DH DL þ vh vl l2DL þ l2 ð2a a2 Þvh c0 DH DL compared to EPND ¼ (vl c0)2DL and WND ¼ (lvh + (1 l)vl c0)2DL, we can easily conclude that EPFI ¼ EPND þ D1 þ D2 þ D3 W FI ¼ W ND þ D1 þ D3 □ Proof of Lemma 2. The proof is straightforward since we can show that the second order condition of the above objective function is non-negative. It means that the objective function is a convex function. The optimal solution of the maximization □ problem would exist on the boundary. That is, it is either 0 or min{Oi, Di}. Proof of Proposition 3. In this stage, the supplier determines the capacity level to maximize his future revenue less the capacity investment. That is, pðO; D; K; pe Þ ¼ m1 þ m2 po ðO1 þ O2 Þ c0 K ¼ pe Do ðO; D; K; pe Þ þ vl ðminðK; DÞ Do ðO; D; K; pe ÞÞþ vh ðDo ðO; D; K; pe Þ KÞþ c0 K Applying the outcomes from Tables 1–6, we can derive the profit function with parameters O; D; K, and pe. Taking expectation over the realized demand D, we obtain the expected profit function for each O; pe , and K. Maximizing those the expected profit function by choosing K, we can obtain the optimal capacity as stated □ in Proposition 3 as a function of O and option strike price pe. Proof of Lemma 3. From the result of 1. When pe rvh ðvh vl ÞDDH , optimal capacity K* ¼ 2DL for all the possible configurations of O. No options will be exercised for all possible realization of D. Therefore, the option has no value. In other words, fo ðOi ; DH ; pe Þ ¼ fo ðOi ; DL ; pe Þ ¼ fo ðOi ; 0; pe Þ ¼ 0 H h h l DL 2. When vh ðvh vl Þ2DD H DL bpe < v ðv v ÞDH , if the customer has bought Oi ¼ DL, she will never exercise the options no matter what type the other customer is. Therefore, at the first stage, the value of the options contract would be 0 and the customer should not purchase any options with a positive price L

Capacity Management and Price Discrimination under Demand Uncertainty

215

3. When vh ðvh vl ÞDH2DþDL bpe < vh ðvh vl Þ2DD H DL : if the customer has bought Oi ¼ DL units of options, she will only exercise it when D ¼ ðDH ; DH Þ and Doi 6¼ DH . Hence if the other customer is of high type, she is always better off if the other customer has purchased Oi ¼ DH rather than Oi ¼ DL units of options. However, given Oi ¼ DH, customer i will never exercise her options and the value of the options is 0. If the other customer is of low type, the customer’s expected utility from exercising the options are L

H

(a) Oi ¼ DH : ui ðDH Þ ¼ a2 ðvh pe ÞDH þ ð1 a2 ÞDL po DH L H 2 L L (b) Oi ¼ DL : ui ðDL Þ ¼ a2 ðvh pe ÞDL þ ðvh vl Þ2DDH D DL þ ð1 a ÞD po D h l L (c) Oi ¼ 0 : ui ð0Þ ¼ ðv v ÞD Þui ð0Þþpo D D From the results above, we can show that uuiiðD ðDH Þui ð0Þþpo DL > DL , meaning that the customer is better off by purchasing Oi ¼ DH units of options. Following the same calculation, we can show that DL is not the optimal choice when pe > vl. □ H

H

H

Proof of Proposition 4. We need to discuss the profit according to the different pe segments: 1. When pe r vh ðvh vl Þ DDH , no options will be exercised, po ¼ 0 and EP ¼ (vl c0)2DL H L h h l DL 2. When vh ðvh vl ÞD2DþD we have K ðDH ; DH Þ ¼ H bpe < v ðv v ÞDH , vh pe 2DH from Proposition 3. The supplier’s profit vh vl L

EP ¼ Ep þ 2lpo DH

vh pe H 2 2 l 2 l L ¼ l ð2a a Þv c0 h 2D þ ð1 a Þv 2D v vl

þ 2lð1 lÞ a2 ðpe vl ÞDH þ ðvl c0 Þ2DL þ ð1 lÞ2 ðvl c0 Þ2DL

þ 2l Euhi ðDH Þ Euhi ð0Þ which we can show dEP dpe < 0. H L * H H 3. When vl bpe < vh ðvh vl ÞD2DþD H , we have optimal capacity K (D ,D ) ¼ H L * H * L D + D and K (D ,0) ¼ K (0,0) ¼ 2D from Proposition 3. One can get EP ¼ ðvl c0 Þ2DL þ 2alðvh vl ÞðDH DL Þ þ l2 ð2a a2 Þvh c0 ðDH DL Þ: In this case, dEP dpe ¼ 0. In summary, we can conclude that the optimal option exercise price pe should be H L □ vh ðvh vl ÞD2DþD H Proof of Proposition 5. The proof is straightforward from the proof of Proposition 4. □

216

F. Fang and A. Whinston

References Afeche P (2006) Incentive-compatible revenue management in queueing systems: optimal strategic delay and other delaying Tactics. Working paper, The Kellogg School of Management, Northwestern University, Evanston, IL Armstrong M, Vickers J (2001) Competitive price discrimination. Rand J Econ 32:1–27 Barnes-Schuster D, Bassok Y, Anupindi R (2002) Coordination and flexibility in supply contracts with options. Manuf Serv Options Manage 43:171–207 Birge JR (2000) Option methods for incorporating risk into linear capacity planning models. Manuf Serv Operation Manage 2:19–31 Boyaci T, Ray S (2006) The impact of capacity costs on product differentiation in delivery time, delivery reliability, and prices. Prod Oper Manage 15:179–198 Cachon GP, Fisher M (2000) Supply chain inventory management and the value of shared information. Manage Sci 46:1032–1048 Cachon GP, Lariviere MA (1999) An equilibrium analysis of linear, proportional and uniform allocation of scarce capacity. IIE Trans 31:835–849 Cachon GP, Lariviere MA (2001) Contracting to assure supply: how to share demand forecasts in a supply chain. Manage Sci 47:629–646 Cachon GP (2002) Supply coordination with contracts. In: Graves S, Kok T (eds) Handbooks in operations research and management science. North-Holland, Amsterdam Chen F (2003) Information sharing and supply chain coordination. In: Graves SC, De Kok AG (eds) Handbooks in operations research and management science: supply chain management design, coordination and operation. Elsevier, Amsterdam Corsetti G, Dedola L (2005) A macroeconomic model of international price discrimination. J Int Econ 67:129–155 Deshpande V, Schwartz L (2002) Optimal capacity choice and allocation in decentralized supply chains. Technical report, Krannert School of Management, Purdue University, West Lafayette, IN Geng X, Wu R, Whinston AB (2007) Profiting from partial allowance of ticket resale. J Mark 71:184–195 Guo Z, Fang F, Whinston AB (2006) Supply chain information sharing in a macro prediction market. Decis Support Syst 42:1944–1958 Gupta A, Stahl DO, Whinston AB (1996) An economic approach to network computing with priority classes. J Organ Comput Electron Commer 6:71–95 Gupta A, Stahl DO, Whinston AB (1999) The economics of network management. Commun ACM 42:57–63 Huchzermeier A, Cohen MA (1996) Valuing operational flexibility under exchange rate risk. Oper Res 44:100–113 Iyer AV, Deshpande V, Wu Z (2003) A postponement model for demand management. Manage Sci 49:983–1002 Kleindorfer PR, Wu DJ (2003) Integrating long-and-short term contracting via business-to-business exchanges for capacity intensive industries. Manage Sci 49:1597–1615 Lariviere M (1999) Supply chain contracting and coordination with stochastic demand. In: Tayur S, Ganeshan R, Magazine M (eds) Quantitative models for supply chain management. Kluwer, Boston Li L (2002) Information sharing in a supply chain with horizontal competition. Manage Sci 48:1196–1212 Lin L, Kulatilaka N (2007) Strategic Growth Options in Network Industries. In: Silverman B (eds) Advances in Strategic Management. Emerald Group Publishing Limited., pp. 177–198 Martı´nez-de-Albe´niz V, Simchi-Levi D (2005) A portfolio approach to procurement contracts. Prod Options Manage 14:90–114 Mas-Colell A, Whinston MD, Green JR (1995) Microeconomic theory. Oxford University Press, USA

Capacity Management and Price Discrimination under Demand Uncertainty

217

Maskin E, Riley J (1984) Monopoly under incomplete information. Rand J Econ 15:171–196 Mortimer JH (2007) Price discrimination, copyright law, and technological innovation: evidence from the introduction of DVDs. Q J Econ 122:1307–1350 Noldeke G, Schmidt KM (1995) Option contracts and renegotiation: a solution to the hold-up problem. Rand J Econ 26:163–179 ¨ zer O, Wei W (2004) Inventory control with limited capacity and advance demand information. O Oper Res 52:988–1000 Sethi SP, Yan H, Zhang H (2004) Quantity flexible contracts in supply chains with information updates and service-level constraints. Decis Sci 35:691–712 Sodhi M (2004) Managing demand risk in tactical supply chain planning. Prod Oper Manage 14(1):69–79 Tirole J (1988) The theory of industrial organization. MIT, Cambridge, MA Varian HR (1985) Price discrimination and social welfare. Am Econ Rev 75:870–875

.

Dynamic Procurement, Quantity Discounts, and Supply Chain Efficiency Feryal Erhun, Pinar Keskinocak, and Sridhar Tayur

Abstract We study a model with a single supplier and a single buyer who interact multiple times before the buyer sells her product in the end-consumer market. We show that when the supplier uses a wholesale price contract, even under perfect foresight, the supplier, the buyer, and the end-consumers benefit from multiple trading opportunities versus a one-shot procurement agreement. Keywords Advance capacity procurement • Incremental discounts • Strategic interactions • Supply chain coordination

quantity

1 Introduction This chapter studies the benefits of trading more than once while procuring/selling capacity. Consider a simple model with one supplier and one buyer. The buyer produces a product by using the capacity she buys from an uncapacitated supplier. Before the buyer’s selling season begins, there are N periods in which the buyer can procure capacity. The supplier uses a simple wholesale price contract; he

A significant part of the materials in this invited chapter is from the following original article: Erhun F, Keskinocak P, Tayur S (2008) Dynamic procurement, quantity discounts, and supply chain efficiency. Prod Oper Manage 17(5):1–8. F. Erhun (*) Department of Management Science and Engineering, Stanford University, Stanford, CA, USA e-mail: [email protected]edu P. Keskinocak School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, GA, USA e-mail: [email protected] S. Tayur Tepper School of Business, Carnegie Mellon University, Pittsburgh, PA, USA e-mail: [email protected] T.-M. Choi and T.C. Edwin Cheng (eds.), Supply Chain Coordination under Uncertainty, International Handbooks on Information Systems, DOI 10.1007/978-3-642-19257-9_9, # Springer-Verlag Berlin Heidelberg 2011

219

220

F. Erhun et al.

charges a unit capacity price of wn in period n ¼ 1; . . . ; N, which he determines dynamically. Once the buyer procures the capacity, she can produce the product with no additional cost and sell it in an end-consumer market, where the market price of the product is determined by the market clearance assumption. Dynamic procurement, i.e., simple wholesale price contracts repeated over time (possibly with different prices), is a commonly observed practice in a vertical channel. The typical justification for multiple procurement trades is risk hedging. In order to manage the demand risk, a buyer may prefer to procure capacity dynamically over time after receiving some update on demand. Other commonly observed reasons for dynamic procurement include spreading payments over a period of time, minimizing potential capacity risks (supplier’s or buyer’s), supplier’s decreasing cost over time which may translate to lower prices (e.g., as in the electronics industry), and forward buying. We discuss yet another potential impact of dynamic procurement, i.e., as a tool to influence future prices. In a vertical setting, we show that risk hedging is not the only justification for multiple trades. We derive a Pareto improving rationale for the use of additional trading periods in the case of deterministic demand (i.e., when the commonly known and intuitive benefits due to risk hedging are not present) and wholesale price contracts, where all participants (supplier, buyer, and end-consumers) benefit. The additional trading periods inherently create the equivalent of a non-linear pricing scheme, which makes the performance of the decentralized supply chain approach that of a centralized supply chain when the number of trading periods is sufficiently high.

2 Literature Review The paper that is closest to our work is by Allaz and Vila (1993). The authors study a deterministic model where two Cournot duopolists trade in forward markets for delivery in a single period. The authors conclude that even though producers are worse off by forward trading, in equilibrium they will trade forward. In the limit, as the number of forward markets goes to infinity, the competitive outcome is achieved in a duopoly setting. In our model, we look at vertical interactions, as opposed to the horizontal competition of Allaz and Vila. In a setting similar to ours, Anand et al. (2008) study a dynamic model of a procurement contract between a supplier and a buyer in a two-period, uncapacitated, deterministic demand game. The authors eliminate all the classical reasons for inventories, yet show that the buyer’s optimal strategy in equilibrium is to carry inventories, and the supplier is unable to prevent this. The inventories arise for “strategic” reasons. Keskinocak et al. (2003, 2008) extend Anand et al.’s model to limited capacity and limited capacity with backordering, respectively. Research on quantity discounts also relates to our problem. We refer readers to Benton and Park (1996) and Munson and Rosenblatt (1998) for extensive reviews, and to Dolan (1987) for a detailed survey of different variants of the quantity

Dynamic Procurement, Quantity Discounts, and Supply Chain Efficiency

221

discount problem from a marketing research standpoint. Jeuland and Shugan (1983) show that profit sharing mechanisms with quantity discounts can coordinate the supply chain. Following their work, many researchers study the role of quantity discounts as a channel coordination mechanism under different settings; e.g., Weng (1995) and Chen et al. (2001) combine channel coordination with price-sensitive demand and operating cost, Ingene and Parry (1995) introduce competing retailers, Raju and Zhang (2005) study channel coordination with a dominant retailer, and Chen and Roma (2010) consider a single manufacturer offering quantity discounts to competing retailers. Another stream of literature study quantity discounts to improve operational efficiency (Crowther 1964; Monahan 1984; Dada and Srikanth 1987). The dynamic procurement model that we study in this chapter falls in to this category by effectively creating an incremental quantity discount mechanism. Unlike the papers in the literature, the terms of trade are set by both the supplier and the buyer. Beside these two streams of literature that is closely related to our problem, there are three other streams that are related in spirit (1) timing of purchase commitments (2) two-period procurement and risk allocation, and (3) multi-period price and capacity adjustment. These streams consider multiple procurement opportunities in settings where the buyer can accrue additional demand information by postponing his procurement decision. The timing of purchase commitments has been the subject of many studies in the operations management literature. Iyer and Bergen (1997) study a supply chain with a single supplier and a single buyer to compare a traditional system, where the buyer places her order early on, to Quick Response (QR), where the buyer collects demand information before she places her order. The authors assume that the buyer pays the same wholesale price in either case and show that QR is not always Pareto improving. However, quantity discounts and volume commitments across products make QR Pareto improving. Ferguson (2003) and Ferguson et al. (2005) investigate an end-product buyer’s choice of when to commit to an order quantity when there is a demand information update during the supplier’s leadtime. The former paper assumes either all or none of the demand uncertainty is resolved, while the latter relaxes this assumption. The authors find that the buyer is not always better off delaying her quantity commitment and the supplier may prefer delayed commitment depending upon the amount of demand uncertainty resolved during the information update. Taylor (2006) studies a problem similar to the one in Ferguson (2003), however, he considers the sale timing of a supplier. The supplier may sell either early, i.e., well in advance of the selling season, or late, i.e., close to the selling season. Taylor shows that, in considerable generality, the supplier’s profit is greater when he sells late. In a duopolistic environment, Spencer and Brander (1992) identify conditions on demand variability under which the buyers would prefer to postpone their quantity decisions. Cvsa and Gilbert (2002) introduce a supplier to Spencer and Brander’s model and investigate how the supplier can influence the form of competition in the downstream market by offering a precommitment opportunity. In all of these papers, the buyer is limited to one mode of commitment (i.e., early or delayed). However, the buyer may prefer to use both

222

F. Erhun et al.

modes of commitment, which is the subject of the literature on two-period procurement and risk allocation. The literature on two-period procurement and risk allocation allows the buyer two purchasing opportunities: One is before and the other one is after demand realization. Gurnani and Tang (1999) analyze the trade-off between a more accurate demand information and a potentially higher procurement cost at the second period. Donohue (2000) studies a buy-back contract in a two-period setting where the procurement cost at the second period is higher and shows that this contract can coordinate the system. Cachon (2004) and Dong and Zhu (2007) study push (early commitment), pull (delayed commitment), and advance-purchase discount (purchase at a discounted price before the season and at a regular price during the selling season) to study how the inventory ownership impacts supply chain efficiency. Guo et al. (2009) study a three-tier supply chain with two outsourcing structures (delegation and control) to investigate how an OEM can use a two-wholesaleprice contract to increase the available upstream capacity. Erhun et al. (2008) and Li and Scheller-Wolf (2010) extend this literature to a vertical setting by considering the supplier’s pricing decisions as well. Erhun et al. study a capacitated two-tier supply chain and assume that the wholesale price is set by the supplier and the procurement quantity by the buyer. The authors investigate the impact of timing of the decisions and of additional demand information on the supplier’s pricing and the buyer’s procurement decisions. Li and Scheller-Wolf (2010) consider a supply chain composed of a buyer and multi-suppliers with private cost information. The buyer first offers a push or pull contract, and then selects the supplier through a wholesale price auction. The authors numerically find that a push system is more preferable by the buyer if the suppliers’ number is large and the demand level is high, while a pull system is more preferable if demand has high uncertainty and the suppliers’ cost is large. The literature on multi-period price and capacity adjustment seeks answers for when and how much to adjust the price or capacity or both in a dynamically changing environment. In particular, Burnetas and Gilbert (2001) consider a multi-period newsvendor model to study the trade-off between a more accurate demand information and increasing procurement costs. The authors numerically demonstrate that the broker tends to cluster his procurements just before price increases. Elmaghraby and Keskinocak (2003) and Van Mieghem (2003) provide a literature review on dynamic pricing and capacity investment and adjustment issues, respectively.

3 Main Model We study a model where there are N possible periods for capacity procurement before the buyer’s production/selling season begins. The supplier and the buyer maximize their profits. The supplier’s decisions are the wholesale prices for each period, wn, ðn ¼ 1; . . . ; N Þ. The buyer’s decisions are the procurement quantities

Dynamic Procurement, Quantity Discounts, and Supply Chain Efficiency

223

for each period, qn, ðn ¼ 1; . . . ; N Þ, and the production quantity, QN. (We set Q0 ¼ 0 and q0 ¼ 0.) The market is characterized by a linear inverse demand function P(QN) ¼ a bQN, where a is the market potential, b is the price sensitivity, P(QN) is the per-unit market price of the product for QN. We assume that the buyer’s unit cost of production is zero. However, the analysis of a positive, constant production cost (c < a) case is trivial. Since (a bQ)Q cQ ¼ ((a c) bQ)Q, we can simply modify the demand intercept a such that a^ ¼ a c, and the analysis follows. The sequence of events in each period n of the N-period gameis as follows (1) Given previous capacity procurements, qj ; j ¼ 1; . . . ; n 1 , the supplier price wn. (2) Given previous capacity procurements determines the capacity qj ; j ¼ 1; . . . ; n 1 and the current capacity price (wn), the buyer determines her procurement quantity qn. (3) In the last period N, the buyer chooses her production quantity QN and procures extra capacity, if necessary. The market clears only once at the end of the N-th period; i.e., there is only a single selling opportunity to end-consumers. We use backward induction and obtain the pure-strategy Subgame Perfect Nash Equilibrium (SPNE). Proposition 1 characterizes this equilibrium. Proposition 1. The unique pure-strategy SPNE for the N-period dynamic procurement model is the following. For n ¼ 1; . . . ; N 1, let n ¼ N n. Then, a a 2 nþ1 2 n 2 q1 ¼ qn ; wnþ1 ¼ wn ; ; w1 ¼ KN N ; qnþ1 ¼ 4Nb 2 2 n 2 nþ1 QN1 2kþ1 where K1 ¼ 1 and KN ¼ k¼1 The production 2kþ2. PN a a a . QN ¼ n¼1 qn ¼ 2b 4bKN . As N tends to infinity, QN tends to 2b

quantity

is

From Proposition 1, the production quantity for the N-period model is a a QN ¼ 2b 4b KN . As N increases, KN decreases, and the production quantity increases. Therefore, the double marginalization (DM) effect decreases, the efficiency increases and approaches that of the centralized solution. Similar to the argument of Allaz and Vila, a higher N does not necessarily imply that the capacity procurement is over a longer horizon, but rather that procurement occurs more frequently. Even though our model does not include a discount factor due to our interpretation of these N periods, the results of the main model are not sensitive to a discount factor. When we incorporate a discount factor d 1 to our analysis, ^n ¼ dNn wn , n ¼ 1; . . . ; N: The quantities maintain wholesale prices become w their original values. Figure 1 summarizes the prices and quantities in different periods a for the SPNE. 3a 5a The last period’s capacity price decreases & & & & 0 , while the first 2 8 16 9a 75a % % as N increases. The total period capacity price increases a2 % a 16 5a128 11a a production quantity, QN, increases 4b % 16b % 32b % % 2b ; and the market of the product (following the relationship P(QN) ¼ a bQN) decreases price 3a 11a 21a a . Even though the quantity in the first period decreases & & & & 4 16 32 2

224

F. Erhun et al.

n=1

1

a 2

2

9a 16

←⎯ ⎯⎯

3a 8

3

75a 128

←⎯ ⎯ ⎯ 5

15a 32

←⎯ ⎯ ⎯ 3

5a 16

4

1225a 2048

⎯ ←⎯ ⎯ 7

525a 1024

←⎯ ⎯⎯ ⎛ 5⎞

105a 256

…

…

Prices

N

n=2

n=4 …

n=3

n=1

Quantities n=3

n=2

n=4 …

a 8b

⎯⎯ → ⎯ 3

3a 16b

a 12b

⎯⎯ ⎯ → 5

5a ⎯⎯ ⎯→ ⎛×3 ⎞ ⎜ ⎟ 48b ⎝ 2⎠

5a 32b

a 16b

⎯⎯ ⎯ → 7

7a 96b

⎯⎯ ⎯ → 5

35a 384b

…

…

…

…

⎛× ⎞ ⎜ ⎟ ⎝ 6⎠

⎛× ⎞ ⎜ ⎟ ⎝ 4⎠

11a 32b ⎯⎯ ⎯ → 3 ⎛× ⎞ ⎜ ⎟ ⎝ 2⎠

35a 256

93a 256b …

⎛× ⎞ ⎜ ⎟ ⎝ 4⎠

5a 16b

…

⎛× ⎞ ⎜ ⎟ ⎝ 2⎠

…

35a 128

⎜× ⎟ ⎝ 2⎠

a 4b

…

←⎯ ⎯⎯ ⎛ 3⎞

…

⎜× ⎟ ⎝ 4⎠

…

⎛× ⎞ ⎜ ⎟ ⎝ 6⎠

⎛× ⎞ ⎜ ⎟ ⎝ 2⎠

…

⎛× ⎞ ⎜ ⎟ ⎝ 4⎠

…

⎛×3 ⎞ ⎜ ⎟ ⎝ 2⎠

…

a 4b

QN

Fig. 1 Capacity prices and quantities for different N values under dynamic procurement

100 90 80

% of Profits

70 60 50 40 30 20 10 0

1

3

DM Loss 25.00 9.77

5 6.06

7 4.39

9

11

13

15

17

19

3.44

2.83

2.40

2.09

1.85

1.65

Buyer

25.00 31.64 33.38 34.18 34.64 34.94 35.15 35.30 35.42 35.52

Supplier

50.00 58.59 60.56 61.43 61.92 62.23 62.45 62.61 62.73 62.83

Number of Periods, N

Fig. 2 The distribution of profits between the supplier, the buyer, and the double marginalization effect versus the number of periods (N)

a as N increases q1 ¼ 4Nb , the buyer procures capacity in each period. That is, trading occurs in all N periods. The buyer is willing to procure capacity at a given period (at a higher price) because she knows that by doing so the best response for the supplier is to lower the price in the subsequent periods. For a fixed N, the quantity that the buyer procures in each period increases and the capacity price decreases over time. As N increases, the double marginalization effect decreases and the supplier’s and the buyer’s profits both increase (see Fig. 2). Dynamic procurement not only increases the supply chain efficiency, but also naturally allocates the surplus to the supplier and the buyer such that both parties benefit. Independent of the values of a and b, the supplier’s profit converges to approximately 64% of the total profits, and the buyer’s profit converges to

Dynamic Procurement, Quantity Discounts, and Supply Chain Efficiency

225

approximately 36% of the total profits in the presence of additional capacity procurement periods. Even for small values of N, dynamic procurement decreases the inefficiency considerably. For example, for N ¼ 3, the inefficiency is already less than 10% (compared to 25% for N ¼ 1). In our analysis, we assume that N is exogenously determined, i.e., it is an input to the game. We show that as N increases, so do the profits of both players. Hence, the buyer and the supplier can jointly decide on an appropriate N value a priori, based on the marginal benefit of each additional trading period and the possible cost of each trade.

4 Extensions Our main result has two parts (1) as N increases, all participants (supplier, buyer, and end-consumers) strictly benefit and (2) as N goes to infinity, the performance of the decentralized supply chain approaches that of the centralized supply chain. However, we made several simplifying assumptions in our model. Hence we next discuss the implications of these assumptions on our main result and how they can be relaxed.

4.1

Limited Capacity

In this section, we consider the case where the supplier has a capacity of K units that he can sell throughout N periods. Let QN correspond to the total production quantity of an N-period uncapacitated game. For the N-period limited capacity a case, when the capacity is “tight,” i.e., CS b 4b , or when the capacity is “abundant”, i.e., CS ⩾ QN, the results are straightforward and intuitive. In the first case, as the capacity is tight, the supplier does not change his price through the game, so the N-period game is equivalent to a single-period game. When CS ⩾ QN, the problem is equivalent to an unlimited capacity game (Proposition 1). What happens in between these extremes is more interesting. Our main result is as follows. Proposition 2. The SPNE for the N-period capacitated model for CS < QN is as follows: Let N 2 f1; 2; ; Ng be such that QN 1

N X

qj ;

qN ¼

j¼NN þ2

wNN þ1 ¼

NY 1

i¼1

2i þ 1 wN ; 2i

wN ¼ 2

a 2b

a

CS ;

2 bCS ;

qNi ¼

2i qNiþ1 ; 2i þ 1

wNi ¼

2i þ 1 wNiþ1 : 2i

226

F. Erhun et al.

When CS < QN, depending on the capacity, the supplier and the buyer play an N*-period game (N* ⩽ N), the supplier sells all the capacity, and the supply chain is coordinated. Here, period N* corresponds to the smallest n (n 2 f1; 2; . . . ; Ng) for which the supplier’s capacity CS will not be enough for an n-period uncapacitated game. Contrary to the unlimited capacity case, increasing the number of trading periods beyond N* does not increase the profits of either player. The total profits for the buyer and the supplier can be maximized in a finite number of trading periods, N*, which depends on the capacity of the supplier.

4.2

Alternative Price-Sensitive Demand Functions

For most of the insights, we believe that the linear inverse demand function is not a critical assumption. When demand is price-sensitive, i.e., can be modeled by a downward-sloping demand curve, the intuition is as follows: When demand is price-sensitive, the purchasing power of each additional end-consumer is lower. Hence, if the buyer can reduce her marginal cost for each additional unit, she can profitably sell to a larger set of end-consumers. Dynamic procurement allows the supplier to charge lower prices as the number of units purchased by the buyer increases (as in the case of an incremental quantity discount), and hence, it allows the buyer to profitably sell to those end-consumers with lower valuations. Clearly, under a different demand curve, the split of profits between the buyer and the supplier will be different, but dynamic procurement will continue to occur as long as demand is price-sensitive. In order to verify our intuition, we study a multiplicative demand function where D(P) ¼ aP2 and an exponential demand function where D(P) ¼ a exp(P) for a two-period model. (For the multiplicative model we can no longer assume that the production cost of the buyer is zero, hence, for both cases we assume that the unit production cost is c.) Both multiplicative and exponential demand functions lead to similar dynamics to the linear inverse demand function. Therefore, even though we cannot verify the second part of our main result, we continue to show that all participants (supplier, buyer, and endconsumers) strictly benefit as N increases.

4.3

Newsvendor Setting

To test the robustness of our results to price-sensitive demand assumption, we study a setting where a newsvendor procures capacity N times before the demand is realized (we assume that there is no forecast update between periods). We analyze the case where demand is uniformly distributed between 0 and u, the market price is fixed at p per unit, and the buyer’s production cost is c < p per unit. Proposition 3 shows that dynamic procurement continues to increase the quantity that the buyer procures as the supplier is willing to decrease the wholesale price over time.

Dynamic Procurement, Quantity Discounts, and Supply Chain Efficiency

227

Proposition 3 (Newsvendor setting with uniform demand). The unique purestrategy SPNE for the N-period dynamic procurement model is the following. For n ¼ 1; . . . ; N 1, let n ¼ N n. Then, p c 2 nþ1 2 n u p c 2 ; w1 ¼ KN N qn ; wnþ1 ¼ wn ; ; qnþ1 ¼ q1 ¼ p 2 2 n 2 nþ1 2N QN1 2kþ1 The production quantity is where K1 ¼ 1 and KN ¼ k¼1 2kþ2. pc PN pc KN p . QN ¼ n¼1 qn ¼ u 1 2 p . As N tends to infinity, QN tends to u Furthermore, as N goes to infinity, the performance of the decentralized supply chain approaches that of the centralized supply chain. Therefore, our main result continues to hold in a newsvendor-setting with uniform demand distribution. Finally, the uniform distribution leads to the same profit split as in the linear inverse demand case.

4.4

Effort-Dependent Demand

In order to understand the impact of dynamic procurement in case of effort-dependent demand, we study an N-period model where demand is a function of the sales effort in addition to price, i.e., D(P, e) ¼ a P þ e (cost of the effort is e2). Proposition 4 shows that dynamic procurement continues to benefit both parties. Proposition 4 (Effort-dependent demand). The unique pure-strategy SPNE for the N-period dynamic procurement model is the following. For n ¼ 1; . . . ; N 1, let n ¼ N n. Then, a a 2 nþ1 2 n QN ; qn ; wnþ1 ¼ wn ; e ¼ q1 ¼ ; w1 ¼ KN2 N ; qnþ1 ¼ 2 3N 2 2 n 2 nþ1 QN1 2kþ1 The production quantity is where K1 ¼ 1 and KN ¼ k¼1 2kþ2. P a QN ¼ Nn¼1 qn ¼ 2a K . As N tends to infinity, QN tends to 2a 3 3 N 3. Furthermore, as N tends to infinity, both the effort level and the production quantity tend to those of the centralized supply chain. Due to dynamic procurement, the buyer can afford to invest more in the sales effort (as her average procurement cost decreases) and decrease the price of the product; therefore, demand for the product increases. The division of the profits, as well as the improvement of the system performance, mimic those of the setting with linear inverse demand function.

4.5

Information Asymmetry

In practice, the buyer is closer to the end-consumer market and may have more information about the demand compared to the supplier. Hence, we consider

228

F. Erhun et al.

1.5 Increase in the supplier’s 1 expected 0.5 profit 0 0

1 0.8 0.6 0.4

0.2 0.4

α

al /ah

0.2

0.6 0.8 1

0

Fig. 3 The increase in the supplier’s expected profits due to the additional period (ah ¼ 10)

information asymmetry between the supplier and the buyer regarding the market potential. According to the supplier, the market potential can either be “high”, ah, with a probability of a, or “low”, al, with a probability of 1 a. The buyer, on the other hand, knows the exact value of the market potential. (For simplicity, we assume that bi ¼ 1, i ¼ l, h.) Under asymmetric information, depending on the relative values of al and ah, the buyer and the supplier may engage in dynamic procurement. However, despite improving the system performance for a large range of parameters, the additional trading period is not always beneficial, i.e., the result in part (i) can be reversed under asymmetric information. Figure 3 illustrates these facts by plotting the increase in the supplier’s expected profits due to the additional period for different values of al and a. When al is relatively low compared to ah, the supplier does not benefit from the additional period. For higher values of al, there are values of a for which dynamic procurement improves the system performance. Hence, under asymmetric information the additional trading period may continue to enable dynamic procurement depending on the relative values of al and ah. However, our main result may also be reversed under asymmetric information.

4.6

Competition

In this section, we study the impact of dynamic procurement under competition. In this model, we restrict ourselves to two periods and study two competing buyers. Using the capacity they procure from the supplier, the buyers produce similar products, which they sell to end-consumers at the end of the second period. Before production takes place, the buyers can procure capacity in both periods. Firms simultaneously choose their procurement quantities in the first period; these become common knowledge and then firms simultaneously decide how much

Dynamic Procurement, Quantity Discounts, and Supply Chain Efficiency

229

additional capacity to procure in the second period before the market clears. Erhun (2007) studies this setting in detail (when there are more than two periods, more than two buyers, etc.) We present one of their results below to show how dynamic procurement extends to the situations with competition. The following proposition presents the unique SPNE for the dynamic procurement model under competition. Proposition 5. The unique SPNE for the dynamic procurement model is symmetric and is as follows. 9a (i) The wholesale prices for first and second periods are w1 ¼ 31a 60 and w2 ¼ 20, respectively. a . (ii) The buyers’ first period procurement quantity is: q1;1 ¼ q1;2 ¼ 30b 3a . (iii) The buyers’ second period procurement quantity is: q2;1 ¼ q2;2 ¼ 20b

Dynamic procurement continues to increase supply chain efficiency under competition. When N ¼ 1, the supply chain inefficiency is around 11%. With dynamic procurement, this inefficiency decreases to 7% when N ¼ 2. Furthermore, dynamic procurement naturally allocates the surplus to the supplier and the buyers such that they all benefit.

4.7

Demand Uncertainty with Information Update

Even though our goal in this chapter is to characterize a potential impact of dynamic procurement other than mitigating demand risk, as our final extension, we also discuss the dynamics of dynamic procurement when mitigating demand risk is an option. We consider a two-period extension of our main model where the endconsumer market can be in one of two states (indexed by i): “high” demand state, i ¼ h, or “low” demand state, i ¼ l. The probability that the demand market will be in state i is fi, i ¼ l, h, such that fh + fl ¼ 1. The buyer and the supplier learn the state of the demand sometime during the planning horizon. Therefore, the information update splits the planning horizon into two periods, possibly of unequal lengths. The buyer can procure capacity in both periods before the selling season begins. We assume that bh ¼ bl ¼ 1; i.e., the market is characterized by a linear inverse demand function Pi(Qi) ¼ ai Qi, where Pi(Qi) is the price of the product when Qi is the quantity sold by the buyer in the end-consumer market when the demand is in state i and ai is the market potential in state i. We assume that 0 < al ⩽ ah, that is, the market potential is positive in both states and higher under the high demand state. In this two-period model, the supplier announces the first period wholesale price at the beginning of the first period. He announces the second period wholesale price at the beginning of the second period after the demand state is revealed. The buyer chooses the first period procurement quantity before the demand state is revealed. She chooses the second period procurement quantity after the demand state is revealed. The market clears at the end of the second period.

230

F. Erhun et al.

Erhun et al. (2008) study this setting in detail (when there is limited capacity, more than two demand states, etc.). We present one of their results below to show how dynamic procurement extends to the situations with uncertain demand: Proposition 6. When demand is uncertain, the prices and quantities are set as follows: 8 < 9ðfh ah þfl al Þ; fh ah þfl al 16 8 ðw1 ; q1 Þ ¼ : 9fh ah ; ah 16 8 ðw2;i ; q2;i Þ ¼

if fh <

3al ah al

2

otherwise

þ a q þ ai i 1 q1 ; ; 2 4 2

i ¼ h; l:

Depending on the demand state, dynamic prices may increase (advance purchase discount) or decrease (markdown) over time. Dynamic procurement works for the supplier even under uncertain demand; the supplier’s increase with dynamic procurement. However, for the buyer, the value of dynamic procurement depends on when the supplier sets the price in a single procurement situation. If the supplier sets to price after the uncertainty is revealed (no-commitment model), then the buyer’s expected profits are also higher. Since both players benefit, it is a natural consequence that dynamic procurement eliminates supply chain inefficiencies compared to the no-commitment case. If the supplier sets to price before the uncertainty is revealed (early commitment model), the buyer’s expected profits may either increase or decrease under dynamic procurement. Especially when the difference between the market potentials of the high and low demand states is considerable, the buyer becomes worse off with dynamic procurement compared to the early commitment model. However, when the supplier chooses his capacity as well as the prices, for a wide-range of parameters, dynamic procurement is the best alternative for all parties, including the end customers.

5 Conclusion Dynamic procurement is commonly used to mitigate demand risk. However, our research shows that this may not be the only reason why companies use dynamic procurement. During our discussions with a consulting firm for a major manufacturer of finished goods, we observed a strong indication that the manufacturer was able to impact its raw material costs by using a multiple period sourcing approach. Interestingly, there was very little uncertainty in the finished good demand for this particular manufacturer and the supply of the material was constrained. Based on a data analysis over a 4-year horizon, the consulting firm concluded that higher inventory levels at the manufacturer showed strong correlations with reduced sourcing costs on a per-unit basis. A likely reason that the multiple period sourcing

Dynamic Procurement, Quantity Discounts, and Supply Chain Efficiency

231

helped to reduce prices was expressed by many of the manufacturer’s purchasers as “being able to walk away from the table.” The raw material supplier tended to know the inventory position of the manufacturer and understood the economics of the manufacturer’s business such that she could leverage her supply when the manufacturer had immediate shortage concerns. When the manufacturer felt strong enough to walk away, the negotiating position was reversed. In this paper, we analyzed this phenomenon with a stylized model. Similar to the efficiency result of Allaz and Vila, our main result states that, as the number of trading periods increases, the total output of the supply chain increases and approaches that of the centralized supply chain. Contrary to their result, we show that all the parties benefit from multiple trading periods. In equilibrium, dynamic procurement is similar to an incremental quantity discount where the supplier sets the prices and the buyer sets the breakpoints (Fig. 4). It can be viewed as a sequence of bilateral negotiations between the supplier and the buyer, and it provides incentives to both parties to increase the total supply chain profits. There are several directions for future research. In our model, the number of trading periods is known in advance. However, the situation where this information is not common knowledge, either to the buyer and/or to the supplier, would be interesting to analyze. Another possibility is to extend Erhun et al. (2008) and model a setting where the uncertainty is revealed to either player partially. Studying models with signalling or screening would be particularly beneficial to understand the dynamics when the uncertainty revelation is only partial. Finally, further studying a more general model where the wholesale prices can be negotiated and the products can be sold via N possible periods to competing buyers under partial demand revelation would be interesting.

a

b 1500 60

Unit Cost

Total Cost

55 1000

500

50 45 40

0

35 0

10

20 30 Quantity

40

0

10

20

30

40

Quantity

Fig. 4 The total and unit capacity costs versus quantities under dynamic procurement when N ¼ 20 (P(Q20) ¼ 100 Q20)

232

F. Erhun et al.

Acknowledgements The first author was partially supported by NSF Award DMI-0400345 and the second author was supported by NSF Career Award DMII-0093844. The authors would like to express their deepest gratitude to two anonymous reviewers for their constructive comments and suggestions.

References Allaz B, Vila JL (1993) Cournot competition, forward markets and efficiency. J Econ Theory 59 (1):1–16 Anand K, Anupindi R, Bassok Y (2008) Strategic inventories in vertical contracts. Manage Sci 54 (10):1792–1804 Benton WC, Park S (1996) A classification of literature on determining the lot size under quantity discounts. Eur J Oper Res 92(2):219–238 Burnetas A, Gilbert S (2001) Future capacity procurements under unknown demand and increasing costs. Manage Sci 47(7):979–992 Cachon GP (2004) The allocation of inventory risk in a supply chain: push, pull, and advancepurchase discount contracts. Manage Sci 50(2):222–238 Chen R, Roma P (2010) Group buying of competing retailers. Prod Oper Manage 20(2):181–197 Chen F, Federgruen A, Zheng Y (2001) Coordination mechanism for a distribution system with one supplier and multiple retailers. Manage Sci 47(5):693–708 Crowther J (1964) Rationale for quantity discounts. Harv Bus Rev 42(2):121–127 Cvsa V, Gilbert SM (2002) Strategic commitment versus postponement in a two-tier supply chain. Eur J Oper Res 141(3):526–543 Dada M, Srikanth KN (1987) Pricing policies for quantity discounts. Manage Sci 33 (10):1247–1252 Dolan RJ (1987) Quantity discounts: managerial issues and research opportunities. Mark Sci 6 (1):1–23 Dong L, Zhu K (2007) Two-wholesale-price contracts: push, pull, and advance purchase discount contracts. Manuf Serv Oper Manage 9(3):291–311 Donohue KL (2000) Efficient supply contracts for fashion goods with forecast updating and two production modes. Manage Sci 46(11):1397–1411 Elmaghraby W, Keskinocak P (2003) Dynamic pricing in the presence of inventory considerations: research overview, current practices, and future directions. Manage Sci 49 (10):1287–1309 Erhun F (2007) Dynamic pricing in Cournot duopoly with two production periods. Working paper, Department of Management Science and Engineering, Stanford University, Stanford, CA Erhun F, Keskinocak P, Tayur S (2008) Dynamic procurement in a capacitated supply chain facing uncertain demand. IIE Trans 40(8):733–748 Ferguson ME (2003) When to commit in a serial supply chain with forecast updating. Nav Res Logist 50(8):917–936 Ferguson ME, DeCroix GA, Zipkin PH (2005) Commitment decisions with partial information updating. Nav Res Logist 52(8):780–795 Guo P, Niu B, Wang Y (2009) Two-wholesale-price contract in a three-tier supply chain. Working paper, Hong Kong Polytechnic University, Hung Hom, Kowloon Gurnani H, Tang CS (1999) Note: optimal ordering decisions with uncertain cost and demand forecast updating. Manage Sci 45(10):1456–1462 Ingene CA, Parry ME (1995) Channel coordination when retailers compete. Mark Sci 14 (4):360–377 Iyer AV, Bergen ME (1997) Quick response in manufacturer-retailer channels. Manage Sci 43 (4):559–570

Dynamic Procurement, Quantity Discounts, and Supply Chain Efficiency

233

Jeuland AP, Shugan SM (1983) Managing channel profits. Mark Sci 2(3):239–272 Keskinocak P, Charnsirisakskul K, Griffin P (2003) Supply chain procurement with inventory and backordering options. Working paper, School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, GA Keskinocak P, Charnsirisakskul K, Griffin P (2008) Strategic inventory in capacitated supply chain procurement. Managerial Decis Econ 29(1):23–36 Li C, Scheller-Wolf A (2010) Push or pull? Auctioning supply contracts. Prod Oper Manage 20(2): 198–213 Monahan JP (1984) A quantity discount pricing model to increase vendor profits. Manage Sci 30 (6):720–726 Munson CL, Rosenblatt MJ (1998) Theories and realities of quantity discounts: an exploratory study. Prod Oper Manage 7(4):352–369 Raju J, Zhang ZJ (2005) Channel coordination in presence of a dominant retailer. Mark Sci 24 (2):254–262 Spencer BJ, Brander JA (1992) Pre-commitment and flexibility: applications to oligopoly theory. Eur Econ Rev 36(8):1601–1626 Taylor T (2006) Sale timing in a supply chain: when to sell to the retailer. Manuf Serv Oper Manage 8(1):23–42 Van Mieghem JA (2003) Capacity management, investment, and hedging: review and recent developments. Manuf Serv Oper Manage 5(4):269–302 Weng ZK (1995) Channel coordination and quantity discounts. Manage Sci 41(9):1509–1522

.

Coordination of the Supplier–Retailer Relationship in a Multi-period Setting: The Additional Ordering Cost Contract Nicola Bellantuono, Ilaria Giannoccaro, and Pierpaolo Pontrandolfo

Abstract Coordinating supply chains by adopting a centralized decision making approach, which is theoretically desirable, is often practically infeasible, if not ineffective: the high number of involved companies in a supply chain, the lack of adequate contractual power concentrated in the hand of only a few of them, the difficulty to gather all the relevant information by the unique/few decision maker/s, are some of the many reasons preventing supply chains from implementing such a centralized coordination approach. Supply contracts have been proposed in the literature as an alternate way to face such a problem: they let the chain’s partners to autonomously make decision, but at the same time guide them to behave coherently among each other as well as with the chain’s goal. Designing a contract is quite a challenging task, especially under the hypothesis of multi-period settings, which is the assumption considered in this chapter. In the majority of cases, multi-period supply contracts are inherently complex (e.g. many parameters that need to be frequently updated), therefore difficult to be implemented, as well as often designed under hypotheses barely realistic (e.g. null order costs). We propose a supply contract for a two-stage supply chain (supplier–retailer) in a multi-period setting, which tries to overcome such drawbacks. The proposed contract is based on two key mechanisms: additional ordering cost for retailer and price discount offered by the supplier to the retailer. A numerical analysis is finally conducted to identify the conditions that allow the best performance to be achieved. N. Bellantuono (*) • P. Pontrandolfo Dipartimento di Ingegneria dell’Ambiente e per lo Sviluppo Sostenibile, Politecnico di Bari, via De Gasperi s.n., 74100 Taranto, Italy e-mail: [email protected]; [email protected] I. Giannoccaro Dipartimento di Ingegneria Meccanica e Gestionale, Politecnico di Bari, viale Japigia 182, 70125 Bari, Italy e-mail: [email protected] T.-M. Choi and T.C. Edwin Cheng (eds.), Supply Chain Coordination under Uncertainty, International Handbooks on Information Systems, DOI 10.1007/978-3-642-19257-9_10, # Springer-Verlag Berlin Heidelberg 2011

235

236

N. Bellantuono et al.

Keywords Multi-period setting • Supply chain coordination • Supply contracts

1 Introduction A supply chain is a network of organizations that are involved in the different processes and activities to produce value in the form of products and services in the hands of the ultimate consumer (Christopher 1992). The coordination of this network is a key issue in supply chain management. System efficiency is assured when the supply chain is globally managed by a single decision maker who optimizes the performance of the whole system (channel coordination). This approach is usually referred to as centralized strategy (Federgruen 1993). Coordination problem becomes complex when different decision makers coexist within the supply chain, each taking decisions by pursuing their own goals, which are likely to be conflicting against each other (Schneeweiss 2003). These locally rational behaviours result in global inefficiency of the supply chain (Whang 1995). To improve the overall performance, the decision makers should be pushed to behave in the interest of the global supply chain rather than in their own ones. Such a problem is referred to as alignment of incentives in decentralized supply chains (Narayanan and Raman 2004). To align incentives in supply chains, supply contracts should be adopted. They formally rule the transaction between the actors and force them to pursue channel coordination. These mechanisms are based on transfer payment schemes that rule how to split the savings (or the increase in revenues) and let risks be fairly shared (Tsay et al. 1999; Cachon 2003). Transfer payment schemes are designed to increase the system-wide profit so as to make it closer – possibly equal – to the profit resulting from a centralized control (channel coordination). They modify the decision makers’ goals, bringing them to take the same decisions that the single decision maker would take. A further important issue for the contract design is the so-called win-win condition, which occurs if the contract makes every actor gain a profit higher than the one he or she would gain in default of the contract, i.e. under a decentralized setting. Indeed, if the win–win condition is not satisfied, the actor would not be prompted to adopt the contract (Giannoccaro and Pontrandolfo 2004). A number of supply contracts have been developed in the literature. They include quantity/volume discounts (Monahan 1984; Li and Liu 2006), buy back or return policies (Pasternack 1985; Emmons and Gilbert 1998), backup agreements (Eppen and Iyer 1997), allocation rules (Cachon and Lariviere 1999), quantity flexibility contracts (Tsay 1999; Wang and Tsao 2006), and revenue sharing contracts (Giannoccaro and Pontrandolfo 2004; Cachon and Lariviere 2005). For a complete review of supply contract the reader is referred to Tsay et al. (1999), Cachon (2003), and Tang (2006). Most of these contracts address the problem of coordinating supply chains under a single-period setting, namely assuming that the items that flow along the supply

Coordination of the Supplier–Retailer Relationship in a Multi-period Setting

237

chain are perishable or have a quickly decreasing demand. In these cases, the selling season is assumed short enough to impel both new orders be issued once demand has revealed and stocks in excess be held until the following selling season. This assumption permits the adoption of models deriving from the classical newsvendor model (Wadsworth 1959; Hadley and Whitin 1963). On the contrary, few studies have analyzed the same problem under a multi-period setting. In this case, further parameters (e.g. the ordering and inventory costs) must be taken into account, so making more complex not only the design of the contract but also its implementation in practice. The aim of this work is thus to develop an supply contract to coordinate a twostage supply chain under a multi-period setting, which is effective as well as simple to be implemented. In particular, we consider a supply chain in which a supplier provides in a random lead time a single product to a retailer, who in turn serves the market characterized by uncertain demand. Both incur ordering and inventory holding costs. Moreover, when final demand cannot be immediately satisfied, the retailer incurs backorder cost. The contract is designed so as to achieve both channel coordination and win–win condition and is mainly based on two mechanisms (1) the additional ordering cost that the retailer pays to the supplier and (2) the price discount that the supplier gives to the retailer. In the following section a review of contracts for multi-period settings is presented. Then, the supply chain model is described and the contract is designed. A numerical analysis is finally carried out to illustrate how the supply contract works and to identify the scenarios where the contract is more effective.

2 Supply Contracts in Multi-period Settings: A Review Since the 1980s, several studies have been carried out to design contracts for supply chain management, yet only lately research has addressed also multi-period settings. This section reviews the literature on multi-period contracts and describes the most significant contract models. The review proves useful to derive insights and suggestions supporting the design of an innovative contract. The quantity-flexibility multi-period contract by Tsay and Lovejoy (1999) derives from the homonymous contract for the single-period setting (Tsay 1999), which is modified mainly through the adoption of a rolling horizon: at the beginning of each selling season, the retailer and the supplier agree on (1) the wholesale price of the product, (2) the demand forecast for each season up to the planning horizon, and (3) the minimum and maximum quantity for the delivery. In the following seasons, actors update the demand forecast and the contract parameters, to reduce the allowable demand fluctuation for every future season as it becomes closer. Tsay and Lovejoy (1999) argue that their contract enables actors to share the risk related to demand uncertainty: the retailer is guaranteed for a minimum delivery of goods and so she recedes from the rationing game (Lee et al. 1997), i.e. the practice

238

N. Bellantuono et al.

to communicate overestimated forecasts in order to secure the possible delivery of more units than expected as demand. In turn, the supplier is kept from the risk that actual retailer’s orders will be lower than he predicts. Nevertheless, the need to continually update contract parameters restricts the adoption of the quantityflexibility contract to firms willing to cooperate and having human resources able to deal with so complex schemes. Steady-state contracts are simpler to be implemented; indeed, once they have been agreed, they are valid also in future periods without any need for parameters’ updating. Recently, literature has focused on this class of contracts (for a review, see Cachon 2003) and pointed out which clauses can be useful to coordinate the SC. They may include: – Additional penalties for the backorder at each stage of the SC – Grants to boost holdings – Time lag between physical and financial flows Cachon and Zipkin (1999) recur to the game theory to affirm that a supply chain can be coordinated by a contract that uses a linear money transfer from an actor to another to align the Nash equilibrium (i.e. the equilibrium in a decentralized setting) to the global optimum. By adopting such a contract, the retailer is encouraged to increase her stock; the supplier, in turn, increases the punctuality of his deliveries to avoid retailer’s stockouts. A contract similar to that by Cachon and Zipkin (1999) is the additional backlog penalty contract (Lee and Whang 1999): by a shortage reimbursement contract, the supplier pays the retailer when he cannot meet her orders and receives money from her when she has a stockout. Such a mechanism urges actors to increase their holding stock, which otherwise in a decentralized setting would be lower then what the optimal solution prescribes. Lee and Whang (1999) emphasize three properties of the proposed contract, which indeed are common to many others: cost conservation, incentive compatibility, and information decentrability. Cachon (2003) shows that the shortage reimbursement contract substantially agrees with the methodology by Chen (1999) to coordinate systems with a delayed receipt of orders. Also the contract by Porteus (2000), based on the so called responsibility tokens, is similar to the shortage reimbursement contract. In fact, it results in the same transfer payment between actors, although it assigns to the supplier a penalty per late delivered unit, instead of a penalty per occurrence of a – partial or total – late delivery. Lee and Whang (1999) observe that a method to coordinate supply chains based on time discounting consists in abandoning the hypothesis that physical transfers and related financial flows are concurrent. In particular, authors describe the consignment contract, by which retailers give money to the suppliers only once products are sold to the final customers. The steady-state contracts do not need to continually update the parameters, differently from the rolling horizon contracts. Unfortunately, to our knowledge the steady-state contracts so far developed hold under hypotheses that are barely realistic. Their most critical aspect refers to the ordering cost, which in the literature

Coordination of the Supplier–Retailer Relationship in a Multi-period Setting

239

on supply chain inventory management is generally assumed as null (Axs€ater 1990; Axs€ater 1993; Cachon and Zipkin 1999; Lee and Whang 1999) or proportional to order quantity (Shang and Song 2003). In both cases, the optimal replenishment quantity becomes qi ¼ 1 and the search of optimal policies collapses to the simple search – for each actor – of the order point that minimizes the sum of expected holding and backorder costs. By analyzing each cost factor that adds up to the ordering cost, one can demonstrate the ineffectiveness of these assumptions: in fact, the ordering cost is a step-function of the order quantity and thus it can be assumed as constant for a wide interval of values for the latter.

3 The Model Consider a serial two-stage supply chain consisting of a retailer, who faces the market demand, and a supplier, who provides items to the retailer.1 Both actors are risk-neutral, i.e. their utility functions are proportional to the profit (Schweitzer and Cachon 2000), and financial flows occur simultaneously to the corresponding physical flows. At both stages inventories are managed by a continuous review system ruled by the pair ðqi ; ri Þ, the former being the order quantity ðqi > 0Þ and the latter the reorder point ðri > 0Þ. The unmet demand is fully backordered (Fig. 1). The instantaneous demand is a continuous random variable. All the notations are summarized in Table 1. For the sake of simplicity, we prefer to assume that the supplier’s lead time is null, so that the latter can issue his orders at the same time he receives the retailer’s order. In fact, this assumption is usually adopted when it is wished to neglect the effect of suppliers external to the supply chain. This assumption also implies that the demand during the supplier’s lead time is null ðm2 ¼ s2 ¼ 0Þ, as well as the

external supplier

L2

supplier

L1

retailer

(stage 2)

(stage 1)

A2 h2

A1 h1 b

market

d

Fig. 1 The supply chain model

1 For the sake of clarity, we denote the retailer as actor 1 and the supplier as actor 2. Moreover, we use the pronoun “she” for the retailer and “he” the supplier.

240 Table 1 Model notations

N. Bellantuono et al.

Variable D 1 2 mi si Ai hi b qi ri k n ’(k) F(k) () Ci() C() d * c a d

Description Expected annual demand Retailer’s subscript Supplier’s subscript Mean of the demand during the i-th actor’s lead time Standard deviation of the demand during the i-th actor’s lead time Ordering cost at the i-th stage Annual holding cost per unit at the i-th stage Backorder cost per unit Order quantity at the i-th stage Reorder point at the i-th stage Safety factor Nested factor p.d.f. of the standard normal distribution c.d.f. of the standard normal distribution Expected shortage per replenishment cycle Expected annual cost of the i-th actor Expected annual cost of the supply chain Decentralized setting superscript Centralized setting superscript Contract setting superscript Additional ordering cost (contract parameter) Discount per unit sold (contract parameter)

supplier’s reorder point ðr2 ¼ 0Þ and his expected backorder stock. The retailer’s lead time, in turn, is a random positive variable; demand during the retailer’s lead time is normally distributed and its mean m1 and standard deviation are both known and denoted as m1 and s1 , respectively. The probability density function and cumulative distribution function of the standard normal distribution are respectively denoted as ’ðkÞ and FðkÞ. Thus, the retailer’s order point can be expressed in terms of the safety factor, as follows: k¼

r1 m1 : s1

(1)

Therefore, the supplier’s expected annual cost consists in the sum of expected ordering and holding cost: the former is proportional to the expected number of orders per year, whereas the latter is proportional to the units that he holds on average. We assume also that the holding cost of the pipeline stock (units in transit from the supplier to the retailer) is paid by the former. The retailer, in turn, is affected by ordering, holding, and backorder cost. The latter is assumed proportional to the number of units backlogged, irrespective of the time for which the backorder lasts.

Coordination of the Supplier–Retailer Relationship in a Multi-period Setting

3.1

241

Decentralized Setting

Each actor autonomously makes inventory policy decisions, aimed at minimizing his/her own cost. Hence, two separate optimization problems have to be solved (1) finding the pair ðqd1 ; kd Þ that minimizes the retailer’s cost, and (2) identifying the parameter qd2 that minimizes the supplier’s cost, given the retailer’s policy. Under the assumptions above described, the retailer’s expected annual cost is calculated as follows (Hadley and Whitin 1963; Silver et al. 1998)2: C1 ðq1 ; kÞ ¼ A1

hq i D D 1 þ s1 k þ b ðkÞ; þ h1 2 q1 q1

(2)

where the quantity between square brackets is the expected net inventory and: ðkÞ ¼ s1 ½’ðkÞ þ kFðkÞ k

(3)

is the expected shortage per replenishment cycle, i.e. the unmet demand between two consecutive orders [see Appendix A.2]. The optimal retailer’s policy ðqd1 ; kd Þ minimizing (2) can be obtained by the iterative procedure described in Hadley and Whitin (1963) using the following equations [see Appendix A.3]: qd1

rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2D ¼ ½A1 þ bðkd Þ; h1

(4)

and h1 qd1 kd ¼ F1 1 : bD

(5)

To solve the supplier’s optimization problem, we assume that he adopts a nested policy, which allows the computational complexity of the problem to be reduced at the expense of a slight possible decrease of the solution effectiveness (Schwarz and Schrage 1975; Roundy 1985; Axs€ater and Rosling 1993). When a nested policy is used, a stage can issue a order only when the downstream stage does the same. This implies that when the supplier makes a order, his order quantity will be: q2 ¼ nq1 ;

(6)

being the nested factor n a positive integer. 2

The (2) is an approximation but consistent with a wide stream of the literature on inventory management (Hadley and Whitin 1963; De Bodt and Graves 1985; Silver et al. 1998; Mitra and Chatterjee 2004). See Appendix A.1.

242

N. Bellantuono et al.

Therefore, the supplier’s inventory problem is reduced to finding the positive integer nd that minimizes his expected annual cost: C2 ðn; q1 Þ ¼ A2

D n1 þ h2 q1 þ m1 ; nq1 2

(7)

where the quantity between square brackets is the expected net inventory at the supplier’s stage and includes the expected pipeline stock, i.e. the expected quantity in transit from the supplier to the retailer. To this aim, the following procedure is suggested: 1. Compute: qﬃﬃﬃﬃﬃﬃﬃﬃ n^ ¼

2A2 D h2 : d q1

(8)

2. If n^ is an integer, then nd ¼ n^; otherwise: nd ¼ arg min C2 ðn; qd1 Þ; n2fn1 ;n2 g

(9)

where n1 and n2 are the positive integers that surround n^.

3.2

Centralized Setting

Under a centralized setting, the optimal inventory policy is the one that minimizes the expected annual cost of the whole supply chain, whose formula is given by referring to the model by De Bodt and Graves (1985). This model adopts a nested policy using an echelon perspective. The echelon perspective requires to compute (1) the echelon stock of each stage, i.e. the sum of the stock at the stage and all the downstream stages, including the pipeline stock, (2) the echelon holding cost (i.e. the incremental inventory cost at a given stage with reference to the upstream stage), and (3) the echelon order point (i.e. the sum of the order point at the considered stage and those at all the upstream stages). Under the current hypotheses, the expected total supply chain cost is given by De Bodt and Graves (1985). By substituting the echelon cost expressions with the correspondent installation ones, it follows that: hq i A2 D n1 D 1 þ s1 k þ h2 Cðq1 ; k; nÞ ¼ A1 þ þ h1 q1 þ m1 þ b ðkÞ; (10) n q1 2 2 q1 wherein the expected shortage per replenishment cycle ðkÞ is given by (3) and the nested factor n is a positive integer. The retailer’s and supplier’s expected annual costs are given by (2) and (7), respectively.

Coordination of the Supplier–Retailer Relationship in a Multi-period Setting

243

The centralized inventory problem is to find the optimal policy ðq1 ; k ; n Þ that minimizes (10). To solve it, a heuristics based on the continuous relaxation of the problem is proposed, consisting of the following steps (see Appendix A.4): 1. Assume ðkÞ ¼ 0 and compute:

rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ A2 h1 h2 : n~ ¼ A1 h2

(11)

2. Compute: sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2D A2 q1 ð~ nÞ ¼ A1 þ þ bðkÞ n~ h1 þ ð~ n 1Þh2

(12)

3. Compute: 1

kðq1 Þ ¼ F

h1 q1 ð~ nÞ 1 bD

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2A2 D=h2 n~ðq1 Þ ¼ q1

(13)

(14)

4. Iterate steps 2–3 until a suitable approximation is obtained. 5. If n~ðq1 Þ is an integer, then n ¼ n~ðq1 Þ; otherwise denote the positive integers that surround n~ as n1 and n2 . 6. For both n1 and n2 iteratively use (12) and (13) until convergence to compute q1 ðn1 Þ and k ðn1 Þ and, respectively, q1 ðn2 Þ and k ðn2 Þ. 7. Compute (10). The optimal value for the nested factor is: n ¼ arg min C n; q1 ðnÞ; k ðnÞ ; (15) n2fn1 ;n2 g

where q1 ðnÞ and k ðnÞ are the correspondent conditionally optimal values of the other two variables, as determined at step 6.

4 The Additional Ordering Cost Contract The additional ordering cost contract aims to push retailer to make larger orders than she will do under a decentralized setting, so as to led the supply chain to behave like in a centralized fashion. In fact, by comparing the decentralized and the centralized settings, we notice that the inefficiency of the decentralized setting is due to the fact that the retailer makes smaller and more frequent orders. The additional ordering cost contract is then based on a transfer payment from the supplier to the retailer, so defined:

244

N. Bellantuono et al.

FTðq1 ; a; dÞ ¼ dD aA1

D : q1

(16)

It is ruled by two parameters (1) the penalty that the supplier gives to the retailer for each order she issues ðaÞ, and (2) the discount that the suppliers grants to the retailer for each unit sold ðdÞ. Through a suitable design of the values for a and d both the channel coordination and the win–win condition are assured. The retailer’s and the supplier’s expected annual costs under the contract setting are given by the equations, respectively: Cc1 ðq1 ; k; a; dÞ ¼ C1 ðq1 ; kÞ FTðq1 ; a; dÞ hq i D D 1 þ s1 k þ b ðkÞ dD ¼ ð1 þ aÞA1 þ h1 2 q1 q1

(17)

and: Cc2 ðn; q1 ; a; dÞ ¼ C2 ðn; q1 Þ þ FTðq1 ; a; dÞ D n1 D þ h2 ¼ A2 q1 þ m1 þ dD aA1 : nq1 2 q1

(18)

Thus, as in the decentralized setting even under the contract two separate optimization problems have to be solved (1) the retailer’s problem is to identify the pair ðqc1 ; kc Þ that minimizes (17), and (2) the supplier’s problem is to find the positive integer nc that minimizes (18). In particular, from (17) it follows: rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2D c q1 ðaÞ ¼ ½ð1 þ aÞA1 þ bðkc Þ: (19) h1 and 1

k ¼F c

h1 qc1 1 : bD

(20)

Proposition 1. The channel coordination is achieved if the actors agree on an additional ordering cost contract where: a¼

h1 A 2 n

ðn 1Þh2 ½A1 þ bðk Þ : A1 ½h1 þ ðn 1Þh2

(21)

(Proof: See Appendix A.5). Proposition 2. Once the channel coordination is achieved, there exists a range of values for d which assure the win–win condition. (Proof: See Appendix A.6). Observation 1. Channel coordination does not depend on d. Observation 2. The annual expected costs of both actors linearly depend on the value of d.

Coordination of the Supplier–Retailer Relationship in a Multi-period Setting

245

5 Numerical Analysis For illustrative purposes, in this section a numerical analysis is provided to give a measure of the inefficiency of the decentralized approach and to show how the additional ordering cost contract works. Results prove that the contract coordinates the channel and assures a win–win condition. As a measurement of the inefficiency of the decentralized setting, the competition penalty (Cachon and Zipkin 1999) is defined as follows: C qd1 ; kd ; nd C q1 ; k ; n CP ¼ 100 : C q1 ; k ; n

(22)

The higher CP, the higher the penalty in terms of increase of the supply chain expected cost. Data used in the numerical analysis are shown in Table 2. The latter consists of 27 scenarios designed by varying the value ðs1 =m1 Þ, the ratios between the ordering costs ðA1 =A2 Þ, and the annual holding costs per unit ðh2 =h1 Þ. The sensitivity analysis aims to identify the scenarios where the contract is more effective. Table 3 shows the retailer’s, the supplier’s, and the system-wide expected annual costs both in the decentralized and in the centralized setting, as well as the corresponding competition penalty in all scenarios. As we expected, CP is always positive, which means that the centralized setting provides better system-wide performances than the decentralized one. Furthermore, the retailer’s performance gets worse moving from the decentralized to the centralized setting, so explaining why the retailer has no incentive to agree on system-wide optimal policy and why a supply contract is thus necessary. The data presented in Table 3 are analyzed in Table 4, where for each value of s1 , A1 , and h2 the means of the competition penalty obtained for the three levels of the other two variables are reported. Results show that CP is positively affected by h2 and to a smaller extent by A1 , whereas the effects of s1 on CP are in general negligible, in spite of the differences in the expected costs. Therefore, the contract proves very useful especially when the holding costs per unit at both stages are similar, irrespective of the demand variability or the difference between retailer’s Table 2 Values used in the numerical analysis.

Variable D m1 s1 A1 A2 h1 h2 b

Levels 1 1 3 3 1 1 3 1

Values 1,000 100 10, 20, 30 5, 20, 35 50 1 0.5, 0.7, 0.9 10

246

N. Bellantuono et al.

Table 3 Retailer’s, supplier’s and system-wide expected annual costs in decentralized and centralized settings for different values of s1, A1, and h2, given D ¼ 1,000, m1 ¼ 100, A2 ¼ 50, h1 ¼ 1, and b ¼ 10 A1 h2 Decentralized setting Centralized setting CP (%) s1 10

5

20

35

20

5

20

35

30

5

20

35

0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9

C

C1

375.01 426.05 470.80 447.82 488.20 528.57 497.95 543.99 563.99 400.12 452.24 495.10 470.64 511.39 552.14 520.67 564.39 584.39 425.35 477.41 519.57 493.43 534.57 575.71 543.37 584.77 604.77

126.59 126.59 126.59 224.18 224.18 224.18 287.71 287.71 287.71 153.07 153.07 153.07 248.28 248.28 248.28 310.79 310.79 310.79 179.43 179.43 179.43 272.32 272.32 272.32 333.82 333.82 333.82

Table 4 Mean competition penalty for different values of s1, A1, and h2, given D ¼ 1,000, m1 ¼ 100, A2 ¼ 50, h1 ¼ 1, and b ¼ 10

A1 Mean (CP) h2 Mean (CP) s1 mean (CP)

C2 248.42 299.45 344.21 223.65 264.02 304.39 210.24 256.28 276.28 247.05 299.17 342.03 222.35 263.10 303.86 209.88 253.60 273.60 245.91 297.98 340.14 221.11 262.25 303.39 209.56 250.95 270.95

C 369.68 413.79 443.93 440.86 465.96 485.96 483.71 503.71 523.71 394.93 438.17 466.15 464.26 487.70 507.70 505.07 525.07 545.07 420.14 462.50 488.32 487.61 509.41 529.41 526.39 546.39 566.39

5 3.56% 0.5 1.93% 10 4.91%

C1 132.56 145.72 204.95 227.97 263.75 263.75 312.21 313.63 313.63 159.01 170.83 228.93 251.76 286.91 286.91 333.31 336.17 336.17 184.63 195.87 252.85 275.50 310.01 310.01 354.36 358.67 358.67

C2 236.36 268.07 238.98 212.89 202.21 222.21 170.08 190.08 210.08 235.92 267.34 237.22 212.49 200.80 220.80 168.90 188.90 208.90 235.51 266.63 235.47 212.12 199.40 219.40 167.73 187.73 207.73

20 5.00% 0.7 5.16% 20 4.83%

1.44 2.96 6.05 1.58 4.77 8.77 2.94 8.00 7.69 1.31 3.21 6.21 1.37 4.86 8.75 3.09 7.49 7.21 1.24 3.22 6.40 1.19 4.94 8.75 3.23 7.02 6.78

35 5.94% 0.9 7.40% 30 4.75%

and supplier’s ordering costs. Moreover, as ordering costs at both stages become similar, the competition penalty increases ceteris paribus, thus the possible benefit deriving from the adoption of the contract grows. Finally, keeping equal the ratios A1 =A2 and h2 =h1 , an increase in demand variability results in a CP slightly decreasing. However, it does not mean that the additional ordering contract

Coordination of the Supplier–Retailer Relationship in a Multi-period Setting

247

becomes ineffective or unnecessary: indeed, since the expected annual cost increase in demand variability, even if CP reduces the savings that can be obtained through the contract are significant. From Table 5, which gives the optimal values for q1 ; k; and n in both settings, we can see that in all the scenarios the optimal retailer’s order quantity is higher in the centralized policy than in the decentralized one. This justifies why the additional ordering cost contract has been designed so as to make the retailer increase her order quantity, by a penalty paid whenever she issues an order. Table 6 illustrates the value of a that allows the channel coordination be achieved in all the scenarios and the corresponding range for d where also the win–win condition is satisfied. In particular, below the minimum value of such an interval the contract is not convenient for the retailer (namely, it increases her expected cost compared to the decentralized setting), whereas above the maximum value it is not convenient for the supplier. Table 5 Optimal values for q1 ; k; and n in decentralized and centralized settings for different values of s1, A1, and h2, given D ¼ 1,000, m1 ¼ 100, A2 ¼ 50, h1 ¼ 1, and b. ¼ 10 A1 h2 Decentralized setting Centralized setting s1 10

5

20

35

20

5

20

35

30

5

20

35

0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9

qd1

kd

nd

q1

k

n

103.5 103.5 103.5 203.7 203.7 203.7 268.4 268.4 268.4 107.1 107.1 107.1 207.5 207.5 207.5 272.3 272.3 272.3 110.8 110.8 110.8 211.4 211.4 211.4 276.3 276.3 276.3

2.3136 2.3136 2.3136 2.0461 2.0461 2.0461 1.9294 1.9294 1.9294 2.3006 2.3006 2.3006 2.0385 2.0385 2.0385 1.9231 1.9231 1.9231 2.2876 2.2876 2.2876 2.0308 2.0308 2.0308 1.9168 1.9168 1.9168

4 4 3 2 2 2 2 1 1 4 4 3 2 2 2 2 1 1 4 3 3 2 2 2 2 1 1

149.0 190.0 335.6 247.5 378.2 378.2 416.4 416.4 416.4 150.8 192.2 339.6 250.0 382.3 382.3 420.5 420.5 420.5 152.6 194.4 343.7 252.6 386.4 386.4 424.7 424.7 424.7

2.1728 2.0748 1.8308 1.9643 1.7766 1.7766 1.7320 1.7320 1.7320 2.1680 2.0701 1.8255 1.9599 1.7716 1.7716 1.7273 1.7273 1.7273 2.1632 2.0654 1.8201 1.9555 1.7667 1.7667 1.7227 1.7227 1.7227

3 2 1 2 1 1 1 1 1 3 2 1 2 1 1 1 1 1 3 2 1 2 1 1 1 1 1

248

N. Bellantuono et al.

Table 6 Optimal value for a and minimum and maximum value for d that let the additional ordering cost contract achieve channel coordination and win–win condition for different values of s1, A1, and h2, given D ¼ 1,000, m1 ¼ 100, A2 ¼ 50, h1 ¼ 1, and b ¼ 10 A1 h2 a dmin dmax s1 10 5 0.5 1.1253 0.0437 0.0498 0.7 2.4723 0.0842 0.0964 0.9 10.0000 0.2273 0.2542 20 0.5 0.4844 0.0429 0.0499 0.7 2.5000 0.1718 0.1940 0.9 2.5000 0.1718 0.2144 35 0.5 1.4286 0.1446 0.1602 0.7 1.4286 0.1460 0.1863 0.9 1.4286 0.1460 0.1863 20 5 0.5 1.0597 0.0411 0.0463 0.7 2.4137 0.0805 0.0946 0.9 10.0000 0.2231 0.2520 20 0.5 0.4685 0.0410 0.0473 0.7 2.5000 0.1694 0.1931 0.9 2.5000 0.1694 0.2139 35 0.5 1.4286 0.1414 0.1599 0.7 1.4286 0.1443 0.1836 0.9 1.4286 0.1443 0.1836 30 5 0.5 1.0040 0.0381 0.0433 0.7 2.3537 0.0770 0.0919 0.9 10.0000 0.2189 0.2501 20 0.5 0.4522 0.0390 0.0448 0.7 2.5000 0.1671 0.1922 0.9 2.5000 0.1671 0.2134 35 0.5 1.4286 0.1383 0.1596 0.7 1.4286 0.1426 0.1810 0.9 1.4286 0.1426 0.1810

6 Concluding Remarks This work has proposed an innovative contract to manage supplies in decentralized two-stage supply chains characterized by: random, independent demand and lead time; infinite planning horizon; continuous review of inventory; total backorder of the unmet demand. The contract is innovative because it takes into account ordering costs, which are usually neglected in the literature on multi-period supply contracts. It ensures both the system-wide efficiency and the win–win condition. Furthermore, the proposed contract is straightforward to be implemented, since it requires that the actors agree on two parameters only, which control how costs are split up among the actors. In particular, the contract is ruled by the parameter a, which is the penalty that the retailer imposes to the supplier for each order, and the parameter d, which specifies the discount per unit that the supplier grants to the retailer.

Coordination of the Supplier–Retailer Relationship in a Multi-period Setting

249

Finally, the contract has two interesting properties (1) the parameter a is enough to drive the system to efficiency; (2) the parameter d has a linear effect on the expected cost of each actor. As a result, each actor is able to assess the benefits that the contract provides not only to him/her but also to his/her counterpart. In this way typical cautious behaviour that characterize the negotiation phase on the contract parameters should be mitigated. We believe that there are several directions to which this field of study can be extended. Further research would be addressed to the application of the additional ordering cost contract to more complex supply chains: in particular, it can be interesting to analyze how to extend its application field to distribution supply chain, characterized by arborescent topologies, as well as to supply chains having more than two stages. Another possible extension of this research would consist in analyzing if the actors perceive as fair the agreement of the contract: to this aim the research could encompass both on field studies and laboratory experiments. Finally, the performance of contract can be compared to the ones assured by other coordination schemes. Acknowledgments This work has been supported by Regione Puglia (APQ PS025 - ICT supporting logistics services: a model of organized market).

Appendix: Proofs and Discussions Approximations in Equation (2) Two approximations are made in (2). They refer to the computations of the expected on hand inventory and the expected annual shortage, which respectively affect the expected annual inventory cost and backorder cost. We discuss both approximations in the followings. Let us denote the retailer’s lead time as L1 and the probability density function of the demand during L1 as f ðxjL1 Þ. By definition, the expected on hand inventory: OHðq1 ; r1 Þ ¼

q1 þ 2

ð r1

ðr1 xÞf ðxjL1 Þdx

(23)

0

is equal to the expected net inventory plus the expected backorder stock. However, if the backorder cost per unit is high, the expected backorder stock is negligible compared to the expected net inventory. Thus, the expected on hand inventory can be assumed equal to: NIðq1 ; r1 Þ ¼

q1 þ 2

ð þ1 0

ðr1 xÞf ðxjL1 Þdx ¼

q1 þ r 1 m1 : 2

(24)

250

N. Bellantuono et al.

If the demand during the retailer’s lead time is normally distributed, by recurring to the safety factor defined in (1), the expected net inventory can be also expressed as: NIðq1 ; kÞ ¼

q1 þ ks1 : 2

(25)

The expected annual shortage is equal to the number of replenishments per year ðD=qi Þ times the expected shortage per replenishment cycle ðkÞ. The approximation made in (2) consists in assuming that the latter is equal to the expected backorder stock when an order arrives, which is exact only if all backorders are satisfied within one cycle.

Proof of Equation (3) The expected shortage per replenishment cycle is: ðr1 Þ ¼

ð þ1

ðx r1 ÞfðxjL1 Þdx:

(26)

r1

By assuming the demand during the retailer’s lead time normally distributed, being z ¼ ðx m1 Þ=s1 and k as in (1), we obtain that x ¼ m1 þ s1 z and r1 ¼ m1 þ s1 k, and observe that dx ¼ s1 dz. Therefore, (26) becomes: ðkÞ ¼

ð þ1

s1 ðz kÞ’ðzÞdz ¼ s1 ½’ðkÞ þ kFðkÞ k:

(27)

k

Proof of Equations (4) and (5) To minimize (2), we impose the first order conditions: 8 @ D h1 D > > < 0 ¼ @q C1 ðq1 ; kÞ ¼ A1 q þ b q ðkÞ 2 1 1 1 ; > @ D > : 0 ¼ C1 ðq1 ; kÞ ¼ h1 s1 þ b s1 ½FðkÞ 1 @k q1

(28)

where we have observed that, in case of normally distributed demand, since: @’ðkÞ @ 1 k2 =2 p ﬃﬃﬃﬃﬃ ﬃ e ¼ k’ðkÞ; ¼ @k @k 2p

(29)

Coordination of the Supplier–Retailer Relationship in a Multi-period Setting

251

then: @ ðkÞ ¼ s1 ½FðkÞ 1: @k

(30)

Rearranging (28), both (4) and (5) derive.

Discussion on the Heuristics for the Centralized Setting The continuous relaxation of the problem consist in finding the minimum of the following equation: h i ~ 1 ; k; n~Þ ¼ A1 þ A2 þ bðkÞ D þ h1 q1 þ s1 k þ h2 n~ 1 q1 þ m1 : (31) Cðq q1 n~ 2 2 It is equal to (10) except for the variable n~, which is a positive real number instead of a positive integer as n. The first order condition consists in imposing that the first derivatives of (31) are null: @ ~ A2 D 1 Cðq1 ; k; n~Þ ¼ A1 þ þ bðkÞ 2 þ ½h1 þ ðn 1Þh2 0¼ @q1 n~ q1 2 0¼

(32)

@ ~ bD Cðq1 ; k; n~Þ ¼ h1 þ ½FðkÞ 1 @r1 q1

(33)

@ ~ A2 D h2 q1 Cðq1 ; k; n~Þ ¼ þ 2 @ n~ q1 n~2

(34)

0¼

which respectively result in (12)–(14). Rearranging (14) and combining it with (12), we obtain: sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ A2 h1 h2 n~ ¼ : A1 þ bðkÞ h2

(35)

Assuming ðkÞ ¼ 0, from (35) we derive (11), which can be used as starting value for the heuristics. It can be observed that (11) is equal to the optimal choice for n in the continuous relaxation of the deterministic problem, as described in Silver et al. (1998). By recursively calculating (12)–(14), the optimal solution for the relaxed problem (31) is obtained. To derive the one for the original problem, as in De Bodt and Graves (1985) we conjecture that (10) is unimodal in q1 ðnÞ; k ðnÞ; n .

252

N. Bellantuono et al.

Proof of Proposition 1 The achievement of the channel coordination implies that the actors autonomously define their policies so as to allow the expected annual system-wide cost be minimized. By summing (17) and (18), it can be observed that the expected annual system-wide cost is the same as the one in default of the contract – see (10) – and does not depend on the contract parameters. Therefore, a sufficient condition is to assure that: qc1 ¼ q1 ;

(36)

k1c ¼ k1 ;

(37)

nc ¼ n :

(38)

and

To prove (37), it is enough to observe that (13) and (20) have the same analytical expression, and are identical if (36) holds. Moreover, combining (12) and (20), (36) can be written as: rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2D 2D A2 c ½ð1 þ aÞA1 þ bðk Þ ¼ A1 þ þ bðk Þ : n h1 h1 þ ðn 1Þh2

(39)

When (20) holds, (39) can be rearranged so as to obtain (21). Once both (36) and (37) are satisfied, to obtain the channel coordination it is enough to choose nc so as to satisfy (38), too.

Proof of Proposition 2 The win–win condition is assured if each actor incurs lower cost under the contract setting than under the decentralized setting. Thus, it follows that:

Cc1 qc1 ; kc ; a; d C1 qd1 ; kd : Cc2 qc1 ; nc ; a; d C2 qd1 ; nd

(40)

Let us remember that, when the actors agree on the additional ordering cost contract, the expected annual retailer’s and supplier’s costs can be expressed in terms of the cost they sustain in default of the contract and the transfer payment, as shown in (17) and (18). Therefore, (40) can be written also as:

Coordination of the Supplier–Retailer Relationship in a Multi-period Setting

8 d d D > > < C1 q1 ; k dD þ aA1 q C1 q1 ; k 1 d d ; > D > : C2 q1 ; n þ dD aA1 C2 r1 ; n q1

253

(41)

which can be rearranged so as to obtain: C1 q1 ; k C1 qd1 ; kd C2 qd1 ; nd C2 q1 ; n aA1 aA1 þ d þ : q1 q1 D D

(42)

To prove that such a range for d is defined in a consistent domain, we consider the first and third members of (42), which can be rearranged to obtain: C1 q1 ; k þ C2 q1 ; n C1 qd1 ; kd þ C2 qd1 ; nd :

(43)

C q1 ; k ; n C qd1 ; kd ; nd :

(44)

which means:

The inequality above is always true by definition.

References Axs€ater S (1990) Simple solution procedures for a class of two echelon inventory problems. Oper Res 38(1):64–69 Axs€ater S (1993) Exact and approximate evaluation of batch-ordering policies for two level inventory systems. Oper Res 41(4):777–785 Axs€ater S, Rosling K (1993) Notes: installation vs. echelon stock policies for multilevel inventory control. Manage Sci 39(10):1274–1280 Cachon GP (2003) Supply chain coordination with contracts. In: Graves S, de Kok T (eds) Handbooks in operations research and management science: supply chain management. North-Holland, Amsterdam Cachon GP, Lariviere MA (1999) Capacity choice and allocation: strategic behaviour and supply chain performance. Manage Sci 45(8):1091–1108 Cachon GP, Lariviere MA (2005) Supply chain coordination with revenue sharing contracts: strength and limitations. Manage Sci 51(1):30–44 Cachon GP, Zipkin PH (1999) Competitive and cooperative inventory policies in a two stage supply chain. Manage Sci 45(7):936–953 Chen F (1999) Decentralized supply chains subject to information delays. Manage Sci 45(8): S221–S234 Christopher M (1992) Logistics and supply chain management. Pitman, London De Bodt MA, Graves SC (1985) Continuous review policies for a multi-echelon inventory problem with stochastic demand. Manage Sci 31(10):1286–1299 Emmons H, Gilbert S (1998) Return policies in pricing and inventory decision for catalogue goods. Manage Sci 44(2):276–283

254

N. Bellantuono et al.

Eppen GD, Iyer AV (1997) Backup agreements in fashion buying: the value of upstream flexibility. Manage Sci 43(11):1469–1484 Federgruen A (1993) Centralized planning models for multi-echelon inventory systems under uncertainty. In: Graves S, Rinnooy Kan A, Zipkin P (eds) Logistics of production and inventory. North Holland, Amsterdam, pp 133–173 Giannoccaro I, Pontrandolfo P (2004) Supply chain coordination by revenue sharing contracts. Int J Prod Econ 89:131–139 Hadley G, Whitin TM (1963) Analysis of inventory systems. Prentice-Hall, Englewood Cliffs, NJ Lee H, Padmanabhan V, Whang S (1997) Information distortion in a supply chain. The bullwhip effect. Manage Sci 43(4):546–558 Lee H, Whang S (1999) Decentralized multi-echelon supply chains: incentives and information. Manage Sci 45(5):633–639 Li J, Liu L (2006) Supply chain coordination with quantity discount policy. Int J Prod Econ 101:89–98 Mitra S, Chatterjee AK (2004) Echelon stock based continuous review (R, Q) policy for fast moving items. Omega 32:161–166 Monahan JP (1984) A quantitative discount pricing model to increase vendor profits. Manage Sci 30(6):720–726 Narayanan VG, Raman A (2004) Aligning incentives in supply chains. Harv Bus Rev 82 (11):94–102 Pasternack BA (1985) Optimal pricing and returns policies for perishable commodities. Mark Sci 4(2):166–176 Porteus E (2000) Responsibility tokens in supply chain management. Manuf Serv Oper Manage 2:203–219 Roundy R (1985) 98%-effective integer-ratio lot-sizing for one-warehouse multi-retailer systems. Manage Sci 31(11):1416–1430 Shang KH, Song JS (2003) Newsvendor bounds and heuristic for optimal policies in serial supply chains. Manage Sci 49(5):618–638 Schneeweiss C (2003) Distributed decision making – a unified approach. Eur J Oper Res 150:237–252 Schwarz LB, Schrage L (1975) Optimal and system myopic policies for multi-echelon production/ inventory systems. Manage Sci 21(11):1285–1294 Schweitzer ME, Cachon GP (2000) Decision bias in the newsvendor problem with a known demand distribution: experimental evidence. Manage Sci 46(3):404–420 Silver EA, Pyke DF, Peterson R (1998) Inventory management and production planning and scheduling. Wiley, New York Tang CS (2006) Perspectives in supply chain risk management. Int J Prod Econ 103(2):451–488 Tsay AA (1999) Quantity-flexibility contract and supplier–customer incentives. Manage Sci 45 (10):1339–1358 Tsay AA, Lovejoy WS (1999) Quantity flexibility contracts and supply chain performance. Manuf Serv Oper Manage 1(2):89–111 Tsay AA, Nahmias S, Agrawal N (1999) Modeling supply chain contracts: a review. In: Tayur S, Ganeshan R, Magazine M (eds) Quantitative models for supply chain. Kluwer, Norwell, MA, pp 299–336 Wadsworth GP (1959) Probability. In: Wadsworth GP (ed) Notes on operations research. Technology Press, Cambridge, MA Wang Q, Tsao D (2006) Supply contract with bidirectional options: the buyer’s perspective. Int J Prod Econ 101:30–52 Whang S (1995) Coordination in operations: a taxonomy. J Oper Manage 12:412–422

Use of Supply Chain Contract to Motivate Selling Effort Samar K. Mukhopadhyay and Xuemei Su

Abstract Selling of a product is often delegated by the Original Equipment Manufacturers (OEM) to another firm called sales agent. The OEM needs to devise a mechanism to motivate the agent to exert higher marketing effort in order to boost her sales revenue. She also needs to design a profit allocation scheme, a complex task because of the fact that she has incomplete information about the agent’s marketing cost. In this chapter, two important contract forms are analyzed, compared and the OEM’s strategy are developed. Closed form solutions have been derived for three decision variables: marketing effort, order quantity and retail price for both forms of contracts. The revelation principle has been applied in that derivation which find inefficiency and “distribution distortion” due to information asymmetry. We show that the two contract forms perform differently, and each party’s preference toward a particular contract form is linked with the total reservation profit level and/or the sales agent’s cost type. We find that full trading opportunity, as in the full information case, cannot be achieved by any of the two contracts and the OEM suffers due to information deficiency. The chapter also identifies guidelines for the OEM to exert higher control or be more flexible. Further research avenues are also identified. Keywords Distribution channel • Game theory • Retail contracts • Sales agent • Supply chain

S.K. Mukhopadhyay (*) Graduate School of Business, Sungkyunkwan University, Jongno-Gu, Seoul 110–745, South Korea e-mail: [email protected] X. Su College of Business Administration, California State University Long Beach, 1250 Bellflower Blvd, Long Beach, CA 90840, USA e-mail: [email protected] T.-M. Choi and T.C. Edwin Cheng (eds.), Supply Chain Coordination under Uncertainty, International Handbooks on Information Systems, DOI 10.1007/978-3-642-19257-9_11, # Springer-Verlag Berlin Heidelberg 2011

255

256

S.K. Mukhopadhyay and X. Su

1 Introduction A common practice among Original Equipment Manufacturers (OEM) is to delegate the sales of the product to another firm variously called sales agent, franchisee, or sales representative. The motivation from the OEM’s point of view is to concentrate their effort to product design and manufacturing while leaving the sales and marketing to another firm with suitable expertise. This mode is one of the prominent methods in product distribution (Kaufmann and Dant 2001). This is especially true in global distribution when the OEM wants to introduce their products in foreign markets. In that situation, a local firm with its knowledge of local market would be invaluable and in some cases, inevitable. The sales agent provides services like presale advice, after sales service and advertising. These services are selling efforts that would enhance demand for the product. In the industrial goods market, this will also include customer information sessions, product demonstrations, trade shows and so on. It is, therefore, common for an OEM to use incentives to increase an agent’s effort (Lafontaine and Slade 1997). These incentives are formalized in sales contracts between the OEM and the sales agent. Two common types of contracts used in supply chain franchising are the Franchise Fee (FF) contract and the Retail Price Maintenance (RPM) contract. The FF contract is characterized by a variable wholesale price per unit and a fixed franchise fee. Thus, the FF contract is a two-part-tariff contract. The agent is free to set the retail price. When the RPM contract is employed, it is the OEM that sets the retail price and also the order quantity. Then a cost-plus payment from the agent to the OEM is specified. We study these two types of contracts in this chapter. Other forms of contracts that are used in a supply chain are revenue-sharing contract (Foros et al. 2009; Cachon and Lariviere 2005), quantity discounts contract (Raju and Zhang 2005), channel rebates contract (Taylor 2002), buy-back contract (Zhao et al. 2010), quantity flexibility contract (Krishnan et al. 2004; Tsay 1999) and optimal contracts via mechanism design (Laffont and Martimort 2000; Watson 2007). This chapter investigates how an OEM can use the FF and RPM contracts to motivate the sales agent to put in more efforts which in turn increases the demand for her product and thus her revenue. To design an effective contract, a number of parameters are needed to be specified. Note that the agent’s sales effort cannot be effectively monitored and therefore cannot be put in the contract as a parameter. We also recognize the fact that the agent’s cost of selling effort is only known to himself. So the OEM designs the contract without this information. We will devise the optimal contract design by the OEM under this information asymmetry and will identify the conditions under which one type of contract is preferred over the other. Our model includes “reservation profits” for both the OEM and the sales agent. The reservation profit of each party is the level of the profit they expect from their respective outside opportunities. The sales agent, therefore, would refuse to enter into a contract with the OEM if the expected profit under any contract is less than his reservation profit. The same is true for the OEM. As will be seen later, we

Use of Supply Chain Contract to Motivate Selling Effort

257

uncover an important role for the total of two reservation profits. We find that given the information asymmetry suffered by the OEM, her preference to one contract form or the other depends on the total reservation profits. Also, when the total is within certain ranges, the OEM has a dominant strategy because her preference does not depend on the agent’s cost. This book chapter is based on the authors’ original work of Mukhopadhyay et al. (2009). Some of the research work that studied contract design in the presence of an agent’s service effort are cited below. Desai and Srinivasan (1995) investigate a franchising channel where an informed principal (the contract designer) signals the market demand to an agent whose effort cannot be monitored. Unlike their paper, we assume that the contract designer is less informed and a screening game is played. Desiraju and Moorthy (1997) study how the requirements set by the manufacturer on retail price or service or both may improve the working of a distribution channel. The agent’s service is not contractible in our study. Blair and Lewis (1994) investigate optimal retail contracts that can be used by the manufacturer to encourage dealer promotion, and conclude that the optimal contract exhibits a form of resale price maintenance and quantity fixing. Our study develops new insights for helping an OEM to make a judicious choice between two contract forms under different conditions. Chen et al. (2010) study the coordination mechanism for the supply chain with leadtime consideration and price-dependent demand. Zhu and Mukhopadhyay (2009) study contract design in call-center outsourcing where the agent determines the service level. Dukes and Liu (2010) study the effects of retailer in-store media (ISM) on distribution channel relationships. They show that ISM is important in coordinating a distribution channel on advertising volume and product sales. Information asymmetry is considered in research by Gal-Or (1991). In that study the retailer has private information about demand and retailing cost. Buyer’s marginal cost is private information in the study by Ha (2001). Better information as value to the supplier is characterized by Corbett et al. (2004). Supplier’s cost is private information in Gan et al. (2005) who find that supply chain coordination can be achieved only when the supplier’s reservation profit decreases with production cost. Co-ordination can be achieved, as Krishnan et al. (2004) find, when buy backs can be combined with promotional cost-sharing agreements. In Cakanyildirim et al. (2008), production cost is private information. The retailer designs a menu of contracts specifying the order quality and profit percentage. Yang et al. (2009) study a manufacturer that faces a supplier privileged with private information about supply disruptions. Information asymmetry is also studied by Mukhopadhyay et al. (2008) and Su et al. (2010) in dual-channel distribution. Our study includes the effect of the OEM’s incentive to motivate the agent’s effort to increase sales. Agent compensation literature typically includes moral hazard (selling effort not observable to the firm) and adverse selection (the salesperson has superior information about the market prior to contracting with the firm). Kreps (1990) and Laffont and Martimort (2001) devise a menu of contracts offered to the agent as a typical solution to these types of problems. Laffont and Tirole (1986) and Gibbons (1987) show that in some cases a menu of linear contracts would be optimal. Chen (2005)

258

S.K. Mukhopadhyay and X. Su

studies Gonik’s (1978) scheme and compares it with a menu of linear contracts in a model where the market information possessed by the salesforce is important for the firm’s production and inventory-planning decisions. Ayra et al. (2009) study a Quasi-Robust multiagent model in which the mechanism must be designed before the environment is as well understood as is usually assumed. Wu et al. (2008) argue that people not only care about outcomes, but also about the process that produces these outcomes. They analytically show why fair process is not always used even though fair process enhances both employee motivation and performance. A comprehensive review of salesforce compensation problems can be found in Coughlan (1993). An emerging research stream studies contracting in a complex supply chain with multiple manufacturers and/or multiple retailers. Cui et al. (2008) proposes a trade promotion model that can price discriminate between a dominant retailer and small independents. Krishnan and Winter (2010) study a channel where a manufacturer distributes a product through retailers who compete on both price and fill rate. Cachon and Kok (2010) study a contracting scenario where multiple manufacturers compete for a retailer’s business, and conclude that the same contractual form can exhibit quite different properties from that seen in a one-manufacturer supply chain. Majumder and Srinivasan (2008) show that contract leadership, as well as the position in the supply chain network, affect the performance of the entire supply chain. This chapter is organized as follows. Section 2 introduces our model and derives contracts under full information. Section 3 derives the contracts under asymmetric information, compares the two forms of contracts and discusses the OEM’s strategies. Section 4 concludes the chapter including some avenues for the future research.

2 Contract Under Full Information Our supply chain consists of an OEM who sells her product through a sales agent. The sales agent uses a selling effort denoted by e aimed at increasing demand. We assume that the cost of exerting marketing effort is a convex, increasing function of e, say 12 ke2 . The constant k denotes the agent’s cost type and reflects how efficiently the agent conducts the marketing effort. The OEM’s unit production cost is s. The reservation profits of the OEM and the agent are pM and pR respectively. The reservation profits are lowest level of profits expected by the parties and represent the amount of profits that can be obtained from outside opportunities. Thus, neither party would enter the contract if the expected profit is below their respective reservation profit. The sales agent can choose to either sign the contract or reject it. Negotiation is not allowed. The demand function is: q ¼ a bp þ e

(1)

Where p is the retail price, a is the base demand that depends on factors not included in our model, and b is the sensitiveness of demand with respect to price.

Use of Supply Chain Contract to Motivate Selling Effort

259

Notice that the sales effort e positively impacts the demand. a and b are constants and common knowledge. The linear demand function is widely used in the literature (Desiraju and Moorthy 1997; Lal 1990; Gal-Or 1991).

2.1

The FF and RPM Contracts

In this subsection, we will model the two types of contracts. 2.1.1

FF Contract

In the Franchise Fee (FF) contract, the OEM specifies the unit wholesale price w and a fixed fee L, paid by the agent. Wimmer and Garen (1997) and Federal Trade Commission’s Guide (2005) gives a comprehensive guide of franchising and its fees. L can possibly be negative in which case it is the OEM who makes the payment to the agent, presumably to subsidize the agent’s marketing effort. Positive L, though, is more common. The agent’s total payment to the OEM is wq + L for an order quality q. The OEM’s profit is, pM ¼ ðw sÞq þ L

(2)

1 pR ¼ ðp wÞq ke2 L 2

(3)

The agent’s profit is,

2.1.2

RPM Contract

In Retail Price Maintenance (RPM) Contract, the OEM specifies the retail price. These types of contracts are widely adopted in practice, for example, in the fashion and luxury goods industry, companies such as Gucci set the retail price of their goods for sale through both vertically integrated and independent retailers. Nike requires its retailers to not sell their shoes below a suggested retail price (Gurnani and Xu 2006). RPM contracts are characterized by three parameters: the retail price p, the order quantity q, and a cost plus payment amount R. Thus the total payment to OEM is s q þ R, where s q covers the OEM’s total production cost, and R is her profit. If the agent decides to accept the contract, each party’s profit is pM ¼ R

(4)

1 pR ¼ pq ke2 sq R 2

(5)

260

2.2

S.K. Mukhopadhyay and X. Su

The Full Information Case

In this section, we derive the optimal contract under full information. Here, the agent’s cost type k is common knowledge. The OEM maximizes her profit by maximizing the total channel profit and letting the agent earn his reservation profit pR , thereby extracting the rest of the channel profit for herself. The joint profit maximization function is: 1 Max pT ðp; q; e; kÞ ¼ ðp sÞq ke2 p;q;e 2

(6)

Equation (6) is maximized to obtain the optimum values of the decision variables as follows: e ¼

ða bsÞ ; 2bk 1

q ¼

bkða bsÞ ; 2bk 1

p ¼

kða þ bsÞ s ; 2bk 1

pT ¼

kða bsÞ2 2ð2bk 1Þ

These optimum values are the “first best” solutions, because other solutions, due to information asymmetry, would be inferior to these solutions. With full information, the OEM can maximize the channel profit by specifying that the sales agent adopt the fist-best solutions for marketing effort level, sales level and retail price. Then, the OEM can extract the whole channel profit by specifying L (in case of FF contract) or R (in case of RPM contract). It is intuitive that these first best solutions are all decreasing in k. This means that a cost-inefficient agent cannot provide optimal marketing effort, leading to optimal sales, and costumers would not likely pay a high price for a low service level. An inefficient sales agent, therefore, brings sluggish channel profit. Obviously, the two parties will enter into a contract only if pM þ pR pT , which mandates that the agent’s cost type k bT . bT , a threshold value, is called the cutoff point, and is derived as: bT ¼

2ðpM þ pR Þ 4bðpM þ pR Þ ða bsÞ2

We will use, “1” and “2” as subscripts or superscripts to represent FF contract and RPM contract respectively. Table 1 shows the solutions for both contract forms. Table 1 Equilibrium results under complete information

FF contract w¼s L ¼ pT pR b1 ¼ bT pT pR if k bT p1M ¼ pM if bT k p1R ¼ pR

RPM contract p2 ¼ p ; q2 ¼ q R ¼ pT pR b2 ¼ bT pT pR if k bT p2M ¼ pM if bT k p2R ¼ pR

Use of Supply Chain Contract to Motivate Selling Effort

261

3 The Asymmetric Information Case In this section, we consider the problem of contract design when k is unknown to the OEM. She only has a prior knowledge that k is somewhere within the range [k; k], with a distribution denoted by FðkÞ. f ðkÞ is the probability density function. Without knowing the exact value of k, the OEM has no way of determining optimal values of the parameters w, L, and R. In these cases, it is customary to offer a “menu” of contracts. This menu is a set of options for the agent to choose from based on his cost type k, known only to himself.

3.1

FF Contract Menu

The OEM’s menu of contracts consists of a number of tuples {w(k),L(k)}, each item consisting of parameters w and L for a given value of k. By virtue of the “revelation principle” (Myerson 1979), the OEM hopes that the agent would declare the true value of k because the menu is designed in such a way that a truthful revelation of the information would yield highest profit for the agent. Define pR ðk~jkÞ as the profit ~ LðkÞg ~ from the of an agent who is of a cost type k and chooses a contract fwðkÞ; menu. The agent solves the problem: ~ wðkÞÞ ~ q/ ðwðkÞÞ ~ 1 k e/ ðwðkÞÞ ~ 2 LðkÞ ~ R1 : Max pR ðk~jkÞ :¼ ðp/ ðwðkÞÞ 2 k~ ~ 2 kða bwðkÞÞ ~ ¼ LðkÞ 2ð2kb 1Þ Where “k” is the true cost type, and k~ is the announced cost type by the agent. ~ as obtained by using the First p ; q/ and e/ are the agent’s best responses to wðkÞ Order Condition (FOC) on (3). Revelation principle requires that pR ðk~jkÞ be concave in k~ and achieves the maximum at k~ ¼ k. Only then it will be to the agent’s interest to reveal k. Depending on the range of k and the value of b, some common types of distributions like uniform, beta and truncated normal meet this requirement. We, therefore, can write the OEM’s problem as /

ð b1 M1 Max

wðkÞ;LðkÞ

k

pM ðk; qðkÞÞf ðkÞdk þ

ðk b1

pM f ðkÞdk

(7)

S:t: IC : qðkÞ ¼ arg max pR ðk; qÞ

(8)

1 IR : pR ðk; qðkÞÞ ¼ ðp wðkÞÞqðkÞ ke2 LðkÞ pR 2

(9)

262

S.K. Mukhopadhyay and X. Su

pM ðk; qðkÞÞ ¼ ðwðkÞ sÞqðkÞ þ LðkÞ

(10)

The OEM profit in (7) depends on the quantity q ordered by the agent, which in turn depends on the agent’s cost type k. Constraint (8) is called the agent’s “Incentive-Compatibility” constraint. This constraint ensures that, an agent with cost type k will choose q to maximize its profit. Constraint (9) represents the agent’s “Individual Rationality” constraint. This states that the agent’s profit must be no less than his reservation profit pR for him to agree to trade. The quantity b1 in the objective function (7) is a value of k 2 ½k; k, such that when k ¼ b1 either of the two parties hits their respective reservation profit. Therefore, for the values k > b1 , no contract is signed between the two parties and the OEM would earn her reservation profit (pM ) elsewhere. As pM is a decreasing function of k (see Corollary 1c,) we would have pM ðk; qðkÞÞ pM for k b1 . The formulation given in (7) through (10) fits the optimal control formulation with variable endpoint conditions and salvage value (see Kamien and Schwartz 1981, pp. 143–148). We use the methodology therein to solve the problem M1 . The solution to this problem is given in proposition 1. Proofs of all propositions, unless otherwise stated, are given in the Appendix. We use the notation x?½l; u :¼ max fl; min fx; ugg as the projection of x on the interval [l; u]. Proposition 1. Under asymmetric information the optimal values of the OEM’s parameters in the franchise fee contract is given by: w¼sþ

ða bsÞFðkÞ bFðkÞ þ bkð2bk 1Þf ðkÞ

L(k) is given by the solution of @L bk2 ða bsÞf ðkÞ @w ¼ @k FðkÞ þ kð2bk 1Þf ðkÞ @k Lðb1 Þ

satisfies

Lðb1 Þ ¼ pM

8k k b1

ða bsÞ2 b21 Fðb1 Þf ðb1 Þ 2ðFðb1 Þ þ b1 ð2bb1 1Þf ðb1 ÞÞ2

The resulting cutoff point is given by b1 ¼ b0 ?½k; k where b0 is the solution of: 2ðpM þ pR Þ b2 f ðbÞ ¼ FðbÞ þ bð2bb 1Þf ðbÞ ða bsÞ2 The second column of Table 2 gives a summary of equilibrium results for FF contract. Corollary 1 summarizes major properties of the equilibrium results.

Use of Supply Chain Contract to Motivate Selling Effort

263

Corollary 1. Under the FF contract with asymmetric information, (a) As k increases, w increases, and L decreases (b) At equilibrium, e1 ; q1 and p1 are all decreasing in k, and e1 < e ; q1 < q ; p1 > p , for any k 2 ðk; k. (c) pM ðkÞ and pR ðkÞ are decreasing in k, until the cutoff point b1 ; where b1 bT . (d) For any k 6¼ k, the agent’s profit is higher and the OEM’s profit is lower compared to their counterparts under full information. Total channel profit is lower than that under full information. We observe the following from Corollary 1. A higher fixed fee is associated with a lower unit wholesale price and vice versa. As seen in Fig. 1. As also reported by Wimmer and Garen (1997), factors that increase the franchisee’s effort (e here), would lower the recurring fee (w here), and increase the franchise fee (L here). The insight here is that a cost-efficient sales agent (with low k) will enjoy a discounted wholesale price, would exert higher marketing effort to gain higher demand, and can charge customers a higher price. All of these actions would contribute to higher profits for both the OEM and the agent. This is an important finding of this chapter. We also see that the cut-off point, if it exists, is unique. It is possible for the franchise fee to go negative for a high cost type agent. In that case, the “franchise fee” is from the OEM to the sales agent meaning that the OEM subsidizes an inefficient sales agent, or simply because the effort required is costly, and the OEM offers to cover part of the investment. Proposition 1 requires the revelation principle to work. The design of the menu of contract must ensure that w is increasing in k, and L is decreasing in k. This will make w > s (the production cost), giving rise to the double marginalization problem. This double marginalization phenomenon was first identified by Spengler (1950). In our case, double marginalization is reflected as higher retail price, lower 250

L

200

150

100

50

0 2.00

2.17

2.35

2.52

2.69

2.87 k

Fig. 1 Optimal w and L for varying k

3.04

3.21

3.39

3.56

w

264

S.K. Mukhopadhyay and X. Su

Table 2 Results of FF and RPM contracts under asymmetric information FF contract: fwðkÞ; LðkÞg Unit transfer price Retail price Marketing effort Sales Total channel profit

RPM contract: fpðkÞ; qðkÞ; RðkÞg

ða bsÞFðkÞ bFðkÞ þ bkð2bk 1Þf ðkÞ ða bsÞðFðkÞ þ bk2 f ðkÞÞ p1 ¼ s þ bFðkÞ þ bkð2bk 1Þf ðkÞ ða bsÞkf ðkÞ e1 ¼ FðkÞ þ kð2bk 1Þf ðkÞ ða bsÞbk2 f ðkÞ q1 ¼ FðkÞ þ kð2bk 1Þf ðkÞ

sþ

p1T ¼

s ða bsÞðFðkÞ þ kf ðkÞÞ 2bðFðkÞ þ kf ðkÞÞ f ðkÞ ða bsÞf ðkÞ e2 ¼ 2bðFðkÞ þ kf ðkÞÞ f ðkÞ ða bsÞbðFðkÞ þ kf ðkÞÞ q2 ¼ 2bðFðkÞ þ kf ðkÞÞ f ðkÞ p2 ¼ s þ

ða bsÞ2 k2 f ðkÞð2FðkÞ þ kð2bk 1Þf ðkÞÞ 2ðFðkÞ þ kð2bk 1Þf ðkÞÞ

2

p2T ¼

ða bsÞ2 ðFðkÞ þ kf ðkÞÞ 4bðFðkÞ þ kf ðkÞÞ 2f ðkÞ

ða bsÞ2 ðk þ zðkÞÞ 4bðk þ zðkÞÞ 2 ð 1 k ða bsÞ2 dx pR 2 k ð2bðx þ zðxÞÞ 1Þ2 ðk 1 ða bsÞ2 p2R ¼ pR þ dx 2 k ð2bðx þ zðxÞÞ 1Þ2 p2M ¼

Profit of the OEM

p1M ¼

Profit of the sales agent p1R ¼

ða bsÞ2 k2 FðkÞf ðkÞ ðFðkÞ þ kð2bk 1Þ f ðkÞÞ2 2 3

ða bsÞ k ð2bk 1Þ f ðkÞ

þL

2

2ðFðkÞ þ kð2bk 1Þ f ðkÞÞ2

L

sales, and less marketing effort compared to the first-best. The inefficiency caused by information asymmetry is further reflected on the cutoff point. Since b1 bT , the FF contract cannot fully explore all the trading opportunities as presented under the full information case. Part (d) of the corollary shows that the OEM is worse off and the sales agent is better off under asymmetric information. This phenomenon is called information rent, that is, the benefit earned due to holding private information. To visualize this, we use the results in the Table 2 to draw Fig. 2, to show the variation of p1M pM ; p1R pR and p1T pT with respect to k respectively. Of these values, p1R pR measures the information rent. As we can see from Fig. 2, the lower the cost type, the higher the information rent. Gal-Or (1991) refers to this as a “distributional distortion”, since it relates to the distribution of the surplus between the OEM and the agent. Obviously, the channel as a whole is worse off compared to the full information case. Figure 2 shows that p1T pT is negative and decrease as k increases, but almost close to zero, which means the channel’s profit loss due to information asymmetry is trivial. We will further discuss it in Sect. 3.3.

3.2

The RPM Contract Menu

In the RPM contract, the menu consists of a tuple fpðkÞ; qðkÞ; RðkÞg. Each item on the menu is intended for an agent of a specific cost type. The profit of the agent of cost type k declaring a cost type k~ is given by ~ sÞ qðkÞ ~ 1 keðkÞ ~ ¼ pT ðpðkÞ; ~ qðkÞ; ~ kÞ RðkÞ ~ ~ 2 RðkÞ pR ðk~jkÞ ¼ ðpðkÞ 2

Use of Supply Chain Contract to Motivate Selling Effort

265

250 200 150 100 50 0 2.00 -50

2.17

2.35

2.52

2.69

2.87 k

3.04

3.21

3.39

3.56

-100 -150 -200 Series 2

-250

Series 1

Series 3

Fig. 2 Information rent under FF contract1

Where pT is defined in function (6). The OEM’s problem M2 of designing the optimal contract is: ð b2 M2

max

pðkÞ;qðkÞ;RðkÞ

RðkÞf ðkÞdk þ

k

ðk b2

pM f ðkÞdk

~ pR ðkjkÞ ~ S:t: pR ðkÞ pR ðk~jkÞ and pR ðkÞ

(11)

k k; k~ k;

pR ðkÞ pR qðkÞ 0 Lemma 1 provides a characterization of problem M2, to be used for deriving the optimal contract menu. Lemma 1. A solution fpðkÞ; qðkÞ; RðkÞg is feasible for problem ðM2 Þ if and only if Rk (a) pR ðkÞ ¼ pR þ 12 k eðxÞ2 dx (b) eðkÞ is decreasing in k (c) qðkÞ 0 The design problem M2 can then be reformulated as: ðk max

pðkÞ;qðkÞ0

k

ðpT ðpðkÞ; qðkÞ; kÞ

zðkÞ eðkÞ2 ÞdFðkÞ 2

Data series 1, 2, 3 denote p1M pM ; p1R pR and p1T pT respectively.

1

(12)

266

S.K. Mukhopadhyay and X. Su

We need two constraints: (1) qðkÞ is nonnegative and (2) eðkÞ is decreasing in k (Proof is provided in the Appendix). zðkÞ is defined as zðkÞ ¼ FðkÞ=f ðkÞ. If we ignore the feasibility requirements, we can maximize the integrand in function (12) for each k like in (6) with a cost factor k þ zðkÞ for the agent. So, we substitute pðkÞ ¼ p ðk þ zðkÞÞ and qðkÞ ¼ q ðk þ zðkÞÞ where p and q are the first-best solutions. To have a feasible solution, we need eðk þ zðkÞÞ to be decreasing in k and zðkÞ ¼ FðkÞ=f ðkÞ increasing in k. We now consider the contract menu ^ k k kg with f^ pðkÞ; q^ðkÞ; RðkÞ q^ðkÞ ¼ q ðk þ zðkÞÞ p^ðkÞ ¼ p ðk þ zðkÞÞ 1 ^ ¼ pT ð^ pðkÞ; q^ðkÞ; kÞ RðkÞ 2

ðk k

e^ðxÞ2 dx pR

The optimal solutions are shown in Proposition 2. Proposition 2. If zðxÞ ¼ FðxÞ=f ðxÞis increasing in x, ^ (a) f^ pðkÞ; q^ðkÞ; RðkÞg is the optimal solution to M2 ^ðkÞs (b) p^ðkÞ; q^ðkÞ; and e^ðkÞ, are all decreasing in k, and e^ðkÞ ¼ pkþzðkÞ < e ðkÞ; q^ðkÞ < q ðkÞ; p^ðkÞ < p ðkÞ for any k 2 ðk; k (c) pM ðkÞ and pR ðkÞ are decreasing in k until the cutoff point. (d) The agent’s profit is higher and the OEM’s profit is lower compared to their counterparts under full information. Total supply chain profit is lower than that under full information. (e) The cutoff point b2 ¼ b0 ?½k; k where b0 is the solution of: b þ zðbÞ 2bðb þ zðbÞÞ 1

ðk b

dx ð2bðx þ zðxÞÞ 1Þ

2

¼

2ðpM þ pR Þ ða bsÞ2

The results are shown in the third column of Table 2. Under RPM contract, the OEM makes all the decisions using a cost factor of k þ zðkÞ instead of k. Compared to the first best solution, the OEM is worse off and the agent is better off; the agent orders less and exerts less marketing effort; and the retail price is lower. The channel as a whole is also worse off, for every cost type k. We show the variation of p2M pM , p2T pT and p2R pR with respect to k in Fig. 3. Like earlier, the information rent decreases with the agent’s cost type k. Both parties’ profits and the total channel profit are monotonically decreasing in k till k ¼ b2 . The cutoff point b2 bT . This shows that the RPM contract cannot fully explore all the trading opportunities compared to the full information case. There are two insights from the monotonic property: a low cost type agent benefits both the agent and the OEM, and therefore the channel; and, that the cutoff point is unique.

Use of Supply Chain Contract to Motivate Selling Effort

267

60

40

20

0 2.00

2.17

2.35

2.52

2.69

2.87 k

3.04

3.21

3.39

3.56

-20

-40

-60

Series1

Series2

Series3

Fig. 3 Information rent under RPM contract2

3.3

FF and RPM Contracts Compared

In the above analysis, we found that, under information asymmetry, both FF and RPM contract forms induce less marketing effort provisions, realize less sales, and generate lower channel profit compared to the full information case. Also, neither contract explores all the trading opportunities. Under full information, the trade stops only when both are earning no more than their respective reservation profits. However, under asymmetric information, the trade stops as soon as either party hits their respective reservation profit first. We will now compare these two contract forms, using subscripts 1 and 2 to denote the result of FF and RPM contracts respectively. Also, to simplify notations we use p2 ; q2 , and e2 for the RPM contract, instead of p^ðkÞ; q^ðkÞ; and e^ðkÞ respectively. The comprehensive comparison of the two forms of contracts is given in Corollary 2, while Fig. 4 shows a visual portray. Corollary 2. (a) Marketing level comparison: e e1 e2 . FF Contracts exerts more marketing effort than RPM contracts and both are less than the first best effort. (b) Price comparison: p1 p p2 . Price is highest in FF contract. RPM contract price is lower than the first best price. (c) Sales level comparison: q q2 q1 . FF sales level is lowest of all. First best sales level is highest. Data series 1, 2, 3 denote p2M pM ; p2R pR and p2T pT respectively.

2

268

S.K. Mukhopadhyay and X. Su

50

p1

e1

q1

p2

e2

q2

45 40 35 30 25 20 15 10 5 0 2.00

2.17

2.35

2.52

2.69

2.87 k

3.04

3.21

3.39

3.56

Fig. 4 Optimal p, e and q for two contract forms

The equalities only hold when k ¼ k. An agent with a cost type k 6¼ k, given an FF contract, will price the product higher, provide higher marketing effort, but sell less compared to when given a RPM contract. Some more observations follow. 1. The FF contract provides a better mechanism to motivate marketing effort provisions. The sales agent is free to choose the marketing effort level and set the retail price. This flexibility motivates him to exert more marketing effort compared to that of an RPM contract, and enables him to charge higher retail price. This flexibility becomes even more critical when the agent is of a high cost type. Recall that, in the RPM contract, the OEM specifies that e2 ðkÞ ¼ e ðk þ zðkÞÞ which is greatly distorted down from the first-best level when k is high because zðkÞ increases in k and e decreases in k. In the FF contract, it is true that a high cost type agent will face a high unit wholesale price, but the detrimental effect of the increase in unit wholesale price is softened when the increase in the unit wholesale price is combined with the offer of a more dramatic decrease in franchise fee. As a result, the agent still has the room to exert reasonable level of marketing effort. The decrease in marketing effort is not seen as dramatic as it is in the RPM contract. Relatively speaking, FF contract provides the agent higherpowered incentives. 2. It is notable that higher retail price in an FF contract results from not only the higher marketing effort as seen above but also from “double marginalization”. The double marginalization hurts the channel profit and the OEM’s profit as well. In contrast, double marginalization is avoided in the RPM contract because the OEM acts like a central planner and dictates a retail price and order quantity to the agent. However, with incomplete information, this centralized decision is

Use of Supply Chain Contract to Motivate Selling Effort

269

not trouble free. With RPM contract, both the retail price and the marketing effort level are distorted down from the first-best solutions ðp2 < p and e2 < e Þ for inducing truthful information reporting. These distortions hurt the channel profit too. Further discussions will follow Corollary 3. 3. It seems counter-intuitive that the FF contract realizes lower sales level with more marketing effort compared to the RPM contract. The high retail price in the FF contract will shed light on this. It is to the benefit of the agent to increase profit through pricing higher, instead of selling more units under the FF contract. There are two reasons for this. One is that customers would like to pay higher price given better services. The other is the sales agent finds it harder to induce more demand through price cut, which is limited by the double marginalization problem. Next, we examine the effect on the channel profit for each of the two contract forms. Corollary 3 summarizes the results and Fig. 5 visually depicts the channel profit under each of the two contract forms. Corollary 3. With information asymmetry, the total channel profit for each contract form is equal at k ¼ k, and decreases monotonically with k. For any k > k, the total channel profit of an FF contract is higher than that of the RPM contract. The difference in channel profit increases as k increases. Note that the conclusion drawn in Corollary 3 is based on the assumptions presented earlier in Sects. 3.1 and 3.2. Those assumptions guarantee that the profit functions are concave and revelation principle can be applied. Corollary 3 shows that for any cost type k, the RPM contract generates less channel profit and this worsens when the agent is of a high cost type. This is because the way the RPM contract is designed. In a RPM contract, the OEM acts like a central planner and directly specifies the agent’s order quantity and retail price. The marketing effort

Profit--FF

Profit--RPM

500

450

400

350 2.00

2.17

2.35

2.52

2.69

2.87 k

3.04

Fig. 5 Channel profits under FF contract and RPM contract

3.21

3.39

3.56

270

S.K. Mukhopadhyay and X. Su

level is indirectly specified by the OEM through specifying order quantity and retail price according to (1). However, with asymmetric information, this tight control comes with a cost. It requires distorting price, output and marketing effort level to induce the agent’s truthful information reporting. Specifically, q^ðkÞ ¼ q ðk þ zðkÞÞ; p^ðkÞ ¼ p ðk þ zðkÞÞ; and e^ðkÞ ¼ e ðk þ zðkÞÞ. Note that q ; p and e are decreasing in k, and zðkÞ is increasing in k. As a result, e^ðkÞ, q^ðkÞ and p^ðkÞ will be quite off the first-best solutions of e ðkÞ, q ðkÞ and p ðkÞ if the agent is of a high cost type. As a result, the channel profit will be greatly reduced. Figure 3 displays that the channel profit loss, p2T pT ; is substantial when the agent is of a high cost type. By comparing Fig. 2 with Fig. 3, we can see that the channel profit loss is minimal with the FF contract. We do recognize that the channel profit gets hurt as double marginalization problem is introduced into the design of the FF contract. But because the agent has the freedom to choose its retail price, the agent has the motivation to exert more marketing effort so as to charge customers a higher price. In addition, due to the arrangement that the fixed fee charge decreases as the agent’ cost type k increases, even a high cost type retailer is encouraged to exert a reasonable amount of marketing effort. With this flexibility in design, the FF contract can better align the agent’s interests with that of the channel.

3.4

Profit Allocation Mechanisms and the OEM’s Strategy

We treat the allocation of channel profit as a two-phase process. Note that no matter which contract form is offered, each party has to get at least their reservation profit before they would enter a contract. First, each party takes their reservation profit (pM or pR ) out of the total channel profit; Second, the two parties share the rest of the channel profit, the allocation of which is dependent on the contract form. This profit allocation mechanism in FF contract is analytically complex. We do the analysis for the RPM contract. The term “allocable profit” is defined as the profit in excess of the total of reservation profits i.e. Allocable profit ¼ total channel profit ðpM þ pR Þ The optimal strategy for contract offering is guided by the value of the allocable profit and, therefore, the total reservation profit. This observation is one of the main contributions of this chapter. Recall that the agent’s profit under the RPM contract is, 1 pR ðkÞ ¼ pR þ 2

ðk

eðxÞ2 dx ðLemma 1ðaÞÞ

(13)

k

Ðk Then the agent’s marginal utility from entering the RPM contract is 12 k eðxÞ2 dx. For a particular k, the realized total channel profit is fixed (refer to Table 2), and the

Use of Supply Chain Contract to Motivate Selling Effort

271

Ðk agent’s share of the allocable profit, 12 k eðxÞ2 dx, is also a fixed amount. If pM þ pR is small, the allocable profit is large and the OEM’s share of allocable profit will be large by restricting the agent’s share. Therefore, the RPM contract is favorable to the OEM. When pMÐ þ pR is large, the allocable profit is small. But the OEM still k has to allocate 12 k eðxÞ2 dx to the agent. This makes the RPM contract less attractive to the OEM. We now study the question of what contract the OEM would prefer based on an agent’ cost type for a fixed pM þ pR . For a low k, the realized channel profit, and hence the allocable profit would be large. The agent would benefit because equilibrium eðkÞ is larger for a smaller k, and the agent’s share of allocable profit is higher according to (13). But as the allocable profit generated by this cost efficient agent is large, the OEM may benefit even more. Thus, the RPM contract is favorable to the OEM when k is small, and becomes less attractive when k is large. However, this discussion holds only when pM þ pR is moderately small. If pM þ pR is very large, the allocable profit is very small, leaving the OEM almost nothing even with a cost efficient agent. This is summarized below. Observation 1. For a given combination of ðpM þ pR ; kÞ, where k is a value below cutoff points, and for a large allocable profit, the profit allocation mechanism of a RPM contract is favorable to the OEM. The OEM should offer RPM contract to the agent. For small allocable profits, the OEM should offer FF contract to the agent. When k is high enough so that either b1 or b2 comes into play, the monotonic property of either party’s profit with respect to k will be interrupted. Each party’s preference toward a certain contract type may change accordingly. Figure 6 summarizes the impact of cutoff points, pM þ pR and k on the OEM’s choice of a certain contract form. Figure 6 depicts the OEM’s preferred contract forms as a function of pM þ pR . It plots the tracks of kM , b1 and b2 where kM is a possible cost type of the agent, at which the OEM switches its preference between the two contract forms. Below kM , the combinations of ðpM þ pR ; kÞ result in adequate allocable profit, which makes the RPM contract attractive to the OEM. Above the tracks of b1 and b2 , no trading is possible either because the total reservation profit is too high, or because the agent is too inefficient, or both. For the combinations of ðpM þ pR ; kÞ which are between the tracks kM and b1 , the FF contract is more attractive to the OEM. For the small area above the track of b1 but below the track of b2 , the OEM prefers RPM contract because no trading is possible for FF contract.

3.5

The OEM’s Dominant Strategy

The OEM’s choice of a certain contract form are dependent on the different combinations of ðpM þ pR ; kÞ and the cutoff points. However, the OEM does not know the agent’s cost type k at the time of deciding about the contract form. Recall the sequence of events. First, the OEM chooses a contract form, FF or RPM.

272

S.K. Mukhopadhyay and X. Su

pM + pR

Z

hY,Z

Y

hX,Y

X k

k increase

k

k

Preference is RPM contract

Track of kM

Preference is FF contract

Track of b1

Indifferent between the two contract forms

Track of b2

Fig. 6 The OEM’s preferred contract form

Second, for the selected contract form, the OEM provides a menu of contracts, each of which is intended for an agent of a particular cost type k. Finally, k is revealed when the agent selects one contract from that menu. It would, therefore, be practically meaningful to investigate if the OEM can make a choice between the two contract forms before knowing the agent’s cost type k. Noting that the total reservation profit pM þ pR is common knowledge, the area in Fig. 6 is divided into three regions X; Y and Z with two thresholds of X;Y and Y;Z . The threshold X;Y is the value of pM þ pR where kM ¼ k and Y;Z is the value of pM þ pR where kM ¼ k. The values of X;Y and Y;Z can be uniquely determined and X;Y < Y;Z . Proposition 3 summarizes the OEM’s dominant strategy. Proposition 3. If pM þ pR X;Y , the OEM prefers the RPM contract regardless of the value of k; If pM þ pR Y;Z , the OEM prefers the FF contract or no contract, regardless of the value of k. Proposition 3 provides some clear cut strategies for the OEM even when the agent’s marketing cost information is not known. For moderate level of total

Use of Supply Chain Contract to Motivate Selling Effort

273

reservation profit, i.e., X;Y < pM þ pR Y;Z the OEM’s optimal strategy depends on the unknown k, so any choice has some risks.

4 Conclusion and Further Research When an OEM is dependent on the sales agent for the marketing of her product, she needs to devise a mechanism to motivate the agent to exert higher marketing effort in order to boost her sales revenue. She also needs to design a profit allocation scheme, a complex task because of the fact that she has incomplete information about the agent’s marketing cost. In this chapter, two important contract forms are analyzed, compared and the OEM’s strategy are developed. Closed form solutions have been derived for three decision variables: marketing effort, order quantity and retail price for both forms of contracts. The revelation principle has been applied in that derivation which find inefficiency and “distribution distortion” due to information asymmetry. Full trading opportunity, as in the full information case, cannot be achieved by any of the two contracts and the OEM suffers due to information deficiency. The operational differences of the two contract forms and marginal guidelines are fully identified in this chapter. For example, the FF contract motivates more marketing effort and generates more channel profit. This chapter identifies the role of total reservation profit for selecting a suitable contract form. Figure 6 highlights the guidelines in selecting a contract form for varying k and different values of the total reservation profit. It also identifies ranges where the OEM’s strategy, surprisingly, does not depend on the value of k, thereby making the lack of cost information irrelevant. Double marginalization is a concern in supply chain coordination literature. This chapter finds some further insight into the problem. It is present in the FF contract, but not in the RPM contract. But, it has been proved that, in FF contract, channel profit is always higher, and so is the OEM’s profit under certain conditions. Therefore, double marginalization need not to be viewed as detrimental. As per the RPM contract is concerned, the OEM has higher control of even determining the agent’s order quantity and retail price. But it still does not guarantee higher profit for her – a fact that is counter intuitive. In this form of contract, larger value of k results in reduced channel profit and the allocable profit, and it hurts the OEM more than the agent. The chapter identifies guidelines for the OEM to exert higher control or to be more flexible. We now identify avenues for further research. We have identified regions in Fig. 6 where the OEM’s choice is not clear cut and a risk is involved in selecting a contract form. A coordination plan can be developed for such cases. Two possible directions could be to devise an incentive plan for the agent to divulge private information, and to offer both contract forms with the provision of compensation for the OEM. Contract forms, other than the two studied here, can also be examined and designed. A further research area will be examining multiple agents.

274

S.K. Mukhopadhyay and X. Su

Appendix Proof of Proposition 1 We can rewrite the OEM’s problem as, bð1

M1

mðkÞdk þ Fðb1 Þ

Max

wðkÞ;LðkÞ;b1 k

s:t: L ðkÞ ¼ g1 ðkÞ w ðkÞ ¼ g2 ðkÞ where mðkÞ :¼ ððw sÞq þ LÞf ðkÞ ¼ ððw sÞ bkðabwÞ 2bk1 þ LÞf ðkÞ g1 ðkÞ :¼

bkða bwÞ u1 2bk 1

g2 ðkÞ :¼ u1 ¼ w

and

Fðb1 Þ :¼ pM ð1 FðkÞÞ

Using the multiplier equations

@mðkÞ @g1 @g2 þ l2 Þ ¼ f ðkÞ þ l1 @L @L @L ) l1 ¼ FðkÞ

l1 ¼ ð

@mðkÞ @g1 @g2 þ l2 Þ þ l1 @w @w @w bkða bwÞ b2 k b2 k ¼ f ðkÞ ð Þ þ f ðkÞ ðw sÞ l1 u1 2bk 1 2bk 1 2bk 1

(14)

l2 ¼ ð

(15)

Using the optimality conditions @mðkÞ @g1 @g2 bkða bwÞ þ l2 ¼ 0 þ l1 þ l2 ¼ 0 ) l1 @u1 2bk 1 @u1 @u1

(16)

Because both pM and pR are decreasing in k (we will verify it later), IRM and IRR need only hold at k ¼ b1 . They will then be satisfied at all k b1 . Write: Kðb1 Þ :¼

b1 ða bwÞ2 Lðb1 Þ pR 0 2ð2bb1 1Þ

The transversality conditions then require that there exists p such that: l1 ðb1 Þ ¼

@F @K þp ¼ p @L @L

(17)

Use of Supply Chain Contract to Motivate Selling Effort

l2 ðb1 Þ ¼

275

@F @K a bw þp ¼ pbb1 @w @w 2bb1 1

mðb1 Þ þ l1 ðb1 Þg1 ðb1 Þ þ l2 ðb1 Þg2 ðb1 Þ pM f ðb1 Þ þ p

(18) @K ¼0 @b1

p 0; Kðb1 Þ 0; pKðb1 Þ ¼ 0

(19) (20)

Taking derivative on both sides of (16) and using (14)

) l2 ¼ FðkÞð

bða bwÞ

b2 ku1 bkða bwÞ Þ f ðkÞ þ 2 2bk 1 2bk 1 ð2bk 1Þ

(21)

Solving (21) and (15) ) w¼sþ

ða bsÞFðkÞ bFðkÞ þ bkð2bk 1Þf ðkÞ

From (19) ððwðb1 Þ sÞ

bb1 ða bwðb1 ÞÞ p ða bwðb1 ÞÞ2 þ Lðb1 ÞÞf ðb1 Þ pM f ðb1 Þ ¼0 2bb1 1 2 ð2bb1 1Þ2 2

@K 1 ÞÞ ¼ ðabwðb and p ¼ F) (Note that @b 2ð2bb1 1Þ2 1 Plug in wðb1 Þ

) Lðb1 Þ ¼ pM

ða bsÞ2 b21 Fðb1 Þf ðb1 Þ 2ðFðb1 Þ þ b1 ð2bb1 1Þf ðb1 ÞÞ2

(22)

or f ðb1 Þ ¼ 0 FðkÞ=f ðkÞ is increasing in k (one of the assumptions), so f ðb1 Þ ¼ 0 can only occur at b1 ¼ k or b1 ¼ k. For k < b1 < k, using p ¼ F(b1 Þ > 0 gives Kðb1 Þ ¼ 0, which, combined with (22) )

2ðpM þ pR Þ b21 f ðb1 Þ ¼ Fðb1 Þ þ b1 ð2bb1 1Þf ðb1 Þ ða bsÞ2

Proof of Corollary 1 Part (b) e e1 ¼ ¼

ða bsÞ ða bsÞkf ðkÞ 2bk 1 FðkÞ þ kð2bk 1Þf ðkÞ ða bsÞFðkÞ >0 ð2bk 1ÞðFðkÞ þ kð2bk 1Þf ðkÞÞ

276

S.K. Mukhopadhyay and X. Su

q q1 ¼ ¼

p p 1 ¼

ða bsÞbkFðkÞ >0 ð2bk 1ÞðFðkÞ þ kð2bk 1Þf ðkÞÞ

kða þ bsÞ s ða bsÞðFðkÞ þ bk2 f ðkÞÞ ðs þ Þ 2bk 1 bFðkÞ þ bkð2bk 1Þf ðkÞ

¼

q1

bkða bsÞ ða bsÞbk2 f ðkÞ 2bk 1 FðkÞ þ kð2bk 1Þf ðkÞ

ða bsÞðbk 1ÞFðkÞ <0 bð2bk 1ÞðFðkÞ þ kð2bk 1Þf ðkÞÞ

bkða bwÞ 0 bða bwÞ b2 k w ¼ð < 0ðsince w > 0Þ Þk ¼ 2 2bk 1 2bk 1 ð2bk 1Þ

Proof of Corollary 1 Part (c)

p1M ¼ ðw sÞq þ L ¼ ðw sÞ

bkða bwÞ þL 2bk 1

@p1M bðw sÞ ¼ ða bw þ bkð2bk 1Þ wÞ < 0ðsince w > 0Þ 2 @k ð2bk 1Þ

1 kða bwÞ2 L p1R ¼ ðp wÞq ke2 L ¼ 2ð2bk 1Þ 2 @p1R ða bwÞ2 bkða bwÞ ¼ < 0ð recall that L ¼ wÞ 2 @k 2bk 1 2ð2bk 1Þ

Since both p1M and p1R are decreasing in k, p1T ¼ p1M þ p1R is decreasing in k.

Proof of Corollary 1 Part (d) Under complete information, pM ¼ pT pR Under asymmetric information p1M ¼ p1T p1R As has been approved, the agent’s profit is monotonically decreasing in k, until hitting its reservation profit pR . However, under complete information, the agent

Use of Supply Chain Contract to Motivate Selling Effort

277

always earns its reservation profit pR . Hence, the agent’s profit is always better off under asymmetric information, i.e., p1R > pR For any k > k, p1T pT ¼ Hence p1M < pM

kða bsÞ2 FðkÞ2 2ð2bk 1ÞðFðkÞ þ kð2bk 1Þf ðkÞÞ2

<0

Proof of Lemma 1 ~2 eðkÞ ~ ~ ~ pR ðkjkÞ ¼ pR ðkÞ þ ðk kÞ 2 Let ~ þ ðk~ kÞ pR ðk~jkÞ pR ðkÞ ) pR ðkÞ

~ eðkÞ pR ðkÞ 2 2

(23)

For the same reason, 2

~ eðkÞ pR ðkÞ ~ pR ðkÞ þ ðk kÞ 2

(24)

2 ~ eðkÞ ~ ðk~ kÞ eðkÞ pR ðkÞ pR ðkÞ 2 2

(25)

From (23) and (24), ðk~ kÞ

2

Divided by k~ k, and take limitation k~ ! k ) pR ðkÞ ¼ eðkÞ 2 Part (a)

2

pR ðkÞ ¼

ðk k

pR ðkÞdk ¼

ðk k

pR ðkÞdk

ðk k

pR ðkÞdk ¼ pR þ

Part (b) 2 ~2 from (25): ðk~ kÞ eð2kÞ ðk~ kÞ eðkÞ 2 ~ 2 eðkÞ2 ) eðkÞ ~ eðkÞ when k~> k ) eðkÞ 2 2 ~ eðkÞ ~ ~ when k < k ) eðkÞ eðkÞ ) eðkÞ This finishes the proof that eðkÞ is decreasing in k

1 2

ðk k

eðkÞ2 dk

(26)

278

S.K. Mukhopadhyay and X. Su

Derivation of Equation (12) The objective function (11) can be written as: ðk

RðkÞdFðkÞ ¼

k

ðk

where ðk ðk

ðpT ðpðkÞ;qðkÞ;kÞ pR ðkÞÞdFðkÞ ¼

k

1 2

eðxÞ2 dxdFðkÞ ¼

k k

ðk ðk k k

ðk

ðk

pT ðpðkÞ;qðkÞ;kÞdFðkÞ

k

eðxÞ2 dxdFðkÞ pR

FðkÞeðxÞ2 dk

k

¼

ðk

ðpT ðpðkÞ;qðkÞ;kÞ

1 FðkÞ eðkÞ2 ÞdFðkÞ pR 2 f ðkÞ

ðpT ðpðkÞ;qðkÞ;kÞ

zðkÞ eðkÞ2 ÞdFðkÞ pR 2

k

¼

ðk k

where zðkÞ ¼

FðkÞ f ðkÞ

Proof of Proposition 2, Part (c)

p2R ¼

e^ðkÞ2 ða bsÞ2 ¼ <0 2 2ð2bðk þ zðkÞÞ 1Þ2

2

ðabsÞ ð1þzðkÞÞ p2T ¼ 2ð2bðkþzðkÞÞ1Þ 2 < 0 total supply chain profit is decreasing in k p2M

¼

p2T

p2R

ða bsÞ2 1 þ zðkÞ 1 ð ¼ Þ 2 2 ð2bðk þ zðkÞÞ 1Þ2 ð2bðk þ zðkÞÞ 1Þ

¼

ða bsÞ2 zðkÞ 2ð2bðk þ zðkÞÞ 1Þ2

< 0:

where zðkÞ ¼

FðkÞ and zðkÞ > 0; f ðkÞ

The OEM’s profit is also monotonically decreasing in k.

Use of Supply Chain Contract to Motivate Selling Effort

279

Proof of Proposition 2, Part (e) Since the retailer earns a profit strictly higher than pR for any k 6¼ k the cutoff point should be where the OEM’s profit hits pM if there is one on the support ½k; k. Let p2M

ð ða bsÞ2 ðb þ zðkÞÞ 1 k ða bsÞ2 ¼ dx pR ¼ pM 2ð2bðb þ zðkÞÞ 1Þ 2 b ð2bðx þ zðxÞÞ 1Þ2 ðk 2ðpM þ pR Þ b þ zðkÞ 1 ) dx ¼ 2 2bðb þ zðkÞÞ 1 ða bsÞ2 b ð2bðx þ zðxÞÞ 1Þ

References Ayra A, Demski J, Glover J, Liang P (2009) Quasi-robust multiagent contracts. Manage Sci 55(5):752–762 Blair BF, Lewis TR (1994) Optimal retail contracts with asymmetric information and moral hazard. Rand J Econ 25(2):284–296 Cachon GP, Kok AG (2010) Competing manufacturers in a retail supply chain: on contractual form and coordination. Manage Sci 56(3):571–589 Cachon GP, Lariviere MA (2005) Supply chain coordination with revenue-sharing contracts: strengths and limitations. Manage Sci 51(1):30–45 Cakanyildirim M, Feng Q, Gan X, Sethi SP (2008) Contracting and coordination under asymmetric production cost information. SSRN Working Paper Series. http://ssrn.com/abstract¼1084584 Chen F (2005) Salesforce incentives, market information, and production/inventory planning. Manage Sci 51(1):60–75 Chen H, Chen Y, Chiu CH, Choi TM, Sethi SP (2010) Coordination mechanism for the supply chain with leadtime consideration and price-dependent demand. Eur J Oper Res 203(1):70–80 Corbett CJ, Zhou D, Tang CS (2004) Designing supply contracts: contract type and information asymmetry. Manage Sci 50(4):550–559 Coughlan AT (1993) Salesforce compensation: a review of MS/OR advances. In: Eliashberg J, Lilien GL (eds) Handbooks in OR&MS, vol 5. Elsevier, Amsterdam, pp 611–651 Cui TH, Raju JS, Zhang ZJ (2008) A price discrimination model of trade promotions. Mark Sci 27(5):779–795 Desai PS, Srinivasan K (1995) Demand signaling under unobservable effort in franchising: linear and nonlinear price contracts. Manage Sci 41(10):1608–1623 Desiraju R, Moorthy S (1997) Managing a distribution channel under asymmetric information with performance requirements. Manage Sci 43(12):1628–1644 Dukes A, Liu Y (2010) In-store media and distribution channel coordination. Mark Sci 29(1): 94–107 Federal Trade Commission (2005) Consumer guide to buying a franchise. Purch World 37(4):17 Foros Ø, Hagen KP, Kind HJ (2009) Price-dependent profit sharing as a channel coordination device. Manage Sci 55(8):1280–1291 Gal-Or E (1991) Vertical restraints with incomplete information. J Ind Econ 39(5):503–516 Gan X, Sethi SP, Yan H (2005) Channel coordination with a risk-neutral supplier and a downsiderisk-averse retailer. Prod Oper Manage 14(1):80–89 Gibbons R (1987) Price-rate incentive schemes. J Labor Econ 5(4):413–429 Gonik J (1978) Tie salesmen’s bonuses to their forecasts. Harv Bus Rev 56(3):116–122

280

S.K. Mukhopadhyay and X. Su

Gurnani H, Xu Y (2006) Resale price maintenance contracts with retailer sales effort: effect of flexibility and competition. Nav Res Logistics 53(5):448–463 Ha AY (2001) Supplier–buyer contracting: asymmetric cost information and cutoff level policy for buyer participation. Nav Res Logistics 48:41–64 Kamien MI, Schwartz N (1981) Dynamic optimization: the calculus of variations and optimal control in economics and management. North-Holland, New York Kaufmann PJ, Dant RP (2001) The pricing of franchise rights. J Retail 77:537–545 Kreps D (1990) A course in microeconomic theory. Princeton University Press, Princeton, NJ Krishnan H, Winter A (2010) Inventory dynamics and supply chain coordination. Manage Sci 56(1):141–147 Krishnan H, Kapuscinski R, Butz DA (2004) Coordinating contracts for decentralized supply chains with retailer promotional effort. Manage Sci 50(1):48–62 Laffont JJ, Martimort D (2000) Mechanism design with collusion and correlation. Econometrica 68(2):309–342 Laffont JJ, Martimort D (2001) The theory of incentives: the principal-agent model. Princeton University Press, Princeton, NJ Laffont JJ, Tirole J (1986) Using cost observation to regulate firms. J Polit Econ 94(3):614–641 Lafontaine F, Slade ME (1997) Retail contracting: theory and practice. J Ind Econ 45(1):1–25 Lal R (1990) Manufacturer trade deals and retail price promotions. J Mark Res 27(4):428–444 Majumder P, Srinivasan A (2008) Leadership and competition in network supply chains. Manage Sci 54(6):1189–1204 Mukhopadhyay SK, Zhu X, Yue X (2008) Optimal contract design for mixed channels under information asymmetry. Prod Oper Manage 17(6):641–650 Mukhopadhyay SK, Su X, Ghose S (2009) Motivating retail marketing effort: optimal contract design. Prod Oper Manage 18(2):197–211 Myerson RB (1979) Incentive compatibility and the bargaining problem. Econometrica 47:61–74 Raju J, Zhang ZJ (2005) Channel coordination in the presence of a dominant retailer. Mark Sci 24(2):254–304 Spengler JJ (1950) Vertical integration and antitrust policy. J Polit Econ 58:347–352 Su X, Wu L, Yue X (2010) Impact of introducing a direct channel on supply chain performance. Int J Electron Bus 8(2):101–112 Taylor TA (2002) Supply chain coordination under channel rebates with sales effort effects. Manage Sci 48(8):992–1007 Tsay A (1999) Quantity flexibility contract and supplier-customer incentives. Manage Sci 45(10):1339–1358 Watson J (2007) Contract, mechanism design, and technological detail. Econometrica 75(1):55–81 Wimmer BS, Garen JE (1997) Moral hazard, asset specificity, implicit bonding, and compensation: the case of franchising. Econ Enquiry 35(3):544–555 Wu Y, Loch CH, Heyden LV (2008) A model of fair process and its limits. Manuf Serv Oper Manage 10(4):637–653 Yang Z, Aydin G, Babich V, Beil DR (2009) Supply disruptions, asymmetric information, and a backup production option. Manage Sci 55(2):192–209 Zhao Y, Choi TM, Cheng TCE, Sethi SP, Wang S (2010) Buyback contracts with a stochastic demand curve. SSRN Working Paper Series. http://ssrn.com/abstract¼1559475 Zhu X, Mukhopadhyay SK (2009) Optimal contract design for outsourcing: pricing and quality decisions. Int J Revenue Manage 3(2):197–217

Price and Warranty Competition in a Duopoly Supply Chain Santanu Sinha and S.P. Sarmah

Abstract This chapter analyzes the coordination and competition issues in a twostage distribution channel where two different retailers compete on their retail price and warranty policy to sell two substitutable products in the same market. The demand faced by each retailer not only depends on its own price and warranty duration, but also on the price and warranty duration set by the other. Mathematical models have been developed to analyze the dynamic competition and coordination mechanism for three different cases where retailers compete (1) exclusively on price; (2) exclusively on warranty duration; (3) both price and warranty duration. The mathematical models show that under price/warranty competition, the steady state equilibrium is dynamically stable in nature under certain condition(s). Further, it has been shown that the channel profit for each case is higher under coordination than that of under competition and the maximum channel profit is achieved when retailers coordinate each other to adopt a centralized policy to set both price and warranty duration. However, it has been observed that though coordination enhances overall supply-chain profitability, it may make consumers worse-off due to higher product prices. The model is illustrated with suitable numerical examples. Keywords Competition • Coordination • Game-theory • Pricing • Stability • Supply-chain management • Warranty

S. Sinha Complex Decision Support Systems, Tata Consultancy Services, Akruti Trade Centre, MIDC, Andheri (E), Mumbai 400093, India e-mail: [email protected] S.P. Sarmah (*) Department of Industrial Engineering and Management, Indian Institute of Technology, Kharagpur 721302, India e-mail: [email protected] T.-M. Choi and T.C. Edwin Cheng (eds.), Supply Chain Coordination under Uncertainty, International Handbooks on Information Systems, DOI 10.1007/978-3-642-19257-9_12, # Springer-Verlag Berlin Heidelberg 2011

281

282

S. Sinha and S.P. Sarmah

1 Introduction The landscape of business environment has experienced significant changes in recent years. Among many factors, globalization of business, increased market competition, awareness of customers, and increased demand for the value added products/services have largely contributed to the change in the shift. The changing face of business environment has compelled academic researchers and industry leaders to rethink about how to manage business operations more efficiently and effectively. Since, the scope for improvement within an organization is restricted with limited resources; the researchers and practitioners are looking for newer alternatives. In this sense, the importance of integrating business activities, both inside and outside the organization’s boundary have been realized by all. The concept of integrating business functions beyond the organization’s boundary has led to the development of the theory and practices of Supply Chain Management (SCM) and one of the important issues in SCM is coordination. There is a growing body of academic researchers and practitioners from a variety of disciplines who focus on different issues of supply chain coordination and strive to establish potential coordination mechanisms to eliminate sub-optimization within a supply chain and enhance overall performance. An important finding from the existing body of literature is that in most of the coordination models, the buyers are assigned to the supplier(s) exogenously, i.e. products are considered independent. However, when there are many vendors in the market who can supply similar type of the product to the buyers, there is a pricecompetition among the vendors. In real-life business, there are many substitutable products in different market place where the respective vendors/retailers have to compete the others to sell the products. For example, Pepsi and Coca-Cola in soft drink market, Sotheby’s and Christie’s in diamond auctions; Kodak and Fuji-film in motion picture film stock market; ABC, CBS, and NBC in US television (before FOX); GM, Ford and Chrysler in auto industry (before the 1970s); etc. Under such scenario, development of coordination mechanism and analysis of competition is an important area of study (Sinha and Sarmah 2010). There are numerous papers on monopolistic and duopolistic competition in marketing and operations management literature; for example, Moorthy (1988), Choi (2003), Yao and Liu (2005), etc. The authors have studied several issues on price competition in supply-chain distribution channels under different contexts. In most of these models, demand of a product is assumed to be a function both its own price as well as the price of the other. However, one critical observation is that in addition to price, consumers also look for additional “value” from the various nonprice attributes, such as quality, service, delivery flexibility, etc. In this sense, the suppliers may also consider the non-price attribute(s) as a competitive tool in their marketing strategy and tend to compete on both price and non-price attribute(s). Several researchers have brought many dimensions of such competition, for example, quality (Banker et al. 1998), service (Tsay and Agrawal 2000), delivery frequency (Ha et al. 2003), etc.

Price and Warranty Competition in a Duopoly Supply Chain

283

The end consumers’ value perception and purchase decisions are also significantly influenced by warranty duration. This is perfectly related with the growing consumption pattern of FMCG, engineering instruments, manufacturing/electronics products where customers not only look for comparable prices but also associated product value/risk. Thus, offering free repair/replacement during the warranty/protection period often enhances the purchase decision of buyers. Many manufacturers/retailers offer warranties to the end consumers in different forms to boost up the overall sales demand. For example, automobile manufacturer Hyundai is well known for their extensive warranty coverage. GM and Ford have extended the powertrain warranty for their 2007 vehicles from 3 years/36,000 miles to 5 years/100,000 miles and 5 years/60,000 miles respectively (Scherer 2006). This has resulted in higher sales, bringing greater profit (Connelly 2006). Following GM, several imported brands have offered broader powertrain coverage on their 2007 vehicles (Hu 2008). Mitsubishi has started offering warranty coverage over 10 years or 100,000 miles. Similarly Suzuki vehicles have powertrain warranty coverage for 7 years or 100,000 miles (Scherer 2006). Similar to automobile industry, FMCG industry have also applied various forms of warranty to escalate the overall market demand. Similarly, several other firms have also used warranty as a marketing weapon to boost up the product demand – along with pricing strategy. However, the issue of price and warranty competition among competing vendors is still an unaddressed research question which requires further analysis. In this chapter, we consider the issues of price and warranty competition between two different retailers. The two competing retailers obtain a product from a common manufacturer, add some value on it, and finally sell it to the market. Three different cases are considered here where the retailers compete (1) exclusively on price; (2) exclusively on warranty duration; (3) both price and warranty duration. For each case, we have developed the steady state equilibrium for dynamic competition and system-wide solution under integrated/coordination mechanism. The main features that make a distinction of this work from the existing related literature is that the formulations and equilibrium strategies of our models explicitly depend on the pricing and warranty policy of competing retailers. We have considered here the inherent dynamics associated with the process of price adjustment while modeling competition. Static modeling of retail price competition can derive the equilibrium but the adjustment of retail price to equilibrium does not occur instantaneously. Like most of the dynamic economic systems, the mechanism of dynamic adjustment is an iterative process converging to equilibrium over a period of time. This chapter analyzes the stability of such equilibrium. Here, the term “stability” means that whether the process of dynamic adjustment of price/warranty duration will eventually converge to equilibrium over a period of time and there is no further divergence from that “fixed/stable” point. We have derived the conditions for the equilibrium to be dynamically stable. This chapter has been organized as follows. A brief review of literature is included in Sect. 2. Section 3 includes the notation and modeling assumptions. The mathematical models are developed in Sect. 4. Section 5 illustrates the dynamics of price, warranty, and simultaneous price and warranty competition.

284

S. Sinha and S.P. Sarmah

The impact of coordination has been analyzed in Sect. 6. Further, numerical illustration has been included in Sect. 7. Finally, the conclusion of the chapter is given in Sect. 8 along with a few possible future research directions.

2 Review of Literature In this section we have provided a brief review of literature related to the work. From the perspective of economic theory, a large number of research papers are available on market competition. Most of the papers deal with either quantitycompetition (Cournot 1938) or price-competition (Bertrand 1883) and their primary focus is on applying game theory to derive equilibria under varied assumptions. On the other hand, in marketing and operations management literature, there are many papers on monopolistic and duopolistic competition. The aspects of coordination and competition have received a considerable amount of attention from the researchers. Moorthy (1988) has considered two identical firms competing on quality and price and analyzed the role of consumer preferences, firms’ costs and price competition in determining a firm’s equilibrium product strategy. Rao (1991) has developed a modelling framework to derive the equilibrium in a duopoly market where the members compete on price and promotions. Competition between direct and indirect channels has been analyzed by Choi (2003). Further, Yao and Liu (2005) have developed competitive equilibrium pricing policies under the Bertrand and the Stackelberg competition model between a mixed e-tail and retail distribution channel. They have shown that introduction of e-tail into a manufacturing distribution system not only generates competitive pricing and payoffs, but also encourages cost-effective retail services. They have also proposed a strategic approach for the manufacturer to add an e-tail channel. However, most of these models have focused only on price competition. Apart from price competition, there are several other models where the authors have studied the aspect of non-price competition. For example, Ha et al. (2003) have considered a supply chain in which two suppliers compete on price and delivery frequency to supply to a customer. They have shown that the customer is better off under delivery competition, while the suppliers are better off under price competition. However, the model did not consider any coordination aspects within the channel and the demand was assumed to be price-independent. Banker et al. (1998) have studied both price and quality competition and addressed the question of how quality is influenced by competitive intensity in an oligopoly market. So (2000) has studied the aspect of price and delivery time as the competition attributes and illustrated how different firms and market characteristics might affect the price and delivery time competition in the market. Tsay and Agrawal (2000) have studied a distribution system in which a manufacturer supplies a common product to two independent retailers, who in turn use service as well as retail price to directly compete for end customers. They have examined the drivers of each firm’s strategy, and the consequences for total sales, market share, and profitability. Finally, it has

Price and Warranty Competition in a Duopoly Supply Chain

285

been shown that the relative intensity of competition with respect to each competitive dimension plays a key role, as does the degree of cooperation between the retailers. Another important non-price attributes is warranty policy – which is also a very popular marketing tool. Most common form of warranty policy may include free replacement of failed product, coverage of parts/labor work, repair of failed products within a specified interval – known as warranty duration. As a common practice in industry, warranties have received the attention of researchers from many diverse disciplines. Authors like, Menezes and Currim (1992) and Padmanabhan (1993) have justified how application of warranty policy could be a marketing tool to differentiate from competitors. A comprehensive review of related research in different application domains can be found in Blischke and Murthy (1996). Several researchers have explored various aspects of warranties, such as warranty type, product failures during warranty period, warranty claims, warranty costs, and warranty logistics. A review has been provided by Hu (2008). However, though there is a stream of literature that has focused exclusively on design of warranty policy (Murthy 2006; Wu et al. 2009), there are not much research on warranty competition. Further, most of the competition models have dealt with deriving static equilibrium, and not studied the aspects of dynamic process adjustment. Thus, in order to address the issues, in this chapter, we extend the aspects of price and warranty competition in a duopoly market. We also extend the analysis to include the dynamics of such competition under various scenarios.

3 Notation and Assumptions The following notations are used to develop the mathematical model (where i ¼ 1, 2) Di Demand of product i pi Unit retail price of manufacturer i ti Warranty length of product i Unit repair cost of product i ci w Per unit wholesale price of the manufacturer pi Profit of retailer i

3.1

Demand Function

In this study, a two-stage distribution channel is considered where a manufacturer distributes a single product to two different retailers. The retailers add some value to the product and sell it to the end customers. The end products are substitute to each other. Further, in addition to product prices of the product itself and those of

286

S. Sinha and S.P. Sarmah

related substitute products, demand also depends on a variety of other non-price factors such as quality, delivery, service, etc. In this study we consider warranty length as another factor that influences demand. In this sense, the demand function is assumed to take the following linear form (Banker et al. 1998): Di ¼ a bpi þ gpj þ y ti mtj

ði; j ¼ 1; 2; i 6¼ jÞ

(1)

The price and cross-price elasticity parameters b and g move independently. It is assumed that a; b>0 so that demand for each product declines with the product’s own price. Further, b>jgj so that “own-price effect” always dominates the “crossprice effect”. The assumed demand function is downward slopping, a variation of a class of more general linear demand functions used in many previous studies (McGuire and Staelin 1983; Choi 1991).

3.2

Warranty Cost

Under the warranty policy, retailer i is responsible to repair each failed product during the warranty length ti with no extra charge to the customer. Following Wu et al. (2009), let us consider that N(t) is the number of failures for the product over a warranty duration of t. Thus, assuming the failure time of the products is independently and identically distributed with the cumulative distribution function F(ti), the expected value of N(ti) is given by E½N ðti Þ ¼ Mðti Þ, where M(ti) is the expected number of renewals of the product with warranty length ti. Further, considering Fðti Þ as a Weibull distribution with failure rate li and a n shape parametern, Fðti Þ ¼ 1 eðli ti Þ . Accordingly, the expected number of failures of product with warranty length ti can be obtained as: Mðti Þ ¼ ðli ti Þn (Wu et al. 2009). Thus, the warranty-related cost of product i in the warranty duration can be given as, Ci ¼ ci ðli ti Þn Di .

4 Mathematical Models Let us consider a typical market model with two competing retailers where each retailer procures the same material from the sole manufacturer at a certain wholesale price w, adds some value on it, and finally sells it in the market – as shown in Fig. 1. Depending on the nature of competition between the two retailers, three different cases have been considered here: (i) Price competition where both the retailers compete each other on price only. (ii) Warranty competition where both retailers compete each other on warranty length. (iii) Price and Warranty competition where both retailers compete each other on price and warranty duration simultaneously.

Price and Warranty Competition in a Duopoly Supply Chain

287

Manufacturer

w

Retailer 1

{p1,t1}

Retailer 2

{p2,t2}

D1

D2

MARKET

Fig. 1 The distribution channel

4.1

Equilibrium in Price Competition

The expected profit function of retailer i can be given as: pi ðpi Þ ¼ Di ½pi w ci ðli ti Þn ¼ a bpi þ gpj þ y ti mtj ½pi w ci ðli ti Þn ; ði; j ¼ 1; 2; i 6¼ jÞ The objective of retailer i, Max pi ðpi Þ

(2)

Since, @@ppi 2i ¼ 2b<0, pi ðpi Þ is concave in nature and the optimal solution can be i achieved from first order conditions: @p @pi ¼ a 2bpi þ gpj þ y ti mtj þ wbþ n ci ðli ti Þ b ¼ 0 Or, 2

pi ¼

a þ wb þ gpj þ y ti þ ci ðli ti Þn b mtj 2b

(3)

This is known as the best response function1 of retailer i. The objective of each retailer is to find out the best response function against each other’s retail price and accordingly find out the Pareto-optimal solution. It is obvious that a change in the retail price of a product can influence demand for other product. Since both retailers are profit maximizer, each retailer will adjust her retail price in response to the

1 A retail price p1 ¼ p1 ðp2 Þ is defined as the best response function against retail price p2 if and only if p1 ðp1 ðp2 Þ; p2 Þ p1 ðp1 ; p2 Þ, for any p1 . Similarly, a retail price p2 ¼ p2 ðp1 Þ is defined as the best response function against retail price p1 if and only if p1 ðp1 ðp2 Þ; p2 Þ p1 ðp1 ; p2 Þ, for any p2 (Shy 2003).

288

S. Sinha and S.P. Sarmah

p1(p2)

p2

p2(p1)

{p1*, p2*}

p1

Fig. 2 Best response functions in price competition

other. The process of price-adjustment then leads to further price rise/fall till it reaches the equilibrium (Fig. 2). This equilibrium is a Pareto-optimal solution derived by the intersection(s) of the best response functions beyond which no retailer has an incentive to unilaterally change its strategy given the other’s equilibrium strategy and hence it is a Paretooptimal solution. In this particular case, the Pareto-optimal solution is Nash–Bertrand equilibrium.2 Proposition 1. (a) Under the general retail price competition, both retailers adjust their retail prices in the similar direction of the change (i.e. simultaneously either increase or decrease). (b) Under the general retail price competition, given retailer j sets her retail pricepj , retailer i will increase her retail price pi , if a þ gpj þ y ti mtj þ wb þ ci ðli ti Þn b >pi . 2b Proof. The proof is given in Appendix A. The Nash–Bertrand equilibrium ðp1 ; p2 Þ can be derived by solving the best response functions as given by (3) and accordingly, pi ¼

n 2abþaggmti þ2byti þ2b2 ci ðli ti Þn 2bmtj þgyt2 þbcj g lj tj þ2b2 wþbgw 4b2 l2 (4)

2 A Nash–Bertrand equilibrium point is a pair of retail prices ðp1 ; p2 Þ offered by the retailers, each of which is a best response of the other: p1 ¼ p1 ðp2 Þ and p2 ¼ p2 ðp1 Þ (Shy 2003).

Price and Warranty Competition in a Duopoly Supply Chain

289

Finally, ½pi p ¼ Di ðp1 ; p2 Þðpi w ci ðli ti Þn Þ

(5)

Proposition 2. Under the general retail price competition, the Nash–Bertrand equilibrium increases with failure rate li and warranty length ti . However, the Nash–Bertrand equilibrium increases with shape factor n if 2b2 ci ðli ti Þn lnðli ti Þþ n bcj g lj tj ln lj tj >0. Proof. The proof is given in Appendix B.

4.2

Equilibrium in Warranty Competition

In this case, both retailers are assumed to compete each other on warranty length. Thus, the objective of retailer i, Max pi ðti Þ

(6)

It is straightforward to derive that, @ 2 pi ¼ ci nli n ðti Þn2 ½2y ti þ Di ðn 1Þ @ti 2 Since, ci nli n ðti Þn2 >0, for Di >0 and n 1, @@tip2i <0. Thus, pi ðti Þ is concave in nature and the optimal solution can be achieved from first order conditions: 2

@pi ¼ ci li n ðti Þn1 y ti þ a bpi þ gpj þ y ti mtj n pi y þ wy ¼ 0 @ti Or, ðti Þn1 ¼

ðpi wÞy ; ci li n ðy ti þ Di nÞ

where

Di ¼ a bpi þ gpj þ y ti mtj :

(7)

The intermittent values of ti can be derived by iterative computations and finally, the Nash equilibrium ðt1 ; t2 Þ can be derived by solving, ðti Þn1 ¼

ðpi wÞy ; where Di ¼ a bpi þ gpj þ yti mtj ci li ðy ti þ Di nÞ n

(8)

Accordingly, ½pi w ¼ Di ðt1 ; t2 Þðpi w ci li ti Þ

(9)

290

S. Sinha and S.P. Sarmah

t1(t2)

t2

t2(t1)

{t1*,t2*}

t1

Fig. 3 Best response functions in warranty competition

Following (7), it has been observed that the best response functions ti tj , ði; j ¼ 1; 2; i 6¼ jÞ are nonlinear in nature due to the shape parameter n. Thus, the dynamic adjustment of warranty duration, starting from an initial condition, could further go uphill or downhill to reach the Nash equilibrium ðt1 ; t2 Þ – as shown in Fig. 3 as an example. However, it is to be noted that the slopes and the trajectory of any such best response functions and the number of intersection points (equilibrium) depend on the functional form and type of the profit functions.

4.3

Equilibrium in Price and Warranty Competition

In this case, both the retailers compete each other on warranty length. Thus, the objective of retailer i, Max pi ðpi ; ti Þ

(10)

The Hessian matrix of pi ðpi ; ti Þ is given as, 2

@ 2 pi 6 @pi 2 Hi ¼ 6 4 @ 2 pi @ti @pi

3 @ 2 pi 2b @pi @ti 7 7 ¼ @ 2 pi 5 y þ bci nli n ti n1 @ti 2

y þ bci nli n ti n1 n ci nli ðti Þn2 ½2y ti þ Di ðn 1Þ

Price and Warranty Competition in a Duopoly Supply Chain

291

Here, the members of the principal diagonal are negative; thus, the objective function pi ðpi ; ti Þ is a concave function if jHi j<0, 2 Or, ðy þ bci nli n ti n1 Þ >2bci nli n ðti Þn2 ½2y ti þ Di ðn 1Þ: The Nash equilibrium under simultaneous price and warranty competition ðp~i ; ~ti Þ can be derived by solving the following set of best response functions, 9 a þ wb þ gpj þ y ti þ ci ðli ti Þn b mtj > > = 2b > ðpi wÞy > ; ti n1 ¼ n ci li ðy ti þ Di nÞ

pi ¼

(11)

The Nash equilibrium ðp~1 ; ~t1 Þ and ðp~2 ; ~t2 Þ can be found by simultaneously solving the sets of equations in (11). Finally, ½pi p;w ¼ D~i ðp~i ; ~ti Þðp~i w ci li~ti Þ

(12)

5 Dynamics of Competition In the earlier section, we have derived the Nash–Bertrand equilibrium under three different cases where the retailers compete each other on (1) price, (2) warranty duration, (3) both price and warranty duration. However, the adjustment of initial price or warranty duration gradually leads toward the Nash–Bertrand equilibrium following an iterative process, where at each step; each retailer chooses a policy which maximizes the individual profit based on the expected policy set by her opponent. Hence, at each time period every retailer depends on an expectation of the other retailer’s policy in the next time period to determine the corresponding profit-maximizing policy for that period. This leads to a dynamic adjustment of price and warranty duration which finally reaches to the Nash–Bertrand equilibrium. Here, we analyze the behavior of such dynamic adjustment process of price and warranty competition under two different scenarios (1) naı¨ve expectation and (2) adaptive expectation. In the former case, each player assumes the last values taken by the competitors without estimation of their future reactions in each step. However, in case of adaptive expectation, each retailer revises her beliefs according to the adaptive expectations rules which compute the outputs with weights between last period’s outputs and her reaction function. A related discussion is included in Agiza and Elsadany (2003) and Shone (2001). In this section, we develop mathematical models to capture the scenario where retailers compete each other in (1) price, (2) warranty duration, (3) both price and warranty duration under both naı¨ve and adaptive .expectation. We have investigated the stability condition(s) corresponding to each case. The main objective of the

292

S. Sinha and S.P. Sarmah

models is to investigate the dynamic behavior of a duopoly game with price and warranty competition under different expectation rules. Through a numerical we further explain the movement of the system variables over a period of time.

5.1

Dynamics of Price Competition

If we denote the retail price of product i at time period t by pi ðtÞ, then the retail price pi ðt þ 1Þ for the period ðt þ 1Þ is decided by solving the two optimization problems (Agiza and Elsadany 2003): p1 ðt þ 1Þ ¼ arg max p1 ðp1 ðtÞ; p2 e ðt þ 1ÞÞ p1

p2 ðt þ 1Þ ¼ arg max p2 ðp1 e ðt þ 1Þ; p2 ðtÞÞ;

(13)

p2

where the function pi ð:Þ denotes the profit of the retailer i and pj e ðt þ 1Þ represents the expectation of retailer i about the pricing decision of retailer j, ði; j ¼ 1; 2; j 6¼ iÞ. We consider that the retailers could be naı¨ve or adaptive players – depending on their adjustment process.

5.2

Dynamics of Price Competition with Naı¨ve Expectation

We assume that both the retailers are naı¨ve. With the assumption, we can express the process of duopoly game which can be defined as, 9 a þ wb þ gp2 ðtÞ þ y t1 þ c1 ðl1 t1 Þn b mt2 > > p1 ðt þ 1Þ ¼ = 2b > a þ wb þ gp1 ðtÞ þ y t2 þ c2 ðl2 t2 Þn b mt1 > ; p2 ðt þ 1Þ ¼ 2b

(14)

In equilibrium, pi ðt þ 1Þ ¼ pi ðtÞ and thus we are interested in the solution of the system with non-negative equilibrium points defined as, 9 a þ wb þ gp2 ðtÞ þ y t1 þ c1 ðl1 t1 Þn b mt2 > > = 2b a þ wb þ gp1 ðtÞ þ y t2 þ c2 ðl2 t2 Þn b mt1 > > ; p2 ðtÞ ¼ 2b

p1 ðtÞ ¼

The equilibrium point ðp1 ; p2 Þ has already been derived as,

(15)

Price and Warranty Competition in a Duopoly Supply Chain

pi ¼

293

n 2abþaggmti þ2byti þ2b2 ci ðli ti Þn 2bmtj þgyt2 þbcj g lj tj þ2b2 wþbgw 4b2 g2

It is clear that the Nash equilibrium ðp1 ; p2 Þ is located at the intersection of the two reaction functions. In this study, we are interested in studying the local stability of equilibrium point at ðp1 ; p2 Þ. This can be analyzed by the eigen values of the Jacobian matrix of the system (14) on the complex plane. The Jacobian matrix of (14) at the point ðp1 ; p2 Þ has the following form, 2

@f1 6 @p1 JpN ðp1 ; p2 Þ ¼ 6 4 @f2 @p1

3 @f1 @p2 7 7; @f2 5

(16)

@p2

a þ wb þ gpj ðtÞ þ y ti þ ci ðli ti Þn b mtj ¼ fi 2b 2 g 3 0 6 2b 7 Thus, JpN ðp1 ; p2 Þ ¼ 4 g 5 0 2b We now investigate the local stability The characteristic of Nash equilibrium. @f1 @f2 2 N N N equation is given as, f ðeÞ ¼ e Tr Jp e þ Det Jp ,where Tr Jp ¼ @p þ @p 1 2 @f1 @f2 @f1 @f2 N is the trace and Det Jp ¼ @p1 @p2 @p2 @p1 is the determinant of the Jacobian where, pi ðt þ 1Þ ¼

matrix defined in (16). 2 2 N J Now, Tr JpN ¼ 0 and Det JpN ¼ g 4Det JpN ¼ 2 <0. Thus, Tr p 4b 2 g >0. b2 Since, Tr2 JpN 4Det JpN >0, the eigen values of Nash equilibrium are real. Following a standard stability analysis, the necessary and sufficient condition for the stability of Nash equilibrium at ðp1 ; p2 Þ is that the eigen values of the Jacobian matrix JpN ðp1 ; p2 Þ are inside the unit circle of the complex plane. This is true if and only if the following conditions are hold (Puu 2002; Agiza and Elsadany 2004): 1. 1 Tr JpN þ Det JpN >0 2. 1 þ Tr JpN þ Det JpN >0 3. Det JpN 1<0 2 g Since, Det JpN 1 ¼ 4b 2 þ 1 <0, the third condition is already satisfied. g2 The first and second conditions imply, 1 4b 2 >0. Thus, the dynamic adjustment of retail price eventually reaches to a stable Nash equilibrium for b2 > g4 . 2

294

5.3

S. Sinha and S.P. Sarmah

Dynamics of Price Competition with Adaptive Expectation

Assuming both the retailers to be adaptive, the dynamic equation of the adaptive expectation can be defined as below, pi ðt þ 1Þ ¼ ð1 vi Þpi ðtÞ þ

vi a þ wb þ gpj ðtÞ þ y ti þ ci ðli ti Þn b mtj ; 2b

(17)

ði; j ¼ 1; 2; i 6¼ jÞ Here, vi 2 ½0; 1 is the speed of adjustment of the adaptive players. It can easily be noted that, if vi ¼ 1, it reduces to the form of naı¨ve expectation. This implies that naive expectation is a special case of adaptive expectations behavior. Now, we look for the equilibrium of the system (17) and discuss their stability properties. The fixed points of the map (17) are obtained as nonnegative solutions of the algebraic system by setting pi ðt þ 1Þ ¼ pi ðtÞ. This implies, vi a þ wb þ gpj ðtÞ þ y ti þ ci ðli ti Þn b mtj 2pi b ¼ 0; 2b

(18)

ði; j ¼ 1; 2; i 6¼ jÞ Since, v1 ; v2 > 0; the equilibrium point ðp1 ; p2 Þ can be derived as,

pi ¼

n 2abþaggmti þ2byti þ2b2 ci ðli ti Þn 2bmtj þgyt2 þbcj g lj tj þ2b2 wþbgw 4b2 g2

This shows that, the Nash equilibrium does not change with adaptive expectation; however the speed to reach Nash equilibrium depends on the speed of adjustment. Here, we are interested in studying the local stability of equilibrium point at ðp1 ; p2 Þ which can be analyzed by the eigen values of the Jacobian matrix of the system (17) on the complex plane – as shown below, 2

@g1 6 @p1 JpAD ðp1 ; p2 Þ ¼ 6 4 @g2 @p1

3 @g1 @p2 7 7; @g2 5

(19)

@p2

vi where, pi ðt þ 1Þ ¼ ð1 vi Þpi ðtÞ þ 2b a þ wb þ gpj þ y ti þ ci ðli ti Þn b mtj ¼ gi 2 v1 g 3 ð1 v1 Þ 6 2b 7 Thus, JpAD ðp0 1 ; p0 2 Þ ¼ 4 v2 g 5 ð 1 v2 Þ 2b v2 g2 AD Now, Tr Jp ¼ 2 v1 v2 and Det JpAD ¼ ð1 v1 Þð1 v2 Þ v14b 2 . For stable equilibrium, the following conditions have to be fulfilled,

Price and Warranty Competition in a Duopoly Supply Chain

295

C1. 1 Tr JpAD þ Det JpAD >0 C2. 1 þ Tr JpAD þ Det JpAD >0 C3. Det JpAD 1<0 Proposition 3. The Nash equilibrium is dynamically stable in price competition 2 under adaptive expectation if b2 > g4 . Proof. The proof is given in Appendix C.

5.4

Dynamics of Warranty Competition

Similar to the earlier model of dynamic price competition, the simultaneous game of warranty competition through warranty length adjustment can be represented in the following form, 9 t1 ðT þ 1Þ ¼ arg max p1 ðt1 ðTÞ; t2 e ðT þ 1ÞÞ = t1

t2 ðT þ 1Þ ¼ arg max p2 ðt1 e ðT þ 1Þ; t2 ðTÞÞ ;

(20)

t2

where warranty length offered for product i at time period T and T+1 is represented by ti ðTÞ and ti ðT þ 1Þ respectively. Here, we analyze the adjustment process and predict whether the Nash equilibrium is globally stable. The term “stability”, means whether during the process of dynamic price adjustment, an initial combination of warranty length will eventually converge to equilibrium in the long run without further deviation. The present dynamic system can be represented by (Ferguson and Lim 1998): t_1 ¼ c1 ðt1 t1 Þ

(21)

t_2 ¼ c2 ðt2 t2 Þ;

(22)

where, t_1 ¼ dt1 =dT, t_2 ¼ dt2 =dT and c1 ; c2 > 0 are adjustment coefficients representing the speed of the adjustment. The terms ðt1 ; t2 Þ and ðt1 ; t2 Þ denote the Nash equilibrium and the actual level of warranty length at any time-period (T) respectively. Since the present system is non-linear, we apply the following theorem developed by Olech (1963) to check the dynamic stability of the system given by (21). Theorem 1. Consider an autonomous system x_ ¼ f ðx; yÞ y_ ¼ gðx; yÞ

) (23)

296

S. Sinha and S.P. Sarmah

where x_ ¼ dx=dt, y_ ¼ dx=dt, and ðx; yÞ 2 R2 . The functions f and g are assumed to be of class C1 onR2 . Suppose that there is a unique equilibrium point ðx; yÞ on R2 , i.e., a point such that f ðx; yÞ ¼ 0 and gðx; yÞ ¼ 0. If the following conditions are satisfied, the equilibrium is asymptotically stable in the large (i) Trace J ðx; yÞ fx þ gy <0, for all ðx; yÞ in R2 : for all ðx; yÞ in R2 : (ii) Det J ðx; yÞ fx gy fy gx >0, (iii) Either fx gy 6¼ 0, for all ðx; yÞ in R2 : for all ðx; yÞ in R2 . or fy gx 6¼ 0, where, fx ð@ [email protected]Þf ðx; yÞ and fy , gx , and gy are similarly defined. A proof of the theorem is given in Olech (1963). We have considered this theorem as given and proceed to look for conditions which guarantee the global stability of the mentioned system. It is straight-forward to derive the following differential equations: 2

1 !n1

ðp1 wÞy t_i ¼ ci 4 n ci li y ti þ a bpi þ gpj þ y ti mtj n

3 ti 5;

ði; j ¼ 1; 2; i 6¼ jÞ (24)

Conditions under which the above nonlinear system will be dynamically stable have been derived and shown in the form of the following proposition. Proposition 4. The Nash equilibrium under non-coordinated warranty length competition, is globally asymptotically stable for the following: (a) n 1,

i h ih i h n n n Di >0 and K1 ðy t1 þ D1 nÞn1 þ 1 K2 ðy t2 þ D2 nÞn1 þ 1 > L1 ðyt1 þ D1 nÞn1 h i n L2 ðy t2 þ D2 nÞn1

(b) For n<1

h i n n Di >0, Ki ðy ti þ Di nÞn1 < 1, and K1 ðy t1 þ D1 nÞn1 þ 1 h i h ih i n n n K2 ðy t2 þ D2 nÞn1 þ 1 > L1 ðy t1 þ D1 nÞn1 L2 ðy t2 þ D2 nÞn1

1

yðn þ 1Þ pi y wy n1 nm pi y wy n1 1 and L ¼ where, Ki ¼ : i ðn 1Þ ci li n n1 ci l i n

Proof. The Jacobian matrix of t_i (i ¼ 1, 2) is given as, 2

@ t_1 6 @t 1 J¼6 4 @ t_2 @t1 It is straight-forward to derive,

3 @ t_1 @t2 7 7 ¼ J1 @ t_2 5 J3 @t2

J2 J4

Price and Warranty Competition in a Duopoly Supply Chain

h i9 n @ t_i > ¼ ci Ki ðy ti þ Di nÞn1 þ 1 > = @ti h i n > @ t_i > ; ¼ ci Li ðy ti þ Di nÞn1 @tj

297

ði; j ¼ 1; 2; i 6¼ jÞ

(25)

(I) From stability condition (i), • for n 1 and Di >0, Trace J ðx; yÞ fx þ gy ¼ J1 þ J4 <0 • for n<1 and Di >0, Trace J ðx; yÞ fx þ gy ¼ J1 þ J4 <0 n Ki ðy ti þ Di nÞn1 < 1

if

(II) From stability condition (ii), h ih i n n Det ¼ c1 c2 K1 ðy t1 þ D1 nÞn1 þ 1 K2 ðy t2 þ D2 nÞn1 þ 1 h ih i n n c1 c2 L1 ðy t1 þ D1 nÞn1 L2 ðy t2 þ D2 nÞn1 If Det J ðx; yÞ>0, then, h

ih i h i n n n K1 ðy t1 þ D1 nÞn1 þ 1 K2 ðy t2 þ D2 nÞn1 þ 1 > L1 ðy t1 þ D1 nÞn1 h i n L2 ðy t2 þ D2 nÞn1

(III) Since no element in the matrix J is zero the other conditions of the theorem are also satisfied. □

Hence proved.

5.5

Dynamics of Warranty Competition with Adaptive Expectation

Assuming both the retailers to be adaptive, the dynamics of warranty length adjustment can be defined as below,

ti ðT þ 1Þ ¼ ð1 wi Þti ðTÞ þ wi

ðpi wÞy ci li n ½y ti ðT þ 1Þ þ Di n

1 n1

ði ¼ 1; 2Þ

298

S. Sinha and S.P. Sarmah

Here, wi 2 ½0; 1 is the speed of adjustment of the adaptive players which considers both previous warranty policy and the best response function. Now, we look for the equilibrium of the system (25). The fixed points of the map (25) are obtained as nonnegative solutions of the algebraic system by setting ti ðT þ 1Þ ¼ ti ðTÞ. This implies,

ti ðTÞ ¼

ðpi wÞy ci li n ½y ti ðT þ 1Þ þ Di n

1 n1

ði ¼ 1; 2Þ

(26)

This clearly shows that Nash equilibrium remains unchanged irrespective of the speed of adjustment. Under adaptive expectation, the warranty competition leads to the same Nash equilibrium and the Nash equilibrium is dynamically stable under the conditions shown in Proposition 4.

5.6

Dynamics of Price and Warranty Competition

In this case, we consider that both retailers dynamically adjust both retail price and warranty length in each time period. The retail price pi ðT þ 1Þ and warranty length ti ðT þ 1Þ for the period ðT þ 1Þ is decided by solving the two optimization problems (Agiza and Elsadany 2003): 9 p1 ðT þ 1Þ ¼ arg max p1 ðp1 ðTÞ; p2 e ðT þ 1ÞÞ > > p1 > > > e > p2 ðT þ 1Þ ¼ arg max p2 ðp1 ðT þ 1Þ; p2 ðTÞÞ > = p 2

t1 ðT þ 1Þ ¼ arg max p1 ðt1 ðTÞ; t2 e ðT þ 1ÞÞ > > > t1 > > > > e t2 ðT þ 1Þ ¼ arg max p2 ðt1 ðT þ 1Þ; t2 ðTÞÞ ;

(27)

t2

Here, the function pi ð:Þ denotes the profit of the retailer i and ð:Þj e ðT þ 1Þ represents the expectation of retailer i about the strategy of retailer j, ði; j ¼ 1; 2; j 6¼ iÞ. The dynamic adjustment of retail price and warranty length adjustment can be represented as follows (Ferguson and Lim 1998): 9 p_ 1 ¼ t1 ðp1 p1 Þ > > > p_ 2 ¼ t2 ðp2 p2 Þ = t_1 ¼ t3 ðt1 t1 Þ > > > ; t_2 ¼ t4 ðt2 t2 Þ

(28)

where, p_ i ¼ dpi =dT, t_i ¼ dti =dT and t1 ; t2 ; t3 ; t4 >0 are adjustment coefficients representing the speed of the adjustment. The terms ðp1 ; p2 Þ, ðt1 ; t2 Þ and

Price and Warranty Competition in a Duopoly Supply Chain

299

ðt1 ; t2 Þ, ðp1 ; p2 Þ denote the Nash equilibrium and the actual level of price and warranty length respectively at any time-period (T). We now check the dynamic stability of the system given by (28). The dynamic process can be represented by the following differential equations: p_ i ¼ ti

n 2ab þ ag gmti þ 2byti þ 2b2 ci ðli ti Þn 2bmtj þ gytj þ bcj g lj tj þ 2b2 w þ bgw 2 t_i ¼ t2þi 4

4b2 l2

! pi

3 !1 n1 ðp1 wÞy ti 5 ci li n y ti þ a bpi þ gpj þ y ti mtj n (29)

The Jacobian matrix of the dynamic system (28) is given as, 2

@ p_ 1 6 @p1 6 6 @ p_ 2 6 6 @p1 J ðp; tÞ ¼ 6 6 @ t_1 6 6 @p1 6 4 @ t_2 @p1

@ p_ 1 @p2 @ p_ 2 @p2 @ t_1 @p2 @ t_2 @p2

@ p_ 1 @t1 @ p_ 2 @t1 @ t_1 @t1 @ t_2 @t1

3 @ p_ 1 @t2 7 7 @ p_ 2 7 7 @t2 7 7 @ t_1 7 7 @t2 7 7 @ t_2 5 @t2

(30)

Further derivation of the Jacobian matrix J ðp; tÞ is given in Appendix D. Proposition 5. The Nash equilibrium under price and warranty length competition is globally asymptotically stable for the following: • For n 1: Di >0 and Det ¼ jJ ðp; tÞj>0 n • For n<1: Di >0, Ki ðy ti þ Di nÞn1 < 1, and Det ¼ jJ ðp; tÞj>0

Proof. The proof is straightforward from Theorem 1.

6 Channel Coordination In this section, we consider the different aspects of channel coordination between the retailers. A typical case may occur where both the retailers, understanding their inter-dependence, coordinate each other to set the optimal values of price/warranty duration that maximize the overall system/channel profit and thereby the individual pay-offs. The centralized policy thus includes deciding globally optimal retail price and warranty duration. The retailers can choose to decide system-wide optimal (1) retail price, (2) warranty duration, (3) both retail price and warranty duration.

300

S. Sinha and S.P. Sarmah

However, the current discussion excludes the aspect of sharing of coordination benefit.

6.1

Coordinated Policy to Set Retail Price

The system/channel profit is the total profit of the two retailers as follows, Pch ¼ p1 þ p2 Or, Pch ¼ ða bp1 þ gp2 þ yt1 mt2 Þ½p1 w c1 ðl1 t1 Þn þ ða bp2 þ gp1 þ yt2 mt1 Þ½p2 w c2 ðl2 t2 Þn . From the first order conditions, n @Pch ¼a 2bpi þ 2gpj þ yti mtj þ wb þ ci bðli ti Þn wg cj g lj tj ¼ 0; @pi ði;j ¼ 1; 2;j 6¼ iÞ

(31)

Solving, Ab þ Bg pi cp ¼ 2 2 b g2

(32)

n where, A ¼ a þ yti mtj þ wb þ ci bðli ti Þn wg cj g lj tj and, B ¼ a þ ytj n mti þ wb þ cj b lj tj wg ci gðli ti Þn ; where i; j ¼ 1; 2; i 6¼ j. The super-script “cp” indicates coordinated retail price. Finally, the system profit can be represented as, Pch cp ¼ p1 ðp1 cp ; p2 cp Þ þ p2 ðp1 cp ; p2 cp Þ.

6.2

Coordinated Policy to Set Warranty Duration

The objective of this model is to find out the system-wide optimal warranty duration for both the retailers. Since, the system profit, Pch ¼ p1 þ p2 is the total profit of the two retailers as follows, from the first order conditions, @Pch ¼0 @ti Or, "

1 n #n1 y½pi w ci ðli ti Þn m pj w cj lj tj ; ti ¼ ci li n Di n

ði; j ¼ 1; 2; j 6¼ iÞ (33)

Price and Warranty Competition in a Duopoly Supply Chain

301

The explicit solution of (33) is intractable; however iterative computation can be applied to derive the optimal solutionðt1 cw ; t2 cw Þ. The super-script “cw” indicates coordinated warranty length. The system profit is given as, Pch cw ¼ p1 ðt1 cw ; t2 cw Þ þ p2 ðt1 cw ; t2 cw Þ

6.3

(34)

Global Coordination

We use the term “global coordination” to mention a typical case where the retailers take a centralized decision to set both retail price and warranty duration to maximize system/channel profit. The system profit is the total profit of the two retailers as follows, Pch ðp1 ; p2 ; t1 ; t2 Þ ¼ p1 ðp1 ; p2 ; t1 ; t2 Þ þ p2 ðp1 ; p2 ; t1 ; t2 Þ From the first order conditions, @Pch @Pch @pi ¼ 0 and @ti ¼ 0 9 a þ 2gpj þ y ti mtj þ wb þ c1 bðli ti Þn wg cj g lj tj > > pi ¼ > > = 2b 1 " # n n1 > y½pi w ci ðli ti Þn m pj w cj lj tj > > > ti ¼ ; n ci nli Di

(35)

ði; j ¼ 1; 2; j 6¼ iÞ Solving the above simultaneous equations, the optimal ðpi g ; ti g Þ can be derived. Here, the super-script “g” indicates global coordination policy. Accordingly, Pch g ¼ p1 ðt1 g ; t2 g Þ þ p2 ðt1 g ; t2 g Þ.

7 Numerical Illustration A numerical illustration has been included to validate the mathematical models. The following data are considered for the numerical example. The data are very similar to Banker et al. (1998). a ¼ 1; 000; b ¼ 10; g ¼ 8:8; y ¼ 6; m ¼ 5:4; l1 ; l2 ¼ ½0:5; 6:0; c1 ¼ 2:5; c2 ¼ 2; w ¼ 5; n ¼ ½1:5; 5:0.

302

7.1

S. Sinha and S.P. Sarmah

Price Competition

Let us consider a typical case with, n ¼ 2:5, l1 ¼ l2 ¼ 0:5, t1 ¼ 0:5, and t2 ¼ 1:5. Let us further assume that, at t ¼ 0, retailer 1 and retailer 2 have the following offering in the market: fp1 ð0Þ ¼ 35; t1 ð0Þ ¼ 0:5g and fp2 ð0Þ ¼ 40; t2 ð0Þ ¼ 1:5g. If both the retailers compete only on the retail price, the price equilibrium is achieved at: fp1 ; p2 g ¼ f93:92; 94:63g.

7.2

Dynamics of Price Competition

Let us consider at typical case with, n ¼ 2:5, l1 ¼ l2 ¼ 0:5, t1 ¼ 0:5, and t2 ¼ 1:5. The dynamics of the price competition under naı¨ve expectation has been shown below (Table 1; Fig. 4): Dynamics of price competition with adaptive expectation has been shown below under varying speed of adjustment. It has been found that irrespective of the speed, the dynamical system converges to the same equilibrium. However, with faster speed, the equilibrium is reached faster and if v1 ¼ v2 ¼ 1, the system behaves like a dynamical system under naı¨ve expectation (Fig. 5).

7.3

Price Competition: Sensitivity Analysis

The following table shows the sensitivity of the price equilibrium on different parameters. In this experiment we have assumed the basic initial data: n ¼ 2:5, l1 ¼ l2 ¼ 0:5, t1 ¼ 0:5, and t2 ¼ 1:5 (Table 2). Table 1 Dynamics of price competition with naı¨ve expectation Sr. p1 p2 D1 p1 D2 1 35 40 997 29,829.12 914 2 69.88 68.70 901 58,366.70 934 3 82.51 84.05 909 70,420.82 892 4 89.27 89.61 891 74,993.68 896 5 91.71 92.58 892 77,318.90 888 6 93.02 93.66 889 78,168.52 888 7 93.49 94.23 889 78,618.36 887 8 93.75 94.44 889 78,781.51 887 9 93.84 94.55 889 78,868.59 887 10 93.89 94.59 888 78,900.13 887 11 93.90 94.61 888 78,916.99 887 12 93.91 94.62 888 78,923.09 887 13 93.92 94.62 888 78,926.35 887 14 93.92 94.63 888 78,927.53 887 15 93.92 94.63 888 78,928.17 887

p2 31,109.72 58,604.03 69,636.99 74,916.21 76,867.88 77,888.30 78,259.24 78,456.73 78,528.28 78,566.52 78,580.36 78,587.76 78,590.44 78,591.87 78,592.39

Pch 60,938.83 116,970.73 140,057.81 149,909.89 154,186.78 156,056.82 156,877.60 157,238.24 157,396.88 157,466.64 157,497.35 157,510.85 157,516.79 157,519.41 157,520.56

Price and Warranty Competition in a Duopoly Supply Chain

303

100

pi

75

50

p1 p2

25

0 0

5

10

15

20

25

Time period

Fig. 4 Dynamics of price competition with naı¨ve expectation 100 90 80 70

p1 p2 p1 p2 p1 p2 p1 p2

Pi

60 50 40 30 20

(v (v (v (v (v (v (v (v

= = = = = = = =

0.25) 0.25) 0.5) 0.5) 0.75) 0.75) 1.0) 1.0)

10 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43

Time period

Fig. 5 Dynamics of price competition with adaptive expectation

It shows that, the equilibrium price fp1 ; p2 g increases with increase in li and ti . However, the equilibrium price decreases with increase in n. Further, it has been observed that the channel profit increases with increase in n and decreases with increase in ti .

7.4

Price Coordination

Assuming, n ¼ 2:5, l1 ¼ l2 ¼ 0:5, t1 ¼ 0:5, and t2 ¼ 1:5, under integrated pricing policy, the optimal retail prices can be derived as: p1 cp ¼ 419:30 and p2 cp ¼ 420:05. Accordingly, the demand and profits are as below (Table 3),

304

S. Sinha and S.P. Sarmah

Table 2 Sensitivity analysis: price competition p2 Parameter Value p1 n 1.5 94.15 94.89 2.0 94.01 94.74 2.5 93.92 94.63 3.0 93.86 94.54 3.5 93.82 94.46 4.0 93.78 94.40 4.5 93.76 94.34 5.0 93.74 94.30 li 0.5 93.92 94.63 1.0 95.38 97.54 1.5 98.50 103.75 2.0 103.66 114.01 2.5 111.17 128.96 3.0 121.32 149.15 3.5 134.34 175.08 4.0 150.49 207.21 5.0 192.98 291.77 6.0 250.36 405.97 0.5 93.84 93.84 ti 1.0 94.17 94.14 1.5 94.85 94.77 2.0 95.95 95.78 2.5 97.55 97.24 3.0 99.69 99.21 3.5 102.43 101.72 4.0 105.82 104.84 4.5 109.91 108.59 5.0 114.73 113.01 Table 3 Price coordination p2 cp D1 cp p1 cp 419.30 420.05 498

p1 cp 206,427.95

p1 78,927.79 78,948.91 78,928.59 78,891.70 78,850.10 78,809.26 78,771.50 78,737.59 78,928.59 80,894.41 85,167.67 92,470.06 103,642.42 119,742.08 142,124.85 172,520.80 266,577.51 425,957.74 78,790.74 78,733.86 78,558.83 78,239.11 77,753.07 77,082.58 76,212.36 75,129.73 73,824.48 72,288.85

D2 cp 495

p2 78,487.47 78,529.12 78,592.85 78,661.91 78,728.66 78,789.95 78,844.78 78,893.15 78,592.85 75,738.65 69,827.89 60,586.82 48,300.93 33,915.78 19,160.67 6,681.53 4,546.26 72,296.33 78,808.85 78,836.28 78,840.93 78,817.57 78,761.86 78,669.99 78,538.57 78,364.53 78,145.05 77,877.56

p2 cp 205,227.87

Pch 157,415.26 157,478.03 157,521.44 157,553.60 157,578.76 157,599.21 157,616.28 157,630.74 157,521.44 156,633.07 154,995.56 153,056.88 151,943.35 153,657.86 161,285.52 179,202.33 271,123.78 498,254.06 157,599.59 157,570.14 157,399.76 157,056.68 156,514.93 155,752.57 154,750.93 153,494.26 151,969.52 150,166.40

Pch cp 411,655.82

A comparison between price competition and global coordination has been illustrated through the Fig. 6. This shows that price coordination can generate significantly higher profit as compared to that of under price competition.

7.5

Warranty Competition

We consider a case with, n ¼ 2:5, l1 ¼ l2 ¼ 0:5, p1 ¼ 35, and p2 ¼ 40. Let us further assume that, at t ¼ 0, retailer 1 and retailer 2 have the following offering in

Price and Warranty Competition in a Duopoly Supply Chain

305

450000 400000

Profit

350000

Profit of R1 (Price Competition)

300000 250000

Profit of R2 (Price Competition)

200000

Channel Profit (Price Competition)

150000

Profit of R1 (Price Coord.)

100000

Profit of R2 (Price Coord.) Channel Profit (Price Coord.)

50000 0 1

3

5

7

9

11

13

15

17

19

Time Period

Fig. 6 Price competition vs. price coordination: a comparison Table 4 Dynamics of price competition with naı¨ve expectation Sr. t1 t2 D1 p1 D2 1 0.500 t2 997 29,829.12 914 2 0.299 64.30 1,002 30,025.94 909 3 0.298 81.15 1,002 30,026.00 909 4 0.298 88.76 1,002 30,026.00 909 5 0.298 92.02 1,002 30,026.00 909

p2 31,109.72 31,775.00 31,775.22 31,775.22 31,775.22

Pch 6,0938.83 61,800.94 61,801.22 61,801.22 61,801.22

the market: t1 ð0Þ ¼ 0:5 and t2 ð0Þ ¼ 1:5. If both the retailers compete only on the warranty length, the equilibrium is achieved at:ft1 ; t2 g ¼ f0:30; 0:41g.

7.6

Dynamics of Warranty Competition

Assuming, n ¼ 2:5, l1 ¼ l2 ¼ 0:5, p1 ¼ 35, p2 ¼ 40, t1 ð0Þ ¼ 0:5 and t2 ð0Þ ¼ 1:5, the dynamics of the warranty length competition under naı¨ve expectation has been shown below (Table 4; Fig. 7): Dynamics of warranty competition with adaptive expectation has been shown below under varying speed of adjustment. It has been found that irrespective of the speed, the dynamical system converges to the same equilibrium – similar to price competition. However, with faster speed, the equilibrium is reached faster and if w1 ¼ w2 ¼ 1, the system behaves like a dynamical system under naı¨ve expectation – as observed earlier in the case of price competition (Fig. 8).

7.7

Warranty Competition: Sensitivity Analysis

The following table shows the sensitivity of the warranty equilibrium with respect to different parameters. In this experiment the basic initial data have been assumed as: n ¼ 2:5, l1 ¼ l2 ¼ 0:5, p1 ¼ 35, and p2 ¼ 40. The following table shows the sensitivity of the price equilibrium on different parameters (Table 5).

306

S. Sinha and S.P. Sarmah 1.600 1.400 1.200 t1

t(i)

1.000

t2

0.800 0.600 0.400 0.200 0.000 1

2

3 Time period

4

5

Fig. 7 Dynamics of warranty length competition with naı¨ve expectation

1.6

t1 (v = 0.25) t2 (v = 0.25)

1.4

t1 (v = 0.5) 1.2

t2 (v = 0.5) t1 (v = 0.75)

1.0 ti

t2 (v = 0.75) 0.8

t1 (v = 1.0) t2 (v = 1.0)

0.6 0.4 0.2 0.0 1

2

3

4

5

6

7

8 9 10 Time period

11

12

13

14

15

16

Fig. 8 Dynamics of warranty competition with adaptive expectation

It shows that, the equilibrium price ft1 ; t2 g increases with increase in nand pi and decreases with increase in li . The channel profit is found to increase with increase in li and pi .

7.8

Coordinated Warranty Policy

Assuming, n ¼ 2:5, l1 ¼ l2 ¼ 0:5, p1 ¼ 35, p2 ¼ 40, under integrated warranty policy, the optimal warranty length are: t1 cw ¼ 0 and t2 cw ¼ 0:15. Accordingly, the demand and profits are derived as below (Table 6),

Price and Warranty Competition in a Duopoly Supply Chain

307

Table 5 Sensitivity analysis: warranty competition t2 p1 Parameter Value t1 n 1.5 0.02 0.05 30,053.41 2.0 0.14 0.23 30,035.53 2.5 0.30 0.41 30,026.00 3.0 0.44 0.55 30,022.68 3.5 0.56 0.67 30,022.53 4.0 0.66 0.77 30,023.92 4.5 0.75 0.86 30,026.02 5.0 0.82 0.93 30,028.42 li 0.5 0.30 0.41 30,026.00 1.0 0.09 0.13 30,049.27 1.5 0.05 0.07 30,054.54 2.0 0.03 0.04 30,056.62 2.5 0.02 0.03 30,057.67 25.0 0.23 0.27 19,387.63 pi 35.0 0.31 0.36 28,715.50 45.0 0.37 0.43 37,800.12 50.0 0.41 0.47 42,251.28 60.0 0.47 0.54 50,971.38 70.0 0.53 0.61 59,448.60 80.0 0.59 0.68 67,683.02 90.0 0.64 0.75 75,674.67 100.0 0.70 0.81 83,423.57 125.0 0.84 0.97 101,733.84 150.0 0.97 1.13 118,526.83

p2 31,779.86 31,777.11 31,775.22 31,774.99 31,775.77 31,777.06 31,778.58 31,780.17 31,775.22 31,778.50 31,779.24 31,779.53 31,779.67 19,394.32 28,728.75 37,821.68 42,277.62 51,008.46 59,497.98 67,746.21 75,753.15 83,518.82 101,877.44 118,728.02

Pch 61,833.26 61,812.63 61,801.22 61,797.67 61,798.30 61,800.98 61,804.60 61,808.59 61,801.22 61,827.77 61,833.78 61,836.15 61,837.34 38,781.96 57,444.26 75,621.80 84,528.89 101,979.84 118,946.59 135,429.22 151,427.82 166,942.39 203,611.28 237,254.84

Table 6 Integrated warranty policy t2 cw D1 cw p1 cw t1 cw 0 0.15 1001 30,035.25

p2 cw 31,809.15

Pch cw 61,844.40

D2 cw 909

A comparison between warranty competition and warranty coordination has been illustrated through Fig. 9. This shows that integrated warranty policy can generate higher profit as compared to that of warranty competition.

7.9

Price and Warranty Competition

In this case, we illustrate a case where both retailers simultaneously compete on retail price and warranty length. For this example, we consider n ¼ 1:5 and 2:5, l1 ¼ l2 ¼ 0:5. Let us further assume that, at t ¼ 0, retailer 1 and retailer 2 have the following offering in the market: p1 ð0Þ ¼ 35, p2 ð0Þ ¼ 40 and t1 ð0Þ ¼ 0:5, t2 ð0Þ ¼ 1:5. Accordingly, the equilibrium price and warranty length have been derived as below (Table 7):

308

S. Sinha and S.P. Sarmah

Ch Profit (Warranty Competition)

Profit R2 (Warranty Competition)

Profit R1 (Warranty Competition)

Ch Profit (Warranty Coord.)

70000 60000

Profit

50000 40000 30000 20000 10000 0 1

2

3 Time period

4

5

Fig. 9 Warranty competition vs. warranty coordination: a comparison Table 7 Price and warranty competition ~t1 ~t2 D1 p~2 n p~1 1.5 93.83 0.20 93.89 0.32 887 2.5 93.92 0.67 93.97 0.77 888

p1 78,756.68 78,778.10

D2 888 888

p2 78,784.02 78,823.38

Pch 157,540.70 157,601.48

This shows that channel profit increases with increase in shape parameter n. Below, Fig. 10 represents the dynamics of the equilibrium for n ¼ 1:5; 2:5, l1 ¼ l2 ¼ 0:5 and the initial condition p1 ð0Þ ¼ 35, p2 ð0Þ ¼ 40 and t1 ð0Þ ¼ 0:5, t2 ð0Þ ¼ 1:5.

7.10

Global coordination

Assuming, n ¼ 2:5, l1 ¼ l2 ¼ 0:5, under integrated price and warranty policy, the optimal solutions have been derived as below (Table 8), A comparison between warranty competition and warranty coordination has been illustrated through Fig. 11.

7.11

Global Coordination: Sensitivity Analysis

Here, further experiments have been conducted to study the impact of n, Li, and ci on global coordination. The basic parameters assumed as, n ¼ 2:5, l1 ¼ l2 ¼ 0:5, c1 ¼ 2:5 and c2 ¼ 2. The results have been tabulated in Table 9. It shows that increase in shape parameter n increases channel profit, retail price, and warranty duration. However, increase in failure rate li and repair cost ci decreases channel profit, retail price, and warranty duration.

p1 (n=1.5)

p1 (n=2.5) p2 (n=2.5)

p2 (n=1.5)

309

t1 (n=1.5)

t1 (n=2.5)

t2 (n=1.5)

t2 (n=2.5)

100.0

1.6

90.0

1.4

80.0

1.2

Profit

70.0 60.0

1.0

50.0

0.8

40.0

0.6

30.0

0.4

20.0

Warranty Length

Price and Warranty Competition in a Duopoly Supply Chain

0.2

10.0 0.0

0.0 1

2

3

4

5

6

7 8 9 Time period

10

11

12

13

14

15

Fig. 10 Dynamics of price and warranty competition Table 8 Optimal price and warranty length under global coordination n p1 g t1 g p2 g t2 g D1 g p1 g D2 g p2 g 2.5

419.4

0.59

419.41

0.68

497

205,856.3

497

206,015.22

Pch g 411,871.56

450000 400000 350000 Profit R1 (Price & Warr. Comp.)

Profit

300000

Profit R2 (Price & Warr. Comp.)

250000

Channel Profit (Price & Warr. Comp.)

200000

Profit R1 (Price & Warr. Coord.) Profit R2 (Price & Warr. Coord.)

150000

Channel Profit (Price & Warr. Coord.)

100000 50000 0 1

2

3

4

5

6

7

8

9

10 11 12 13 14 15

Time period

Fig. 11 Price and warranty policy: competition vs. global coordination

8 Conclusion This chapter analyzes the coordination and competition issues in a two-stage distribution channel where two different retailers compete each other on their retail price and warranty policy to sell two substitute products in the same market.

310 Table 9 Sensitivity analysis: global coordination t1 g p2 g t2 g Parameter Value p1 g n 1.5 419.2 0.14 419.26 0.22 2.0 419.3 0.40 419.36 0.50 2.5 419.4 0.59 419.41 0.68 3.0 419.4 0.73 419.44 0.82 3.5 419.4 0.84 419.46 0.92 5 419.5 1.06 419.48 1.12 Li 0.05 428.6 27.29 430.44 31.82 0.1 422.1 8.61 422.71 10.01 0.25 419.8 1.87 419.93 2.17 0.5 419.4 0.59 419.41 0.68 1 419.2 0.19 419.24 0.22 2.5 419.2 0.04 419.18 0.05 2.5 419.4 0.59 419.37 0.59 ci 5 419.3 0.37 419.30 0.37 7.5 419.3 0.28 419.26 0.28 10 419.2 0.23 419.25 0.23 15 419.2 0.18 419.23 0.18

S. Sinha and S.P. Sarmah

D1 g 497 497 497 497 497 497 494 496 497 497 497 497 497 497 497 497 497

D2 g 497 497 497 497 497 498 511 501 498 497 497 497 497 497 497 497 497

p1 g 205,810.7 205,822.7 205,856.3 205,890.3 205,921.6 205,836.9 206,510.6 206,061.0 205,889.5 205,856.3 205,845.6 205,841.9 205,928.7 205,896.2 205,883.1 205,875.7 205,867.5

p2 g 205,901.2 205,970.8 206,015.2 206,047.7 206,072.2 206,279.9 214,038.8 208,402.0 206,395.3 206,015.2 205,895.8 205,852.8 205,928.7 205,896.2 205,883.1 205,875.7 205,867.5

Pch g 411,711.9 411,793.5 411,871.6 411,938.0 411,993.8 412,116.8 420,549.3 414,463.0 412,284.7 411,871.6 411,741.5 411,694.7 411,857.5 411,792.4 411,766.2 411,751.4 411,734.9

The demand faced by each retailer not only depends on its own price and warranty duration, but also on the price and warranty duration set by the other. Mathematical models have been developed to analyze the dynamic competition and coordination mechanism for three different cases where retailers compete (1) exclusively on price; (2) exclusively on warranty duration; (3) both price and warranty duration. The adjustment of initial price or warranty duration during dynamic competition gradually leads toward the Nash–Bertrand equilibrium following an iterative process, where at each step each retailer chooses a policy which maximizes the individual profit based on the expected policy set by her opponent. Further, we analyze the behavior of such dynamic adjustment process of price and warranty competition under two different scenarios (1) naı¨ve expectation and (2) adaptive expectation – depending on the adjustment of expectation function of each retailer. Finally, it has been shown that under non-cooperative price/warranty competition, the steady state equilibrium is dynamically stable in nature under certain conditions. It has been shown here that the channel profit for each case is higher under coordination that of under competition. The channel profit is found to be the maximum under global coordination where retailers adopt centralized policy to set price and warranty duration. However, it has been observed that though coordination enhances overall supply-chain profitability, it may make consumers worseoff due to higher product prices. The model is illustrated with suitable numerical examples. The model can significantly help industry practitioners to visualize and understand the dynamic nature of price and non-price (warranty) competition. It also can predict the overall pay-off in case of centralized or coordinated strategy.

Price and Warranty Competition in a Duopoly Supply Chain

311

Accordingly, a delicate balance between coordination and competition can be achieved in case the existing business model fails to meet profitability expectation. Thus, the industry practitioners can take a pro-active role in choosing the attribute to compete and decide when to coordinate with their competitor. The model could be extended in several directions. Various other forms of demand function could be used to replicate more realistic scenarios. Also, in most of the industrial cases, price/warranty competition takes place under asymmetric information. Further, there could be more number of players in the market; and one interesting dimension towards further research is the “entry” and “exit” decisions of a firm in the market. Finally, all retailers may not set price simultaneously. There are cases where one firm takes the role of a price-leader while the others are followers. Such type of competition model under Stackelberg game framework is also worth mentioning for future research.

Appendix A: Proof of Proposition 1 g i Proof. 1(a). It is straightforward to derive, @p @pj ¼ 2b >0, which shows both retailers change their retail price in the similar direction. (b). It has already been derived,

@pi ¼ a 2bpi þ gpj þ y ti mtj þ wb þ ci ðli ti Þn b ¼ 0 @pi @pi @pi

Given retailer j sets her retail price pj , retailer i will increase her retail price pi , if >0, a þ gpj þ y ti mtj þ wb þ ci ðli ti Þn b >pi . Or, 2b Hence proved. □

Appendix B: Proof of Proposition 2 It is straight forward to derive,

@pi 2b2 ci nðti Þn ðli Þn1 ¼ >0 @li 4b2 g2

@pi gm þ 2by þ 2b2 ci nðli Þn ðti Þn1 ¼ >0 @ti 4b2 g2 Since, g < b, m < y, then gm þ 2by > 0. Hence,

@pi @ti

>0

312

S. Sinha and S.P. Sarmah

n n @pi 2b2 ci ðli ti Þ lnðli ti Þ þ bcj g lj tj ln lj tj ¼ @n 4b2 g2

Thus,

@pi @n

n >0 if 2b2 ci ðli ti Þn lnðli ti Þ þ bcj g lj tj ln lj tj >0

Appendix C: Proof of Proposition 3 Substituting Tr JpAD and Det JpAD in condition C1, C2 and C3, (i)

v1 v2 g2 1 Tr JpAD þ Det JpAD ¼ 1 ð2 v1 v2 Þ þ ð1 v1 Þð1 v2 Þ 4b2 g2 Or, 1 Tr JpAD þ Det JpAD ¼ v1 v2 1 4b 2 . Since, v1 v2 >0, the first condi-

g 2 g tion yields 1 4b 2 >0, or, b > 4 . 2

(ii)

2

v1 v2 g2 1 þ Tr JpAD þ Det JpAD ¼ 1 þ ð2 v1 v2 Þ þ ð1 v1 Þð1 v2 Þ 4b2

g2 Or, 1 Tr JpAD þ Det JpAD ¼ 4 2ðv1 þ v2 Þ þ v1 v2 1 4b 2 . Since,0 g4 . (iii)

v1 v2 g2 Det JpAD 1 ¼ ð1 v1 Þð1 v2 Þ 1 4b2

v2 g Let, f ðvÞ ¼ ð1 v1 Þð1 v2 Þ v14b 2 g2 C ¼ 1 4b . 2 0 The Hessian matrix for f ðvÞ, Hv ¼ C following the first order conditions, v1 ¼ v2

2

if C>0 or, b2 > g4 . □ Hence Proved. 2

1 ¼ ðv1 v2 þ v1 v2 CÞ,

where

C shows that f ðvÞ is concave. Thus, 0 ¼ C1 and f ðvÞ ¼ 1 C . Thus, Max: f ðvÞ<0

gmþ2byþ2b2 c1 nl1 n t1 n1

!

2bmþgyþbc2 gnl2 n t2 n1

!3

where, X1 ¼

p1 y wy p2 y wy ; Y ; Y2 ¼ ½y t2 þ D2 n. ¼ ½ y t þ D n , X ¼ 1 1 1 2 c 1 l1 n c2 l2 n

t1 t1 0 t1 6 7 6 7 4b2 l2 4b2 l2 6 ! !7 6 7 6 2bmþgyþbc1 gnl1 n t1 n1 gmþ2byþ2b2 c2 nl2 n t2 n1 7 6 7 0 t t t2 1 2 6 7 6 7 4b2 l2 4b2 l2 6 7 2 3 6 7 1 J ðp;tÞ ¼ 6

h i h i 7 n1 2n 1 1 n 1 n n t p ywy 6 3 7 1 4 5 X1 n1 Y1 n1 þX1 n1 bnY1 n1 t3 K1 ðyt1 þD1 nÞn1 þ1 t3 ½gnn1 t3 L1 ðyt1 þD1 nÞn1 6 7 n 6 n1 7 c1 l1 6 7 6 7 2 3 6 7 1

h i h i 6 7 n1 1 2n 1 1 n n n p ywy t 2 4 4 5 4 5 n1 n1 n1 n1 n1 n1 n1 X2 Y2 t4 t4 K2 ðyt2 þD2 nÞ t4 L2 ðyt2 þD2 nÞ ðgnÞ þX2 bnY2 þ1 n1 c2 l2 n

2

Appendix D: The Jacobian Matrix J ðp; tÞ

314

S. Sinha and S.P. Sarmah

References Agiza HN, Elsadany AA (2003) Nonlinear dynamics in the Cournot duopoly game with heterogeneous players. Phys A 320:512–524 Agiza HN, and Elsadany AA (2004) Chaotic dynamics in nonlinear duopoly game with heterogeneous players, Applied Mathematics and Computation 149(3):843–860 Agliari A, Gardini L, Puu T (2000) The dynamics of a triopoly Cournot game, Chaos, Solitons and Fractals 11(15):2531–2560 Bertrand J (1883) ‘Theorie mathematique de la Richesse Sociale par Leon Walras and Recherches sur les principes mathematique de la theorie des dicheses par Augustin Coumot’, Journal des Savants, pp. 499–508, translated by Friedman, J.W. (1988). In: Daughety AR (ed) Coumot oligopoly. Cambridge University Press, Cambridge, pp 73–81 Blischke W, Murthy D (1996) Product warranty handbook. Marcel Dekker, New York Choi SC (1991) Price competition in a channel structure with a common retailer, Marketing Science 10(4):271–296 Choi SC (2003) Expanding to direct channel: market converges as entry barrier. J Interact Mark 17(1):25–40 Connelly M (2006) Dealers: extended factory warranties boost profits. Automot News Detroit 81:46 Cournot A (1938) Researches into the mathematical principles of the theory of wealth, translated by Bacon NT (1960). Kelley, New York Ferguson BS, Lim GC (1998) Introduction to Dynamic Economic models. Manchester University Press, Manchester, M13 9NR, UK Ha A, Li L, Ng SM (2003) Price and delivery logistics competition in a supply chain. Manage Sci 49(9):1139–1153 Hu W-T (2008) Supply chain coordination contracts with free replacement warranty. Doctoral dissertation, Drexel University, Philadelphia, PA McGuire TW, Staelin R (1983) An industry equilibrium analysis of downstream vertical integration, Marketing Science, 2 (Spring), 161–191 Menezes MAJ, Currim IS (1992) An approach for determination of warranty length. Int J Res Mark 9:177–196 Moorthy KS (1988) Product and price competition in a duopoly. Mark Sci 7(2):141–168 Murthy NP (2006) Product warranty and reliability. Ann Oper Res 143:133–146 Olech C (1963) On the global stability of an autonomous system on the plane, in Contributions to Differential Equations. In: Lasalle JP, Diaz JB. (Eds.), Vol.1, Wiley, New York Padmanabhan V (1993) Warranty policy and extended service contracts: theory and an application to automobiles. Mark Sci 12:230–248 Rao CR (1991) Pricing and promotions in asymmetric duopoly. Mark Sci 10(2):131–144 Scherer K (2006) GM warranties take big leap. Automot News Detroit 81:30 Shone R (2001) An introduction to economic dynamics. Cambridge University Press, New York, pp 1–5 Shy O (2003) How to price. Cambridge University Press, Cambridge Sinha S, Sarmah SP (2010) Coordination and price competition in a duopoly common retailer supply chain. Comput Ind Eng 59(2):280–295 So KC (2000) Price and time competition for service delivery. Manuf Serv Oper Manage 2(4):392–409 Tsay A, Agrawal N (2000) Channel dynamics under price and service competition. Manuf Serv Oper Manage 2(4):372–391 Wu C-C, Lin P-C, Chou C-Y (2009) Optimal price, warranty length and production rate for free replacement policy in the static demand market. Omega 37(1):29–39 Yao DQ, Liu J (2005) Competitive pricing of mixed retail and e-tail distribution channels. Omega 33:235–247

Supply Chain Coordination for Newsvendor-Type Products with Two Ordering Opportunities Yong-Wu Zhou and Sheng-Dong Wang

Abstract This chapter discusses supply chain coordination issues for a newsvendor-type product with two ordering opportunities. We consider a twoechelon supply chain consisting of one manufacturer and one buyer, where the manufacturer sells his product through the buyer who faces a random demand. The manufacturer does not hold inventory but activates production with a fixed production setup cost to respond to the buyer’s order. At the start of the selling period, the buyer’s first order is delivered. By the end of the sales period, an urgent second order is allowed to meet the willingly-backordered demand if the buyer shares the manufacturer’s setup cost incurred by the second order. We discuss two parties’ optimal order policies in the decentralized setting, and examine the impact of pool schemes of the second setup cost on the decentralized system performance. We show that the decentralized system would perform best if the manufacturer covers utterly the second production setup cost. Also, we prove that under the twice-order framework in the chapter the expected profit of the centralized system is not equal to but greater than the sum of two members’ expected profits in the decentralized system, which is not consistent with our expectation. In order to maximize the expected profit of the channel, two coordinated policies are proposed to achieve perfect coordination: a two-part-tariff policy for the special case that the buyer pays all the manufacturing setup cost, and a revised revenue-sharing contract for the case that two parties share the manufacturing setup cost. Keywords Supply chain coordination • Newsvendor model • Two ordering opportunities • Partial backlogging • Revised revenue-sharing contract Y.-W. Zhou (*) School of Business Administration, South China University of Technology, Guangzhou, Guangdong, P.R. China e-mail: [email protected] S.-D. Wang Department of Mathematics, Hefei Electronic Engineering Institute, Hefei, Anhui, P.R. China e-mail: [email protected] T.-M. Choi and T.C. Edwin Cheng (eds.), Supply Chain Coordination under Uncertainty, International Handbooks on Information Systems, DOI 10.1007/978-3-642-19257-9_13, # Springer-Verlag Berlin Heidelberg 2011

315

316

Y.-W. Zhou and S.-D. Wang

1 Introduction Like our human beings, products have also their own life cycle. From introduction to decline products pass through various stages: introduction, growth, maturity, saturation and decline. Nowadays, with the rapid improvement of technology, the life cycle of products becomes shorter and shorter. It makes more and more products, like USB flash drivers, personal computers, mobile phones, fashion apparel, etc., to have the attributes of fashion or seasonal goods. We call all of short life-cycle products as fashion products or newsvendor-type products. For nonfashion products, like most of fast moving consumer products, we all know that there exists substantial progress in theory and application of supply chain management. But the same success has not been achieved for fashion products. Fisher and Raman (1999) pointed out three possible reasons. First, the demand patterns for these products make the estimation of demand and demand variability extremely difficult. Second, traditional demand forecasting methods usually assume that at least 1 year of demand history is available, which is not possible for short life-cycle products since their life cycles are generally less than 1 year. Finally, the cost of carrying inventory is much higher for short life-cycle products because of the risk of obsolescence. The challenge in managing supply chains for newsvendor-type products is to ensure product availability while keeping leftover products as low as possible. In the past five decades, both researchers and practitioners paid much attention on supply chain management for newsvendor-type products. A lot of models on this topic can be found in literature (see Sect. 2). Most of the existing models, which studied supply chain coordination problems for single-period items under the newsvendor framework, assumed that products could be ordered (or produced) only once during the whole selling period. Consequently, the decision maker is unable to take advantage of the subsequent information that becomes available as the season draws closer or after the season begins. In reality, however, newsvendortype products can be replenished many times in the selling season. For example, in the 1980s Benetton, the fashion retail giant, significantly reduced markdowns and leftover inventory by employing a second order opportunity around the start of the season. This phenomenon leads to a lot of researches on the newsvendor models with two order opportunities. Most of these researches, such as Lau and Lau (1998) and Li et al. (2009b), considered the following twice-order framework: the first order is placed at the start of the preseason and delivered at the start of the selling season; the second order is placed at or after the start of the selling season for subsequent delivery. Besides, there is yet another class of models, in which twice orders are placed before the start of the season. However, the models just mentioned only considered how retailers make their twice-order decisions but neglected the coordination issues of supply chains. The main aim of this chapter is to discuss coordination issues of a manufacturerbuyer supply chain for newsvendor-type products with two ordering opportunities. Unlike the two twice-order frameworks mentioned above, we consider such

Supply Chain Coordination for Newsvendor-Type Products

317

a twice-order framework, in which the first order is placed at the start of the season and the second one (if needed) is placed at the end of the season. This framework is first employed by Weng (2004). Its benefit is mainly in that placing the second order at the end of the season can obtain accurate demand information. Under the framework, we will consider such a situation that a manufacturer supplies a newsvendor-type product to a buyer who faces a stochastic demand. The manufacturer does not hold inventory but activates production with a fixed production setup cost to respond to the buyer’s order. At the beginning of the selling period, the buyer’s first order is delivered from the manufacturer to the buyer in order to meet random demand. Due to the uncertainty of demand, the actual demand during the selling period may deviate from the ordered lot size. By the end of the sales period, if the demand exceeds the ordered quantity, the buyer would have an opportunity to place an urgent second order from the manufacturer to meet the willinglybackordered demand as long as the buyer shares the manufacturer’s second setup cost incurred by the second order. The main questions we concern are the following (1) What is the first order lot size of the buyer and whether should he/she place the second order? (2) Whether should the manufacturer activate an urgent second production in response to the buyer’s second order? (3) How do we design an effective mechanism to coordinate the whole supply chain? The coordination problem considered in this chapter generalizes that in Weng (2004), where the manufacturing setup cost of the second order (if happened) is paid utterly by the buyer and the demands exceeded the first order quantity are all backordered. One of the aims of this chapter is to observe the impact of the fraction that the buyer shares the second setup cost on the decentralized system performance. Our research shows that the performance of the decentralized system decreases as the proportion of the second production setup cost shared by the buyer increases. It implies that the decentralized system performs best if the second setup cost is utterly paid by the manufacturer itself but worst if the second setup cost is paid only by the buyer, like in Weng (2004). Another difference between Weng (2004) and this chapter is that Weng (2004) considered the sum of the two members’ expected profits as the total expected profit of the centralized system, whereas this chapter has shown that the expected profit of the centralized system is always greater than the sum of the two members’ expected profits. Consequently, the quantity discount mechanism shown in Weng (2004) could not coordinate the decentralized channel really to the “centralized” system. We present in the chapter two perfect coordination scenarios: a two-part-tariff policy for the special case that the buyer pays all the manufacturing setup cost, and a revised revenue-sharing contract for the general case that the two parties share the manufacturing setup cost. The two coordination mechanisms optimize the expected profit of the whole supply chain rather than the sum of two parties’ expected profits. Hence, they achieve the perfect coordination of the decentralized system. While our work directly generalizes the work of Weng (2004), it also nicely complements the work of Lau and Lau (1998) and Milner and Kouvelis (2005). The key differences are that (1) they do not allow for partial backorders, (2) they focus on determining the buyer’s twice order policies, whereas we focus on how the buyer

318

Y.-W. Zhou and S.-D. Wang

sets the twice-order policies but also on how to design the perfect coordination mechanism of the whole channel, and (3) two different twice-order frameworks are considered. The remainder of the chapter is organized as the following. Section 2 reviews briefly the related literature. Section 3 describes the problems discussed in the chapter and introduces notations and assumptions needed by developing the model. While Sect. 4 shows the mathematical model for the decentralized system, Sect. 5 develops the mathematical model for the centralized system. Also, Sect. 5 compares the optimal policies between the decentralized and centralized system and discusses the property of the decentralized system performance. Section 6 presents a two-part-tariff model that can achieve perfect coordination of the system under the special case that the buyer pays all the manufacturing setup cost, and a revised revenue-sharing contract that can realize perfect coordination of the channel under the case that two parties share the manufacturing setup cost. In Sect. 7, two numerical examples are shown to illustrate the model. Conclusions are given in Sect. 8.

2 Literature Review Over the past five decades, both researchers and practitioners have had great interest in operations management issues for newsvendor-type products. A lot of models on this topic can be found in publications, of which the earlier ones mainly focused on how to find the buyer’s appropriate ordering policies to maximize (minimize) his or her expected profit (cost) (see, e.g., Whitin 1955; Goodman and Moody 1970; Kabak and Schiff 1978; Ismail and Louderback 1979; Atkinson 1979; Lau 1980; Nahmias and Schmidt 1984). An excellent review about these earlier researches can be seen in Khouja (1999). Due to the globalization of market and competition, the idea of supply chain management becomes very popular. Many researchers have shifted their attention to coordination issues of the supply chain for newsvendor-type products. For example, through the returns policy, Pasternack (1985) presented a supply chain coordination model in which a single manufacturer sold a single newsvendor-type product to a single retailer. He pointed out that it was an effective coordinating policy to take back residual products from the retailer at the end of selling period. Emmons and Gilbert (1998) further studied the role of returns policies in pricing and inventory decisions for catalogue goods. Taylor (2002) employed a one-period model with one supplier and one retailer to explore the rebate policy. Cachon (2003) developed a supply chain coordination model with price dependent demand. Webster and Weng (2008) presented an ordering and pricing model in a manufacturing and distribution supply chain for newsvendor-type products. Arcelus et al. (2008) developed a two-echelon supply chain model to depict the profitability of a secondary market to a profit-maximizing manufacturer, who was offering to the retailer a buyback policy for the unsold merchandise left at the end of

Supply Chain Coordination for Newsvendor-Type Products

319

the selling season. Due to the difficulty of obtaining the analytical results, they resorted to the numerical analysis. Li et al. (2009a) considered the supply chain coordination and decision making issue under consignment contract with revenue sharing. Chen et al. (2010) studied a coordination contract for a supplier–retailer channel where a fashionable product with a stochastic price-dependent demand is produced and sold. They formulated a two-stage optimization problem in which the supplier first decides the amount of capacity reservation, and the retailer then determines the order quantity and the retail price after observing the demand information. Other related studies included Chen et al. (2006), Wong et al. (2009), etc. However, all the researches mentioned above ignored demand information updating. That is, they assumed that the retailer orders only once during the whole selling period (including the lead time). The existing literature concerning the newsvendor model with two ordering opportunities can be divided into three categories. The first category assumed that the retailer’s two ordering opportunities take place before the season. Following the initial order, additional market information is obtained. Then, based on an improved demand forecast, the second-order amount is decided before the selling period. For example, Donohue (2000) formulated an efficient supply contract for fashion goods with forecast updating before the season and two production modes. Choi et al. (2003) developed a two-stage optimal ordering model with Bayesian information updating. Serel (2009) investigated a quick response system in which retailers place separate orders for a product at two different times before the selling season. The second one assumed that the retailer’s first order happens at the beginning of the season and the other within the season. Lau and Lau (1997, 1998) were possibly the first to consider the situation in which a second order is placed within the selling period. Through dynamic programming, Milner and Kouvelis (2005) also proposed a single-period inventory model with two ordering opportunities. Yet they focused on examining the interplay between the value of information and flexibility in their order decisions. Li et al. (2009b) further conducted a comprehensive analysis of the stocking problem with two order opportunities, with focus on elucidating the optimal ordering policy structure, whereas Pan et al. (2009) presented a two-period pricing and ordering model for the dominant retailer in a two-echelon supply chain with demand information updating. However, these researches considered retailers’ ordering and pricing issues only. In the third category, it is assumed that retailers have two ordering opportunities happening at the beginning and the end of the season, respectively. For instance, Weng (2004) developed a newsvendor-type coordination model for a single-manufacturer single-buyer supply chain with two ordering opportunities. Zhou and Li (2007) discussed an issue similar to that in Weng (2004) by considering the buyer’s inventory cost but neglecting the buyer’s backorder cost. Wang et al. (2010) developed a supply chain coordination model for newsvendor-type products with two ordering opportunities. In their model, they allowed the manufacturer to use two different production modes to respond to the retailer’s twice orders, and employed the general functions of different forms to model the expected market demand that is dependent on price and advertising expenditure.

320

Y.-W. Zhou and S.-D. Wang

3 Notations and Assumptions Consider a two-echelon supply chain where a manufacturer sells a newsvendor-type product through a buyer, who faces a stochastic demand. The manufacturer does not hold inventory of finished products but activates production with a fixed production setup cost and the negligible manufacturing or assembling time to respond to the buyer’s order. At the beginning of the selling period, an initial order is delivered from the manufacturer to the buyer. Then, the buyer uses the initial order delivered to meet random demand. Due to the uncertainty of demand, the actual demand during the selling period may deviate from the ordered lot size. By the end of the sales period, if the demand exceeds the ordered quantity, the buyer would have an opportunity to place an urgent second order from the manufacturer to meet the willingly-backordered demand as long as the buyer shares the manufacturer’s second setup cost incurred by the second order. Our key questions are the following (1) What is the optimal lot size of the first order? (2) Whether should a second order be placed and an urgent second production activated? (3) What is the impact of pool schemes of the urgent setup cost on the decentralized system performance? (4) How do we design coordination mechanism of the supply chain? A number of industries fall into our model. For example, in the mobile telecom industry, each mobile phone manufacturer (such as Motorola, Nokia or Sumsung) may sell through its downstream telecom service companies who use its equipment. For a new-model mobile phone, telecom service providers would tend to purchase an initial lot size of the new equipment from the mobile phone manufacturer, and then bind some special service with the new-style mobile phone. While the newstyle mobile phone is out of stock during the selling period, part of customers may be willing to wait till the next order is delivered due to their preference to the service and brand. The model we will present in the chapter can be applied to this case. In order to develop our model, the following notations and assumptions are used. Some other notations are given where they are needed. Assumptions 1. The buyer faces a random demand. 2. Each of the two parties has complete information about the other’s cost structure. 3. The manufacturer does not hold inventory and the production time for any lot size is neglected. 4. The buyer has complete demand information before she places her second order and all excess demand (after the first order) is observed but partially backlogged. 5. Both the manufacturer and the buyer share the second production setup cost once the second production happened at the end of the selling season. 6. Among many alternative objectives, we use the common expected profit function as the objectives of two parties. Notations x The random demand for the buyer with increasing concave CDF F(x), PDF f(x) and mean m

Supply Chain Coordination for Newsvendor-Type Products

p c r b

b k Q ti si l

v

321

The buyer’s unit sales price The buyer’s unit purchase price, or the manufacturer’s wholesale price The buyer’s salvage value (e.g., unit discount sales price) The backlogging rate (0 b 1), which means a fraction b of the excess demand during the stock out period is backordered, and the remaining fraction (1 b) is lost The buyer’s unit backorder cost The buyer’s unit shortage cost for the lost sales The buyer’s (first) order quantity The buyer’s ordering and transportation cost for the i-th (i ¼ 1,2) order The manufacturer’s production setup cost for the i-th (i ¼ 1,2) setup The proportion of the manufacturer’s second production setup cost shared by the buyer once the second production happened. This can be explained as a compensation for the manufacturer’s loss incurred by changing the production plan or an incentive for the manufacturer to activate the second production when the buyer place a second order, where 0l1 The manufacturer’s unit production cost

To avoid unrealistic and trivial cases, assume that the following relationship is kept: p > c > v > r and p + k c b > 0.

4 Decentralized Order Policies Consider first the decentralized decision-making system. In this setting, the profitmaximizing buyer would determine independently the initial order quantity Q for the whole sales period, and whether to place a second order to satisfy the part of the unfilled demand at the end of the selling period if he is required to share the setup cost incurred by his second order. The manufacturer would then determine whether to activate a second production in response to the buyer’s second order. Since an order of Q units is placed by the buyer at the start of the sales period, then by the end of the selling period, if demand x exceeds the order quantity Q, there would be a shortage of (x Q) units, of which b(x Q) units could be backordered. Therefore, at the end of the sales period, the buyer will have two alternatives: either place a second order of b(x Q) units, or do not. If the buyer chooses the former, her or his profit will be (p c b)b(x Q) k(1 b) (x Q) t2 ls2. In contrast, if the buyer selects the latter, the corresponding profit is k(x Q). Generally, the buyer is willing to place a second order if the former can have the buyer get more profits than does the latter, i.e., ðp c bÞbðx QÞ kð1 bÞðx QÞ t2 ls2 kðx QÞ or bðx QÞ qb ¼ ðt2 þ ls2 Þ=ðp þ k c bÞ: It means that qb is the buyer’s threshold quantity beyond which the buyer places a second order. On the other hand, as pointed out by Weng (2004), the manufacturer

322

Y.-W. Zhou and S.-D. Wang

would be usually willing to satisfy the second order if it yields positive profit. Hence, the manufacturer is willing to activate a second production in response to the buyer’s second order as long as (c v)b(x Q) (1 l)s2 0 or b(x Q) qm ¼ (1 l)s2/(c v). It shows that qm is a threshold quantity beyond which the manufacturer makes a second production. Generally speaking, qb and qm are probably unequal. That is to say, the manufacturer may not be willing to activate a second production while the buyer is willing to place a second order. Likewise, the buyer may not be willing to place a second order yet while the manufacturer is willing to activate a second run production. In what follows, for convenience, we first derive the expected profits of both the manufacturer and the buyer respectively in terms of two cases: qb qm and qb qm. We then develop the buyer’s optimal ordering policy in the decentralized system.

4.1

Case with qb qm

In this case, by the end of the selling period, there might be four situations pertaining to the practical demand x. Figure 1 shows a sketch of the four situations. Situation 1: The practical demand x does not exceed the order quantity Q. In such a situation, the buyer has (Q x) units left at the end of the selling period. Hence, the buyer’s and manufacturer’s profits will be respectively given by BP1 ðx; QÞ ¼ px þ rðQ xÞ cQ t1 ;

MP1 ðx; QÞ ¼ ðc vÞQ s1 :

Situation 2: Q < x < Q + qb/b. Under this situation, the backordered demand b(x Q) is less than the buyer’s threshold quantity qb. So the buyer does not place a second order and the demand of (x Q) units is lost. Thus, the profits of both parties can be expressed as BP1 ðx; QÞ ¼ ðp cÞQ kðx QÞ t1 ;

MP1 ðx; QÞ ¼ ðc vÞQ s1 :

Situation 3: Q + qb/b x < Q + qm/b. In this situation, the backordered demand b(x Q) exceeds the buyer’s threshold quantity but does not reach the manufacturer’s. It means that the buyer is willing to place a second order but the manufacturer is not willing to activate a second production if the buyer orders only b(x Q) units. If to want the second order satisfied, the buyer has to order at least qm units. However, the order quantity qm is larger than the backlogged demand, which means that the buyer would bear the loss in purchasing costs of unsold items of (qm b(x Q)) units. If the buyer places a second order of qm units, the buyer’s profit is ðp c bÞbðx QÞ kð1 bÞðx QÞ ðc rÞ½qm bðx QÞ t2 ls2 : 1

2 Q

3 Q+ qb /b

Fig. 1 Four possible situations of the practical demand

4 Q+ qm /b

x

Supply Chain Coordination for Newsvendor-Type Products

323

In contrast, if the buyer does not place a second order, the corresponding profit is k(x Q). It is obvious that the buyer is willing to place a second order of qm units if the difference of these profits is nonnegative, i.e., (p c b)b(x Q) which k(1 b)(x Q) (c r)[qm b(x Q)] t2 ls2 k(x Q), is equivalent to b(x Q) q0 ¼ [t2 þ ls2 þ (c r)qm]/(p þ k r b) or x Q þ q0/b. Therefore, when Q þ qb/b x < Q þ q0/b, there is no second transaction happened and both parties’ profits still are BP1(x, Q)¼(p c)Q k(x Q) t1 and MP1(x, Q) ¼ (c v)Q s1; whereas when Q þ q0/b x < Q þ qm/b, the buyer will place a second order of qm units and the manufacturer is also willing to reproduce qm units. Then, when Q þ q0/b x < Q þ qm /b, the profits of the buyer and manufacturer will respectively be BP1 ðx; QÞ ¼ ðp cÞQ þ ðp c bÞbðx QÞ kð1 bÞðx QÞ ðc rÞ½qm bðx QÞ t1 t2 ls; MP1 ðx; QÞ ¼ ðc vÞ½Q þ qm s1 ð1 lÞs2 : Situation 4: x Q+qm/b. That is, the backlogged demand b(x Q) is larger than both partners’ threshold quantities. Thus, the buyer will place a second order with quantity b(x Q), and the manufacturer will also quickly activate a second production in order to ensure the ordered items satisfied in time. Hence, under such a situation, both parties’ profits are given by BP1 ðx; QÞ ¼ ðp cÞQ þ ðp c bÞbðx QÞ kð1 bÞðx QÞ t1 t2 ls2 : MP1 ðx; QÞ ¼ ðc vÞ½Q þ bðx QÞ s1 ð1 lÞs2 : Based on the above analysis, one easily derived that the buyer’s expected profit is: BP1 ðQÞ ¼

ðQ 0

þ

BP1 ðx; QÞf ðxÞdx þ

ð Qþqm =b Qþq0 =b

ð Qþq0 =b Q

BP1 ðx; QÞf ðxÞdx þ

¼ ðp þ k c bÞ½Q þ b þ ðp þ k rÞ

ðQ

BP1 ðx; QÞf ðxÞdx

ð þ1

ð þ1 Qþq0 =b

Qþqm =b

BP1 ðx; QÞf ðxÞdx

ðx QÞf ðxÞdx

ðx QÞf ðxÞdx þ bQ km

0

t1 ðt2 þ ls2 Þ½1 FðQ þ q0 =bÞ ð Qþqm =b ½qm bðx QÞf ðxÞdx: ðc rÞ Qþq0 =b

(1)

324

Y.-W. Zhou and S.-D. Wang

The manufacturer’s expected profit is given by MP1 ðQÞ ¼

ð Qþq0 =b 0

þ

MP1 ðx; QÞf ðxÞdx þ

ð þ1 Qþqm =b

ð Qþqm =b Qþq0 =b

MP1 ðx; QÞf ðxÞdx

MP1 ðx; QÞf ðxÞdx:

¼ ðc vÞ½Q þ b

ð þ1 Qþq0 =b

(2)

ðx QÞf ðxÞdx

s1 ð1 lÞs2 ½1 FðQ þ q0 =bÞ ð Qþqm =b þ ðc vÞ ½qm bðx QÞf ðxÞdx Qþq0 =b

4.2

Case with qb qm

Similarly, when qb qm, if by the end of the selling period the practical demand x is less than or equal to Q, there are (Q x) units left at the end of the sales period. Hence, the buyer does not need to place a second order. When Q < x < Q þ qb/b, the backlogged demand b(x Q) is less than the buyer’s threshold quantity. So the buyer has no incentive to place a second order. If x Q þ qb/b, the backordered demand b(x Q) is larger than the threshold quantities of both parties. Thus, the buyer will place a second order with quantity b(x Q) whereas the manufacturer will also activate a second production for the buyer’s second order. Figure 2 describes graphically the buyer’s second order decision. Under the above twice-ordering strategy, one can obtain that the buyer’s expected profit is: BP2 ðQÞ ¼ ðp þ k c bÞ½Q þ b þ ðp þ k r Þ

ðQ

ð þ1 Qþqb =b

ðx QÞf ðxÞdx (3)

ðx QÞf ðxÞdx

0

þ bQ km t1 ðt2 þ ls2 Þ½1 FðQ þ qb =bÞ:

No shortage

Q

No second order

Q+ qm /b

Fig. 2 The buyer’s second decision

Place a second order of b(x-Q) units Q+ qb /b

x

Supply Chain Coordination for Newsvendor-Type Products

And, the manufacturer’s expected profit is given by ð þ1 ðx QÞf ðxÞdx s1 ð1 lÞs2 MP2 ðQÞ ¼ðc vÞ½Q þ b Qþqb =b

325

(4)

½1 FðQ þ qb =bÞ: Summarizing the above two cases will give the expected profits of the buyer and the manufacturer in the decentralized system respectively as MP1 ðQÞ qb qm BP1 ðQÞ qb qm and MPðQÞ ¼ : (5) BPðQÞ ¼ BP2 ðQÞ qb qm MP2 ðQÞ qb qm From (5), one can derive the property of the manufacturer’s expected profit function, MP(Q). Property 1. The manufacturer’s expected profit under the decentralized system, MP(Q), is a monotone increasing function with respect to Q. Proof. (i) If qb qm, the first-order derivative of MP1(Q) with respect to Q will be MP01 ðQÞ ¼ ðc vÞ½1 b þ bFðQ þ qm =bÞþ½ð1 lÞs2 ðc vÞqm f ðQ þ q0 =bÞ ¼ ðc vÞ½1 b þ bFðQ þ qm =bÞ > 0; which means that MP1(Q) is a monotone increasing function of Q. (ii) If qb qm, the first-order derivative of MP2(Q) with respect to Q is MP02 ðQÞ ¼ ðc vÞ½1 b þ bFðQ þ qb =bÞþ½ð1 lÞs2 ðc vÞqb f ðQ þ qb =bÞ:

(6)

Since f 0 (x) < 0, then FðQ þ qb =bÞ > ðQ þ qb =bÞf ðQ þ qb =bÞ:

(7)

Substituting (7) into (6) gives MP0 2(Q) > (c v)(1 b) þ [(c v)bQ þ (1 l)s2]f(Q þ qb/b) > 0. Hence, MP2(Q) is also a monotone increasing function of Q. □ Based on Property 1, we present the buyer’s optimal ordering policies in the decentralized system in Theorem 1. Theorem 1. For any increasing concave CDF F(.), the buyer’s unique optimal ordering policy, Qb, that maximizes the buyer’s expected profit is given by Qb1 ; qb qm Qb ¼ ; Qb2 ; qb qm

326

Y.-W. Zhou and S.-D. Wang

where Qb1 and Qb2 are respectively given by ðp þ k rÞFðQb 1 Þ þ ðp þ k r bÞbFðQb 1 þ q0 =bÞ ðc rÞbFðQb 1 þ qm =bÞ þ ðp þ k cÞð1 bÞ þ bb ¼ 0 ðp þ k rÞFðQb 2 Þ þ ðp þ k c bÞbFðQb 2 þ qb =bÞ þ ðp þ k cÞð1 bÞ þ bb ¼ 0:

(8)

(9)

Proof. (i) For the case with qb qm, taking the first- and second-order derivatives of BP1(Q) shown in (1) with respect to Q will respectively give

BP01 ðQÞ ¼ ðp þ k rÞFðQÞ þ ðp þ k r bÞbFðQ þ q0 =bÞ ðc rÞbFðQ þ qm =bÞ þ ðp þ k cÞð1 bÞ þ b

(10)

BP001 ðQÞ¼ðpþkrÞ½f ðQÞbf ðQþq0 =bÞbbf ðQþq0 =bÞðcrÞbf ðQþqm =bÞ: (11) Since F00 (x) < 0, i.e., f 0 (x) < 0, f(Q) > f(Q + q0/b). Hence, from (11) one has BP00 1(Q) < 0. Additionally, from (10) one easily derives that limQ !þ1 BP01 ðQÞ ¼ ðc rÞ < 0 and limQ !0þ BP01 ðQÞ ¼ ðp þ k r bÞbFðq0 =bÞ ðc rÞbFðqm =bÞ þ ðp þ k cÞð1 bÞ þ bb: It is obvious that if (p þ k r b)bF(q0/b) (c r)bF(qm/b) þ (p þ k c) (1 b) þ bb 0, then BP0 1(Q) 0, i.e., BP1(Q) is a monotone decreasing function of Q. Thus, the buyer’s optimal order quantity will be Qb1 ¼ 0, which implies that no business happens between the manufacturer and the buyer. In order to avoid such unrealistic and trivial cases, we assume in the subsequent analysis that if qb qm, ðp þ k r bÞbFðq0 =bÞ ðc rÞbFðqm =bÞ þ ðp þ k cÞð1 bÞ þ bb > 0: (12) Hence, there exists a unique positive root Qb1 to equation BP0 1(Q) ¼ 0. Since BP00 1(Q) < 0, BP1(Q) reaches its maximum at Qb1. (ii) If qb qm, the first- and second-order derivatives of BP2(Q) given in (3) with respect to Q will be respectively BP02 ðQÞ ¼ ðpþk rÞFðQÞþðpþk cbÞbFðQþqb =bÞþðpþk cÞð1bÞþbb

Supply Chain Coordination for Newsvendor-Type Products

327

and BP002 ðQÞ ¼ ðp þ k rÞf ðQÞ þ ðp þ k c bÞbf ðQ þ qb =bÞ: Since F00 (x) < 0 and 0 b 1, one has BP00 2(Q) < 0. It is easy to check that limQ !þ1 BP02 ðQÞ ¼ ðc rÞ < 0 and limQ !0þ BP02 ðQÞ ¼ ðp þ k c bÞbFðqb =bÞ þ ðp þ k cÞð1 bÞ þ bb > 0: Thus, there exists a unique positive root Qb2 to BP0 2(Qb2) ¼ 0. Hence, Qb2 is the maximum point of BP2(Q). □ Substituting Qb1 into (1) and (2) if qb qm and Qb2 into (3) and (4) if qb qm will give the optimal expected profits of the buyer and the manufacturer, BP(Qb) and MP(Qb).

5 Centralized Order Policy Consider now a situation where both the manufacturer and the buyer are willing to cooperate to pursue the centralized optimal ordering policy. Hence, unlike in the decentralized channel, the objective in this setting is to maximize the expected total profit of the system. In the subsequent analysis, we first formulate the expected total profit of the system. As described in Sect. 4, the second transaction between the two parties in the decentralized system will occur only if both parties have profits higher than those in the case without the second order (or production). In the centralized system, however, even if the second transaction results in the decrease of one party’s profit, it will still occur if it can lead to the increase of the channel’s profit. That is to say, the occurrence of the second transaction will be subject only to the following condition ðp c bÞbðx QÞ kð1 bÞðx QÞ t2 þ ðc vÞbðx QÞ s2 kðx QÞ or bðx QÞ qc ¼ ðt2 þ s2 Þ=ðp þ k v bÞ: It indicates that qc is the threshold quantity of the centralized system, beyond which the second transaction will occur.

328

Y.-W. Zhou and S.-D. Wang

Similarly, for the centralized system, if the practical demand during the selling period is x, the profit of the system is given by 8 px þ rðQ xÞ vQ t1 s1 ; if x Q > > > > > < ðp vÞQ kðx QÞ t1 s1 ; if Q < x < Q þ qc =b JPðx; QÞ ¼ > ðp vÞQ þ ðp v bÞbðx QÞ kð1 bÞðx QÞ > > > > : t1 t2 s1 s2 ; if x Q þ qc =b Hence, the expected total profit of the system, JP(Q), will be given by ð þ1 JPðQÞ ¼ ðp þ k v bÞ½Q þ b ðx QÞf ðxÞdx þ ðp þ k rÞ

ðQ

Qþqc =b

ðx QÞf ðxÞdx

(13)

0

þ bQ km t1 s1 ðt2 þ s2 Þ½1 FðQ þ qc =bÞ: Maximizing JP(Q) will give Theorem 2. Theorem 2. For any increasing concave CDF F(.), the unique optimal ordering policy, QJ, for the centralized system is given by ðp þ k rÞFðQJ Þ þ ðp þ k v bÞbFðQJ þ qc =bÞ þ ðp þ k vÞð1 bÞ þ bb ¼ 0: (14) Proof. Taking the first- and second-order derivatives of JP(Q) with respect to Q, we obtain JP0 ðQÞ ¼ ðp þ k rÞFðQÞ þ ðp þ k v bÞbFðQ þ qc =bÞ þ ðp þ k vÞð1 bÞ þ bb JP00 ðQÞ ¼ ðp þ k rÞf ðQÞ þ ðp þ k v bÞbf ðQ þ qc =bÞ: Since f(Q) > f(Q + qc/b) and p + k r > p + k v b, we have JP00 (Q) < 0. It is easy to check that limQ!0+JP0 (Q) ¼ (p + k v b)bF(qc/b) + (p + k v)(1 b) + bb > 0 and limQ!+1JP0 (Q) ¼ (v r) < 0. Hence, there exists a unique positive root QJ to JP0 (QJ) ¼ 0, and JP(Q) reaches its maximum at QJ. □ As to a two-echelon supply chain for newsvendor-type products with a single order opportunity, a common fact is that the expected profit of the centralized system is exactly equal to the sum of two members’ expected profits in the decentralized system. Many researchers like Taylor (2002), Cachon (2003), etc., have presented a lot of effective coordination mechanisms by employing successfully this fact. Weng (2004) applied directly this common fact to a supply chain for newsvendor-type products under the twice-order framework defined in this chapter. Then, he presented a quantity discount scheme that could maximize the expected

Supply Chain Coordination for Newsvendor-Type Products

329

profit of his so-called centralized channel. However, Theorem 3 shows that the above common fact does not hold in the supply chain under the twice-order framework considered in the chapter. We find out that the expected profit function [shown in (8)] of the centralized channel is always greater than the sum of the expected profits of two members in the decentralized system. Before giving Theorem 3, we need to show the following lemma. Lemma 1. (i) If qb qm, then qc q0; (ii) if qb qm, then qc qb. Proof. (i) If qb qm, one has qc q0 ¼ ðt2 þ s2 Þ=ðp þ k v bÞ ½t2 þ ls2 þ ð1 lÞs2 ðc rÞ=ðc vÞ=ðp þ k r bÞ ¼ ½t2 þ ls2 ð1 lÞs2 ðp þ k c bÞ=ðc vÞðv rÞ=½ðp þ k v bÞ ðp þ k r bÞ: Since qb qm, t2 þ ls2 (1 l)s2(p þ k c b)/(c v). From assumptions presented in Sect. 2 that p > c > v > r and p þ k c b > 0, one can easily derive p þ k r b > p þ k v b > p þ k c b > 0. Hence, we have qc q0. (ii) If qb qm, one can get (t2 + ls2)/(p + k c b) (1 l)s2/(c v). Therefore, one has qc ¼ ½t2 þ ls2 þ ð1 lÞs2 =ðp þ k v bÞ ½t2 þ ls2 þ ðt2 þ ls2 Þðc vÞ=ðp þ k c bÞ=ðp þ k v bÞ ¼ qb : □ Lemma 1 means that, the threshold value beyond which the centralized system implements the second transaction is always less than or equal to its counterpart in the decentralized system. From (1)–(4) and Lemma 1, one can derive Theorem 3. Theorem 3. The expected profit of the centralized channel is always greater than the sum of the expected profits of two members in the decentralized system. Proof. For the case of qb qm, due to (1) and (2), one can easily derive the expected total profit of the decentralized system as BP1 ðQÞ þ MP1 ðQÞ ¼ ðp þ k v bÞ½Q þ b þ ðp þ k rÞ

ðQ

ð þ1 Qþq0 =b

ðx QÞf ðxÞdx

ðx QÞf ðxÞdx

0

þ bQ km ðt1 þ s1 Þ ðt2 þ s2 Þ½1 FðQ þ q0 =bÞ ð Qþqm =b ðv rÞ ½qm bðx QÞf ðxÞdx Qþq0 =b

(15)

330

Y.-W. Zhou and S.-D. Wang

Due to (13), the expected profit for the centralized system can be expressed as JPðQÞ ¼ ðp þ k v bÞ½Q þ b þ ðp þ k rÞ

ðQ

ð þ1 Qþq0 =b

ðx QÞf ðxÞdx þ b

ð Qþq0 =b Qþqc =b

ðx QÞf ðxÞdx

ðx QÞf ðxÞdx þ bQ km ðt1 þ s1 Þ

0

ðt2 þ s2 Þ½1 FðQ þ q0 =bÞ ðt2 þ s2 Þ½FðQ þ q0 =bÞ FðQ þ qc =bÞ (16) Combining (15) and (16) gives JPðQÞ ¼ BP1 ðQÞ þ MP1 ðQÞ þ f ðxÞdx þ ðv rÞ

ð Qþq0 =b Qþqc =b

ð Qþqm =b Qþq0 =b

½ðp þ k v bÞbðx QÞ t2 s2 (17)

½qm bðx QÞf ðxÞdx

Similarly, for the case of qb qm, combining (3) and (4) with (13), the expected profit for the centralized system can be rewritten as JPðQÞ ¼ BP2 ðQÞ þ MP2 ðQÞ ð Qþqb =b ½ðp þ k v bÞbðx QÞ t2 s2 f ðxÞdx þ

(18)

Qþqc =b

From (17) and (18), one can observe that the expected profit function of the centralized system is not simply equal to, but larger than, the sum of the expected profit functions of the two partners in the system. □ In the following, we give the explanation of this phenomenon. In fact, under the case of qb qm, as analyzed earlier, if the practical demand x is less than Q þ qc/b (of course, also less than Q þ q0/b due to Lemma 1), the buyer, no matter whether in the centralized or decentralized system, does not place a second order. If the practical demand x is greater than Q þ qm/b(of course, also greater than Q + qc/b due to Lemma 1), a second order of b(x Q) units is implemented in both the centralized and the decentralized system. It implies that, under such two situations, the profit for the centralized system is just equal to the sum of two parties’ profits in the decentralized system. However, if the practical demand x satisfies Q þ qc/b x < Q þ q0/b, the centralized system is willing to activate a production to supply a second order of b(x Q) units. This second transaction brings the system the profit of (p v b)b(x Q) k(1 b) (x Q) t2 s2. In contrast, the second transaction will not occur in the decentralized system. Consequently, the sum of two parties’ profits is equal to k (x Q). The difference of these two profits is equal to (p þ k v b)b(x Q) t2 s2, which represents the increment in the channel profit yielded by two

Supply Chain Coordination for Newsvendor-Type Products

331

partners’ cooperation when the practical demand x falls into [Q þ qc/b, Q þ q0/b]. If the practical demand x satisfies Q þ q0/b x < Q þ qm/b, the second order of b(x Q) units will occur in the centralized system, which brings the system the profit of (p v b)b(x Q) k(1 b)(x Q) t2 s2. In the decentralized system, however, the second order of qm units will happen. The sum of two members’ profits resulted from the second transaction is given by ðp c bÞbðx QÞ kð1 bÞðx QÞ ðc rÞ½qm bðx QÞ t2 þ ðc vÞqm s2: The difference of the two profits is equal to (v r)[qm b(x Q)], which denotes the increment in the system profit incurred by the cooperation between the manufacturer and the buyer when the practical demand x belongs to [Q + q0/b, Q + qm/b]. Thus, the third and fourth terms in (17) exactly represents the expected increment of the system profit when two members in the channel are willing to make a decision jointly. Similarly, one can also obtain the intuitive explanation of (18). The above analysis has revealed that if we simply consider the sum of the expected profit of two parties in the considered decentralized system as the jointly decision-making objective or the expected profit of the centralized system, then this cooperative-looking system does not actually reach perfect coordination or complete cooperation state. The main reason is that in the cooperative-looking centralized system, the second order decision made by the buyer is still based on the buyer’s benefit rather than on the channel’s benefit. Hence, we refer to this type of cooperation as incomplete cooperation (ic for brevity) and this cooperative-looking system as an ic system. Let JPic(Q) be the expected profit of this ic system. Then, one has JPic1 ðQÞ; qb qm JPic ðQÞ ¼ JPic2 ðQÞ; qb qm where JPic1(Q) ¼ BP1(Q) þ MP1(Q) and JPic2(Q) ¼ BP2(Q) þ MP2(Q). It is obvious from (17) and (18) that JP(Q) JPic(Q) for any given Q. Theorem 4 shows the optimal ordering policies under the ic system just mentioned. Theorem 4. For any increasing concave CDF F(.), the unique optimal ordering policy, QJic, that maximizes the sum of the expected profit of two members, i.e., JPic(Q), will be given by ic1 QJ ; qb qm QJ ic ¼ Qic2 J ; q b qm where QJic1 and QJic2 satisfy respectively ðp þ k rÞF QJ ic1 þ ðp þ k r bÞbF QJ ic1 þ q0 =b ðv rÞbF QJ ic1 þ qm =b þ ðp þ k vÞð1 bÞ þ bb ¼ 0

(19)

332

Y.-W. Zhou and S.-D. Wang

ðp þ k rÞF QJ ic2 þ ðp þ k v bÞbF QJ ic2 þ qb =b þ ðp þ k vÞð1 bÞ þ bb ½ðp þ k v bÞqb ðt2 þ s2 Þ f QJ ic2 þ qb =b ¼ 0: (20) Proof. (i) If qb qm, taking the first- and second-order derivatives of JPic1(Q) with respect to Q, we have JP0ic1 ðQÞ ¼ ðp þ k rÞFðQÞ þ ðp þ k r bÞbFðQ þ q0 =bÞ ðv rÞbFðQ þ qm =bÞ þ ðp þ k vÞð1 bÞ þ bb; JP00ic1 ðQÞ ¼ ðp þ k rÞ½f ðQÞ bf ðQ þ q0 =bÞ bbf ðQ þ q0 =bÞ ðv rÞ bf ðQ þ qm =bÞ: Since f(Q) > f(Q + q0/b), one has JP00 ic1(Q) < 0. It is easy to check that limQ !þ1 JP0ic1 ðQÞ ¼ ðv rÞ < 0 and limQ !0þ JP0ic1 ðQÞ ¼ ðp þ k r bÞbFðq0 =bÞ ðv rÞbFðqm =bÞ þ ðp þ k vÞ ð1 bÞ þ bb: From (12), one can derive ðp þ k r bÞbFðq0 =bÞ ðv rÞbFðqm =bÞ þ ðp þ k vÞð1 bÞ þ bb ¼ ðp þ k r bÞbFðq0 =bÞ ðc rÞbFðqm =bÞ þ ðp þ k cÞð1 bÞ þ bb þ ðc vÞbFðqm =bÞ > ðc vÞbFðqm =bÞ > 0; which implies limQ!0+ JP0 ic1(Q) > 0. Hence, there exists a unique positive root QJic1 to equation JP0 ic1(QJic1) ¼ 0, at which JPic1(Q) reaches its maximum. (ii) If qb qm, the first- and second-order derivatives of JPic2(Q)with respect to Q will be JP0ic2 ðQÞ ¼ ðp þ k rÞFðQÞ þ ðp þ k v bÞbFðQ þ qb =bÞ þ ðp þ k vÞð1 bÞ þ bb ½ðp þ k v bÞqb ðt2 þ s2 Þf ðQ þ qb =bÞ; JP00ic2 ðQÞ ¼ ðp þ k rÞf ðQÞ þ ðp þ k v bÞbf ðQ þ qb =bÞ ½ðp þ k v bÞqb ðt2 þ s2 Þf 0 ðQ þ qb =bÞ:

Supply Chain Coordination for Newsvendor-Type Products

333

Since f(Q) > f(Q + qb/b), from Lemma 1 one knows that qc qb. Thus, one can derive JP00ic2 ðQÞ ðv þ b rÞf ðQ þ qb =bÞ ½ðp þ k v bÞqc ðt2 þ s2 Þf 0 ðQ þ qb =bÞ ¼ ðv þ b rÞf ðQ þ qb =bÞ 0: Due to limQ !þ1 JP0ic2 ðQÞ ¼ limQ !þ1 fðv rÞ ½ðp þ k v bÞqb ðt2 þ s2 Þ f ðQ þ qb =bÞg < ðv rÞ < 0 and limQ !0þ JP0ic2 ðQÞ ¼ ðp þ k v bÞbFðqb =bÞ þ ðp þ k vÞð1 bÞ þ bb ½ðp þ k v bÞqb ðt2 þ s2 Þf ðqb =bÞ > ðp þ k v bÞqb f ðqb =bÞ þ ðp þ k vÞð1 bÞ þ bb ½ðp þ k v bÞqb ðt2 þ s2 Þf ðqb =bÞ > 0; there exists a unique positive root QJic2 to equation JP0 ic2(QJic2) ¼ 0, at which JPic2(Q) reaches its maximum. □ Before comparing optimal ordering policies under both decentralized and centralized system, we introduce Lemma 2. Lemma 2. If qb qm, then ½FðQ þ q0 =bÞ FðQ þ qc =bÞ=½FðQ þ qm =bÞ FðQ þ qc =bÞ ðv rÞ= ðp þ k r bÞ: Proof. From the integral mean value theorem, one has ð Qþq0 =b

FðQ þ q0 =bÞ FðQ þ qc =bÞ ¼

Qþqc =b

FðQ þ qm =bÞ FðQ þ qc =bÞ ¼

f ðxÞdx ¼ f ðx1 Þðq0 qc Þ=b;

ð Qþqm =b Qþqc =b

f ðxÞdx ¼ f ðx3 Þðqm qc Þ=b

and ð Qþqm =b Qþqc =b

f ðxÞdx ¼

ð Qþq0 =b Qþqc =b

f ðxÞdx þ

ð Qþqm =b Qþq0 =b

f ðxÞdx

¼ f ðx1 Þðq0 qc Þ=b þ f ðx2 Þðqm q0 Þ=b; where x1 2 ðQ þ qc =b; Q þ q0 =bÞ, x2 2 ðQ þ q0 =b; Q þ qm =bÞand x3 2 ðQ þ qc = b; Q þ qm =bÞ.

334

Y.-W. Zhou and S.-D. Wang

Thus, one has f(x3)(qm qc) ¼ f(x1)(q0 qc) + f(x2)(qm q0). Since f(x) is a decreasing function, f(x3)(qm qc) < f(x1)(q0 qc + qm q0) ¼ f(x1)(qm qc) due to x1 < x2. It leads to f(x3) < f(x1). Noting that q0 may be equal to qc, one can obtain ½FðQ þ q0 =bÞ FðQ þ qc =bÞ=½FðQ þ qm =bÞ FðQ þ qc =bÞ ¼ f ðx1 Þðq0 qc Þ=½f ðx3 Þðqm qc Þ ðq0 qc Þ=ðqm qc Þ:

(21)

ðv rÞðp þ k c bÞ ðqm qb Þ and ðp þ k v bÞðp þ k r bÞ ðpþk cbÞ ðpþk cbÞ ðqm qb Þþðq0 qc Þ ¼ qm qc ¼ qm q0 þq0 qc ¼ pþk r b pþk vb ðqm qb Þ, one easily derives Due to q0 qc ¼

ðq0 qc Þ=ðqm qc Þ ¼ ðv rÞ=ðp þ k r bÞ: From (21) and (22), it is obvious to have Lemma 2. Based on Lemma 2, one can obtain the following results.

(22) □

Theorem 5. (i) QJic QJ, QJic > Qb; (ii) JP(QJ) JPic(QJic) > BP(Qb) + MP (Qb). Proof. (i) We first prove QJic QJ. If qb qm, one can get from Lemma 2 that ðp þ k r bÞFðQ þ q0 =bÞ ðv rÞFðQ þ qm =bÞ ðp þ k v bÞ FðQ þ qc =bÞ: Noting (17) and (23), one has 0 ¼ ðp þ k rÞF QJ ic1 þ ðp þ k r bÞbF QJ ic1 þ q0 =b ðv rÞbF QJ ic1 þ qm =b þ ðp þ k vÞð1 bÞ þ bb ðp þ k rÞF QJ ic1 þ ðp þ k v bÞbF QJ ic1 þ qc =b

(23)

(24)

þ ðp þ k vÞð1 bÞ þ bb; which is equivalent to JP0 (QJic1) 0. In addition, from the proof of Theorem 2, one can see that JP0 (QJ) ¼ 0 and JP00 (Q) < 0. Hence, it is clear to have QJic1 QJ . If qb qm, then from Lemma 1 one has qc qb. Letting G(q) ¼ the left side of (13), one can derive dGðqÞ=dq ¼ ½ðp þ k v bÞq t2 s2 f 0 QJ ic2 þ q=b =b: When qc q qb, it is obvious to have (p + k v b)q t2 s2 (p + k v b)qc t2 s2¼0. Since f 0 (x) < 0, dG(q)/dq 0 if qc q qb, which

Supply Chain Coordination for Newsvendor-Type Products

335

means that G(q) is a monotone increasing function of q in [qc, qb]. Therefore, one has G(qc) G(qb) ¼ 0, i.e., ðp þ k rÞF QJ ic2 þ ðp þ k v bÞbF QJ ic2 þ qc =b þ ðp þ k vÞð1 bÞ þ bb 0;

(25)

which is just equivalent to JP0 (QJic2) 0. This together with JP0 (QJ) ¼ 0 and JP00 (Q) < 0 will give QJic2 QJ. The proof of QJic QJ is completed. Next, we prove QJic > Qb. If qb qm, one can know from (13) that the buyer’s optimal order quantity Qb1 satisfies: ðp þ k rÞFðQb 1 Þ þ ðp þ k r bÞbFðQb 1 þ q0 =bÞ ðc rÞbFðQb 1 þ qm =bÞ þ ðp þ k cÞð1 bÞ þ bb ¼ 0; which gives ðp þ k rÞFðQb 1 Þ þ ðp þ k r bÞbFðQb 1 þ q0 =bÞ ðv rÞb FðQb 1 þ qm =bÞ þ ðp þ k vÞð1 bÞ þ bb > 0:

(26)

(26) can be rewritten as JP0 ic1(Qb1) > 0. Additionally, from the proof of Theorem 4 one has JP00 ic1(Q) < 0. It implies that JP0 ic1(Q) is monotone decreasing. Hence, after noting JP0 ic1(QJic1) ¼ 0, one can derive QJic1 > Qb1. Similarly, one can prove QJic2 > Qb2. The proof is omitted. (ii) From Theorem 4, we know that JPic(Q) reaches its maximum at QJic. Since QJic > Qb, JPic(QJic) > BP(Qb) + MP(Qb). In addition, (17) and (18) have implied that JP(Q) > BP(Q) + MP(Q) ¼ JPic(Q) for any Q (>0). Hence, it is obvious to have JPic(QJic) < JP(QJic) < JP(QJ). Theorem 5 indicates that, for the considered two-echelon supply chain, the cooperation between two parties in decision making, even the aforesaid incomplete cooperation, will lead to an increase in the system’s expected profit, and that the buyer’s optimal order quantity in the ic setting is greater than the counterparts in both centralized and decentralized setting.

5.1

Property of the ic and Decentralized System Performance

From the definitions of qb, qm, and q0, one can easily obtain that all these threshold quantities have to do with l. It implies that JPic(Q) depends on l as well. Property 2 shows the monotonity of JPic(Q, l) with respect to l.

336

Y.-W. Zhou and S.-D. Wang

Property 2. For any given Q, JPic(Q, l) is a decreasing function with respect to l. Proof. (i) Prove that JPic1(Q, l) is a monotone decreasing function of l. For a given Q, the first-order partial derivative of JPic1(Q, l) with respect to l is @JPic1 ðQ; lÞ[email protected] ¼ ½ðp þ k v bÞq0 þ t2 þ s2 þ ðv rÞðqm q0 Þ ½f ðQ þ q0 =bÞ=b dq0 =dl

(27)

þ ½FðQ þ qm =bÞ FðQ þ q0 =bÞdqm =dl From definitions of qm and q0, one has ðp þ k v bÞq0 þ t2 þ s2 þ ðv rÞðqm q0 Þ ¼ ½t2 þ ls2 þ ðc rÞqm þ ðv rÞqm þ t2 þ s2 ¼ ðc vÞqm þ ð1 lÞs2 ¼ 0: (28) Substituting (28) into (27) will give @JPic1 ðQ; lÞ[email protected] ¼ ½FðQ þ qm =bÞ FðQ þ q0 =bÞdqm =dl:

(29)

Since q0 qm (see definitions of q0 and qm) and dqm/dl ¼ s2/(c v), (29) means ∂JPic1(Q, l)/∂l 0, i.e., JPic1(Q, l) is a decreasing function of l. (ii) Prove that JPic2(Q, l) is a decreasing function of l. For a given Q, the first-order partial derivative of JPic2(Q, l) about l is @JPic2 ðQ; lÞ[email protected] ¼ ½ðp þ k v bÞqb þ t2 þ s2 ½f ðQ þ qb =bÞ=b dqb =dl: (30) Noting that qb qm and definitions of qb and qm, one can get ðp þ k v bÞqb þ t2 þ s2 ¼ ðp þ k c bÞqb ðc vÞqb þ t2 þ ls2 þ ð1 lÞs2 ¼ ðc vÞqb þ ð1 lÞs2

(31)

ðc vÞqm þ ð1 lÞs2 ¼ 0: Since dqb/dl ¼ s2/(p þ k c b) > 0, substituting (31) into (30) gives ∂JPic2(Q, l)/∂l 0. Namely, JPic2(Q, l) is a monotone decreasing function of l. □ Property 2 indicates that the bigger the value of l, the smaller the expected profit of the ic system. That is to say, the expected profit of the ic system depends on how two parties share the manufacturer’s second production setup cost. Especially, if the manufacturer independently pays all of the second production setup cost, i.e., l ¼ 0, the expected profit of the ic system will be always maximal for any given order quantity Q. In contrast, the buyer’s payment for all of the second setup cost will lead to the minimal expected profit of the ic system. Therefore, the best option

Supply Chain Coordination for Newsvendor-Type Products

337

for the ic system is to let the manufacturer pay all of the second setup cost. This is exactly opposite to the conclusion announced by Weng (2004) that “the general results obtained on the effect of coordination do not depend on how the manufacture’s production setup cost in the second order is allocated between two parties (whether it is paid by the buyer, paid by the manufacturer, or shared by both parties )” (Weng 2004, p. 151). A direct corollary of Property 2 is the following. Corollary 1. The sum of the optimal expected profits of two parties in the decentralized system decreases as l increases. Proof. As known in Sect. 4, the sum of the optimal expected profits of both entities is BP(Qb) þ MP(Qb), which is exactly equal to JPic(Qb, l). Hence, Property 2 also means Corollary 1. □ This corollary explains that sharing the manufacturer’s second production setup cost other than utterly paid by the retailer can increase the decentralized system performance. Moreover, the decentralized system would perform best if the manufacturer covers the second production setup cost completely.

5.2

A Special Case

If l ¼ 1 and b ¼ 1, all excess demand is completely backordered, and the second production setup cost is utterly paid by the buyer. It means that the threshold value that the manufacturer is willing to activate a second production in the decentralized system will be equal to zero, i.e., qm ¼ 0. Under such a case, the expected profit of the ic system, denoted as JPic(Q, l, b), becomes JPic ðQ; l ¼ 1; b ¼ 1Þ ¼ BP2 ðQ; l ¼ 1; b ¼ 1Þ þ MP2 ðQ; l ¼ 1; b ¼ 1Þ; which is just equal to the system’s expected profit (2.7) defined in Weng (2004). For notational convenience, let JPw(Q) denote the system’s expected profit and QJw be the corresponding optimal coordinated ordering quantity in Weng (2004). Then, JPw(Q) ¼ JPic(Q, l ¼ 1, b ¼ 1). Hence, Corollary 2 can also be derived directly from Property 2. Corollary 2. JPw ðQJw Þ JPic ðQJ ic ; l; b ¼ 1Þ: Proof. From the analysis presented in the second paragraph in Sect. 5.2, one has JPw(QJw) ¼ JPic(QJw, l ¼ 1, b ¼ 1). And Theorem 4 means JPic(QJw, l ¼ 1, b ¼ 1) JPic(QJic, l ¼ 1, b ¼ 1). Since JPic(Q, l, b) is a monotone decreasing function with respect to l, one can get JPic(QJic, l ¼ 1, b ¼ 1) JPic(QJic, l, □ b ¼ 1), where 0 l 1. Hence, one has JPw(QJw) JPic(QJic, l, b ¼ 1).

338

Y.-W. Zhou and S.-D. Wang

Corollary 2 further verifies that for any l (0 l < 1), the optimal expected profit of the system in Weng (2004) is always less than that of the ic system in the present model, of course, also less than that of the system under complete cooperation in this chapter.

6 Possible Perfect Coordination Scenarios By designing a simple quantity discount policy, Weng (2004) realized coordination of the ic system under the special case with l ¼ 1 and b ¼ 1. Following the way in Weng (2004), one can also achieve coordination of the ic system under the case with 0 l 1 and 0 b 1 but cannot realize perfect coordination of the system, even if under the special case with l ¼ 1 and b ¼ 1. Maybe, a more complicated quantity discount policy could achieve perfect coordination of the whole channel, but designing such a policy is out of our ability. Then, we pay our attention to a widely-used effective coordination mechanism: two-part tariff (hereafter, TPT for brevity), characterized by a two-tuple parameter (ct, K) in which the manufacturer sells the product to the buyer at the unit wholesale price ct ¼ v and charges the buyer a fixed franchise fee K. For the special case with l ¼ 1, it can be easily shown that for any given K, the buyer’s optimal order quantity would be Qt ¼ QJ, the counterpart of the centralized system, if the buyer accepts the TPT. Hence, the TPT achieves perfect coordination of the channel. Thus, as long as the manufacturer sets a suitable K-value that makes both parties’ benefits greater than before, both parties would accept the TPT that realizes perfect coordination of the channel. However, for the general case with l 6¼ 1, a common TPT mechanism is not able to achieve perfect coordination of the chain. Next, we move to another widely-used effective coordination mechanism: Revenue-Sharing Contract (hereafter, RSC for brevity), proposed by Cachon and Lariviere (2000). It is described by two parameters (cr, F), i.e., the manufacturer charges the buyer a unit wholesale price cr, lower than the unit marginal cost v, in exchange for a percentage (1 F) of the buyer’s revenue. Unfortunately, we find out that RSC also fails to coordinate the supply chain presented in our model. However, a revised revenue-sharing contract (hereafter, RRSC for brevity) would be able to complete perfect coordination of the supply chain. Before describing the RRSC, we need define the buyer’s generalized revenue as follows: Definition 1. Buyer’s generalized revenue ¼ Buyer’s revenue (Buyer’s shortage cost + Buyer’s backorder cost) The considered RRSC is characterized by three-tuple parameters (cr, F, T). Parameters cr and F are used to achieve the supply chain coordination, whereas parameter T is adopted to split the expected profit of the coordinated system between two parties. In such a RRSC, the manufacturer charges the buyer a unit wholesale price cr so that the threshold quantities of both the manufacturer and the

Supply Chain Coordination for Newsvendor-Type Products

339

buyer are equal to the counterpart of the centralized system, then selectively requires from the buyer a percentage (1 F) of his generalized revenue to keep the buyer’s optimal order quantity consistent with the centralized system’s, and finally gives the buyer a return profit T to compensate the buyer’s possible loss for accepting the RRSC. As explained above, under the RRSC, the optimal wholesale price should be chosen to make the threshold quantities of both parties equal to one of the centralized system. That is, cr satisfies qb ¼ qm ¼ qc, which leads to the manufacturer’s optimal wholesale price as: cr ¼ v þ ð1 lÞs2 ðp þ k v bÞ=ðt2 þ s2 Þ:

(32)

It is clear from (32) that this optimal wholesale price is larger than the unit marginal cost v, which is opposite to the counterpart in common RSC. If the buyer places an order of Q units at the wholesale price cr* given by (32), the generalized revenue of the buyer will be GRðQÞ ¼ ðp þ k cr bÞ½Q þ b

ðQ

ð þ1 Qþqc =b

ðx QÞf ðxÞdx þ ðp þ k rÞ

ðx QÞf ðxÞdx þ bQ km:

(33)

0

Thus, for any return profit T set by the manufacturer, the expected profits of the buyer and the manufacturer under the RRSC are respectively BPr ðQ; F; T Þ ¼ FGRðQÞ t1 ðt2 þ ls2 Þ½1 FðQ þ qc =bÞ þ T;

(34)

MPr ðQ; F; T Þ ¼ ð1 FÞGRðQÞ þ MCr ðQÞ s1 ð1 lÞs2 ½1 FðQ þ qc =bÞ T; (35) where MCr ðQÞ ¼ ðcr vÞ½Q þ b

ð þ1 Qþqc =b

ðx QÞf ðxÞdx:

Due to f 0 (x) < 0 and cr* > v > r, one can easily get that BPr(Q,F,T) is a concave function about Q. Hence, to achieve supply chain coordination, the manufacturer should select a F so that under the RRSC the buyer’s optimal order quantity Qr is just equal to the optimal order quantity QJ of the centralized system. That is, ðQ;F;TÞ * this F should satisfy @[email protected] Q¼QJ ¼ 0, which gives the optimal fraction, F , of the generalized revenue kept by the buyer as F ¼ ðt2 þ ls2 Þf ðQJ þ qc =bÞ=½ðt2 þ ls2 Þf ðQJ þ qc =bÞ þ BCr ðQJ Þ;

(36)

340

Y.-W. Zhou and S.-D. Wang

where BCr ðQÞ ¼ ðp þ k rÞFðQÞ ðp þ k cr bÞbFðQ þ qc =bÞ ðp þ k cr Þð1 bÞ bb: From (14) and cr* > v > r, it is not difficult to show that 0 < F* < 1, which means that the F* is indeed a feasible fraction of the generalized revenue. Thus, for any given T, the RRSC, (cr*, F*, T), had actually achieved perfect coordination of the supply chain if it were accepted to implement. However, whether this RRSC can be implemented would depend on whether both parties gain more expected profits under the RRSC or what values the parameter T takes. Suppose that the manufacturer is willing to offer the RRSC only if her expected profit under the RRSC increases by e 100% (e 0) as compared to her original expected profit (MP(Qb)), and that the buyer is willing to accept the RRSC only when it can let the buyer’s expected profit increased by d 100% (d 0). Then, it is easy to show that the values of T available to both parties should satisfy Tmin T Tmax, where Tmax is the manufacturer’s largest endurable return profit and Tmax ¼ ð1 F ÞGRðQJ Þ þ MCr ðQJ Þ s1 ð1 lÞs2 ½1 FðQJ þ qc =bÞ (37) ð1 þ eÞMPðQb Þ Tmin is the buyer’s smallest acceptable return profit and Tmin ¼ ð1 þ dÞBPðQb Þ F GRðQJ Þ þ t1 þ ðt2 þ ls2 Þ½1 FðQJ þ qc =bÞ (38) It can be easily derived from (13), (33), (37) and (38) that Tmin T Tmax is equivalent to ð1 þ dÞBPðQb Þ þ ð1 þ eÞMPðQb Þ JPðQJ Þ

(39)

Thus, if (39) holds, the manufacturer certainly offers the buyer a return profit so that the buyer keeps his reservation profit only, because she, as the designer of the contract, will always want to capture the lion’s share of the channel profit. So, the optimal return profit set by the manufacturer is T* ¼ Tmin. Furthermore, whether (39) holds will depend on what values of d and e (required by the retailer and the manufacturer, respectively) take. For example, for some e specified by the manufacturer, if dmax ¼ {JP(QJ) (1 + e) MP(Qb)}/BP(Qb) 1 0, then (39) will hold as long as the value of d required by the retailer does not exceed dmax. If dmax < 0, (39) does not hold for any d 0. This implies that the manufacturer has asked for a too big e. To sum up, we have the following. Theorem 6. (i)The necessary condition that there exists any feasible RRSC is given by (39). (ii) If the necessary condition is satisfied, then the optimal RRSC that can achieve perfect coordination of the channel will be (cr*, F*, T*), where cr*, F* and T* are given by (32), (36) and (38), respectively.

Supply Chain Coordination for Newsvendor-Type Products

341

7 Numerical Examples In order to illustrate the model, we show a numerical example for each of two cases: qb < qm and qb > qm. Example 1. Case with qb < qm The parameters of the model are listed below: p ¼ 10, c ¼ 6.5, v ¼ 3, b ¼ 1, k ¼ 4, r ¼ 0.5, t1 ¼ 50, t2 ¼ 150, s1 ¼ 200, s2 ¼ 300, l ¼ 0, b ¼ 1 and d ¼ 0. The random demand x is assumed to follow the exponential distribution with m ¼ 150. Following the model presented in this chapter, one can obtain that (1) the optimal first order quantity in the decentralized setting is Qb1 ¼ 45.3, the expected profits of the manufacturer and the buyer are respectively MP(Qb1) ¼ 177.8 and BP (Qb1) ¼ 203.0, and the sum of two parties’ expected profits is 380.8; (2) the optimal first order quantity of the centralized system is QJ ¼ 133.6, the optimal RRSC is (cr*, F*, T*) ¼ (9.7, 0.062, 325.2), which achieves perfect coordination of the supply chain and enhances the system’s expected profit to JP(QJ) ¼ 466.1. However, for the same values of parameters given in Example 1, the coordinated order policy in Weng (2004) enhanced the system’s expected profit only to JPw(167.7) ¼ 460.6. Example 2. Case with qb > qm The parameters of the model are listed below: p ¼ 11, c ¼ 7, v ¼ 1.5, k ¼ 1, b ¼ 0.5, t2 ¼ 260. Other parameters are kept the same as in Example 1. Based on the presented solution procedure in this chapter, the following can be obtained (1) In the decentralized setting, the optimal first order quantity is Qb2 ¼ 39.2, the expected profits of the manufacturer and the buyer are respectively MP(Qb2) ¼ 457.0 and BP(Qb2) ¼ 295.5, and the sum of two parties’ expected profits is 752.5; (2) the optimal first order quantity of the centralized system is QJ ¼ 229.4, the optimal RRSC is (cr*, F*, T*) ¼ (6.9, 0.054, 390.4), which can perfectly coordinate the whole channel and enhance the system’s expected profit to JP(QJ) ¼ 945.6. However, for the same values of parameters given in Example 2, the coordinated order policy in Weng (2004) enhanced the system’s expected profit only to JPw(195.7) ¼ 929.2.

8 Conclusions Most of the literature on coordination issues of the supply chain with single period products assumed that only one order happened during the whole period. However, in practice, buyers probably choose to place more than once order in the selling period because they know more exact information about demand as time moves ahead. In this chapter, we further generalize the newsboy-type order coordination issue considered by Weng (2004) for a two-echelon supply chain with two ordering opportunities, and extend it to cover the case with two-party-shared second setup

342

Y.-W. Zhou and S.-D. Wang

cost and partial backlogging. We prove that the ic system and decentralized system would perform best if the manufacturer covers utterly the second production setup cost. We find out that the expected profit of the centralized system is not always equal to the sum of two members’ expected profits in the decentralized system, which is not consistent with our intuitive expectation and those in the existing related literature, like Cachon (2003), Weng (2004), Zhou and Li (2007), etc. In order to achieve perfect coordination of the considered channel, we try three widely-used effective mechanisms: simple quantity discount, two-part tariff and revenue-sharing contract. Consequently, both simple quantity discount and revenue-sharing contract is not able to achieve the channel’s perfect coordination. Neither can the two-part tariff except the special case that the buyer pays all the manufacturing setup cost. The chapter then presents a RRSC policy that completes the perfect coordination of the supply chain. Worthwhile to mention is that for simplicity the chapter only considers a constant backordered fraction of the unfilled demand in the sales period. In reality, however, this backordered fraction may probably influence the buyer’s expected profit directly. In that case, it would be beneficial for the buyer to choose a suitable second order quantity. This problem will be considered in our future research. Other possible extensions of the model include: considering multiple manufacturers, multiple buyers with price or quantity competition, random demand with unknown probability distribution, etc.

References Arcelus FJ, Kumar S, Srinivasan G (2008) Evaluating manufacturer’s buyback policies in a singleperiod two-echelon framework under price-dependent stochastic demand. Omega 36(5): 808–824 Atkinson AA (1979) Incentives, uncertainty and risk in newsboy problem. Decis Sci 10:341–357 Cachon G, Lariviere M (2000) Supply chain coordination with revenue sharing: strengths and limitations. Working paper, University of Pennsylvania, Philadelphia, PA Cachon G (2003) Supply chain coordination with contracts. In: de Kok AG, Graves SC (eds) Handbooks in operations research and management science, Chap 6, vol 11. Elsevier, Amsterdam Chen H, Chen J, Chen Y (2006) A coordination mechanism for a supply chain with demand information updating. Int J Prod Econ 103:347–361 Chen H, Chen Y, Chiu CH, Choi TM, Sethi S (2010) Coordination mechanism for the supply chain with leadtime consideration and price-dependent demand. Eur J Oper Res 203(1):70–80 Choi TM, Li D, Yan H (2003) Optimal two-stage ordering policy with Bayesian information updating. J Oper Res Soc 54:846–856 Donohue KL (2000) Efficient supply contracts for fashion goods with forecast updating and two production modes. Manage Sci 46(11):1397–1411 Emmons H, Gilbert SM (1998) The role of returns policies in pricing and inventory decisions for catalogue goods. Manage Sci 44(2):276–283 Fisher M, Raman A (1999) Managing short life-cycle products. Ascet 1 Goodman DA, Moody KW (1970) Determining optimal price promotion quantities. J Mark 34: 31–39 Ismail B, Louderback J (1979) Optimizing and satisfying in stochastic cost-volume-profit analysis. Decis Sci 10:205–217

Supply Chain Coordination for Newsvendor-Type Products

343

Kabak I, Schiff A (1978) Inventory models and management objectives. Sloan Manage Rev 10: 53–59 Khouja M (1999) The single-period (news-vendor) problem: Literature review and suggestions for future research. Omega 27(5):537–553 Lau HS (1980) The newsboy problem under alternative optimization objectives. J Oper Res Soc 31:525–535 Lau HS, Lau AH (1997) Reordering strategies for a newsboy-type product. Eur J Oper Res 103: 557–572 Lau AH, Lau HS (1998) Decision models for single-period products with two ordering opportunities. Int J Prod Econ 55:57–70 Li S, Zhu Z, Huang L (2009a) Supply chain coordination and decision making under consignment contract with revenue sharing. Int J Prod Econ 120(1):88–99 Li J, Chand S, Dada M, Mehta S (2009b) Managing inventory over a short season: models with two procurement opportunities. Manuf Serv Oper Manage 11(1):174–184 Milner JM, Kouvelis P (2005) Order quantity and timing flexibility in supply chains: the role of demand characteristics. Manage Sci 51(6):970–985 Nahmias S, Schmidt C (1984) An efficient heuristic for the multi-item newsboy problem with a single constraint. Nav Res Logistics Q 31:463–474 Pan K, Lai KK, Liang L, Leung SCH (2009) Two-period pricing and ordering policy for the dominant retailer in a two-echelon supply chain with demand uncertainty. Omega 37:919–929 Pasternack BA (1985) Optimal pricing and return policies for perishable commodities. Mark Sci 4(2):166–176 Serel D (2009) Optimal ordering and pricing in a quick response system. Int J Prod Econ 121(2):700–714 Taylor TA (2002) Supply chain coordination under channel rebates with sales effort effects. Manage Sci 48(8):992–1007 Wang SD, Zhou YW, Wang JP (2010) Supply chain coordination with two production modes and random demand depending on advertising expenditure and selling price. International Journal of Systems Science, 2010, 41(10):1257–1272 Webster S, Weng ZK (2008) Ordering and pricing policies in a manufacturing and distribution supply chain for fashion products. Int J Prod Econ 114(2):476–486 Weng ZK (2004) Coordinating order quantities between the manufacturer and the buyer: a generalized newsvendor model. Eur J Oper Res 156:148–161 Whitin TM (1955) Inventory control and price theory. Manage Sci 2:61–68 Wong WK, Qi J, Leung SYS (2009) Coordinating supply chains with sales rebate contracts and vendor-managed inventory. Int J Prod Econ 120(1):151–161 Zhou Y, Li DH (2007) Coordinating order quantity decisions in the supply chain contract under random demand. Appl Math Model 31(6):1029–1038

.

Part III

Channel Power, Bargaining and Coordination

.

Bargaining in a Two-Stage Supply Chain Through Revenue-Sharing Contract Jing Hou and Amy Z. Zeng

Abstract We focus on a bargaining problem between one supplier and one retailer that are coordinated by a revenue-sharing contract. The suppler is assumed to have the ability to influence the retailer’s profit by setting his/her target inventory level, which in turn determines the lead time. We examine the cases under which either the supplier or the retailer is dominant in the bargaining process. The key contract parameter, the acceptable range of the revenue-sharing fraction for the two players, and the maximum amount of monetary bargain space are obtained under explicit and implicit information, respectively. Numerical illustrations of the contracts for various scenarios are given to shed more insights. Keywords Dominance and bargaining • Nonlinear optimization • Supply chain coordination • Supply contracts

1 Introduction Revenue sharing mechanism has been applied extensively in various industries, such as internet service (e.g., He and Walrand 2005), airline (e.g., Zhang et al. 2010), and virtual enterprises (e.g., Chen and Chen 2006) – to name a few, as an efficient vehicle to achieve coordination, because it is relatively straightforward

J. Hou (*) Business School, Hohai University, Nanjing, Jiangsu 211100, China e-mail: [email protected] A.Z. Zeng School of Business, Worcester Polytechnic Institute, Worcester, MA 01609, USA e-mail: [email protected] T.-M. Choi and T.C. Edwin Cheng (eds.), Supply Chain Coordination under Uncertainty, International Handbooks on Information Systems, DOI 10.1007/978-3-642-19257-9_14, # Springer-Verlag Berlin Heidelberg 2011

347

348

J. Hou and A.Z. Zeng

for the decision makers to implement and manage the contract. The primary objective of the revenue-sharing contract is to align the two parties’ interests and actions by having the retailer share a portion of his/her revenue with the supplier. As a result, the supplier’s effort and willingness to collaborate should increase. Two desirable outcomes are expected from the revenue-sharing mechanism, namely higher profit level for the entire chain, and a “win-win” situation for each chain member. The classic problem of a revenue-sharing contract is how to determine the revenue-sharing fraction for better coordination outcomes. This contract parameter is determined under various decision-making configurations, one of which can be characterized by the power inequality of negotiation in the bargaining process. In a two-stage supply chain consisting of a single manufacturer (or supplier) and a single retailer, if the supplier has the ability to influence the retailer’s decision on revenue-sharing fraction, then he/she may receive larger increase in profit resulted from the coordination mechanism. On the other hand, if the retailer is dominant, then the revenue-sharing fraction may be set to satisfy the retailer’s requirements. The major contribution of this paper lies in the area where we obtain the key parameters of the revenue-sharing contract and the bargaining space of a singlesupplier-single-retailer supply chain with the consideration of a dominant player. The retailer’s profit depends upon the lead time that is affected by the supplier’s finite target inventory level. The contract requires the supplier to hold larger inventory level to achieve system optimization and also a “win–win” condition for two players. The dominant player (either the supplier or the retailer) in the bargaining process requires more increase in profit. In both situations, the ranges of the revenue-sharing fraction as well as the maximum monetary value that the two parties can bargain are obtained. The impacts of the explicit and implicit information about the supplier’s inventory holding cost on the decisions are also examined. As will be discussed in the literature review, the problem studied in this paper has not been fully addressed in the literature. Our numerical examples show that significant improvements can be accomplished by the proposed contract and the bargaining method. The remainder of the paper is organized as follows. Section 2 summarizes the literature related to revenue-sharing contract and the different ways of distributing the profit among the supply chain entities. In Sect. 3, we review the results from basic centralized and decentralized optimizations from our previous work, which will provide foundation for subsequent analysis. Section 4 examines the joint revenue-sharing and bargaining decisions between the two parties by taking into account of dominance and the kind of knowledge the retailer has about the supplier’s inventory holding cost structure. The ranges of the key contract parameter and monetary bargain space are derived and numerical examples are given. Finally, we provide concluding remarks and directions for future research in Sect. 5.

Bargaining in a Two-Stage Supply Chain Through Revenue-Sharing Contract

349

2 Literature Review Revenue-sharing contracts have attracted considerable attention. An extensive literature review can be found in Cachon and Lariviere (2000) and Yao et al. (2008). We herein focus on most recent examples of studies that have been published in the literature. Giannoccaro and Pontrandolfo (2004) propose a revenuesharing model that aims at coordinating a three-stage supply chain. The model increases the system efficiency as well as the profits of all the chain members by fine tuning the contract parameters. In analyzing a special two-stage supply chain where the revenue decreases with the lead time and increases with inventory, Gupta and Weerawat (2006) design a revenue-sharing contract to maximize the centralized revenue by choosing an appropriate inventory level. Chen et al. (2007) study the performance of the supply chain with one supplier and multiple buyers under deterministic price-sensitive customer demand. Yao et al. (2008) investigate a revenue-sharing contract for coordinating a supply chain comprising one manufacturer and two competing retailers who face stochastic demand before the selling season. Linh and Hong (2009) discuss how the revenue sharing fraction and the wholesale price are to be determined in revenue sharing contract in order to achieve channel coordination and a win–win outcome for a single retailer and a single wholesaler. Giannoccaro and Pontrandolfo (2009) model the negotiation process among the supply chain actors by adopting agent-based simulation, taking into account the contractual power and the collaboration among the SC actors. A number of researchers have recently demonstrated the effectiveness of revenue sharing contract in supply chain coordination by comparing or integrating it with other contract types. For example, Li and Hua (2008) and Li et al. (2009) have examined the coordination effectiveness of consignment contract with revenue sharing for decentralized supply chains. Bellantuono et al. (2009) present a model in which the supply chain partners participate in two different programs – a revenue sharing contract between the supplier and the retailer, and an advanced booking discount program offered by the retailer to the customers. Pan et al. (2010) discuss and compare the results of a wholesale price contract or a revenue-sharing contract under different channel power structures to check whether it is beneficial for manufacturers to use revenue-sharing contracts under different scenarios. Ouardighi and Kim (2010) compare the possible outcomes under a wholesale price contract and a revenue-sharing contract when studying a non-cooperative dynamic game in which a single supplier collaborates with two manufacturers on design quality improvements for their respective products. Lin et al. (2010) compare the revenue sharing contract with the insurance contract, under which the supplier shares the risk of overstock and under-stock with the retailer, improving the efficiency of the supply chain with a newsvendor-type product. In sum, the studies in this category do not consider revenue-sharing as a single coordination mechanism; rather as part of the supply chain collaboration methodology or an alternative to other contracts.

350

J. Hou and A.Z. Zeng

Bargaining and cooperation have been always playing a key role in profit allocation in a supply chain. For example, Jia and Yokoyama (2003) propose a scheme based on Game theory to decide the profit allocation of each independent power producers in the coalitions rationally and impartially. Guardiola et al. (2007) study the coordination of actions and the allocation of profit in supply chains under decentralized control in which a single supplier supplies several retailers with goods for replenishment of stocks. Nagarajan and Sosˇic´ (2008) use cooperative bargaining models to find allocations of the profit pie between supply chain partners. And the problem of how to split the additional profit among the supply chain entities in a revenue-sharing contract has been the subject of many recent researchers. In a study by Chauhan and Proth (2005) where the customer demand depends upon the retail price, a new approach is proposed to maximize the centralized profit by sharing the profit proportional to the risk among the partners. In the work of Jaber and Osman (2006), a simple profit-sharing contract is proposed in such a way that the profit is distributed proportionally to each partner’s investment amount. Rhee et al. (2010) propose a new way of generalizing contract mechanisms to multi-stage settings, where one supply chain entity takes the lead in negotiating a single contract with all other entities simultaneously. Two special cases are discussed – one in which all entities receive the same absolute increase in profit; and one in which all members receive the same relative increase in profit. In addition, Sucky (2006) considers the bargaining problem of a two-stage supply chain where the buyer has no access to the supplier’s complete information. To reduce the system-wide cost, the order quantity is treated as a variable and the coordination mechanism with the buyer being dominant is derived and compared with that under complete information. Inspired by the work of Rhee et al. (2010), our research assumes that either the supplier or the retailer is dominant in the bargaining process and requires more increase in profit. We derive the accepted range of revenue sharing fraction by both parties as well as the associated bargain space. In this paper we restrict our attention to a supply chain that consists of one supplier and one retailer, which are separate and independent organizations, actively seeking favorable opportunities to coordinate. We extend the study of Hou et al. (2009) by considering the situations under which one of the two supply chain members is dominant in the bargaining process. For both cases, we develop the key contract parameters and discuss the range of the monetary amount that can be shared between the two parties under explicit and implicit information about the supplier’s inventory cost, respectively.

3 The Basic Models We study the coordination issue between a supplier and a retailer in a two-stage supply chain that produces and sells one single product. The basic assumptions for this paper are identical to those made in our previous study (Hou et al. 2009) and are

Bargaining in a Two-Stage Supply Chain Through Revenue-Sharing Contract

351

briefly summarized here. The demand rate is known as l and the demand process is stationary and follows Poisson distribution. The supplier’s production cost is cs per unit and average unit inventory holding cost is h. The retailer’s unit cost is denoted as cr. Furthermore, the retailer’s unit profit of the final product, p, is assumed to be sensitive to the lead time. Similar to the study by Gupta and Weerawat (2006), we study the situation under which the supplier has the ability to influence the retailer’s average unit revenue p by setting his/her target inventory level b and by knowing the relationship between the lead time L and the target inventory level through the following expression: pðbÞ ¼ p0 bLðbÞ

(1)

where the parameter, b, is a scale factor, and p0 is the retailer’s largest possible unit revenue achieved in an ideal situation with the highest acceptable sales price by the end-users and the supplier’s shortest lead time. Both parameters ðb; p0 Þcan be estimated and hence can be assumed to be known. Since the retailer’s order lead time is determined by the time the supplier spends on production, transportation, and transaction, it is evident that the more inventory available at the supplier’s site, the shorter the lead time will be. Intuitively, the retailer’s average lead time is a decreasing function of the supplier’s target inventory level, that is, L0 ðbÞ < 0. The lead time is also limited by various factors besides the supplier’s target inventory level; for example, when the lead time is less influenced by the inventory, it is more determined by the other factors such as transportation time and order processing time. Hence, as the inventory level increases, the change rate of the lead time decreases. Therefore, the lead time, L(b), demonstrates the properties of a function that is decreasing but convex with respect to the inventory level (b), which means that L0 ðbÞ < 0 and L00 ðbÞ > 0. The assumed convexity of the lead-time function does simplify the subsequent analyses, but is also general enough to include many possible types of relationships between the inventory and lead time. Besides, we make an assumption that when there is no stock available at the supplier’s site, i.e., the lead time reaches its maximum, the customer will lower the acceptable price to an extent that the unit profit for the retailer becomes zero. As a result, the specific expression of L(b) is given as follows: LðbÞ ¼ lmax kbm ¼ p0 =b kbm ;

8 k > 0; 0 < m 1:

(2)

Where k is interpreted as a scale factor, and m is a known exponent. The values of both can be estimated based on sales history. In addition, lmax is the maximum lead time if there is no stock available when an order is placed, and LðbÞ > 0 holds for all values of b. In what follows, we first review the optimal planning parameters obtained from our previous study (Hou et al. 2009), which will be used as a basis for extensions in this paper.

352

3.1

J. Hou and A.Z. Zeng

The Centralized Planning Model

The goal of a centralized supply chain is to set the target inventory level so that the chain’s total expected profit, P0rþs , calculated in (3), is maximized. P0rþs ¼ lbkbm hb lðcs þ cr Þ

(3)

The first part of formula (3) is the revenue obtained by selling the final products. The second part is the inventory holding cost occurred at the supplier’s, while the last one indicates the supplier’s production cost and the retailer’s cost. It is easy to show that (3) is a concave function with respect to b and the supplier’s optimal inventory level is found as follows: b0

¼

h lbmk

1 m1

(4)

Note that the profit given at the above inventory quantity reaches to the highest point for the supply chain.

3.2

The Decentralized Profit-Sharing Model

In a decentralized supply chain, both players act independently and make decisions to maximize their respective profits. In this situation, the retailer determines a fraction (a) to share the sales revenue with the supplier, and then the supplier decides his/her target inventory level (b) based on the given revenue-sharing fraction. Denote (a*, b1*) as the optimal decisions and ðPr ; Ps Þ as the profits for the retailer and the supplier respectively, we summarize the results of this situation obtained by Hou et al. (2009) as follows: a ¼ m b1

Ps ðhja

¼

h lbm2 k

h ¼ mlbk lbm2 k

; b1 Þ

Pr ðhja

; b1 Þ

(5)

1 m1

m m1

< b0

h

(6)

h lbm2 k

h ¼ ð1 mÞlbk lbm2 k

m m1

1 m1

lcs

lcr

(7)

(8)

Bargaining in a Two-Stage Supply Chain Through Revenue-Sharing Contract

353

It is seen that the two methods provide somewhat different results. The objective now is to find an acceptable set of (a, b) to enhance the profitability of the supplier and the retailer. Note that a complete list of symbols and notations used throughout the paper is provided in Appendix 1.

4 The Bargaining Decision Under Dominance It is intuitive that a higher fraction of revenue offered by the retailer could motivate the supplier to hold a larger inventory, and as a result, a larger amount of revenue for the supply chain. Therefore, we want to see how they can work together to determine the revenue-sharing fraction so that the profits of both parties can increase to the levels they are able to achieve in a decentralized supply chain. The analysis will be performed in the following two situations (1) The supplier is the leader in the bargaining process, and we use the subscript “s” to label related notation, and (2) The retailer is the leader, and we use subscript “r” for all relevant symbols. Whoever is dominant in the supply chain requires larger increased profit from the new revenue sharing contract. Both situations are analyzed with explicit information and implicit information about the supplier’s inventory holding cost, h, respectively.

4.1 4.1.1

Supplier-Dominant Bargaining With Explicit Information

As the target inventory level of the supplier is set, which is b0 shown in (4), we only need to identify the new revenue-sharing fraction that will enable such an inventory quantity. Since increased inventory level causes higher inventory holding cost for the supplier but results in more revenue for the retailer, the dominant member (supplier) would require larger benefit from the coordination. Thus, we will first determine the new revenue-sharing fraction a in the presence of explicit information on the supplier’s inventory cost, h, and then discuss the range of the monetary amount that can be shared between the dominant supplier and the retailer. In a supplier-dominant supply chain, the range of a is given in the following statement: Proposition 4.1. In a supplier-dominant supply chain, to attract the two-stage supplier to hold a larger inventory level b0 and to achieve higher profits for both parties than those in the case of decentralized planning, the retailer’s new share of revenue, a, has the following range of values:

354

J. Hou and A.Z. Zeng

1 2m m 1 1 m m m1m m1m þ 2m1m þ 1 < as < 1 þ m1m m1m ; 2

8 0 < m < 1 (9)

Moreover, the range reaches the maximum when m ¼ 0.25. Proof. See Appendix 2. We refer to the right hand side of (9) as the upper limit of a, that is, 1

m

1m m1m ; aU s ¼1þm

(10)

and the left hand side of (9) as the lower limit, aLs ¼

1 2m m 1 m m1m m1m þ 2m1m þ 1 : 2

(11)

Now that the range of the revenue-sharing fraction is determined, we examine the range of monetary value that the retailer could share with the supplier. We denote such a monetary space as ð0; DPÞ, and give the values of the space in the following statement. Observation 4.1. In a supplier-dominant two-stage supply chain, there exist two possible scenarios when the monetary value for the two parties to share is found. The two scenarios are differentiated by a specific value of m* that is determined by the input parameters, ðl; b; k; h; cr cs Þ, as follows: Case (i): If 0 < m m* [except some values in m < m* when cr > cs, which fall into Case (ii)], then the range of the monetary amount that can be shared between the two parties is given by ð0; DPs1 Þ, where m DPs1 ¼ 0:5Ph ð1 mÞ 1 ð1 þ mÞm1m :

(12)

Case (ii): If m < m < 1 [plus those values of m < m* when cr > cs in Case (i)], there exists a revenue-sharing fraction, ah , where aLs < ah < aU s , at which the two parties’ new profits are identical: ah ¼

1 þ m lmðcs cr Þ : þ 2 2hb0

(13)

Therefore, a monetary quantity that allows the supplier’s new profit to be no less than the retailer’s is found as ð0; DPs2 Þ, where m

DPs2 ¼ Ph ðm 1Þðm1m 0:5Þ 0:5lðcs cr Þ

(14)

Note that in both (12) and (14), the factor, Ph , has the following expression:

h Ph ¼ lbk lbmk

m m1

(15)

Bargaining in a Two-Stage Supply Chain Through Revenue-Sharing Contract

355

Proof. See Appendix 3. It is seen from Observation 4.1 that retailer’s new revenue sharing fraction to attract the supplier to hold a higher inventory quantity falls into an interval, which is also true for the dollar amount. Hence, the final choice of the revenue-sharing fraction will be reached through bargaining between the two parties.

4.1.2

With Implicit Information

The supplier’s inventory cost, h, plays a critical role in decision making. In reality, the supplier may choose not to reveal the actual value of h to the retailer because he either considers it a piece of private information or has difficulty estimating the exact value. As such, the supplier may only tell the retailer a range of his cost structure in such a way that h1 < h < h2 . We call such a situation where the retailer has no specific value about the supplier’s cost structure as “decision-making under implicit information”. In this section, we will examine how to obtain the range of the revenue-sharing fraction and the impacts of the key input parameters on such a decision-making situation. Given the range of the supplier’s cost structure, h 2 ½h1 ; h2 , it is not difficult to show that the two parties’ profit functions are monotonically decreasing with the growth of the cost. Therefore, h1 and h2 represent the best and worst scenario, respectively, and we will only need to consider the new value of the contract parameter at these two limits. Suppose that the upper limit, h2, can be written as a function of the lower limit, that is, h2 ¼ ð1 þ dÞh1 ; where d > 0:

(16)

Table 1 reports two sets of the supplier’s inventory quantities based on the ; b results obtained from the previous sections: b 01 02 in centralized planning, and b11 ; b12 in decentralized coordination, as well as the profit functions of the two parties. Proposition 4.2. In a supplier-dominant two-stage supply chain, to coordinate with the supplier under the case of implicit information where the supplier provides an interval of the inventory cost, h1 h h2 , the retailer would select a revenuesharing fraction from the following range, aLs a aU s , where aLs ¼

1 2m m 1 m m1m m1m þ 2m1m þ 1 ; 2 1

m

1m m1m ; aU s ¼1þm

8 0<m<1

8 0 < m < 1:

However, the monetary bargain space to be negotiated between the two parties differs in the following way:

356

J. Hou and A.Z. Zeng

Table 1 Parameters in the gaming under implicit information Parameters hi, i ¼ 1, 2 a ¼ m a 1 1 m1 m1 b1 h1 h2 b11 ¼ ; b12 ¼ ; b11 > b12 2 2 lbm k lbm k 1 1 b0 h1 m1 h2 m1 b01 ¼ ; b02 ¼ ; b01 > b02 lbmk lbmk m m1 Retailer’s profit hi Pr ðhi ; a ; b1i Þ ¼ ð1 mÞlbk lcr lbm2 k m m1 hi Pr ðhi ; ai ; b0i Þ ¼ ð1 aÞlbk lcr lbmk m 1 m1 m1 Supplier’s profit hi hi Ps ðhi ; a ; b1i Þ ¼ mlbk hi ; lcs km2 bl km2 bl m 1 m1 m1 hi hi Ps ðhi ; ai ; b0i Þ ¼ albk hi lcs lbmk lbmk

Case (i): 0 < m m* [except some values of m < m* when cr > cs, which fall into Case (ii)]: As the supplier’s unit inventory holding cost range expands, which is captured by d (h2 h1 ¼ dh1 ), the gap between the maximum amount of income, ½1 GsDP , that the retailer would share with the supplier increases with d " ½1 GsDP ðdÞ

1 1 1þd

¼ As1

# m 1m

; where

m As1 ¼ 0:5ð1 mÞ 1 ð1 þ mÞm1m lbk

h1 lbmk

(17) m m1

(18)

Case (ii): m < m < 1 [plus some values of m < m* when cr > cs in Case (i)]: There exists a revenue-sharing fraction, ah , where aL < ah < aU , at which the two parties’ new profits are identical: ah ¼

1þm lmðcs cr Þ : þ 2 2hb0

(19)

Furthermore, as the supplier’s cost interval increases, the gap between the ½2 maximum amount of income, GsDP , that the retailer could share with the supplier increases with d " ½2 GsDP ðdÞ

¼ As2

1 1 1þd

# m 1m

; where

(20)

Bargaining in a Two-Stage Supply Chain Through Revenue-Sharing Contract

m

As2 ¼ ðm 1Þðm1m 0:5Þ lbk

h1 lbmk

m m1

:

357

(21)

Proof. See Appendix 4. As clearly stated in Proposition 4.2, it is interesting to see that even if the supplier provides an interval of the cost information rather than a specific value, the range of the fraction of revenue that the retailer will offer to encourage the retailer to hold larger quantity of inventory level remains the same; however, the amount of monetary value to be shared with the supplier varies as the range of the supplier’s cost value widens.

4.2

Retailer-Dominant Bargaining

The preceding section studies the key decision-making parameters when the supplier is dominant. In this section, we examine how the same parameters are determined in the opposite situation – the retailer is the dominator.

4.2.1

With Explicit Information

In this retailer-dominant supply chain in which the retailer has the explicit data about the inventory holding cost, h, we have found that the range of the revenuesharing fraction, a, can be described in the following proposition. Proposition 4.3. In a retailer-dominant two-stage supply chain, to attract the supplier to hold a larger inventory level b0 and to achieve higher profits for both parties than those in the case of decentralized planning, the retailer’s new share of revenue, a, satisfies the following range:

1 2m 2m m 1 m þ m1m m1m < ar < 0:5 m m1m m1m þ 2m1m þ 1 ; 8 0 < m < 1 (22)

Proof. See Appendix 2. Denote the right hand side of (22) as the upper limit of ar , that is, 2m m 1 L 1m m1m þ 2m1m þ 1 ; aU ¼ a ¼ 0:5 m m r s

(23)

and the left hand side of (22) as the lower limit, 1

2m

aLr ¼ m þ m1m m1m :

(24)

358

J. Hou and A.Z. Zeng

Comparing the conclusions in Proposition 4.3 with those in Proposition 4.1, we can see that the range of the revenue-sharing fraction remains unchanged; i.e., L U L aU r ar ¼ as as , and whoever is the leader, the range of the fraction reaches the maximum at m ¼ 0.25. The range of monetary value that the retailer could share with the supplier if the retailer is the leader is discussed in the following statement. Observation 4.2. In a retailer-dominant supply chain, two possible scenarios exist when finding the monetary amount for the two parties to share. The difference is dependent upon a specific range of (m1*, m2*) that is determined by the input parameters, ðl; b; k; h; cr cs Þ; as follows: Case (i): If m 2 = ðm1 ; m2 Þ and cr > cs , or cr cs , the range of the monetary amount that can be shared between the two parties is given by ð0; DPr1 Þ, where m DPr1 ¼ 0:5Ph ð1 mÞ 1 ð1 þ mÞm1m

(25)

Case (ii): If m 2 ðm1 ; m2 Þ and cr > cs , then there exists a revenue-sharing fraction, ah , where aLr < ah < aU r , at which the two parties’ new profits are identical. Therefore, a monetary quantity that allows the retailer’s new profit to be no less than the supplier’s is found as ð0; DPr2 Þ, where 1

2m

DPr2 ¼ Ph ð0:5 0:5m m1m þ m1m Þ þ 0:5lðcs cr Þ:

(26)

Proof. See Appendix 5. According to Observation 4.1 and 4.2, in the case where the supplier is dominant, only when cr–cs > d and m < m* can the supplier’s profit always be larger than the retailer’s (as shown in Appendix 3). This means that, to gain an advantage over the retailer, the supplier must lower his/her unit production cost (cs) to be less than the retailer’s unit cost (cr), and the supplier’s inventory level has minimal impact on the lead time, which is captured by the parameter, m. These two requirements are fairly stringent, and hence, it will be more difficult for the supplier to bargain. When the retailer is dominant, the condition under which the retailer’s profit is always higher than the supplier’s requires that only cr < cs is true. This means that the retailer only needs to ensure his/her unit cost (cr), lower than the supplier’s unit production cost (cs). This constraint is less stringent than that in the other case, and thus, it is easier for the retailer to gain advantage. The reason for this phenomenon is that, to achieve supply chain optimization, increased inventory level causes higher inventory holding cost for the supplier but results in higher revenue for the retailer, and thus makes it easier for the retailer to obtain higher profit than the supplier.

4.2.2

With Implicit Information

With the supplier’s holding cost information switching from a single value to an interval, the properties of the contract parameters in the retailer-dominant decision making are given in Proposition 4.4.

Bargaining in a Two-Stage Supply Chain Through Revenue-Sharing Contract

359

Proposition 4.4. In a retailer-dominant two-stage supply chain, to coordinate with the supplier under the case of implicit information where the supplier provides an interval of his inventory cost, h1 h h2 , the retailer would select a revenuesharing fraction from the following range, aLr a aU r , where 1

2m

aLr ¼ m þ m1m m1m ;

8 0<m<1

2m m 1 1m m1m þ 2m1m þ 1 ; aU r ¼ 0:5 m m

8 0<m<1

However, the monetary bargain space to be negotiated between the two parties differs in the following way: (i) When m 2 = ðm1 ; m2 Þ and cr > cs , or cr cs , and as the supplier’s average unit inventory holding cost range increases, the gap between the maximum amount ½1 of income, GrDP , that the retailer would share with the supplier increases with d " ½1 GrDP ðdÞ

where

¼ Ar1

1 1 1þd

m 1m

Ar1 ¼ 0:5ð1 mÞ 1 ð1 þ mÞm

# m 1m

;

h1 lbk lbmk

(27)

m m1

;

(28)

and (ii) When m 2 ðm1 ; m2 Þ and cr > cs , then there exists a revenue-sharing fraction, ah , where aLr < ah < aU r , at which the two parties’ new profits are identical: ah ¼

1 þ m lmðcs cr Þ þ 2 2hb0

Furthermore, as the supplier’s cost interval increases, the gap between the ½2 maximum amount of income, GrDP , that the retailer could share with the supplier increases with d " # m 1m 1 ½2 ; (29) GrDP ðdÞ ¼ Ar2 1 1þd where

m h1 m1 1 2m Ar2 ¼ 0:5 0:5m m1m þ m1m lbk : lbmk

Proof. See Appendix 4.

(30)

360

J. Hou and A.Z. Zeng

4.3

Numerical Examples

We provide two numerical examples to illustrate the range of a and the impact of the supplier’s holding cost range on the amount of revenue the retailer would share L U with the supplier. One Lexample shows the situation when ar < ah < as , and the U L U other considers ah 2 as ; as or ar ; ar . The values of the cost parameters, whose dimensions are all $/unit, for both cases are: b ¼ 2; l ¼ 500; k ¼ 0:5; h1 ¼ 0:5 Example 1. m ¼ 0.4, cr ¼ 6; cs ¼ 5. Based on these given parameters, we can easily derive values of a set of parameters when h1 ¼ 0.5 as follows: the L U as ; as ¼ ð0:6023; 0:6743Þ; aLr ; aU a ¼ 0:4; r ¼ ð0:5302; 0:6023Þ; b0 ¼ 21; 715; b1 ¼ 4; 716; Pr ðh1 ; a ; b1 Þ ¼ $5; 842; and Ps ðh1 ; a ; b1 Þ ¼ $1; 037: Furthermore, we have found that ah ¼ 0:6908; which exceeds the larger upper limit 0.6743, implying that the two members’ profits will never be the same within the sharing-fraction range. We then have calculated the two parties’ new profits, Pr ðh1 ; a; b0 Þ and Ps ðh1 ; a; b0 Þ, against the range of a (from 0.5302 to 0.6743), and display the results in Fig. 1. The profit values in the case of decentralized planning are also shown for benchmarking purpose. Note that the amount of capital that can be shared or bargained, DPs , for the dominant supplier and DPr for the dominant retailer are both about $1,954, which are also labeled in Fig. 1. 12000

10000

Pr (h1,a, b1*) 8000

P

Pr (h1,a *, b1* )

6000

DPs =

4000

Ps (h1,a,b*0 )

$1,954 DPr = $1,954

2000

Ps (h1,a *, b1* )

0 0.53 L

ar

0.56

0.6 U

0.64 L

ar, as a

Fig. 1 Example 1 – profits of the two parties: m ¼ 0.4; k ¼ 0.5; cr ¼ 6; cs ¼ 5

0.67 U

as

Bargaining in a Two-Stage Supply Chain Through Revenue-Sharing Contract

361

Fig. 2 The gap between the maximum amount of income for h1 < h < h2. (m ¼ 0.4, cr ¼ 6; cs ¼ 5)

With h1 fixed, we vary the range of h by increasing d from 0 to 1 at an increment of 0.2, and then calculate the maximum amount of revenue in dollars that could be bargained between the two parties. The results are shown in Fig. 2, which implies that as the upper limit of the supplier’s cost increases, the gap between the maximum amounts of income increases. Furthermore, the values of the gaps between supplier-dominant and retailer-dominant situations are identical; in particular, the gap is about $289,000 when h2 increases to twice as much as of h1 (i.e., d ¼ 1). 25; cs ¼ 5. Similarly, we have: a ¼ 0:45; aLs ; aU Example 2. m ¼ 0.45, s ¼ Lcr ¼ ð0:6463; 0:7138Þ; ar ; aU b0 ¼ 66; 684; b1 ¼ 15; 613; r ¼ ð0:5788; 0:6463Þ; Pr ðh1 ; a ; b1 Þ ¼ $8; 704; and Ps ðh1 ; a ; b1 Þ ¼ $7; 042: Unlike the previous example, ah ¼ 0:6575 is within aLs ; aU s ¼ ð0:6463; 0:71383Þ: The results are depicted in Fig. 3. Clearly, the amount of monetary capital that can be shared or bargained for the dominant supplier, DPs ¼ $4; 174; is smaller than for the dominant retailer, DPr ¼ $5; 003: Figure 4 shows the gap between the maximum amounts of income for this example. Unlike the previous case, this example sees a dramatic difference between the value of the gaps for supplier-dominant bargaining and retailer-dominant bargaining (About $764,000 when h2 increases to 200% of h1). This implies that the larger the bargain space (DP) is, the faster the gap between the maximum amounts of income ðGDP ðdÞÞ increases with the range of the inventory holding cost information ðdÞ. As such, the supplier would want to provide his cost information as specifically and exactly as possible in order to receive more sharing from the retailer.

362

J. Hou and A.Z. Zeng

Fig. 3 Example 2 – the profits of the two parties: m ¼ 0.45; k ¼ 0.5; cr ¼ 25; cs ¼ 5

Fig. 4 The gap between the maximum amount of income for h1 < h < h2 (m ¼ 0.45, cr ¼ 25; cs ¼ 5)

5 Concluding Remarks This paper studies the bargaining problem in a supplier–retailer supply chain based on revenue sharing. We assume that the retailer’s unit revenue is sensitive to the lead time, which is affected by the supplier’s target inventory level. A revenue-sharing

Bargaining in a Two-Stage Supply Chain Through Revenue-Sharing Contract

363

based coordination mechanism is constructed to select the supplier’s target inventory level and the revenue-sharing fraction to maximize the entire supply chain profit and to achieve a “win–win” condition. We obtain the range of the monetary amount that can be shared between the two parties under the situation where either the supplier or the retailer is dominant in the bargaining process. The impact of the supplier’s cost structure is also examined. This research leads to some subtle and important implications that can guide practices. First of all, the range of the revenue-sharing fraction that can be accepted by both parties remains the same in supplier-dominant and retailer-dominant situations. Secondly, in addition to the retailer’s unit cost (cr) and the supplier’s unit production cost (cs), the monetary amount that can be shared between the two parties depends heavily on the input parameter, m, the exponent associated with the supplier’s inventory level (b), which affects the supplier lead time, which in turn affects the profits of the two parties. Thirdly, in a retailer-dominant situation, it is easy for the retailer to obtain higher profit than the supplier; on the other hand, if the supplier is dominant, then it is hard for the supplier to gain such benefit. Finally, the format of the supplier’s cost structure (a single value versus an interval) does not affect the range of the revenue-sharing fraction that the retailer would use to coordinate the supplier. Furthermore, the larger the bargain space is, the faster the gap between the maximum amounts of capital increases with the range of the cost information. Several directions for future research can stem from our paper. First, given the optimal range of revenue-sharing fraction, how to distribute the additional profit among the supply chain entities would be an interesting issue. Second, the situation where the retailer chooses not to reveal his/her actual revenue to the counterpart is a problem that the supplier may encounter in the bargaining process. Finally, it will be worthwhile to investigate the supplier’s inventory decision if the supplier has a capacity constraint, as well as to consider the average lead time as a general function of the supplier’s inventory level. Acknowledgment This work is partially supported by the Scientific Research Foundation of Graduate School of Southeast University (YBJJ1003). We are also thankful for the helpful suggestions provided by the anonymous referees.

Appendix 1: The List of Symbols Used in the Paper l cs h cr L p

Demand rate (unit) Unit production cost of the supplier ($/unit) Average unit inventory holding cost of the supplier ($/unit) Unit cost of the retailer ($/unit) Lead time Final product’s unit profit of the retailer ($/unit)

364

p0 b b0 b1* P0rþs Pr Ps a* a aL aU ah ½ h 1 ; h2 DPs DPr GsDP

GrDP

J. Hou and A.Z. Zeng

The largest possible unit profit of the retailer ($/unit) Target inventory level of the supplier (unit) Supplier’s optimal target inventory level in a centralized supply chain (unit) Supplier’s optimal target inventory level in a decentralized supply chain (unit) Total expected profit of the supply chain Retailer’s profit function Supplier’s profit function Retailer’s optimal revenue sharing fraction in a decentralized supply chain New revenue-sharing fraction to attract the supplier to hold a larger inventory level (b0 ) and to achieve higher profits for both parties, where aL < a < a U Lower bound of the revenue-sharing fraction Upper bound of the revenue-sharing fraction Revenue-sharing fraction at which the two parties’ new profits are identical Range of the supplier’s inventory holding cost, where h2 ¼ ð1 þ dÞh1 and d > 0 Range of the monetary amount that can be shared between the two parties in a supplier-dominant supply chain Range of the monetary amount that can be shared between the two parties in a retailer-dominant supply chain Gap between the maximum amount of income that the retailer would share with the supplier in a supplier-dominant supply chain with implicit information Gap between the maximum amount of income that the retailer would share with the supplier in a retailer-dominant supply chain with implicit information

Appendix 2: This Appendix Contains the Proof for Proposition 4.1 and 4.3 If the retailer provides a higher share of revenue, a > a , to induce the supplier to hold an inventory level close to b0 , the value of the new revenue sharing fraction should satisfy the following requirements: Pr h; a; b0 > Pr h; a ; b1 ;

(31)

Bargaining in a Two-Stage Supply Chain Through Revenue-Sharing Contract

Ps h; a; b0 > Ps h; a ; b1 :

365

(32)

These two requirements ensure that the profits of both the two parties are increased so they would be willing to cooperate. If the dominant supplier gains more increase in profits, then Ps h; a; b0 Ps h; a ; b1 > Pr h; a; b0 Pr h; a ; b1

(33)

Else if the retailer is the leader, then Ps h; a; b0 Ps h; a ; b1 < Pr h; a; b0 Pr h; a ; b1

(34)

We first consider the case when the supplier is the leader. Since Pr ðh; a; b0 Þ

h ¼ ð1 aÞlbk lbmk

m m1

lcr ;

(35)

and Ps ðh; a; b0 Þ ¼ albk

h lbmk

m m1

1 m1 h h lcs lbmk

(36)

We see that the requirement in (31) implies that the fraction for revenue sharing, a, should take on the following range: 1 m a < 1 þ m1m m1m :

(37)

Similarly, we can derive another range of a based on the requirement in (32) as follows: 1 2m a > m þ m1m m1m :

(38)

And the range of a based on the requirement in (33) should be: 2m m 1 a > 0:5 m m1m m1m þ 2m1m þ 1

(39)

2m m 1 1 2m g1 ðmÞ ¼ 0:5 m m1m m1m þ 2m1m þ 1 m þ m1m m1m 1 m 2m m 1 and g2 ðmÞ ¼ 1 þ m1m m1m 0:5 m m1m m1m þ 2m1m þ 1 , it is easy m 2m to find that gðmÞ ¼ g1 ðmÞ ¼ g2 ðmÞ ¼ 0:5 1 m m1m þ m1m . Given the range Suppose

of m, 0 < m < 1, by plotting the value of g (m) to m (as seen in Fig. 5), we could

366

J. Hou and A.Z. Zeng m 0.01 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95

0.08 0.07 0.06 0.05

g(m)

0.04 0.03 0.02 0.01 0

m = 0.25 0

0.2

0.4

0.6

0.8

1

m

g (m) 0.0178 0.0490 0.0667 0.0753 0.0790 0.0797 0.0784 0.0757 0.0719 0.0675 0.0625 0.0571 0.0513 0.0453 0.0391 0.0327 0.0263 0.0198 0.0132 0.0066

Fig. 5 The plot of g(m) against m

prove that g1 ðmÞ ¼ g2 ðmÞ>0 holds for all values of m within (0, 1), and the function reaches the maximum when m ¼ 0.25. Thus, in this supplier-dominant supply chain, to entice the supplier to hold a larger inventory level b0 ; the retailer’s new share of revenue, a, which meets the requirements in (31), (32) and (33), should satisfy the following range: 2m m 1 1 m 0:5 m m1m m1m þ 2m1m þ 1 < a < 1 þ m1m m1m ;

8 0 < m < 1 (40)

Moreover, the range of a just equals g(m), which reaches the maximum (roughly about 0.0797) when m ¼ 0.25. If the retailer is the leader, since the proof is similar to that above, it is not repeated here. Hence, the proof for the proposition is complete.

Appendix 3: This Appendix Contains the Numerical Proof for Observation 4.1 The bargain space in terms of monetary value is the amount of capital shifted from the retailer to the supplier (as the supplier is the leader in the game, and thus has higher profit level). Hence, the bargain space, ð0; DPÞ, can be determined by the change of profit for the retailer as his share of revenue increases from aLs to aU s .

Bargaining in a Two-Stage Supply Chain Through Revenue-Sharing Contract

367

From Pr h; ah ; b0 ¼ Ps h; ah ; b0 , the fraction ah at which the two parties’ profits become identical can be derived as: ah ¼

1 þ m lmðcs cr Þ : þ 2 2hb0

(41)

We need to make sure that aL < ah < aU ; that is, to satisfy the following requirement, 1 þ m lmðc c Þ 1 2m m 1 1 m s r < 1 þ m1m m1m : m m1m m1m þ 2m1m þ 1 < þ 2 2 2hb0 (42) After some algebra, (42) indicates that the following relationship is required: f1 ðmÞ ¼

lmðcs cr Þ 1 2m m 1 m1m m1m þ 2m1m > 0 2hb0 2

f2 ðmÞ ¼

1 m lmðcs cr Þ 1 m >0 þ m1m m1m 2 2 2hb0

(43)

Since (43) does not offer a closed-form format of m, and 0 < m < 1, we rely on a numerical analysis again to see when the requirement of (43) can be met. The functions, f1(m) and f2(m), are plotted against m and the result are shown in Figs. 6–13 according to three different situations, with the basic parameters setting as l ¼ 500, k ¼ 0.5, b ¼ 2; h ¼ 1.

0.5

0.4

f1(m)

0.3

0.2

0.1

0

Fig. 6 The plot of f1(m) against m for cr ¼ cs

0

0.2

0.4

0.6

m

0.8

1

368

J. Hou and A.Z. Zeng 0.2 0.1

f2(m)

0 -0.1 -0.2 -0.3 -0.4 -0.5

0

0.2

0.4

0.6

0.8

1

m Fig. 7 The plot of f2(m) against m for cr ¼ cs

0.5

0

f1(m)

-0.5

-1

-1.5

-2

0

0.2

0.4

0.6

0.8

1

m Fig. 8 The plot of f1(m) against m for cr > cs (m* ¼ 0.36)

Case 1: cs ¼ cr As shown in Fig. 6, for all m within (0, 1), f1 ðmÞ > 0 holds; while it is seen clearly that when m > 0:5; the value of the function f2 ðmÞ is positive. Therefore, if the input parameter, m, is greater than 0.5 (but less than 1), it is possible for the two parties’ new profits to be identical when the retailer increases his share of revenue from aL to ah , but not yet to aU . Based on the above results, we see that there are two scenarios when the supplier and retailer are coordinating to improve their respective profit from that resulted in the decentralized planning situation (1) if 0 < m < 0.5, then the retailer’s profit is

Bargaining in a Two-Stage Supply Chain Through Revenue-Sharing Contract

369

2.5

2

f2(m)

1.5

1

0.5

0

0

0.2

0.4

0.6

0.8

1

m Fig. 9 The plot of f2(m) against m for cr > cs ðl ¼ 500; k ¼ 0:5; b ¼ 2; h ¼ 1; cs ¼ 5; cr ¼ 10Þ

0.1 0.05 0

f1(m)

-0.05 -0.1 -0.15 -0.2 -0.25 -0.3

0

0.2

0.4

0.6

0.8

1

m Fig. 10 The plot of f1(m) against m for cr > cs (m* ¼ 0.46)

always higher than the supplier’s at aL a aU ; ðiiÞ if 0:5 m < 1: then the two parties reach the same profit level when the revenue-sharing fraction, a ¼ ah ðaL < ah < aU Þ. We now calculate the bargain space for each scenario. In the first scenario where 0 < m < 0.5, the bargain space, DPs1, is the differ ence between the retailer’s profits at the following two points: aL ; b0 and aU ; b0 . Referring to (8) and (11), we can calculate the difference as follows:

370

J. Hou and A.Z. Zeng 0.35 0.3 0.25

f2(m)

0.2 0.15 0.1 0.05 0 -0.05

0

0.2

0.4

0.6

0.8

1

m Fig. 11 The plot of f2(m) against m for cr > cs ðl ¼ 500; k ¼ 0:5; b ¼ 2; h ¼ 1; cs ¼ 5; cr ¼ 6:5Þ

2.5

2

f1(m)

1.5

1

0.5

0

0

0.2

0.4

0.6

0.8

1

m Fig. 12 The plot of f1(m) against m for cr < cs (m* ¼ 0.52)

DPs1 ¼ Pr h; aL ; b0 Pr h; aU ; b0 m m m1 m1 (44) h h ¼ ð1 aL Þlbk ð1 aU Þlbk : lbmk lbmk m m1 h , and substituting aL in (11) to (44), we can be Denoting Ph ¼ lbk lbmk further simplify (44) to the following format:

Bargaining in a Two-Stage Supply Chain Through Revenue-Sharing Contract

371

0.5 0

f2(m)

-0.5 -1 -1.5 -2 -2.5

0

0.2

0.4

0.6

0.8

1

m Fig. 13 The plot of f2(m) against m for cr < cs ðl ¼ 500; k ¼ 0:5; b ¼ 2; h ¼ 1; cs ¼ 9; cr ¼ 5Þ

m DPs1 ¼ 0:5Ph ð1 mÞ 1 ð1 þ mÞm1m :

(45)

In the second scenario where 0:5 m < 1: the two parties’ profits approach to identical before the retailer’s revenue-sharing fraction reaches the upper limit. Since it is unfavorable for the supplier’s profit to be lower than the retailer’s, the monetary bargain space, DPs2 can be computed as DPs2 ¼ Pr h; ah ; b0 Pr h; aU ; b0 m m m1 m1 h h ð1 aU Þlbk ¼ ð1 ah Þlbk lbmk lbmk m

1

¼ Ph ð0:5 0:5m þ m1m m1m Þ 0:5lðcs cr Þ:

(46)

Case 2: cr > cs There is a specific m*, when m > m*, both f1 ðmÞ and f2 ðmÞ are positive. Then the monetary bargain space, DP s2 , is the difference between the retailer’s profits at the following two points: aL ; b0 and ah ; b0 , which can be computed as 1

m

DPs2 ¼ Ph ð0:5 0:5m þ m1m m1m Þ 0:5lðcs cr Þ

(47)

A. When cr – cs > d, where d is determined by other parameters of l; b; k; h, such m* is increasing with the distance between cr and cs, but not necessary larger or smaller than 0.5. For instance, if l ¼ 500, k ¼ 0:5, b ¼ 2, cs ¼ 5, h ¼ 1, then d is about 2, the values of m* are:

372

J. Hou and A.Z. Zeng

cr m*

6 0.11

7 0.25

10 0.36

15 0.42

20 0.45

25 0.47

And in this case when m < m*, the value of the function positive but f1 ðmÞ is f2 ðmÞis 2m m 1 lmðcs cr Þ 1 1m m1m þ2m1m þ1 < negative; i.e., the relationship of 1þm þ < mm 2hb 2 2 0

m

1

1þm1m m1m holds. That means the retailer’s profit is always smaller than supplier’s. Therefore, the bargain space L is the difference U between the retailer’s a ;b a ;b0 : DPs1 ¼0:5Ph ð1mÞ and profits at the following two points: 0 m 1ð1þmÞm1m . B. When cr – cs < d, m* is near 0.5 as the distance between cr and cs is really small. Using the same basic parameters above, m* is 0.47 for cr ¼ 6, and 0.46 for cr ¼ 6.5. But when m < m*, f2 ðmÞ f1 ðmÞ is negative for some values of m, and can be also positive for other values If nega within this range. m tive, then the bargain space is DPs1 ¼ Ph 12 ð1 mÞ 1 ð1 þ mÞm1m ; otherwise, 1 m DPs2 ¼ Ph ð0:5 0:5m þ m1m m1m Þ 0:5lðcs cr Þ. * Case 3: cr < cs . There is a specific m 0.5; its value is increasing slowly with the value of cs – cr. For instance, l ¼ 500, k ¼ 0:5, b ¼ 2, h ¼ 1, cr ¼ 5 the values of m* are: cs m*

6 0.5

7 0.51

8 0.51

9 0.52

10 0.52

11 0.53

13 0.53

15 0.54

20 0.54

When m < m*, the value of the function f2 ðmÞ is negative, while f1 ðmÞ 1 2m m 1 is positive. That is the relationship of m m1m m1m þ 2m2m þ 1 < 1 þ 2 1 m l mðCs Cr Þ þ holds. The retailer’s profit is always larger than m1m m1m < 1þm 2 2hb 0

supplier’s. Similar to m < 0.5, the bargain space is: case 1 when m DPs1 ¼ Ph 12 ð1 mÞ 1 ð1 þ mÞm1m . When m > m*, both f2 ðmÞ and f1 ðmÞ

lmðcs cr Þ are positive, then at ah ¼ 1þm , the two parties reach the same profit 2hb0 2 þ level. The monetary bargain space, DP2 , is the difference between the retailer’s profits at the following two points: aL ; b0 and ah ; b0 , which can be computed as 1 m DPs2 ¼ Ph ð0:5 0:5m þ m1m m1m Þ 0:5lðcs cr Þ. The proof for this observation is then complete.

Appendix 4: This Appendix Shows the Proof for Proposition 4.2 and 4.4 Since the proof for deriving the range of the revenue-sharing fractions is similar to that in Hou et al. (2009) and Appendix 1, it is not repeated here. Rather, we examine the monetary value that the retailer can share with the supplier. The results are summarized in Table 2 for Proposition 4.2 and Table 3 for Observation 4.4, respectively. It is seen that given a range of h½h1 h h1 ð1 þ dÞ instead of

Bargaining in a Two-Stage Supply Chain Through Revenue-Sharing Contract

373

Table 2 The maximums of shared monetary amount and gaps when the supplier is dominant (2) m < m < 1 (1) 0 < m < m L U h1 DPs11 ¼ Pr h1 ; as ; b01 Pr h1 ; as ; b01 DPs21 ¼ Pr h2 ; ah ; b01 Pr h2 ; aU s ; b01 m m m1 m1 h1 h1 lðcs cr Þ F1 ðmÞ F2 ðmÞ ¼ lbk ¼ lbk lbmk lbmk 2 L U U h2 DPs12 ¼ Pr h2 ; as ; b02 Pr h2 ; as ; b02 DPs22 ¼ Pr h2 ; ah ; b02 Pr h2 ; as ; b02 m m h2 m1 h2 m1 lðcs cr Þ F1 ðmÞ F2 ðmÞ ¼ lbk ¼ lbk 2 lbmk lbmk Gap G½1 ðdÞ ¼ DPs11 DPs12 DP " # m 1m 1 ¼ DPs11 1 1þd 1 m F1 ðmÞ ¼ ð1 mÞ 1 ð1 þ mÞm1m 2

½2

GDP ðdÞ ¼ DPs21 DPs22 " # m 1m 1 ¼ DPs21 1 1þd 1 1 1 m F2 ðmÞ ¼ m þ m1m m1m 2 2

Table 3 The maximums of shared monetary amount and gaps when the retailer is dominant (2) m 2 ðm1 ; m2 Þ and cr >cs (1) m 2 = ðm1 ; m2 Þ and cr > cs , or cr cs L U h1 DPr11 ¼ Pr h1 ; ar ; b01 Pr h1 ; ar ; b01 DPr21 ¼ Pr h1 ; ah ; b01 Pr h1 ; aLr ; b01 m m m1 m1 h1 h1 lðcs cr Þ F3 ðmÞ F4 ðmÞ þ ¼ lbk ¼ lbk 2 lbmk lbmk h2 DPr12 ¼ Pr h2 ; aLr ; b02 Pr h2 ; aU DPr22 ¼ Pr h2 ; ah ; b02 Pr h2 ; aLr ; b02 r ; b02 m m m1 m1 h2 h2 lðcs cr Þ F3 ðmÞ F4 ðmÞ þ ¼ lbk ¼ lbk lbmk lbmk 2

Gap G½1 ðdÞ ¼ DPr11 DPr12 DP " # m 1m 1 ¼ DPr11 1 1þd m F3 ðmÞ ¼ 0:5ð1 mÞ 1 ð1 þ mÞm1m

½2

GDP ðdÞ ¼ DPr21 DPr22 " # m 1m 1 ¼ DPr21 1 1þd 1 2m F4 ðmÞ ¼ 0:5 0:5m m1m þ m1m

one value, the gap between the shared monetary values at the two limits is a function of d. Let f ðdÞ ¼ 1

1 1þd

m 1m

:

(48)

As Hou et al. (2009) has proved, f ðdÞ increases as the range of h widens (i.e., h2 becomes bigger). The proof for Proposition 4.2 and 4.4 is then complete.

374

J. Hou and A.Z. Zeng

Appendix 5: This Appendix Shows the Numerical Proof for Observation 4.2 Similar to Appendix 3, from Pr h; ah ; b0 ¼ Ps h; ah ; b0 , the fraction ah at which the two parties’ profits become identical can be derived as: ah ¼

1 þ m lmðcs cr Þ þ : 2 2hb0

(49)

We need to make sure that

1 þ m lmðc c Þ 1 1 2m 2m m 1 s r < þ m m1m m1m þ 2m1m þ 1 : m þ m1m m1m < 2 2hb0 2 (50) After some algebra, (50) indicates that the following relationship is required: lmðcs cr Þ 1 1 1 2m 1m 1m >0 mþm m f1 ðmÞ ¼ 2hb0 2 2 lmðc c Þ 1 2m m 1 s r f2 ðmÞ ¼ >0 (51) m1m m1m þ 2m1m 2 2hb0

Since (51) does not offer a closed-form format of m, and 0 < m < 1, we rely on a numerical analysis again to see when the requirement of (51) can be met. The functions, f1(m) and f2(m), are plotted against m and the result are shown in Figs. 14–19 according to three different situations, with the basic parameters setting as l ¼ 500; k ¼ 0.5, b ¼ 2; h ¼ 1. 0.5

0.4

f1(m)

0.3

0.2

0.1

0

0

0.2

0.4

0.6

m Fig. 14 The plot of f1(m) against m for cr ¼ cs

0.8

1

Bargaining in a Two-Stage Supply Chain Through Revenue-Sharing Contract

375

0

-0.1

f2(m)

-0.2

-0.3

-0.4

-0.5

0

0.2

0.4

0.6

0.8

1

0.6

0.8

1

m Fig. 15 The plot of f2(m) against m for cr ¼ cs 0.5

0

f1(m)

-0.5

-1

-1.5

-2

0

0.2

0.4

m Fig. 16 The plot of f1(m) against m for cr > cs (m* ¼ 0.36)

Case 1: cs ¼ cr As shown in Fig. 14, for all m within (0, 1), f1 ðmÞ > 0 and f2 ðmÞ < 0 1 2m 2m m 1 holds; therefore, m þ m1m m1m Þ < 12 m m1m m1m þ 2m1m þ 1Þ < 1þm 2 þ lmðcs cr Þ L the retailer’s profit is always higher than the supplier’s at a a aU ; 2hb 0

and the bargain spaceDP1 , is the difference between the retailer’s profits at the following two points: aL ; b0 and aU ; b0 . DPr1 ¼ Pr h; aL ; b0 Pr h; aU ; b0 m m m1 m1 h h L U ð1 a Þlbk : (52) ¼ ð1 a Þlbk lbmk lbmk

376

J. Hou and A.Z. Zeng

Fig. 17 The plot of f2(m) against m for cr > cs. ðl ¼ 500; k ¼ 0:5; b ¼ 2; h ¼ 1; cs ¼ 5; cr ¼ 10Þ

2

1.5

f2(m)

1

0.5

0

-0.5

0

0.2

0.4

0.6

0.8

1

m

Fig. 18 The plot of f1(m) against m for cr < cs

2.5

2

f1(m)

1.5

1

0.5

0

0

0.2

0.4

0.6

0.8

1

m m m1 h Denoting Ph ¼ lbk lbmk , and substituting aL in (11) to (52), we can be further simplify (52) to the following format:

1 m DPr1 ¼ Ph ð1 mÞ 1 ð1 þ mÞm1m : 2

(53)

Case 2: cr > cs There is a specific range of (m1*, m2*), only when m is within this range, both f1 ðmÞ and f2 ðmÞ are positive, and the monetary bargain space DP1 is

Bargaining in a Two-Stage Supply Chain Through Revenue-Sharing Contract Fig. 19 The plot of f2(m) against m for cr < cs ðl ¼ 500; k ¼ 0:5; b ¼ 2; h ¼ 1; cs ¼ 9; cr ¼ 5Þ

377

0

-0.5

f2(m)

-1

-1.5

-2

-2.5

0

0.2

0.4

0.6

0.8

1

m

DPr2 ¼ Pr h; ah ; b0 Pr h; aL ; b0 m m m1 m1 h h L ¼ ð1 a Þlbk ð1 ah Þlbk lbmk lbmk 1 1 lðcs cr Þ 1 2m ¼ Ph m m1m þ m1m þ 2 2 2

(54)

Both values of m1* and m2* are increasing with the distance between cr and cs. And the interval between the two values is determined by other parameters of l; b; k; h. For instance, if l ¼ 500, k ¼ 0:5, b ¼ 2, cs ¼ 5, h ¼ 1, the range of (m1*, m2*) are: cr m*

6 (0.03, 0.11)

7 (0.21, 0.25)

10 (0.32, 0.36)

15 (0.38, 0.42)

20 (0.41, 0.45)

25 (0.43, 0.47)

But when m is outside this small range, f2 ðmÞm f1 ðmÞ is negative, and the bargain space is DPr1 ¼ 0:5Ph ð1 mÞ 1 ð1 þ mÞm1m . Case 3: cr < cs As seen from (51), the value of f1 ðmÞ increases with (cs – cr) and f2 ðmÞ decreases with (cs – cr), with other parameters unchanged. From Case 1 we know that, when cs ¼ cr, for all m within (0, 1), f1 ðmÞ > 0 and f2 ðmÞ < 0 holds; therefore, when cr < cs , we also have f1 ðmÞ > 0 and f2 ðmÞ < 0 for all m within (0, 1) (as shown in Figs. 18 and 19). Therefore, similarto case 1, the monetary bargain m space can be computed as DPr1 ¼ 0:5Ph ð1 mÞ 1 ð1 þ mÞm1m . The proof for this observation is then complete.

378

J. Hou and A.Z. Zeng

References Bellantuono N, Giannoccaro I, Pontrandolfo P, Tang CS (2009) The implications of joint adoption of revenue sharing and advance booking discount programs. Int J Prod Econ 121(2):383–394 Cachon G, Lariviere MA (2000) Supply chain coordination with revenue sharing contracts: strengths and limitations. Manage Sci 51(1):30–44 Chauhan SS, Proth JM (2005) Analysis of a supply chain partnership with revenue sharing. Int J Prod Econ 97(1):44–51 Chen J, Chen JF (2006) Study on revenue sharing contract in virtual enterprises. J Syst Sci Syst Eng 15(1):95–113 Chen K, Gao C, Wang Y (2007) Revenue-sharing contract to coordinate independent participants within the supply chain. J Syst Eng Electron 18(3):520–526 Giannoccaro I, Pontrandolfo P (2004) Supply chain coordination by revenue sharing contracts. Int J Prod Econ 89(2):131–139 Giannoccaro I, Pontrandolfo P (2009) Negotiation of the revenue sharing contract: an agent-based systems approach. Int J Prod Econ 122(2):558–566 Guardiola LA, Meca A, Timmer J (2007) Cooperation and profit allocation in distribution chains. Decis Support Syst 44(1):17–27 Gupta D, Weerawat W (2006) Supplier–manufacturer coordination in capacitated two-stage supply chains. Eur J Oper Res 175(1):67–89 He L, Walrand J (2005) Pricing and revenue sharing strategies for internet service providers. Paper presented in Proceedings of IEEE international conference on computer communications (INFOCOM), Miami, FL Hou J, Zeng AZ, Zhao L (2009) Achieving better coordination through revenue sharing and bargaining in a two-stage supply chain. Comput Ind Eng 57(1):383–394 Jaber MY, Osman IH (2006) Coordinating a two-level supply chain with delay in payments and profit sharing. Comput Ind Eng 50(4):385–400 Jia NX, Yokoyama R (2003) Profit allocation of independent power producers based on cooperative Game theory. Int J Electr Power Energy Syst 25(8):633–641 Li S, Hua Z (2008) A note on channel performance under consignment contract with revenue sharing. Eur J Oper Res 184(2):793–796 Li S, Hua Z, Huang L (2009) Supply chain coordination and decision making under consignment contract with revenue sharing. Int J Prod Econ 120(1):88–99 Lin Z, Cai C, Xu B (2010) Supply chain coordination with insurance contract. Eur J Oper Res 205 (2):339–345 Linh CT, Hong Y (2009) Channel coordination through a revenue sharing contract in a two-period newsboy problem. Eur J Oper Res 198(3):822–829 Nagarajan M, Sosˇic´ G (2008) Game-theoretic analysis of cooperation among supply chain agents: Review and extensions. Eur J Oper Res 187(3):719–745 Pan K, Lai KK, Leung SCH, Xiao D (2010) Revenue-sharing versus wholesale price mechanisms under different channel power structures. Eur J Oper Res 203(2):532–538 Ouardighi FE, Kim B (2010) Supply quality management with wholesale price and revenuesharing contracts under horizontal competition. Eur J Oper Res 206(2):329–340 Rhee B, Veen JAA, Venugopal V, Nalla VR (2010) A new revenue sharing mechanism for coordinating multi-stage supply chains. Oper Res Lett 38(4):296–301 Sucky E (2006) A bargaining model with implicit information for a single supplier–single buyer problem. Eur J Oper Res 171(2):516–535 Yao Z, Stephen CH, Leung KKL (2008) Manufacturer’s revenue-sharing contract and retail competition. Eur J Oper Res 186(2):637–651 Zhang A, Fu X, Yang HG (2010) Revenue sharing with multiple airlines and airports. Transp Res B Methodol 8–9(2):944–959

Should a Stackelberg-Dominated Supply-Chain Player Help Her Dominant Opponent to Obtain Better System-Parameter Knowledge? Jian-Cai Wang, Amy Hing Ling Lau, and Hon-Shiang Lau*

Abstract A manufacturer (Manu) supplies a product to a retailer (Reta). The uncertain knowledge of the dominant player (which may be either Manu or Reta) about a system parameter is represented by a subjective probability distribution. At the time when the dominant player is designing the supply or purchase contract, should the dominated player help the dominant player to improve his imperfect system-parameter knowledge? Can the dominant player induce the dominated player to share her superior knowledge by using (or by threatening to use) sophisticated “channelcoordinating” contract formats? It is likely that one would surmise from the literature that the answer to both questions is “yes”. However, this chapter shows that very often the correct answer is “no”. Specifically, for the basic cost and market parameters, we show that the dominated player is (1) always motivated to mislead the dominant player to have a biased mean value for his subjective distribution; and (2) motivated, over a wide range of likely conditions, to increase the variance of the dominant player’s subjective distribution. Moreover, the dominant player cannot narrow this range of confusion-encouraging conditions by using a more sophisticated contract format such as a “menu of contracts.” Our results highlight the need to develop arrangements that can actually motivate a dominated player to share knowledge honestly.

*

Authors contributed equally; names arranged in reverse alphabetical order

J.-C. Wang School of Business, University of Hong Kong, Pokfulam, Hong Kong and School of Management and Economics, Beijing Institute of Technology, Beijing, China e-mail: [email protected] A.H.L. Lau School of Business, University of Hong Kong, Pokfulam, Hong Kong e-mail: [email protected] H.-S. Lau* (*) Department of Management Sciences, City University of Hong Kong, Kowloon Tong, Hong Kong e-mail: [email protected] T.-M. Choi and T.C. Edwin Cheng (eds.), Supply Chain Coordination under Uncertainty, International Handbooks on Information Systems, DOI 10.1007/978-3-642-19257-9_15, # Springer-Verlag Berlin Heidelberg 2011

379

380

J.-C. Wang et al.

Keywords Supply chain contract design • Information sharing

1 Introduction 1.1

Problem Statement

In considering human/organizational interactions, two common notions appear to be intuitively plausible at first glance: 1. It is often “beneficial” to share knowledge and strive for a “bigger pie” for all. 2. There is often some way a player can benefit himself by hiding/distorting the information he is supposed to provide to the other player. Unfortunately, these two notions suggest opposite actions; i.e., sharing knowledge honestly versus hiding/distorting information. Much of the supply chain literature is motivated by the first notion. This chapter emphasizes the validity of the second notion, contrary to what one might surmise from the large supply chain literature on information sharing and channel coordination. Specifically, we consider a supply chain with an upstream “manufacturer” (a male called “Manu”) and a downstream “retailer” (a female called “Reta”). We will consider separately the situations where the dominant player is (1) Manu; and (2) Reta. To facilitate explanation, the first part of this chapter will concentrate on the case in which the dominant player is Manu. Manu will then specify the supply contract. Manu is uncertain about one of the system parameters (say, x), and perceive it as a random variable x~ with subjective probability distribution Fx(•). Reta knows x perfectly, and recognizes that Manu’s x~-knowledge will influence how Manu will specify the supply contract to be offered to Reta. Our question is: from Reta’s perspective, what are the ideal characteristics (or “quality”) of Manu’s x~-perception that would lead Manu to specify a contract that is most advantageous to Reta? The spirit of the current supply chain “movement” suggests that Reta should help to improve the quality of Manu’s x~-perception. In contrast, this chapter summarizes our research results (Wang et al. 2008, 2009) showing that, in most situations, the opposite is true – regardless of what contract format Manu would implement.

1.2 q p C(q)

Summary of Basic Symbols and Relationships The quantity supplied by Manu to Reta and sold by Reta to the retail market The unit retail price set by Reta A supply contract designed and offered by Manu to Reta, requiring Reta to pay Manu $C(q) if Reta wants Manu to supply her q units

Should a Stackelberg-Dominated Supply-Chain Player

m, c x~ PM , P R, P I PC PM, PR PMsub, PRsub

381

The unit variable cost of Manu and Reta, respectively and k (m þ c) A generic random variable with support [xmin, xmax], standard deviation sx, and coefficient of variation kx. x~’s mean is denoted by either mx or x (i.e., bold letter) The profit of, respectively, Manu, Reta, and the “integrated firm” Total channel profit, equals (PM + PR) Expected profit of, respectively, Manu and Reta The subsistence profit of, respectively, Manu and Reta

Reta incurs a unit retail-processing cost c and gets to set the unit retail price p and the purchase quantity q. Given a supply contract C(q) specified by Manu and the (p, q)-decision set by Reta, Manu’s and Reta’s profits are: For Manu : PM ¼ CðqÞ mq; For Reta : PR ¼ ðp cÞq CðqÞ:

(1)

Both players know that for any p Reta sets, the market demand is given by the linear demand curve: q ¼ a bp;

(2)

where parameters (a,b) reflect the basic market-demand characteristics. In this model, Reta’s decision variables are (p,q), and Manu’s decisions are (1) C(q)’s format; and (2) the numerical values of C(q)’s parameters. Besides m, Manu’s “environmental variables” (or “system parameters”) are {a, b, c}. Manu’s knowledge of one of these is imperfect, and perceives it as a random variable (i.e., a~, b~ or c~), with cumulative distribution function (cdf) Fa(•), Fb(•) or Fc(•), respectively.

2 Review and Overview Figure 1 depicts the decision/action sequence of our scenario. At time point A, the dominated player (“Reta” in the current scenario) anticipates that the systemparameter information s/he provides may influence how the dominant player (“Manu”) will specify the supply contract. At time point B the dominant player specifies the supply contract.

2.1

Positioning in the Literature

There exists now a huge literature on supply chain coordination and cooperation; see, e.g., Cachon (2003) and Chen (2003) for excellent reviews. Among others, the following two notions are likely to be learnt from this literature:

382

J.-C. Wang et al.

Reta contemplates the format of Manu’s Fx(•) that will be most beneficial to Reta.

Manu specifies C(q), using his knowledge Fx(•) on x. ˜

Reta orders q units from Manu.

Reta sets the retail price p and sells her stock.

time axis Point A

Point B

Point C

Fig. 1 Schematic diagram of action sequence showing time points A–C

Notion (A): At time point B depicted in Fig. 1, the dominant player with imperfect knowledge of system parameter (say, “a”) can use a series of increasingly sophisticated contract formats to increasingly improve his (and the supply chain’s) expected profit. See, e.g., Corbett et al. (2004), and Liu and C¸etinkaya (2009) about supply chain contract design under “stochastic” and “asymmetric” knowledge scenarios. Notion (B): At time point C depicted in Fig. 1 (after C(q) has been specified), if a player at one supply-chain echelon has better knowledge of the system’s environmental parameters than the player at the other echelon, the better-informed player can often improve the supply chain’s performance by sharing his/her superior information – i.e., the “bigger pie” notion (see, e.g., Lee et al., 2000, Wu and ¨ zer 2010). Note that in our stripped-down cost model Cheng 2008, and Liu and O defined by (1) and (2), by time point C it is too late for Reta to improve channel profit by sharing her superior information with Manu. Our paper extends the earlier related studies in the following aspects: 1. While the overwhelming majority of earlier related studies consider questions ¨ zer raised at time points B or C in Fig. 1 (see, e.g., Ha 2001; Lau and Lau 2005; O and Wei 2006), we consider a question raised at time point A. 2. We use the simplest possible two-echelon structure, summarized by (1) and (2). There is no manufacturing capacity consideration, production lead time, logistics cost, forecasting issue, knowledge transmission cost, etc. Thus, we have removed as much as possible those factors that are most readily identified as motivations for Reta to conceal her superior {a,b,c}-knowledge. The purpose is to minimize any likely confounding effects by factors not directly related to our questions. Among many others, Tirole (1988), Corbett et al. (2004), and Liu and C¸etinkaya (2009) also employ this stylized structure. Regarding Aspect (1) stated above, only a few other supply-chain contractdesign studies have also focused on time point A. For example, Li (2002) examined the following problem: At time point A, are the dominated players willing to sign an information-sharing contract with the dominant player before they learn the true information, noting that once information sharing is agreed, the private information obtained later must be revealed truthfully.

Should a Stackelberg-Dominated Supply-Chain Player

383

That is, although the contract-signing action is at time point A, the possible information acquisition and sharing occurs after time point A. Another example is Taylor and Xiao (2010), who investigated, from the dominant-player’s perspective, which format of Fx(•) would be optimal. In contrast, our study takes the perspective of the dominated player. We now reiterate the difference between our questions and the questions answered in most of the related earlier studies. Consider first the situation where Manu is the dominant player. The earlier studies showed that (1) Manu can use increasingly sophisticated contract formats to give himself increasingly higher expected profits; (2) under a specified supply contract (with specified contract parameter values), often Manu and the channel (and sometimes also Reta) will benefit if Manu becomes better informed about certain system parameters (such as a, b, or c). In contrast, our questions are: 1. Before Manu has finalized a supply contract, is Reta motivated to help Manu to become better informed about certain system parameters? 2. Can Manu motivate Reta to help him become better informed by telling Reta that he will use increasingly more sophisticated supply contract formats? This chapter will show that, under a wide range of plausible situations (hereafter “Situation A”), Reta prefers Manu to be more (rather than less) uncertain about the system’s parameter(s). Moreover, the range of “Situation A” cannot be narrowed significantly by using more sophisticated contract formats. Also, Reta is always motivated to bias Manu’s subjective distributions. For Reta, our results mean that, contrary to what she is likely to conclude from the current supply-chain literature, she should NOT share knowledge honestly but should mislead Manu before Manu finalizes his contract; we also showed what kind of Manu-misperception Reta should aim for. For Manu (and hence the researchers), our results mean that, again contrary to what one might conclude from the literature, the various sophisticated channel coordinating contract formats are unable to induce Reta to share information honestly. Our results therefore also establish the need to find new ways to motivate Reta to share knowledge. To facilitate understanding, the wording in the preceding two paragraphs is for the situation where Manu is the dominant player. For the situation where Reta is the dominant player, simply interchange the terms “Manu” and “Reta”.

2.2

Overview of the Chapter

Sections 3–5 will consider the dominant-Manu case. Section 6 outlines the dominantReta case. The main results are summarized in the concluding Sect. 7.

384

J.-C. Wang et al.

3 The Case of a Dominant Manu: Structure of the Problem 3.1

Supply Contract Formats Considered

We consider four C(q)-formats that Manu may impose (listed below in the order of increasing level of “sophistication”): 1. Price-only contract, label [w]. This is the simplest C(q) format: Manu charges Reta a wholesale price w for each unit she buys from Manu. 2. Franchise Fee contract, label [FF]. Manu requires Reta to pay a specified franchise fee FFM; he then supplies Reta the product at cost (i.e., m/unit). 3. Two-Parts Tariff contract, label [2P]. Manu requires Reta to pay a lump-sum fee L and also charges Reta a wholesale price w for each unit she buys from Manu. 4. Menu of Contracts, label [MC]. Assume for the time being that Manu knows deterministically all parameters except a, and [amin, amax] is the finite support of the subjective probability distribution of Manu’s perceived a˜. The format of Manu’s [MC] is then{[w(adec), L(adec)]|amin adec amax}; i.e., Manu informs Reta that if Reta declares the demand curve’s a-value to be adec, Manu will charge Reta a unit wholesale price w(adec) plus a lump-sum payment L(adec). That is, w(adec) and L(adec) are functions of adec. Among others, Myerson (1979) has shown that, for a given Manu’s initial state of stochastic a-knowledge, Manu can design a w(adec) and a L(adec) such that Reta is forced to reveal the real a-value as adec at time point C of Fig. 1; the resultant [MC] then gives Manu the highest expected profit he can get among all possible contract formats (specified at Point B). Similarly, if Manu knows deterministically all parameters except b or c, the format of Manu’s [MC] will be, respectively, {[w(bdec), L(bdec)]| bmin bdec bmax} or {[w(cdec), L(cdec)]|cmin cdec cmax}. The above four C(q)s are the most popularly considered contract formats. There are of course other formats besides these four. However, as will be justified in Sect. 4.5, considering these four is sufficient to support the conclusions we will be presenting in this chapter.

3.2

Characterizing Manu’s Subjective Distributions

Consider, for example, Manu’s subjective distribution on a˜, with cdf Fa(•), mean ma and standard deviation sa. We consider two aspects of the “quality” of Manu’s a˜-perception: its “bias” (ma a) and its “uncertainty” sa. Manu’s a˜-perception is “perfect” if both bias and uncertainty equal zero. Earlier related studies such as Ha (2001) and Corbett et al. (2004) had to restrict the imperfectly-known parameter’s subjective distribution to be uniform in order to obtain meaningful analytical results. We follow this approach and also obtain analytical results by assuming a uniform Fa(•). We then go one more step and

Should a Stackelberg-Dominated Supply-Chain Player

385

investigate the effects of assuming a more versatile distribution for Fa(•). The gamma distribution is chosen because it can take on a much wider range of coefficients of variations and skewnesses compared to (say) the popular exponential, Erlang or normal (which are special cases of the gamma). For our model, numerical results using a gamma Fa(•) reveal some important behavior unobservable under a uniform-Fa(•) assumption. Of course, the gamma numerical results also confirm the major uniform-based analytical results. In the following sections, numerical results under the gamma assumption will be presented first because they are easier to understand, analytical results for the uniform assumption are then used to provide further support.

3.3

Overview and Preview for the Dominant-Manu Case

Sections 4 and 5 consider, respectively, uncertainties in “a” and “c”. Since fairly similar results are obtained for all three parameters {a, b, c}, we present detailed results only for “a”, while results for “b” are completely omitted. Our results can be briefly summarized as follows: Regardless of what contract format Manu will use, Reta should always try to inflate mb and mc but deflate ma. Regarding the uncertainties, Reta does not want Manu’s (sa, sb, sc) to be too low, but also not too high. To emphasize, Reta’s preferences towards Manu’s m and s are quite different. Regarding the “error” (in m), Reta wants it to be as large as possible (as long as it is in the right direction). Regarding “uncertainty” s, Reta does not want Manu to be either too certain or too uncertain about his estimate. Our conclusions also mean that both Reta and Manu should behave in ways that are quite different from what one might surmise from the current supply chain literature; particularly, they are in stark contrast to the “all or nothing” result in Taylor and Xiao (2010).

3.4

Summary of Basic Benchmark Results

Detailed derivations of the results summarized in this subsection can be found in, e.g., Corbett et al. (2004) and Lau and Lau (2005). Under an “integrated firm” where Manu and Reta are merged into one entity, it is known that the optimal (p,q) decisions and the attainable channel profit are: pI ¼ ða þ bkÞ=ð2bÞ; qI ¼ ða bkÞ=2; and PI ¼ ða bkÞ2 =ð4bÞ; recall k ðm þ cÞ:

(3)

If Manu and Reta are two separate players, each with deterministic knowledge of all the parameters, then the dominant Manu knows that, for any w-value he declares

386

J.-C. Wang et al.

in a [w]-contract, The players’ profits and Reta’s responses (on retail-price and purchase-quantity) are: ðPM Þw ¼ ðw mÞ½a bðw þ cÞ=2; ðPR Þw ¼ ½a bðw þ cÞ2 =ð4bÞ; pw ¼ ða þ bc þ bwÞ=ð2bÞ; qw ¼ ða bc bwÞ=2:

) (4a)

Recognizing the above, Manu maximizes his profit by setting w ¼ ða þ bkÞ=ð2bÞ;

(4b)

leading to the following optimal profits for the players and the channel: PR ¼ ða bkÞ2 =ð16bÞ; PM ¼ ða bkÞ2 =ð8bÞ; PC ¼ PM þ PR ¼ 3ða bkÞ2 =ð8bÞ:

(4c)

Equations (3) and (4) show that PC* < PI*; i.e., [w] does not “coordinate the channel.” In contrast, it is known that, with deterministic parameter knowledge, either [FF] or [2P] enables Manu to not only coordinate the channel (i.e., achieve PC* ¼ PI*), but also acquire absolute power in deciding Reta’s share of ПI* (subject of course to the condition PR PRsub). In the deterministic knowledge context [MC] is irrelevant because it degenerates into [2P]. If Manu does not know all the parameters deterministically, it is known that no contract format enables Manu to achieve the same total channel profit as ПI*. However, in most “stochastic” or “asymmetric” knowledge scenarios, [w], [FF] and [2P] enable Manu to achieve progressively higher expected profit for himself. Ultimately, [MC] is the most powerful contract format for Manu – i.e., an optimized [MC] enables Manu to obtain the largest expected profit for himself. Relative to [w], we will refer to [FF], [2P] and [MC] collectively as “coordination encouraging” contract formats.

4 Dominant Manu is Uncertain About the Market Size a In this section, we will consider in Sect. 4.1 how Reta wants Manu to perceive “a” when both sides know that Manu will offer a price-only ([w]) contract. Then, in Sects. 4.2–4.4 we will consider how Reta wants Manu to perceive “a” when both sides know that Manu will offer, in turn, a franchise fee contract ([FF]), a two-part tariff contract ([2P]) and a menu of contract ([MC]). Under each contract, we first tabulate the numerical results for the situation where Manu’s a priori subjective knowledge of parameter “a” is gamma-distributed; this tabulation enables us to illustrate the main pattern of behavior we are emphasizing in this chapter. This pattern is then confirmed by analytical results we are able to derive for the situation

Should a Stackelberg-Dominated Supply-Chain Player

387

where Manu’s a priori subjective knowledge of parameter “a” is uniformdistributed. Our results consistently show that (1) Reta always prefers Manu to perceive (incorrectly) a deflated ma; moreover, over a wide range of plausible conditions, Reta prefers Manu to be more uncertain about a (i.e., higher sa); and (2) Manu cannot narrow or alter Reta’s range of confusion-preferring conditions by implementing (or threatening to implement) a more sophisticated contract format (e.g., [MC]) instead of a simpler one (e.g., [w]).

4.1

The Price-Only Contract [w]

4.1.1

Problem Statement: Manu’s Knowledge of “a” is Inferior to Reta’s

Manu perceives a˜ with subjective cdf Fa(•). Thus, after setting w, (4a) indicates that ~ M(perM) ¼ (w m)[a˜ b(w þ c)]/2 and Manu will perceive his own profit to be P ~ R(perM) ¼ [a˜ b(w þ c)]2/(4b). Manu will perceive Reta’s profit to be P recognizes that Reta will “play” only if Reta’s profit exceeds PRsub; i.e., Manu ~ R(perM) PRsub; or, equivalently, when a˜ b0 , perceives that Reta will play if P 0 where b ¼ b(w + c) þ √(4bPRsub) is the “cutoff value” (see Ha 2001 for a more ~ M(perM)] under a detailed explanation). Thus, Manu’s problem of maximizing E[P stochastically-perceived a˜ can be written as: ð amax fðw mÞ½a bðw þ cÞ=2g dFa ðaÞ; where b ¼ maxðamin ; b0 Þ: (5) max w

b

Thus, Manu will set the unit wholesale price at w*, where w* is the solution to (5). Then, Reta knows, from her perspective, that if Manu perceives a˜ with cdf F(a), her profit (as perceived by herself) is, PR(perR) ¼ [areal b(w* þ c)]2/(4b). This PR(perR)-expression shows that a higher PR(perR) is brought by a lower w*. Reta’s (and hence our) question is therefore: what kind of a Manu-perceived Fa(•) will lead to a lower w* – hence a higher PR(perR)? 4.1.2

Numerical Results

Table 1 presents the PR(perR)-values for different combinations of c-values and ka-values (or, equivalently, sa-values); recalling that kx x~’s coefficient of variation. Values of other parameters are set at: areal ¼ ma ¼ 5, b ¼ 1, ПRsub ¼ [ma b (c þ m)]2/24, and a˜ is gamma distributed. Without loss of generality, we set m ¼ 1 throughout this chapter. To obtain the PR(perR)-values, first solve (5) numerically for w*, then compute PR(perR) ¼ [areal b(w* þ c)]2/(4b). Table 1 shows that, for any given c-value (i.e., along each c column), PR(perR) decreases as sa increases in the lower (grayed) region where sa is “sufficiently large,” but PR(perR) increases as sa increases in the upper (non-grayed) region where sa is “sufficiently small.” A “Boundary B” separates the grayed and non-grayed regions (or “Situations”).

388

J.-C. Wang et al.

Table 1 ПR(perR) under a price-only contract ([w])

Gamma-distributed a˜, ma ¼ 5, b ¼ 1, and ПRsub ¼ (ma bk)2/(24b) 2.5

ΠR(perR)

2 [w] [FF] [2P] [MC]

1.5 1 0.5 0

2.5

3

3.5

4

4.5

5

5.5

6

6.5

7

7.5 ma

Fig. 2 PR(perR) under gamma-distributed a˜: sa ¼ 0.4, areal ¼ 5, b ¼ 1, c ¼ 1, and PRsub ¼ 0

In other words, over a significant range of plausible conditions, Reta is motivated to increase Manu’s uncertainty in a˜. We have repeated Table 1 computations using a grid system of different combinations of values of the parameters ma, b and ПRsub; their results confirm the pattern illustrated in Table 1 (the same verification approach has been used in all subsequent numerically illustrated patterns to be reported in this chapter). Table 1’s characteristics will be discussed again in greater detail in Sect. 4.1.4 after supporting analytical results are presented below. The “filled diamond” line in Fig. 2 illustrates, for a typical set of (sa, areal, b, c, ПRsub)-parameter values, how PR(perR) increases as ma decreases under a [w] contract. That is, Reta always prefers Manu to perceive (incorrectly) a deflated ma. 4.1.3

Analytical Proofs

The behavior depicted in Table 1 (and other effects) are derived analytically in Appendix 1 of Wang, Lau and Lau (hereafter “WLL”) (2008) for the case of a uniform F(a). The main results are summarized as Lemmas 1A to 1C. Note that one does not need to read these analytical results (and their counterparts in Sects. 4.2.3, 4.3.3 and 4.4.3) in order to follow the basic arguments of this chapter.

Should a Stackelberg-Dominated Supply-Chain Player

389

Corresponding to the slanting “Boundary B” in Table 1, WLL’s (2008) Appendix 1 shows that, for the case of a uniform F(a), there are four “regions” or “Situations,” separated by three boundaries defined as follows. First, the boundary functions saWA, saWB and saWC are derived in (A6), (A9) and (A10) of WLL’s (2008) Appendix A. For example, one boundary function is: pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ saWB ¼ fma bk 5 bPRsub þ

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 pﬃﬃﬃ ½ma bk bPRsub þ 8bPRsub g=ð4 3Þ : (6)

Second, these boundary functions delineate the following four Situations: 1. 2. 3. 4.

Situation Ia: when sa saWA Situation Ib: when saWA < sa saWB Situation II: when saWB < sa saWC Situation III: when sa > saWC

Lemma 1A (Manu’s optimal w-decision). Depending on the “Situation,” Manu’s w* is: Situation Ia (when sa saWA): w* ¼ [ma b(c m)]/(2b). pﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Situation Ib (when saWA < sa saWB): w ¼ ½ma 3sa bc 2 bPRsub =b: Situation II (when saWB < sa saWC):

w ¼

pﬃﬃﬃ 2ðma þ 3sa bcÞ þ bm

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃ 2 ½ma þ 3sa bk þ 12bPRsub =ð3bÞ:

Situation III (when sa > saWC): PR(perR) is less than PRsub, hence no trade occurs. Lemma 1B (Effect of sa on PR(perR)). Since (4) shows that PR(perR) increases as w decreases, one can obtain the following conclusions by simply observing the w*-expressions given in Lemma 1A above: In Situation Ia, w* and hence PR(perR) is constant w.r.t. sa. In Situation Ib, w* decreases and hence PR(perR) increases as sa increases. In Situation II, w* increases and hence PR(perR) decreases as sa increases. Lemma 1C (Effects of ma and PRsub on PR(perR)). Effect of ma: in all Situations, w* increases as ma increases; i.e., PR(perR) decreases as ma increases. Hence Reta is always motivated to mislead Manu into perceiving a smaller ma. Effect of PRsub: in all Situations, w* decreases and hence PR(perR) increases as PRsub increases. Hence Reta is always motivated to convince Manu to recognize an inflated PRsub.

4.1.4

Discussion

Although this chapter is not meant to consider PRsub, the effect of PRsub stated in Lemma 1C is worth noting. On one hand, its Lemma-1C effect appears intuitively reasonable once it is stated; on the other hand, earlier models incorporating PRsub

390

J.-C. Wang et al.

have always assumed that Manu knows and accepts PRsub as it is. Lemma 1C suggests that setting PRsub is an issue that warrants deeper investigations. We return now to our main issue: Manu’s is a˜-knowledge, which we quantify in two aspects: bias (ma a) and uncertainty sa. Reta’s Preference on ma, or a˜ ’s Bias A higher a-value implies a higher “base demand.” Thus, our Lemma-1C result means that Reta will always try to mislead Manu into perceiving as low a ma-value as possible. This is neither counter-intuitive nor intuitively obvious. It is not surprising that a dominated player will want the dominant player (something like a “boss”) to perceive the operating environment as “tougher” than it actually is. Reta’s Preference on sa, or a˜ ’s Level of Uncertainty In contrast to ma’s effects, we show here that sa’s (or, equivalently, ka’s) effects are quite counter-intuitive. Lemma 1B indicates that PR(perR) is a quasi-concave function of sa, attaining its maximum at saWB. This, of course, is the sa (or ka) effect illustrated graphically by Table 1, where the grayed area below Boundary B (the line “══”) corresponds to Situation II, and the white area above Boundary B corresponds to Situation Ib. It can be easily seen from the derivations in Appendix 1 that Situation Ia does not arise if a˜’s probability distribution has a long right-hand tail (as in a gamma distribution), hence Table 1 does not exhibit a Situation-Ia area – in contrast to Lemma-1’s uniform-distribution results. Under Situation Ib, Reta prefers Manu’s sa to be higher; i.e., instead of sharing her a-information, she is actually motivated to confuse Manu and muddy Manu’s a˜-knowledge. This is contrary to the increasingly popular supply chain notion of mutually beneficial information sharing. Nevertheless, Table 1 also depicts that the Situation-Ib area is always above the Situation-II area; i.e., Reta is motivated to increase sa when sa is “too low,” but to reduce sa when sa is “too high” – thus, this aspect of sa’s counter-intuitive effects does fit the intuitively attractive notion of “everything in moderation.” We now study how large the Situation-I area is relative to the Situation-II area. This is facilitated by Table 1’s numerical results. Consider first the case of a uniformly distributed a˜. Define kaBB as the ka-value of Boundary B. Assuming the simpler case of PRsub ¼ 0, (6) indicates that p kaBB ¼ ½1 ðbk=ma Þ=ð2 3Þ; i.e., kaBB should be less that [1/(2√3)], or 0.29. Table 1 depicts, for situations with non-zero PRsub and gamma-distributed (instead of uniform-distributed) a˜, kaBBvalues that are significantly less than 0.29. Thus, for the Table 1 column with c ¼ 1, kaBB 0.12. At this c-value, assuming that ma ¼ areal, (3) gives pI* ¼ (areal + bk)/(2b) ¼ (5 + 2)/2 ¼ 3.5, where k ¼ m+c ¼ 2. Hence the theoretical

Should a Stackelberg-Dominated Supply-Chain Player

391

optimal markup over cost is M ¼ (pI* k)/k ¼ 0.75, which is near the lower end of realistic M values, considering that this is the combined gross profit margin of both Manu and Reta. Thus, columns to the right of the “c ¼ 1” column in Table 1 represent less realistic conditions because they are not sufficiently profitable, whereas columns to the left of the “c ¼ 1” column in Table 1 represent increasingly profitable conditions. In other words, for most realistic combinations of system-parameter values, “Situation I” applies when ka is between 0 and (very roughly) 0.2. Thus, while Situation I is not entirely negligible, it is probably not as prevalent as Situation II. Note, however, that this conclusion will be contradicted in Sect. 5, where knowledge uncertainty in “c” (instead of “a”) is considered.

4.2 4.2.1

The Franchise Fee Contract [FF] Problem Statement: Manu’s Knowledge of “a” is Inferior to Reta’s

We stated in Sect. 3.1 that, under [FF], Manu charges Reta a lump-sum fee FFM but supplies her at cost; i.e., m/unit. If Manu perceives a˜, he then perceives Reta’s profit as, from (3), h i ~ R ðperMÞ ¼ ða~ bkÞ2 =ð4bÞ FFM : P (7a) Since Manu knows that Reta will “play” only if her profit is at least PRsub, ~ R(perM) ¼ PRsub” gives “b” (the “cut-off” a-value below which Reta solving “P “quits”) as pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ (7b) b ¼ bk þ 2 ½bðPRsub þ FFM Þ: Hence Manu’s problem is to set FFM to maximize ð amax FFM dFa ðaÞ:

(8)

maxðamin ;bÞ

From Reta’s perspective, if Manu charges FFM, her profit (as perceived by herself) is PR(perR) ¼ (areal bk)2/(4b) FFM. Thus, Reta prefers a Manuperceived Fa(•) that leads to a lower FFM, and hence a higher PR(perR).

4.2.2

Numerical Results

As counterpart to Table 1, Table 2 presents the PR(perR)-values for different combinations of c-values and ka-values (or, equivalently, sa-values) under [FF] when a˜ is gamma distributed. Values of the other parameters (i.e., ma, b, m and ПRsub) are as for Table 1. The sa-effects on PR(perR) depicted by Table 2 are very

392

J.-C. Wang et al.

similar to those depicted by Table 1. The position of “Boundary B” (marked “══”) remains largely unchanged when one moves from Table 1 to Table 2. That is, implementing [FF] instead [w] does not alter or narrow the range of conditions under which Reta prefers Manu’s sa to be larger.

4.2.3

Analytical Proofs

The behavior depicted in Table 2 (and other effects) are derived analytically in WLL’s (2008) Appendix B for the case of a uniformly-distributed a˜. The main results are summarized as Lemmas 2A to 2C; they parallel Lemmas 1A to 1C given in Sect. 4.1.3 for [w]. Lemma 2A (Manu’s optimal FFM decision). The boundary values saFB and saFC used below are defined in (B4) and (B5) of WLL’s (2008) Appendix B. Then, depending on the “Situation,” Manu’s FFM* is: Situation I (when sa saFB): FFM ¼ ðma

pﬃﬃﬃ 3sa bkÞ2 =ð4bÞ PRsub :

Situation II (when saFB < sa saFC): qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃ pﬃﬃﬃ 2 FFM ¼ fma þ 3sa bk þ ½ma þ 3sa bk þ 12bPRsub g2 =ð36bÞ PRsub :

(9)

(10)

Situation III (when sa > saFC): PR(perR) is less than PRsub, hence no trade occurs. Lemma 2B (Effect of sa on PR(perR)). Since Sect. 4.2.1 showed that PR(perR) increases as FFM decreases, one easily obtains the following conclusions by observing the FFM-expressions given in Lemma 2A above: In Situation I, FFM* decreases and hence PR(perR) increases as sa increases. In Situation II, FFM* increases and hence PR(perR) decreases as sa increases. Therefore, PR(perR) is a quasi-concave function of sa, with its maximum at saFB. Lemma 2C (Effects of ma and PRsub on PR(perR)). The effects of ma and PRsub on PR(perR) under [FF] are identical to those stated in Lemma 1C for the [w] contract. Table 2 ПR(perR) under a Franchise fee contract ([FF])

Gamma-distributed a˜, ma ¼ 5, b ¼ 1, and ПRsub ¼ (ma bk)2/(24b)

Should a Stackelberg-Dominated Supply-Chain Player

4.3 4.3.1

393

The Two-Part Tariff Contract [2P] Problem Statement: Manu’s Knowledge of “a” is Inferior to Reta’s

As stated in Sect. 3.1, under [2P], Manu charges Reta a lump-sum fee L on top of a unit wholesale price w. Similar to the arguments given in Sects. 4.1.1 and 4.2.1 for [w] and [FF], Manu’s problem under [2P] can be formulated as: ð amax ½a bðw þ cÞ þ L dFa ðaÞ; ðw mÞ (11) max w;L;b b 2 subject to

4.3.2

½b bðw þ cÞ2 L PRsub ; amin b amax : 4b

(12)

Numerical Results

As counterpart to Tables 1 and 2, Table 3 presents the PR(perR)-values for different combinations of c- and sa-values under [2P] when a˜ is gamma-distributed. Again, the sa-effects on PR(perR) depicted by Table 3 are very similar to those depicted by Tables 1 and 2, and the comments made in Sect. 4.2.2 for [FF] are also applicable here. 4.3.3

Analytical Proofs

The behavior depicted in Table 3 (and other effects) are derived analytically in WLL’s (2008) Appendix C for the case of a uniform F(a). The main results are summarized as Lemmas 3A to 3C; they parallel Lemmas 1A to 1C given in Sect. 4.1.3 for [w] and Lemmas 2A–2C given in Sect. 4.2.3 for [FF]. Similar to Lemma 1A, the Situation’s boundary functions saTA, saTB and saTC are derived in (C12), (C9) and (C13) of WLL’s (2008) Appendix C. Lemma 3A (Manu’s optimal [2P] decisions for w and L). Depending on the “Situation,” Manu’s optimal [2P] decisions for w and L are: Situation I (when sa min(saTA,saTB)): Table 3 ПR(perR) under a two-part tariff contract ([2P])

Gamma-distributed a˜, ma ¼ 5, b ¼ 1, and ПRsub ¼ (ma bk)2/(24b)

394

J.-C. Wang et al.

pﬃﬃﬃ pﬃﬃﬃ 3sa =b þ m; and L ¼ ðma 2 3sa bkÞ2 =ð4bÞ PRsub :

(13)

Situation IIa (when min(saTA,saTB) < sa saTB): pﬃﬃﬃ pﬃﬃﬃ w ¼ 3sa =b þ m; and L ¼ ðma 2 3sa bkÞ2 =ð4bÞ PRsub :

(14)

w ¼

Situation IIb (when saTB < sa saTC): qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃ pﬃﬃﬃ 2 w ¼ 4ðma þ 3sa bcÞ þ 11bm ðma þ 3sa bkÞ þ 60bPRsub =ð15bÞ; (15a) L ¼

pﬃﬃﬃ ma þ 3sa bk þ

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃ 2 ðma þ 3sa bkÞ þ 60bPRsub =ð100bÞ PRsub : (15b)

Situation III (when sa > saTC): PR(perR) is less than PRsub, hence no trade occurs. Lemma 3B (Effects of sa on PR(perR)). In Situation I, PR(perR) increases as sa increases. In Situation IIa and Situation IIb, PR(perR) decreases as sa increases. Therefore, PR(perR) is a quasi-concave function of sa, with its maximum at min (saTA, saTB). (Note that contrary to its counterparts Lemmas 1B and 2B, Lemma 3B cannot be obtained by merely observing the results given in Lemma 3A; it is derived in WLL’s Appendix 3). Lemma 3C (Effects of ma and PRsub on PR(perR)). The effects of ma and PRsub on PR(perR) under [2P] are identical to those stated in Lemmas 1C and 2C for [w] and [FF].

4.4 4.4.1

A Menu of Contract [MC] Brief Explanation of the Menu of Contracts ([MC]) Format

As explained in Myerson (1979) and Corbett et al. (2004), it is possible for Manu to specify functions w(adec) and L(adec) such that Reta is forced to declare the real a-value as adec, and that the resultant [MC] is the contract format that gives Manu the highest expected profit for a given state of stochastic a-knowledge (see Sect. 3.1).

4.4.2

Numerical Results

As counterpart to Tables 1–3, Table 4 presents the PR(perR)-values for different combinations of c- and sa-values under [MC] when a˜ is gamma distributed. Again, the sa-effects on PR(perR) depicted by Table 4 are very similar to those depicted by

Should a Stackelberg-Dominated Supply-Chain Player

395

Tables 1–3, and the comments made in Sects. 4.2.2 and 4.3.2 for [FF] and [2P] are also applicable here. 4.4.3

Analytical Proofs

The behavior depicted in Table 4 (and other effects) can be proven analytically for the case of a uniformly-distributed a˜. The substance of what amounts to “Lemma 4A” (i.e., the counterpart of Lemmas 1A, 2A and 3A) are detailed in WLL’s Appendix D. Lemmas 4B to 4C stated below are counterparts of the earlier Lemmas 1B/1C, 2B/2C and 3B/3C. Lemma 4B. The definitions of the following critical values saMA, saMB and saMC are given in (D14), (D12) and (D15) of WLL’s Appendix D. Then, depending on the “Situation,” the effects of sa on Reta’s PR(perR) are: Situation I (sa min(saMA,saMB)): PR(perR) increases as sa increases. Situation IIa (min(saMA,saMB) < sa saMB): PR(perR) decreases as sa increases. Situation IIb (saMB < sa saMC): PR(perR) decreases as sa increases. Situation III (sa > saMC): PR(perR) is less than PRsub, hence no trade occurs. Lemma 4C. PR(perR) decreases as ma increases and

Series Editors Peter Bernus, Jacek Błażewicz, Günter J. Schmidt, Michael J. Shaw

For further volumes: http://www.springer.com/series/3795

Titles in the Series M. Shaw, R. Blanning, T. Strader and A. Whinston (Eds.) Handbook on Electronic Commerce ISBN 978-3-540-65882-1

P. Bernus, K. Merlins and G. Schmidt (Eds.) Handbook on Architectures of Information Systems ISBN 978-3-540-25472-0, 2nd Edition

J. Błaz˙ewicz, K. Ecker, B. Plateau and D. Trystram (Eds.) Handbook on Parallel and Distributed Processing ISBN 978-3-540-66441-3

S. Kirn, O. Herzog, P. Lockemann and O. Spaniol (Eds.) Multiagent Engineering ISBN 978-3-540-31406-6

H.H. Adelsberger, Kinshuk, J.M. Pawlowski and D. Sampson (Eds.) Handbook on Information Technologies for Education and Training ISBN 978-3-540-74154-1, 2nd Edition C.W. Holsapple (Ed.) Handbook on Knowledge Management 1 Knowledge Matters ISBN 978-3-540-43527-3 Handbook on Knowledge Management 2 Knowledge Directions ISBN 978-3-540-43848-9 J. Błaz˙ewicz, W. Kubiak, I. Morzy and M. Rusinkiewicz (Eds.) Handbook on Data Management in Information Systems ISBN 978-3-540-43893-9 P. Bernus, P. Nemes and G. Schmidt (Eds.) Handbook on Enterprise Architecture ISBN 978-3-540-00343-4 S. Staab and R. Studer (Eds.) Handbook on Ontologies ISBN 978-3-540-70999-2, 2nd Edition S.O. Kimbrough and D.J. Wu (Eds.) Formal Modelling in Electronic Commerce ISBN 978-3-540-21431-1

J. Błaz˙ewicz, K. Ecker, E. Pesch, G. Schmidt and J. Weglarz (Eds.) Handbook on Scheduling ISBN 978-3-540-28046-0 F. Burstein and C.W. Holsapple (Eds.) Handbook on Decision Support Systems 1 ISBN 978-3-540-48712-8 F. Burstein and C.W. Holsapple (Eds.) Handbook on Decision Support Systems 2 ISBN 978-3-540-48715-9 D. Seese, Ch. Weinhardt and F. Schlottmann (Eds.) Handbook on Information Technology in Finance ISBN 978-3-540-49486-7 T.C. Edwin Cheng and Tsan-Ming Choi (Eds.) Innovative Quick Response Programs in Logistics and Supply Chain Management ISBN 978-3-642-04312-3 J. vom Brocke and M. Rosemann (Eds.) Handbook on Business Process Management 1 ISBN 978-3-642-00415-5 Handbook on Business Process Management 2 ISBN 978-3-642-01981-4 T.-M. Choi and T.C. Edwin Cheng Supply Chain Coordination under Uncertainty ISBN 978-3-642-19256-2

Tsan-Ming Choi

l

T.C. Edwin Cheng

Editors

Supply Chain Coordination under Uncertainty

Editors Tsan-Ming Choi The Hong Kong Polytechnic University Business Division, Institute of Textiles and Clothing Hung Hom, Kowloon Hong Kong SAR [email protected]

T.C. Edwin Cheng The Hong Kong Polytechnic University Department of Logistics and Maritime Studies Hung Hom, Kowloon Hong Kong SAR [email protected]

ISBN 978-3-642-19256-2 e-ISBN 978-3-642-19257-9 DOI 10.1007/978-3-642-19257-9 Springer Heidelberg Dordrecht London New York Library of Congress Control Number: 2011935633 # Springer-Verlag Berlin Heidelberg 2011 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

Preface

Channel coordination is a core subject of supply chain management. It is well-known that a stochastic multi-echelon supply chain system usually fails to be optimal owing to the presence of the bullwhip effect and the double marginalization issue. Motivated by the importance of the topic, over the past decade, much research effort has been devoted to exploring the detailed mechanisms (such as incentive alignment schemes) for achieving supply chain coordination under uncertainty and has generated many fruitful analytical and empirical results. Despite the abundance of research results, there is an absence of a comprehensive reference source that provides state-of-the-art findings on both theoretical and applied research on the subject “under one roof”. In addition, many new topics and innovative measures for supply chain coordination under uncertainty have appeared in recent years and many new challenges have emerged. As a result, we believe it is significant to put together all these interesting works and the respective insights into an edited volume. In view of the above, we co-edit this Springer handbook. The handbook contains five parts, covering (1) introductory materials and review of supply chain coordination; (2) analytical models for innovative coordination under uncertainty; (3) channel power, bargaining, and coordination; (4) technological advancements and applications in coordination; and (5) empirical analysis and case studies. The specific topics covered include the following: – Coordination of Supply Chains with Risk-Averse Agents – A Timely Review on Supply Chain Coordination – A Review of Control Policies for Multi-Echelon Inventory Systems with Stochastic Demand – Supply Chain Models with Active Acquisition and Remanufacturing – Facilitating Demand Risk-sharing with the Innovative Percent Deviation Contract – Value-added Retailer in a Mixed Channel under Asymmetric Information – Capacity Management and Price Discrimination under Demand Uncertainty using Options – Dynamic Procurement and Quantity Discounts in Supply Chains – Coordination in a Multi-period Setting: The Additional Ordering Cost Contract

v

vi

Preface

– Use of Supply Chain Contract to Motivate Selling Effort – Price and Warranty Competition in a Duopoly Supply Chain – Supply Chain Coordination for Newsvendor-type Products with Two Ordering Opportunities – Bargaining in a Two-Stage Supply Chain through Revenue-Sharing Contract – Should a Stackelberg-dominated Supply-chain Player Help her Dominant Opponent to Obtain Better Information – Supply Chain Coordination under Demand Uncertainty using Credit Option – Supply Chain Coordination under Consignment Contract – A Heuristic Approach for Collaborative Planning in Detailed Scheduling – RFID Technology Adoption and Supply Chain Coordination – Possibilistic Mixed Integer Programming Approach for Supply Chain Network Problems – Coordination of Converging Material Flows in Supply Chains under Uncertainty – Bioenergy Systems and Supply Chains in Europe: Conditions, Capacity, and Coordination – Benefits of Involving Contract Manufacturers in Collaborative Planning for Three-Echelon Supply Networks – A Capability-Based Approach for Managing IT Suppliers – Methodology for Assessing Collaboration Strategies and Incentives in the Pulp and Paper Industry We are very pleased to see that this research handbook has generated a lot of new analytical and empirical results with precious insights, which will not only help supply chain agents to understand more about the latest measures for supply chain coordination under uncertainty, but also help practitioners and researchers to know how to improve supply chain performance based on innovative methods. This will be especially meaningful to industries such as fashion apparel and consumer electronics, in which effective supply chain management has been known to be the key to success. We would like to take this opportunity to show our gratitude to Werner A. Mueller and Christian Rauscher for their kind support and advice along the course of carrying out this project. We sincerely thank all the authors who have contributed their decent research to this handbook. We are grateful to the professional reviewers who reviewed the submitted papers and provided us with timely comments and constructive recommendations. We are indebted to our student Pui-Sze Chow for her editorial assistance. We also acknowledge the funding support of the Research Grants Council of Hong Kong under grant number PolyU 5143/07E (General Research Fund) and The Hong Kong Polytechnic University under grant number J-BB6U. Last but not least, we are grateful to our families, colleagues, friends, and students, who have been supporting us during the development of this important research handbook. Tsan-Ming Choi, T.C.E. Cheng The Hong Kong Polytechnic University

Contents

Part I

Introduction and Review

Coordination of Supply Chains with Risk-Averse Agents . . . . . . . . . . . . . . . . . . 3 Xianghua Gan, Suresh P. Sethi, and Houmin Yan Addendum to “Coordination of Supply Chains with Risk-Averse Agents” by Gan, Sethi, and Yan (2004) . . . . . . . . . . . . . . . . . 33 Xianghua Gan, Suresh P. Sethi, and Houmin Yan A Review on Supply Chain Coordination: Coordination Mechanisms, Managing Uncertainty and Research Directions . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 Kaur Arshinder, Arun Kanda, and S.G. Deshmukh Control Policies for Multi-echelon Inventory Systems with Stochastic Demand . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 Qinan Wang Supply Chain Models with Active Acquisition and Remanufacturing . . . 109 Xiang Li and Yongjian Li Part II

Analytical Models for Innovative Coordination under Uncertainty

Facilitating Demand Risk-Sharing with the Percent Deviation Contract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 Matthew J. Drake and Julie L. Swann

vii

viii

Contents

Value-Added Retailer in a Mixed Channel: Asymmetric Information and Contract Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 Samar K. Mukhopadhyay, Xiaowei Zhu, and Xiaohang Yue Capacity Management and Price Discrimination under Demand Uncertainty Using Option Contracts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 Fang Fang and Andrew Whinston Dynamic Procurement, Quantity Discounts, and Supply Chain Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 Feryal Erhun, Pinar Keskinocak, and Sridhar Tayur Coordination of the Supplier–Retailer Relationship in a Multi-period Setting: The Additional Ordering Cost Contract . . . . . 235 Nicola Bellantuono, Ilaria Giannoccaro, and Pierpaolo Pontrandolfo Use of Supply Chain Contract to Motivate Selling Effort . . . . . . . . . . . . . . . . 255 Samar K. Mukhopadhyay and Xuemei Su Price and Warranty Competition in a Duopoly Supply Chain . . . . . . . . . . . 281 Santanu Sinha and S.P. Sarmah Supply Chain Coordination for Newsvendor-Type Products with Two Ordering Opportunities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315 Yong-Wu Zhou and Sheng-Dong Wang Part III

Channel Power, Bargaining and Coordination

Bargaining in a Two-Stage Supply Chain Through Revenue-Sharing Contract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347 Jing Hou and Amy Z. Zeng Should a Stackelberg-Dominated Supply-Chain Player Help Her Dominant Opponent to Obtain Better System-Parameter Knowledge? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379 Jian-Cai Wang, Amy Hing-Ling Lau, and Hon-Shiang Lau Supply Chain Coordination Under Demand Uncertainty Using Credit Option . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403 S. Kamal Chaharsooghi and Jafar Heydari Supply Chain Coordination Under Consignment Contract with Revenue Sharing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 427 Sijie Li, Jia Shu, and Lindu Zhao

Contents

Part IV

ix

Technological Advancements and Applications in Supply Chain Coordination

DEAL: A Heuristic Approach for Collaborative Planning in Detailed Scheduling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 457 J. Benedikt Scheckenbach Inventory Record Inaccuracy, RFID Technology Adoption and Supply Chain Coordination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483 H. Sebastian Heese Possibilistic Mixed Integer Linear Programming Approach for Production Allocation and Distribution Supply Chain Network Problem in the Consumer Goods Industry . . . . . . . . . . . . . . . . . . . . . . 505 Bilge Bilgen Coordination of Converging Material Flows Under Conditions of Uncertainty in Supply Chains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 525 Liesje De Boeck and Nico Vandaele Part V

Empirical Analysis and Case Studies

Bioenergy Systems and Supply Chains in Europe: Conditions, Capacity and Coordination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 545 Kes McCormick Three Is a Crowd? On the Benefits of Involving Contract Manufacturers in Collaborative Planning for Three-Echelon Supply Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563 Henk Akkermans, Kim van Oorschot, and Winfried Peeters Managing IT Suppliers: A Capability-Based Approach . . . . . . . . . . . . . . . . . . 599 Carlos Brito and Mafalda Nogueira Methodology for Assessing Collaboration Strategies and Incentives in the Pulp and Paper Industry . . . . . . . . . . . . . . . . . . . . . . . . . . . . 625 Nadia Lehoux, Sophie D’Amours, and Andre´ Langevin Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 651

.

Contributors

Henk Akkermans Supply Network Dynamics, Department of Information Management, Tilburg University, Warandelaan 2, P.O. Box 90153, 5000 LE, Tilburg, The Netherlands, [email protected] Nicola Bellantuono Dipartimento di Ingegneria dell’Ambiente e per lo Sviluppo Sostenibile, Politecnico di Bari, via De Gasperi s.n, 74100 Taranto, Italy, [email protected] Bilge Bilgen Department of Industrial Engineering, Dokuz Eylul University, 35160 Izmir, Turkey, [email protected] Carlos Brito Faculty of Economics, University of Porto, Rua Roberto Frias, 4200-464 Porto, Portugal, [email protected] S. Kamal Chaharsooghi Industrial Engineering Department, Tarbiat Modares University, Tehran, Iran, [email protected] Sophie D’Amours FORAC, Department of Mechanical Engineering, Pavillon Adrien-Pouliot, Universite´ Laval, Que´bec, Canada, G1V 0A6, sophie. [email protected] Liesje De Boeck Centre for Modeling and Simulation, HUBrussel, Stormstraat 2, 1000 Brussels, Belgium; Research Centre for Operations Management, K.U.Leuven, Naamsestraat 69, 3000 Leuven, Belgium, [email protected] S.G. Deshmukh Department of Mechanical Engineering, Indian Institute of Technology Delhi, New Delhi, 110016, India, [email protected] Matthew J. Drake Palumbo-Donahue Schools of Business, Duquesne University, Pittsburgh, PA 15282, USA, [email protected]

xi

xii

Contributors

Feryal Erhun Department of Management Science and Engineering, Stanford University, Stanford, CA, USA, [email protected] Fang Fang Department of ISOM, College of Business Administration, California State University at San Marcos, 333 S. Twin Oaks Valley Road, San Marcos, CA 92096, USA, [email protected] Xianghua Gan Department of Logistics and Maritime Studies, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong, China, [email protected] edu.hk Ilaria Giannoccaro Dipartimento di Ingegneria Meccanica e Gestionale, Politecnico di Bari, viale Japigia 182, 70125 Bari, Italy, [email protected] H. Sebastian Heese Kelley School of Business, Indiana University, 1309 East Tenth Street, Bloomington, IN 47405, USA, [email protected] Jafar Heydari Industrial Engineering Department, Shiraz University of Technology, Shiraz, Iran, [email protected] Jing Hou Business School, Hohai University, Nanjing, Jiangsu 211100, China, [email protected] hotmail.com Arun Kanda Department of Mechanical Engineering, Indian Institute of Technology Delhi, New Delhi 110016, India, [email protected] Arshinder Kaur Department of Management Studies, Indian Institute of Technology Madras, Chennai 600036, India, [email protected] Pinar Keskinocak School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, GA, USA, [email protected] Andre´ Langevin CIRRELT, Department of Mathematics and Industrial Engineering, E´cole Polytechnique de Montre´al, C.P. 6079, succ. Centre-ville, Montre´al, Canada, H3C 3A, [email protected] Amy Hing Ling Lau School of Business, University of Hong Kong, Pokfulam, Hong Kong, [email protected] Hon-Shiang Lau Department of Management Sciences, City University of Hong Kong, Kowloon Tong, Hong Kong, [email protected] Nadia Lehoux FORAC, Department of Mechanical Engineering, Pavillon AdrienPouliot, Universite´ Laval, Que´bec, Canada, G1V 0A6, [email protected]

Contributors

xiii

Sijie Li Institute of Systems and Engineering, Southeast University, Nanjing, Jiangsu People’s Republic of China, [email protected] Xiang Li Research Centre of Logistics, College of Economic and Social Development, Nankai University, Tianjian 300071, P.R. China, [email protected] Yongjian Li Business School, Nankai University, Tianjin 300071, P.R. China, [email protected] Kes McCormick International Institute for Industrial Environmental Economics (IIIEE), Lund University, Lund, Sweden, [email protected] Samar K. Mukhopadhyay Graduate School of Business, Sungkyunkwan University, Jongno-Gu, Seoul 110–745, South Korea, [email protected] Mafalda Nogueira Management School, Lancaster University, Lancaster, LA1-4YX, UK, [email protected] Winfried Peeters BU HPMS, NXP Semiconductors, High Tech Campus 60, 5656 AG Eindhoven, The Netherlands, [email protected] Pierpaolo Pontrandolfo Dipartimento di Ingegneria dell’Ambiente e per lo Sviluppo Sostenibile, Politecnico di Bari, via De Gasperi s.n, 74100 Taranto, Italy, [email protected] S.P. Sarmah Department of Industrial Engineering and Management, Indian Institute of Technology, Kharagpur 721302, India, [email protected] J. Benedikt Scheckenbach [email protected]

Cranachstr. 16, 50733 Koeln, Germany, benedikt.

Suresh P. Sethi School of Management, SM30, The University of Texas at Dallas, 800W Campbell Road, Richardson, TX 75080-3021, USA, [email protected] Jia Shu Department of Management Science and Engineering, School of Economics and Management, Southeast University, Nanjing, Jiangsu P.R. China, [email protected] Santanu Sinha Complex Decision Support Systems, Tata Consultancy Services, Akruti Trade Centre, MIDC, Andheri (E), Mumbai 400093, India, [email protected] yahoo.com Xuemei Su College of Business Administration, California State University Long Beach, 1250 Bellflower Blvd, Long Beach, CA 90840, USA, [email protected]

xiv

Contributors

Julie L. Swann H. Milton Stewart School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, GA 30332-0205, USA, [email protected] gatech.edu Sridhar Tayur Tepper School of Business, Carnegie Mellon University, Pittsburgh, PA, USA, [email protected] Kim van Oorschot Department of Leadership and Organizational Behaviour, BI Norwegian School of Business, NO-0442 Oslo, Norway, [email protected] Nico Vandaele Research Centre for Operations Management, K.U.Leuven, Naamsestraat 69, 3000 Leuven, Belgium; Faculty of Business and Economics, K.U. Leuven-Campus Kortrijk, Etienne Sabbelaan 53-bus 0000, 8500 Kortrijk, Belgium, [email protected] Jian-Cai Wang School of Business, University of Hong Kong, Pokfulam, Hong Kong; School of Management and Economics, Beijing Institute of Technology, Beijing, China, [email protected] Qinan Wang Nanyang Business School, Nanyang Technological University, Singapore, Singapore 639798, [email protected] Sheng-Dong Wang Department of Mathematics, Hefei Electronic Engineering Institute, Hefei, Anhui, P.R. China, [email protected] Andrew B. Whinston Department of IROM, McCombs School of Business, The University of Texas at Austin, 1 University Station B6000, Austin, TX 78712, USA, [email protected] Houmin Yan Department of Systems Engineering and Engineering Management, The Chinese University of Hong Kong, Shatin, NT, Hong Kong, [email protected] Xiaohang Yue Sheldon B. Lubar School of Business, University of WisconsinMilwaukee, P.O. Box 742, Milwaukee, WI 53201, USA, [email protected] Amy Z. Zeng School of Business, Worcester Polytechnic Institute, Worcester, MA 01609, USA, [email protected] Lindu Zhao Institute of Systems and Engineering, Southeast University, Nanjing, Jiangsu People’s Republic of China, [email protected] Yong-Wu Zhou School of Business Administration, South China University of Technology, Guangzhou, Guangdong, P.R. China, [email protected] Xiaowei Zhu College of Business and Public Affairs, West Chester University of Pennsylvania, West Chester, PA 19383, USA, [email protected]

.

Part I

Introduction and Review

.

Coordination of Supply Chains with Risk-Averse Agents Xianghua Gan, Suresh P. Sethi, and Houmin Yan

Abstract The extant supply chain management literature has not addressed the issue of coordination in supply chains involving risk-averse agents. We take up this issue and begin with defining a coordinating contract as one that results in a Paretooptimal solution acceptable to each agent. Our definition generalizes the standard one in the risk-neutral case. We then develop coordinating contracts in three specific cases (1) the supplier is risk neutral and the retailer maximizes his expected profit subject to a downside risk constraint, (2) the supplier and the retailer each maximizes his own mean-variance trade-off, and (3) the supplier and the retailer each maximizes his own expected utility. Moreover, in case (3) we show that our contract yields the Nash Bargaining solution. In each case, we show how we can find the set of Pareto-optimal solutions, and then design a contract to achieve the solutions. We also exhibit a case in which we obtain Pareto-optimal sharing rules explicitly, and outline a procedure to obtain Pareto-optimal solutions. Keywords Capacity • Coordination • Nash bargaining • Pareto-optimality • Risk averse • Supply chain management

X. Gan (*) Department of Logistics and Maritime Studies, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong e-mail: [email protected] S.P. Sethi School of Management, SM30, The University of Texas at Dallas, 800W. Campbell Road, Richardson, TX 75080-3021, USA e-mail: [email protected] H. Yan Department of Systems Engineering and Engineering Management, The Chinese University of Hong Kong, Shatin, NT, Hong Kong e-mail: [email protected] T.-M. Choi and T.C. Edwin Cheng (eds.), Supply Chain Coordination under Uncertainty, International Handbooks on Information Systems, DOI 10.1007/978-3-642-19257-9_1, # Springer-Verlag Berlin Heidelberg 2011

3

4

X. Gan et al.

1 Introduction Much of the research on decision making in a supply chain has assumed that the agents in the supply chain are risk neutral, i.e., they maximize their respective expected profits. An important focus of this research has been the design of supply contracts that coordinate the supply chain. When each of the agents maximizes his expected profit, the objective of the supply chain considered as a single entity is unambiguously to maximize its total expected profit. This fact alone makes it natural to define a supply chain to be coordinated if the chain’s expected profit is maximized and each agent’s reservation profit is met. A similar argument holds if each agent’s objective is to minimize his expected cost. In this paper we consider supply chains with risk-averse agents. Simply put, an agent is risk averse if the agent prefers a certain profit p to a risky profit, whose expected value equals p. In the literature, there are many measures of risk aversion; see Szeg€o (2004) for examples. Regardless of the measure used, when one or more agents in the supply chain are risk averse, it is no longer obvious as to what the objective function of the supply chain entity should be. Not surprisingly, the issue of coordination of supply chain consisting of risk-averse agents has not been studied in the supply chain management literature. That is not to say that the literature does not realize the importance of the risk-averse criteria. Indeed, there are a number of papers devoted to the study of inventory decisions of a single riskaverse agent. These include Lau (1980), Bouakiz and Sobel (1992), Eeckhoudt et al. (1995), Chen and Federgruen (2000), Agrawal and Seshadri (2000a), Buzacott et al. (2002), Chen et al. (2007), and Gaur and Seshadri (2005). There also have been a few studies of supply chains consisting of one or more risk-averse agents. Lau and Lau (1999) and Tsay (2002) consider decision making by a risk-averse supplier and a risk-averse retailer constituting a supply chain. Agrawal and Seshadri (2000b) introduce a risk-neutral intermediary to make ordering decisions for risk-averse retailers, whose respective profits are side payments from the intermediary. Van Mieghem (2003) has reviewed the literature that incorporates risk aversion in capacity investment decisions. While these papers consider risk-averse decision makers by themselves or as agents in a supply chain, they do not deal with the issue of the supply chain coordination involving risk-averse agents. It is this issue of coordination of supply chains consisting of one or more riskaverse agents that is the focus of this paper. That many decision makers are riskaverse has been amply documented in the finance and economics literature; see, for example, Van Neumann and Morgenstern (1944), Markowitz (1959), Jorion (2006), and Szeg€o (2004). We shall therefore develop the concept of what we mean by coordination of a supply chain, and then design explicit contracts that achieve the defined coordination. For this purpose we use the Pareto-optimality criterion, used widely in the group decision theory, to evaluate a supply chain’s performance. We define each agent’s payoff to be a real-valued function of a random variable representing his profit, and propose that a supply chain can be treated as coordinated if no agent’s payoff can be

Coordination of Supply Chains with Risk-Averse Agents

5

improved without impairing someone else’s payoff and each agent receives at least his reservation payoff. We consider three specific cases of a supply chain (1) the supplier is risk neutral and the retailer maximizes his expected profit subject to a downside risk constraint, (2) the supplier and the retailer each maximizes his own mean-variance trade-off, and (3) the supplier and the retailer each maximizes his own expected utility. We show how we can coordinate the supply chain in each case according to our definition. In each case we do this by finding the set of Paretooptimal solutions acceptable to each agent, and then constructing a flexible contract that can attain any of these solutions. Moreover, the concept we develop and the contracts we obtain generalize the same known for supply chains involving risk-neutral agents. The remainder of the paper is organized as the follows. In Sect. 2 we review the related literature in supply chain management and group decision theory. In Sect. 3 we introduce a definition of coordination of a supply chain consisting of risk-averse agents. In Sect. 4 we characterize the Pareto-optimal solutions and find coordinating contracts for the supply chains listed as the first two cases. In Sect. 5 we first take up the third case using exponential utility functions for the agents, and design coordinating contracts as well as obtain the Nash Bargaining solution. Then we examine a case in which the supplier has an exponential utility followed by a linear utility. Section 6 provides a discussion of our results. The paper concludes in Sect. 7 with suggestions for future research.

2 Literature Review There is a considerable literature devoted to contracts that coordinate a supply chain involving risk-neutral agents. This literature has been surveyed by Cachon (2003). In addition, the book by Tayur et al. (1999) contains a number of chapters addressing supply contracts. In light of these, we limit ourselves to reviewing papers studying inventory and supply chain decisions by risk-averse agents. First we review papers dealing with a single risk-averse agent’s optimal inventory decision. Then we review articles dealing with decision making by risk-averse agents in a supply chain. Chen and Federgruen (2000) re-visit a number of basic inventory models using a mean-variance approach. They exhibit how a systematic mean-variance trade-off analysis can be carried out efficiently, and how the resulting strategies differ from those obtained in the standard analyses. Agrawal and Seshadri (2000a) consider how a risk-averse retailer, whose utility function is increasing and concave in wealth, chooses the order quantity and the selling price in a single-period inventory model. They consider two different ways in which the price affects the distribution of demand. In the first model, they assume that a change in the price affects the scale of the distribution. In the second model, a change in the price only affects the location of the distribution. They show that in comparison to a risk-neutral retailer, a risk-averse retailer will charge a higher price

6

X. Gan et al.

and order less in the first model, whereas he will charge a lower price in the second model. Buzacott et al. (2002) model a commitment and option contract for a risk-averse newsvendor with a mean-variance objective. The contract, also known as a takeor-pay contract, belongs to a class of volume flexible contracts, where the newsvendor reserves a capacity with initial information and adjusts the purchase at a later stage when some new information becomes available. They compare the performance of strategies developed for risk-averse and risk-neutral objectives. They conclude that the risk-averse objective can be an effective approach when the quality of information revision is not high. Their study indicates that it is possible to reduce the risk (measured by the variance of the profit) by six- to eightfold, while the loss in the expected profit is almost invisible. On the other hand, the strategy developed for the expected profit objective can only be considered when the quality of information revision is high. They show furthermore that these findings continue to hold in the expected utility framework. The paper points out a need for modeling approaches that deal with downside risk considerations. Lau and Lau (1999) study a supply chain consisting of a monopolistic supplier and a retailer. The supplier and the retailer employ a return policy, and each of them has a mean-variance objective function. Lau and Lau obtain the optimal wholesale price and return credit for the supplier to maximize his utility. However, they do not consider the issue of improving the supply chain’s performance, i.e., improving both players’ utilities. Agrawal and Seshadri (2000b) consider a single-period model in which multiple risk-averse retailers purchase a single product from a common supplier. They introduce a risk neutral intermediary into the channel, who purchases goods from the vendor and sells them to the retailers. They demonstrate that the intermediary, referred to as the distributor, orders the optimal newsvendor quantity from the supplier and offers a menu of mutually beneficial contracts to the retailers. In every contract in the menu, the retailer receives a fixed side payment, while the distributor is responsible for the ordering decisions of the retailers and receives all their revenues. The menu of contracts simultaneously (1) induces every risk-averse agent to select a unique contract from it; (2) maximizes the distributor’s profit; and (3) raises the order quantities of the retailers to the expected value maximizing (newsvendor) quantities. Tsay (2002) studies how risk aversion affects both sides of the supplier–retailer relationship under various scenario of relative strategic power, and how these dynamics are altered by the introduction of a return policy. The sequence of play is as follows: first the supplier announces a return policy, and then the retailer chooses order quantity without knowing the demand. After observing the demand, the retailer chooses the price and executes on any relevant terms of the distribution policy as appropriate (e.g., returning any overstock as allowed). Tsay shows that the behavior under risk aversion is qualitatively different from that under risk neutrality. He also show that the penalty for errors in estimating a channel partner’s risk aversion can be substantial.

Coordination of Supply Chains with Risk-Averse Agents

7

In a companion paper (Gan et al. 2005), we examine coordinating contracts for a supply chain consisting of one risk-neutral supplier and one risk-averse retailer. There we design an easy-to-implement risk-sharing contract that accomplishes the coordination as defined in this paper. Among these supply chain papers, Lau and Lau (1999) and Tsay (2002) consider the situation in which both the retailer and the supplier in the channel are risk averse. However, neither considers the issue of the Pareto-optimality of the actions of the agents. The aim of Agrawal and Seshadri (2000b) is to design a contract that increases the channel’s order quantity to the optimal level in the risk-neutral case by having the risk-neutral agent assume all the risk. Once again, they do not mention the Pareto-optimality aspect of the decision they obtain. Finally since our definition of coordination is based on the concepts used in the group decision theory, we briefly review this stream of literature. From the early fifties to the early eighties, a number of papers and books appeared that deal with situations in which a group faces intertwined external and internal problems. The external problem involves the choice of an action to be taken by the group, and the internal problem involves the distribution of the group payoff among the members. Arrow (1951) conducted one of the earliest studies on the group decision theory, and showed that given an ordering of consequences by a number of individuals, no group ordering of these consequences exists that satisfies a set of seemingly reasonable behavioral assumptions. Harsanyi (1955) presented conditions under which the total group utility can be expressed as a linear combination of individuals’ cardinal utilities. Wilson (1968) used Pareto-optimality as the decision criterion and constructed a group utility function to find Pareto-optimal solutions. Raiffa (1970) illustrates the criterion of Pareto-optimality quite lucidly, and discusses how to choose a Pareto-optimal solution in bargaining and arbitration problems. LaValle (1978) uses an allocation function to define Pareto-optimality. Eliashberg and Winkler (1981) investigate properties of sharing rules and the group utility functions in additive and multilinear cases.

3 Definition of Coordination of a Supply Chain with Risk-Neutral or Risk-Averse Agents In this section we define coordination of a supply chain consisting of agents that are risk neutral or risk averse. We use concepts developed in group decision theory that deals with situations in which a group faces intertwined external and internal problems. The external problem involves the choice of an action to be taken by the group, and the internal problem involves the distribution of the group payoff among the members. In group decision problems, a joint action of the group members is said to be Pareto-optimal if there does not exist an alternative action that is at least as acceptable to all and definitely preferred by some. In other words, a joint action is Pareto-optimal if it is not possible to make one agent better off without making

8

X. Gan et al.

another one worse off. We call the collection of all Pareto-optimal actions as the Pareto-optimal set. It would not be reasonable for the group of agents to choose a joint action that is not Pareto-optimal. Raiffa (1970) and LaValle (1978) illustrate this idea quite lucidly with a series of examples. A supply chain problem is obviously a group decision problem. The channel faces an external problem and an internal problem. External problems include decisions regarding order/production quantities, item prices, etc. The internal problem is to allocate profit by setting the wholesale price, deciding the amount of a side payment if any, refund on the returned units, etc. Naturally, we can adopt the Pareto-optimality criterion of the group decision theory for making decisions in a supply chain. Indeed, in the risk-neutral case, the optimal action under a coordinating contract is clearly Pareto-optimal. In general, since the agents in the channel would not choose an action that is not in the Pareto-optimal set, the first step to coordinate a channel is to characterize the set. Following the ideas of Raiffa (1970) and LaValle (1978), we formalize below the definition of Pareto-optimality. Let (O; F ; P) denote the probability space and N denote the number of agents in the supply chain, N r2. Let Si be the external action space of agent i; i ¼ 1; . . . ; N, and S ¼ S1 SN . For any given external joint action s ¼ ðs1 ; . . . ; sN Þ 2 S, the channel’s total profit is a random variable Pðs; oÞ; o 2 O. Let E and V denote the expectation and variance defined on (O; F ; P), respectively. Now we define a sharing rule that governs the splitting of the channel profit among the agents. Let Y be the set of all functions from S O to RN . P Definition 1. A function uðs; vÞ 2 Q is called a sharing rule if i ui ðs; vÞ ¼ 1 almost surely. Under the sharing rule uðs; oÞ, agent i’s profit is represented by Pi ðs; v; uðs; vÞÞ ¼ ui ðs; vÞPðs; vÞ; i ¼ 1; . . . ; N: Often, when there is no confusion, we write Pðs; vÞ simply as PðsÞ, uðs; vÞ as uðsÞ, and Pi ðs; v; uðs; vÞÞ as Pi ðs; uðsÞÞ. A supply chain’s external problem is to choose an s 2 S and its internal problem is to choose a function uðsÞ 2 Y. Thus the channel’s total problem is to choose a pair ðs; uðsÞÞ 2 S Y. Now we define the preferences of the agents over their random profits. Let G denote the space of all random variables defined on ðO; F ;PÞ. For X; X0 2 G, the agent i’s preference will be denoted by a real-valued payoff function ui ðÞ defined on G. The relation ui ðXÞ>ui ðX0 Þ, ui ðXÞ

Coordination of Supply Chains with Risk-Averse Agents

ui ðXÞ ¼

9

EðXÞ; if PðXbaÞbb; 1; if PðXbaÞ>b:

Example 3. Suppose agent i has a concave increasing utility function gi : R1 ! R1 of wealth and wants to maximize his expected utility. Then the agent’s payoff function is ui ðXÞ ¼ E½gi ðXÞ; X 2 G. Remark 1. In Raiffa (1970) and LaValle (1978), each agent is assumed to have a cardinal utility function of profit, and his objective is to maximize his expected utility. However, some preferences, such as the one in Example 2, cannot be represented by a cardinal utility function. A point a 2 RN is said to be Pareto-inferior to or Pareto-dominated by another point b 2 RN , if each component of a is no greater than the corresponding component of b and at least one component of a is less than the corresponding component of b. In other words, we say b is Pareto-superior to a or b Pareto-dominates a. A point is said to be a Pareto-optimal point of a subset of RN , if it is not Paretoinferior to any other point in the subset. With these concepts, we can now define Pareto-optimality of a sharing rule uðsÞ and an action pair ðs; uðsÞÞ. Definition 2. Given an external action s of the supply chain, u ðsÞ is a Paretooptimal sharing rule, if ðu1 ðP1 ðs; u ðsÞÞÞ; ; uN ðPN ðs; u ðsÞÞÞÞ is a Pareto-optimal point of the set fðu1 ðP1 ðs; uðsÞÞÞ; ; uN ðPN ðs; uðsÞÞÞÞ; u 2 Yg; where ui ðPi ðs; uðsÞÞÞ is the payoff of the ith agent. Definition 3. ðs ; u ðs ÞÞ is a Pareto-optimal action pair if the agents’ payoffs ðu1 ðP1 ðs ; u ðs ÞÞÞ; ; uN ðPN ðs ; u ðs ÞÞÞÞ is a Pareto-optimal point of the set fðu1 ðP1 ðs; uðsÞÞÞ; ; uN ðPN ðs; uðsÞÞÞÞ; ðs; uðsÞÞ 2 S Yg: Clearly if ðs ; u ðs ÞÞ is a Pareto-optimal action pair, then u ðs Þ is a Paretooptimal sharing rule given s . We begin now with an examination of the Pareto-optimal set in a supply chain consisting of risk-neutral agents. If an external action maximizes the supply chain’s expected profit, then it is not possible to make one agent get more expected profit without making another agent get less. More specifically, we have the following proposition.

10

X. Gan et al.

Proposition 1. If the agents in a supply chain are all risk neutral, then an action pair ðs; uðsÞÞ is Pareto-optimal if and only if the channel’s external action s maximizes the channel’s expected profit. Proof. The proof follows from the fact that in the risk-neutral case, for each s, X

ui ðPi ðs; uðsÞÞÞ ¼

X

EPi ðs; uðsÞÞ ¼ E

X

Pi ðs; uðsÞÞ ¼ EPðsÞ:

Thus, every ðs ; uðs ÞÞ 2 S Y is Pareto-optimal provided s maximizes EPðs Þ. □ Since agents in a supply chain maximize their respective objectives, the agents’ payoffs might not be Pareto-optimal if their objectives are not aligned properly. In this case, it is possible to improve the chain’s performance, i.e., achieve Paretosuperior payoffs. The agents can enter into an appropriately designed contract, under which their respective optimizing actions leads to a Pareto-superior payoff. In the supply chain management literature, a contract is defined to coordinate a supply chain consisting of risk-neutral agents if their respective optimizing external actions under the contract maximize the chain’s expected profit. Then, according to Proposition 1, a coordinating contract is equivalent to a Pareto-optimal action in the riskneutral case. It is therefore reasonable to use the notion of Pareto-optimality to define supply chain coordination in the general case. Definition 4. Supply Chain Coordination. A contract agreed upon by the agents of a supply chain is said to coordinate the supply chain if the optimizing actions of the agents under the contract 1. Satisfy each agent’s reservation payoff constraint. 2. Lead to an action pair ðs ; u ðs ÞÞ that is Pareto-optimal. Besides Pareto-optimality of a contract, we have introduced the individualrationality or the participation constraints as part of the definition of coordination. The constraints ensure that each agent is willing to participate in the contract by requiring that each gets at least his reservation payoff. It is clear that each agent’s reservation payoff will not be less than his status-quo payoff, which is defined to be his best payoff in the absence of the contract. Thus, we need consider only the subset of Pareto-optimal actions that satisfy these participating constraints. The reservation payoff of an agent plays an important role in bargaining, as we shall see in the next section. Now we illustrate the introduced concept of coordination by an example. Example 4. Consider a supply chain consisting of one supplier and one retailer who faces a newsvendor problem. Before the demand realizes, the supplier decides on his capacity first, and the retailer then prices the product and chooses an order quantity. The supplier and the retailer may enter into a contract that specifies the retailer’s committed order quantity and the supplier’s refund policy for returned items. In this channel, the external actions are the supplier’s capacity selection and the retailer’s pricing and ordering decisions. These are denoted as s. The internal

Coordination of Supply Chains with Risk-Averse Agents

11

actions include decision on the quantity of commitment, the refundable quantity, and the refund credit per item. These internal actions together lead to a sharing rule denoted by uðsÞ. Once the contract parameters are determined, the agents in the supply chain choose their respective external actions that maximize their respective payoffs. If ðs; uðsÞÞ satisfies the agents’ reservation payoffs and is Pareto-optimal, then the channel is coordinated by the contract. The definition of coordination proposed here allows agents to have any kind of preference that can be represented by a payoff function satisfying the complete and transitive axioms specified earlier. For example, all of the seven kinds of preferences listed in Schweitzer and Cachon (2000), including risk-seeking preferences, are allowed. Since often in practice, an agent is either risk neutral or risk averse, we restrict our attention to only these two types. Remark 2. Our definition applies also to a T-period case. For this, we define the payoff function of player i as ui ðP1i ðs ; u ðs ÞÞ; P2i ðs ; u ðs ÞÞ; ; PTi ðs ; u ðs ÞÞÞ : GT ! R1 ; where Pti ðs ; u ðs ÞÞ is agent i’s profit in period t.

4 Coordinating Supply Chains Each Pareto-optimal action pair ðs; uðsÞÞ results in a vector of payoffs ðu1 ðP1 ðs; uðsÞÞÞ; ; uN ðPN ðs; uðsÞÞÞÞ; where ui ðPi ðs; uðsÞÞÞ is the payoff of the ith agent. Let C ¼ fðu1 ðP1 ðs; uðsÞÞÞ; ; uN ðPN ðs; uðsÞÞÞÞjðs; uðsÞÞ is Pareto - optimal; ðs; uðsÞÞ 2 S Yg; denote the set of all Pareto-optimal payoffs, and let F C be the subset of Paretooptimal payoffs that satisfy all of the participation constraints. We shall refer to F as Pareto-optimal frontier. We will assume that F is not empty. To coordinate a supply chain, the first step is to obtain the Pareto-optimal frontier F. If F is not a singleton, then agents bargain to arrive at an element in F to which they agree. A coordinating contract is one with a specific set of parameters that achieves the selected solution. A contract is appealing if it has sufficient flexibility. In Cachon (2003), a coordinating contract is said to be flexible if the contract, by adjustment of some parameters, allows for any division of the supply chain’s expected profit among the risk-neutral agents. This concept can be extended to the general case as follows.

12

X. Gan et al.

Definition 5. A coordinating contract is flexible if, by adjustment of some parameters, the contract can lead to any point in F: We shall now develop coordinating contracts in supply chains consisting of two agents: a supplier and a retailer. We shall consider three different cases. In each of these cases, we assume that agents have complete information. In Case 1, the supplier is risk neutral and the retailer has a payoff function in Example 2, i.e., the retailer maximizes his expected profit subject to a downside constraint. In Case 2, the supplier and the retailer are both risk averse and each maximizes his own meanvariance trade-off. In Case 3, the supplier and the retailer are both risk averse and each maximizes his own expected concave utility. We consider the first two cases in this section and the third case in Sect. 5. In each case, let us denote the retailer’s and the supplier’s reservation payoffs as pr r0 and ps r0, respectively. We first obtain F and then design a flexible contract that can lead to any point in F by adjusting the parameters of the contract.

4.1

Case 1: Risk Neutral Supplier and Retailer Averse to Downside Risk

We consider the supplier to be risk neutral and the retailer to maximize his expected profit subject to a downside risk constraint. This downside risk constraint requires that the probability of the retailer’s profit to be higher than a specified level is not too small. The risk neutrality assumption on the part of the supplier is reasonable when he is able to diversify his risk by serving a number of independent retailers, which is quite often the case in practice. When the retailers are independent, the supply chain can be divided into a number of sub-chains, each consisting of one supplier and one retailer. This situation, therefore, could be studied as a supply chain consisting of one risk-neutral supplier and one risk-averse retailer. We say that an action pair ðs; uðsÞÞ is feasible if the pair satisfies the retailer’s downside risk constraint. We do not need to consider a pair ðs; uðsÞÞ that is not feasible since under the pair the retailer’s payoff is 1 and he would not enter the contract. We denote PðsÞ, Pr ðs; uðsÞÞ, and Ps ðs; uðsÞÞ as the profits of the supply chain, the retailer, and the supplier, respectively. Other quantities of interest will be subscripted in the same way throughout the chapter, i.e., subscript r will denote the retailer and subscript s will denote the supplier. Then we have the following result. Theorem 1. If the supplier is risk neutral and the retailer maximizes his expected profit subject to a downside risk constraint, then a feasible action pair ðs; uðsÞÞ is Pareto-optimal if and only if the supply chain’s expected profit is maximized over the feasible set. Proof. ONLY IF: It is sufficient to show that if EPðsÞ is not maximal over the feasible set, then ðs; uðsÞÞ is not Pareto-optimal.

Coordination of Supply Chains with Risk-Averse Agents

13

If EPðsÞ is not the maximal channel profit, then there exists an s0 such that EPðs0 Þ>EPðsÞ. Consider the pair ðs0 ; u0 ðs0 ÞÞ in which Pr ðs0 ; u0 ðs0 ÞÞ ¼ Pr ðs; uðsÞÞ and Ps ðs0 ; u0 ðs0 ÞÞ ¼ Pðs0 Þ Pr ðs; uðsÞÞ, we then get ur ðPr ðs0 ; u0 ðs0 ÞÞÞ ¼ EPr ðs; uðsÞ) and us ðPs ðs0 ; u0 ðs0 ÞÞÞ ¼ EPðs0 Þ EPr ðs; uðsÞÞ: We can see that ur ðPr ðs0 ; u0 ðs0 ÞÞÞ ¼ ur ðPr ðs; uðsÞÞ) and us ðPs ðs0 ; u0 ðs0 ÞÞÞ>us ðPs ðs; uðsÞÞÞ: This means that ðs; uðsÞÞ is Pareto-inferior to ðs0 ; u0 ðs0 ÞÞ, which contradicts with the Pareto-optimality of ðs; uðsÞÞ. IF: Suppose the supply chain’s expected profit is maximized. If ðs; uðsÞÞ is not Pareto-optimal, then according to the definition of a Pareto-optimal action pair, there exists a feasible pair ðs00 ; u0 ðs00 ÞÞ that is Pareto-superior to ðs; uÞ. Since it is Paretosuperior to ðs; uðsÞÞ, it is also feasible. Thus, us ðPs ðs00 ; u00 ðs00 ÞÞÞ þ ur ðPr ðs00 ; u0 ðs00 ÞÞÞ ¼ EPðs00 Þ>EPðsÞ ¼ ur ðPr ðs; uðsÞÞÞ þ us ðPs ðs; uðsÞÞÞ; which contradicts the fact that EPðsÞ is the maximal expected channel profit. □ Let s be an action of the channel that maximizes the channel’s expected profit, and let EPr ðs Þ be the retailer’s payoff. Since the retailer’s and the supplier’s reservation payoffs are pr and ps , respectively, we must impose the participating constraints of the agents on the solutions in C. Thus, EPr ðs Þrpr and EPðs Þ EPr ðs Þrps :

(1)

Together with Theorem 1, we get F ¼ fðEPr ðs Þ; EPðs Þ EPr ðs ÞÞjEPðs Þ ps rEPr ðs Þrpr g: Clearly, if EPðs Þ ps rpr , then F is not empty. In Gan et al. (2005), we show that a retailer, who is subject to a downside risk constraint, may order a lower quantity from a supplier than that desired by the channel under a wholesale, buy-back or revenue-sharing contract. Based on an initial contract, a risk-sharing contract is designed, which stipulates the supplier to offer a full refund on unsold items up to a limited quantity. The contract coordinates the supply chain, and requires that both the supplier and the retailer share the risk. Another coordinating contract is possible when EPr ðs Þ exceeds the retailer’s target profit a, where s is the channel’s optimal action. In this case, a contract that provides a payoff of EPr ðs Þ to the retailer and remainder to the supplier coordinates the supply chain. This contract is of two-part tariffs type as defined, for example, in Chopra and Meindl (2001, p. 160). However, if EPr ðs Þ is less than the retailer’s target profit a, then the contract does not work since the downside risk constraint of the retailer is not satisfied. But the risk-sharing contract in

14

X. Gan et al.

Gan et al. (2005) still works, since the retailer’s downside risk constraint P ðXbaÞbb is always satisfied under that contract.

4.2

Case 2: Mean-Variance Suppliers and Retailers

In this case, both the supplier and the retailer maximize their respective meanvariance trade-offs. First we consider a two-agent scenario and then extend it to the case of N agents. Let the retailer’s payoff function be EPr ðs; uðsÞÞ lr VðPr ðs; uðsÞÞÞ;

(2)

and the supplier’s payoff function be EPs ðs; uðsÞÞ ls VðPs ðs; uðsÞÞÞ:

(3)

We first find all Pareto-optimal sharing rules for any given channel’s external action s. We show that regardless of the selected external action s, the optimal sharing rule has the same specific form. Under this form of a sharing rule, we obtain optimal external actions. This procedure results in a Pareto-optimal ðs; uðsÞÞ. We now solve for the Pareto-optimal set for a supply chain consisting of N agents, and then specialize it for supply chains with two agents. We assume that the ith agent’s payoff function is EPi ðs; uðsÞÞ li VðPi ðs; uðsÞÞÞ:

(4)

To obtain Pareto-optimal sharing rules, we solve max u2Y

s.t: X

X

EPi ðs; uðsÞÞ

i

X

li VðPi ðs; uðsÞÞÞ;

(5)

i

Pi ðs; uðsÞÞ ¼ PðsÞ:

(6)

i

The solution of this problem is given in the following proposition. Proposition 2. A sharing rule u is a solution of the problem (5)–(6) if and only if 1=li PðsÞ þ pi ; i ¼ 1; . . . N; Pi ðs; uðsÞÞ ¼ P j 1==lj almost surely, where

P i

pi ¼ 0.

(7)

Coordination of Supply Chains with Risk-Averse Agents

Proof. Because

P i

15

EPi ðs; uðsÞÞ ¼ EPðsÞ, the problem is equivalent to X

min u2Y

s.t: X

li VðPi ðs; uðsÞÞÞ;

(8)

i

Pi ðs; uðsÞÞ ¼ PðsÞ:

(9)

i

It is easy to see that X

li VðPi ðs; uðsÞÞÞ

i

¼

X

h i X 2 E PðsÞ PðsÞ P ðs; uðsÞÞ i i j 1=lj

li VðPi ðs; uðsÞÞÞ þ P

i

" # X 1 1=li ¼P VðPðs; uðsÞÞÞ þ li V Pi ðs; uðsÞÞ P PðsÞ : j 1=lj j 1=lj i

(10)

Since the second term on the RHS of (10) is nonnegative, we have shown that X

1 VðPðs; uðsÞÞÞ j 1=lj

li VðPi ðs; uðsÞÞÞr P

i

(11)

for any feasible Pi ðs; uðsÞÞ; i ¼ 1; . . . N. Thus, P 11=l VðPðs; uðsÞÞÞ provides a j

j

lower bound for the objective function (8). Note that a u satisfies (7) if and only if 1=li PðsÞ ¼ 0; i ¼ 1; . . . N: Pi ðs; uðsÞÞ P j 1=lj This means that X

li VðPi ðs; uðsÞÞÞ ¼ P

i

1 VðPðsÞÞ j 1=lj

(12)

and X

1 VðPðsÞÞ j 1=lj

li VðPi ðs; uðsÞÞÞ> P

i

for any u not satisfying (7).

□

16

X. Gan et al.

For any optimal sharing rule u given in Proposition 2, the sum of the agents’ payoffs equals 1 VðPðsÞÞ: j 1=lj

EPðsÞ P

By adjusting p0 , the sharing rule u allows for any division of the total payoff among the agents. Therefore, an optimal external action, given u , has to maximize the total payoff, i.e., it must be an action pair of "

# 1 VðPðsÞÞ : max EPðsÞ P s2S j 1=lj

(13)

Next we characterize the set of Pareto-optimal actions by summarizing the results we have got. Theorem 2. An action pair ðs ; u Þ is Pareto-optimal if and only if "

#

1 VðPðsÞÞ s ¼ arg max EPðsÞ P s2S j 1=lj

(14)

1=li PðsÞ þ pi ; i ¼ 1; . . . N; Pi ðs; u ðsÞÞ ¼ P j 1=lj

(15)

and

almost surely. Clearly, if a contract can allocate the channel profit among the N agents proportionally, then the contract along with a side payment scheme can coordinate the supply chain. Moreover, this contract is flexible by adjusting the amounts of side payment. Theorem 3, as a special result of Theorem 2, characterizes the set of Paretooptimal actions for supply chains consisting of one supplier and one retailer. Theorem 3. An action pair ðs ; u ðs ÞÞ is Pareto-optimal if and only if lr ls s ¼ arg max EPðsÞ VðPðsÞÞ s2S lr þ ls

(16)

and Pr ðs ; u ðs ÞÞ ¼

ls Pðs Þ þ p0 ; lr þ ls

(17)

Coordination of Supply Chains with Risk-Averse Agents

Ps ðs ; u ðs ÞÞ ¼

lr Pðs Þ p0 ; lr þ ls

17

(18)

almost surely. It follows from Theorem 3 that under any Pareto-optimal solution, the retailer gets a fixed proportion ls =ðlr þ ls Þ of the channel profit plus p0 and the supplier gets the remaining profit, i.e., lr =ðlr þ ls Þ of the channel profit minus p0 . If lr >ls , i.e., the retailer is more risk-averse than the supplier, then the supplier takes a greater proportion of the channel profit. In other words, the agent with a lower risk aversion takes a higher proportion of the total channel profit than the other one does. The side payment, which is determined by the respective bargaining powers of the agents, determines the agents’ final payoffs. According to Theorem 3, C¼

ls lr 1 uðs Þ þ p0 ; uðs Þ p0 jp0 2 R ; lr þ l s lr þ l s

where uðs Þ represents EPðs Þ lVðPðs ÞÞ: Since the retailer’s and the supplier’s reservation payoffs are pr and ps , respectively, p0 has to satisfy the participating constraints of the agents. Thus, ls lr uðs Þ þ p0 rpr and uðs Þ p0 rps : lr þ ls lr þ l s Then F can be represented by

lr ls lr ls uðs Þ þ p0 ; uðs Þ p0

uðs Þ ps rp0 rpr uðs Þ : l r þ ls lr þ l s lr þ l s l r þ ls

Furthermore, if lr ls uðs Þ ps rpr uðs Þ; lr þ ls lr þ l s i.e., if pr þ ps buðs Þ; then F is not empty. The problem considered thus far is quite general, in the sense that the external action s is rather an abstract one that can include such decisions as order quantity, item price, etc. We next consider a special case. Here the retailer faces a newsvendor problem and makes a single purchase order of a product from the supplier at the beginning of a period, who in turn produces and delivers the order to the retailer before the selling season commences. Let p denote the price per unit,

18

X. Gan et al.

c the supplier’s production cost, v the salvage value, and q the retailer’s order quantity. In this problem, the supply chain’s external action is the retailer’s order quantity q. According to Theorem 3, the coordinating contract should allocate the profit in the same proportion for every realization of the channel profit in the absence of any side payment. We shall call such a sharing rule a proportional sharing rule. Here we only examine buy-back and revenue-sharing contracts. With a buy-back contract, the supplier charges the retailer a wholesale price per unit, but he pays the retailer a credit for every unsold unit at the end of the season. With a revenuesharing contract, the supplier charges a wholesale price per unit purchased, and the retailer gives the supplier a percentage of his revenue. See Pasternack (1985) and Cachon and Lariviere (2005) for details on these contracts. In the following, we see that both buy-back and revenue-sharing contracts allocate the channel profit proportionally. Proposition 3. A revenue-sharing contract allocates the channel profit (a random variable) proportionally. If w ¼ fc;

(19)

the retailer’s share is f and the supplier’s share is 1 f. Proof. Let D denote the demand faced by the retailer. Then the supply chain’s profit is

pD þ ðq DÞv cq pq cq;

if Dbq; if D>q:

(20)

On the other hand, the retailer’s profit is

fpD þ fðq DÞv wq if Dbq; fpq wq; if D>q:

(21)

By using w ¼ fc into (21), we can see that the retailer gets the proportion f of the supply chain’s profit for every realization of the demand. □ Cachon and Lariviere (2005) prove that for each coordinating revenue-sharing contract, there exists a unique buy-back contract that provides the same profit as in the revenue-sharing contract for every demand realization. They show that the buyback contract’s parameters have the form b ¼ ð1 fÞðp vÞ;

(22)

w ¼ pð1 fÞ þ fc;

(23)

where b is the refund to the retailer for each unsold unit, and f is the retailer’s share of the channel profit in the revenue-sharing contract. It is easy to see that the same result holds here as well.

Coordination of Supply Chains with Risk-Averse Agents

19

Proposition 4. A buy-back contract allocates the channel profit (a random variable) proportionally. If the contract parameters satisfy (22) and (23), then the retailer’s fractional share is f and the supplier’s is 1 f under this contract. Nagarajan and Bassok (2008) obtain the Nash Bargaining solution in the riskneutral case. According to their results, if both the retailer and the supplier are risk neutral, the retailer’s share of profit is fPðsÞ ¼ ½PðsÞ ps þ pr =2:

(24)

Under a buy-back or a revenue-sharing contract, the retailer’s problem is max p0 þ s2S

ls ½EPðsÞ lV ðPðsÞÞ: lr þ l s

(25)

For a given p0 , the retailer’s problem (25) becomes max EPðsÞ lVðPðsÞÞ; s2S

(26)

which has been solved by Lau (1980) and Chen and Federgruen (2000). Since the solution is s , the retailer would choose the optimal external action voluntarily. So we can state the following two theorems. Theorem 4. If the parameters of a revenue-sharing contract satisfy w¼

ls c; lr þ l s

(27)

then the revenue-sharing contract along with a side payment p0 to the retailer coordinates the supply chain. The profit allocation is given in (17)–(18). Theorem 5. If the parameters of a buy-back contract satisfy b¼ w¼p

lr ðp vÞ; lr þ ls

lr ls þ c; lr þ ls lr þ ls

(28)

(29)

then the buy-back contract along with a side payment p0 to the retailer coordinates the supply chain. The profit allocation is given in (17)–(18). Note that by adjusting the side payment p0 , the revenue-sharing as well as the buy-back contract can lead to any point in F. Thus, both contracts are flexible. The contracts obtained in Theorems 5 and 4, when lr ¼ ls ¼ 0, reduce to the standard contracts obtained in the risk-neutral case, because the fraction lr =ðlr þ ls Þ can take any value in ½0; 1. In particular, if the supplier is risk neutral and the retailer is risk averse, i.e., ls ¼ 0, the fraction lr =ðlr þ ls Þ ¼ 1, which

20

X. Gan et al.

means that the supplier takes the entire channel profit and gives a side payment to the retailer. In this case, it is Pareto-optimal for the supplier to bear all of the risk. Since the retailer’s profit is a side payment from the supplier, the supplier’s expected profit is the channel’s profit minus that payment. Therefore, the supplier’s payoff is maximized when the channel’s expected profit is maximized. Thus, we have a coordinating contract under which the supplier and the retailer execute s , the retailer gets a constant profit p0 , and the supplier gets the remaining profit. In the following example, we design a coordinating contract according to Theorem 5. We also obtain the optimal ordering quantity and determine the required side payment. Example 5. Consider a supply chain consisting of one retailer and one supplier. The retailer faces a newsvendor problem and makes a single purchase order of a product from the supplier at the beginning of a period, who in turn produces and delivers the order to the retailer before the selling season. Suppose that the demand D is uniformly distributed on some interval, which without loss of generality, can be taken as interval [0,1]. Thus, the distribution function FðxÞ ¼ x for 0bxb1 and FðxÞ ¼ 1 for xr1. We confine the ordering quantity q to be in [0,1]. Let the unit price p be 100, the supplier’s production cost c be 60, and the salvage value v be 20. Let the retailer’s and the supplier’s payoff functions be, respectively, EPr lr VðPr Þ and EPs ls VðPs Þ;

(30)

where lr ¼ 0:05 and ls ¼ 0:01. We assume that the agents have equal bargaining powers in the sense that their payoffs are equal. According to Theorem 3, the retailer’s payoff is ls lr ls EPðqÞ VðPðqÞÞ þ p0 ; lr þ ls lr þ ls

(31)

where PðqÞ is the channel’s profit when the retailer’s ordering quantity is q, 0bqb1. Thus, the retailer’s optimal order quantity is a q that maximizes (31). From Chen and Federgruen (2000), we have EPðqÞ ¼ 40q 80q2 and VðPðqÞÞ ¼ 6400ðq3 =3 q4 =4Þ:

(32)

With this, the retailer’s problem is 160 3 40 4 q þ q ; max 40q 80q2 0b qb 1 9 3

(33)

and q ¼ 0:236. According to Theorem 5, the retailer’s and the supplier’s payoffs are 0:799 þ p0 and 3:99 p0 , respectively. It is easy to see that p0 ¼ 1:598 equalizes their payoffs, as has been assumed.

Coordination of Supply Chains with Risk-Averse Agents

21

5 Coordinating a Supply Chain Consisting of Agents with Concave Utility Functions In this case, we assume that agent i has an increasing concave utility function gi ðÞ of his profit, and wants to maximize his expected utility, i ¼ r; s. Then his payoff function is E½gi ðÞ . To compute the set of Pareto-optimal actions, we first find the Pareto-optimal sharing rules given an external action s. According to the group decision theory literature (Wilson 1968; Raiffa 1970), the problem can be formulated as follows: max ar Egr ðPr ðs; uðsÞÞÞ þ as Egs ðPs ðs; uðsÞÞÞ;

(34)

s.t: Pr ðs; uðsÞÞ þ Ps ðs; uðsÞÞ ¼ PðsÞ;

(35)

u2Y

where ar ; as >0, ar þ as ¼ 1. The specification of ðar ; as Þ is derived from their respective bargaining powers. By varying ar and as , we can get all possible Pareto-optimal sharing rules Cs , denoted as fður ðPr ðs; uðsÞÞÞ; us ðPs ðs; uðsÞÞÞÞju is Pareto - optimal; u 2 Yg:

(36)

Clearly, each point in Cs represents, given s, the agents’ payoffs under a Paretooptimal sharing rule. Then we can get C, which is the set of Pareto-optimal points S of the set s2S Cs . According to Definition 3, any action pair that leads to a point in C is Pareto-optimal. It is well known that the problem of maximizing the expected quadratic utility can be reduced to one of maximizing a mean-variance trade-off. Therefore, when both agents’ utility functions are quadratic, we can coordinate the channel with the contracts developed in Sect. 4.2. Levi and Markowitz (1979) show that a utility function exhibiting constant risk aversion, particularly of the form log x or xa ; 0

5.1

Characterizing the Pareto-Optimal Set

Let the retailer’s and the supplier’s utility functions be, respectively,

22

X. Gan et al.

gr ðxÞ ¼ 1 elr x and gs ðxÞ ¼ 1 els x :

(37)

Then the absolute risk aversion measure for the retailer and the supplier are lr and ls , respectively; see Pratt (1964). We want to find a Pareto-optimal sharing rule for any given channel’s external action s. Raiffa (1970) solves the problem (34)–(35), which implies the following result. Proposition 5. For a given external action s for the channel under consideration, a sharing rule u ðsÞ is a Pareto-optimal sharing rule if and only if Pr ðs; u ðsÞÞ ¼

ls as ls PðsÞ l ln ; lr þ ls ar lr

(38)

Ps ðs; u ðsÞÞ ¼

lr as ls PðsÞ þ l ln ; lr þ ls ar lr

(39)

1 almost surely, where ar ; as >0, ar þ as ¼ 1, and l ¼ lr þl . s Thus we can get

Cs ¼ fður ðPr ðs; u ðsÞÞÞ; us ðPs ðs; u ðsÞÞÞÞjar ; as >0; ar þ as ¼ 1g; where lr as ls lr ls PðsÞ E exp ; ln ur ðPr ðs; u ðsÞÞÞ ¼ 1 exp lr þ ls ar lr lr þ ls

ls ar ls lr ls PðsÞ us ðPs ðs; u ðsÞÞÞ ¼ 1 exp ln E exp : lr þ ls as lr lr þ ls

(40)

(41)

Since both the retailer and the supplier’s payoff functions decrease with lr ls PðsÞ ; E exp lr þ ls it is easy to check that C ¼ fður ðPr ðs ; u ÞÞ; us ðPs ðs; u ÞÞÞjar ; as >0; ar þ as ¼ 1g;

(42)

where s is the solution of the problem lr ls PðsÞ : min E exp s2S l r þ ls

(43)

Coordination of Supply Chains with Risk-Averse Agents

23

Now the supply chain’s external problem has been transformed to problem (43). This problem has been studied in the literature in some special situations. Bouakiz and Sobel (1992) have shown that a base-stock policy is optimal in a multi-period newsvendor problem, when the newsvendor has an exponential utility function. Eeckhoudt et al. (1995) discuss the situation in which the entity faces a newsvendor problem, and they prove that the newsvendor orders less than that in the risk-neutral case. Agrawal and Seshadri (2000a) consider the entity’s price and inventory decision jointly in a newsvendor framework. Remark 3. Although we have got proportional sharing rules for the above case and the second case in Sect. 4, the Pareto-optimal sharing rules usually are not proportional for any two utility functions (Raiffa 1970). Moreover, the Pareto-sharing rules may depend on the channel’s external action. See Wilson (1968), Raiffa (1970), and LaValle (1978) for further details on Pareto-optimal sharing rules. Now we summarize the results in the following theorem. Theorem 6. An action pair ðs ; u ðs ÞÞ is Pareto-optimal if and only if lr ls PðsÞ ; s ¼ arg minE exp s2S lr þ ls

(44)

Pr ðs ; u ðs ÞÞ ¼

ls as ls Pðs Þ l ln ; lr þ ls ar lr

(45)

Ps ðs ; u ðs ÞÞ ¼

lr as ls Pðs Þ þ l ln ; lr þ ls ar lr

(46)

almost surely, where ar ; as >0, ar þ as ¼ 1. It follows from Theorem 6 that under any Pareto-optimal solution, the retailer and the supplier get fixed proportions of channel profit minus/plus a side payment. If lr >ls , i.e., if the retailer is more risk averse than the supplier, then the supplier takes a greater proportion of the channel profit if we ignore the side payment.

5.2

Bargaining Issue

We have got Pareto-optimal payoffs set C in (42). Since ur <1 and us <1 from (40) and (41), we assume that pr <1 and ps <1. Participating constraints of the agents are ur ðPr ðs ; u ðs ÞÞÞrpr and us ðPr ðs ; u ðs ÞÞÞrps :

(47)

Conditions (47) are equivalent to

lr lr ls Pðs Þ as ls lr þls l l Pðs Þ ð1 pr Þ=E exp =ð1 ps Þ: r rE exp r s ar lr lr þ ls lr þ ls (48)

24

X. Gan et al.

Then F can be represented by fður ðPr ðs ;u ðs ÞÞÞ;us ðPs ðs ;u ðs ÞÞÞÞjar ;as >0;ar þ as ¼ 1; and ð48Þ is satisfied:g: (49)

Pðs Þ If ð1 pr Þ=E exp lrllrsþl s

Pðs Þ =ð1 ps Þ, i.e., if rE exp lrllrsþl s

ð1 pr Þð1 ps Þr

lr ls Pðs Þ 2 E exp ; lr þ ls

then F is not empty. Nagarajan and Bassok (2008) use Nash Bargaining concept to deal with bargaining issue in the risk-neutral case. Here we use the same concept to deal with the bargaining issue in a risk-averse case. The approach of Nash (1950) requires that a bargaining solution satisfy the following eight axioms. 1. An agent offered two possible anticipations can decide which is preferable or that they are equally desirable. 2. The ordering thus produced is transitive; if A is better than B and B is better than C then A is better than C. 3. Any probability combination of equally desirable states is just as desirable as either. 4. If A, B, and C, are as in Axiom 2, then there is a probability combination of A and C which is just as desirable as B. This amounts to an assumption of continuity. 5. If 0bpb1 and A and B are equally desirable, then pA þ ð1 pÞC and pB þ ð1 pÞC are equally desirable. Also, A may be substituted for B in any desirability ordering relationship satisfied by B. 6. Let S, which is compact and convex, be the agents’ payoffs set and let cðSÞ denote the bargaining solution point in this set. If a is a point in S such that there exists another point b in S with the property ur ðbÞ>ur ðaÞ and us ðbÞ>us ðaÞ, then a2 = cðSÞ. 7. If the set T contains the set S and cðTÞ is in S, then cðTÞ ¼ cðSÞ: 8. If S is symmetrical with respect to the line ur ¼ us , and ur and us display this, then cðSÞ is a point on the line ur ¼ us . Clearly, exponential utilities in (37) satisfy the first five axioms. The agents’ payoff set S is fður ðPr ðs; uðsÞÞÞ; us ðPs ðs; uðsÞÞÞÞjðs; uðsÞÞ 2 S Yg: We assume that PðsÞ be continuous in s, so that S is compact. Now we prove the convexity of S by showing that its frontier is a concave curve. The Pareto-frontier of this set is given in (49), where

Coordination of Supply Chains with Risk-Averse Agents

25

lr as ls lr ls Pðs Þ E exp ; ur ðPr ðs ; u ðs ÞÞÞ ¼ 1 exp ln lr þ ls ar lr lr þ ls

us ðPs ðs ; u ðs ÞÞÞ ¼ 1 exp

ls ar lr lr ls Pðs Þ E exp : ln lr þ ls as ls lr þ ls

(50)

(51)

From (50) and (51), lr þls lr

ð1 u r Þ

ð1 us Þ

lr þls ls

¼

lr ls Pðs Þ E exp lr þ ls

ðlr þls Þ2 =ðlr ls Þ3

:

(52)

Clearly, the curve represented by (52) is concave since the right side is a constant. Axiom 6 assures that the solution is Pareto-optimal, Axiom 8 expresses the equality of bargaining skills. Nash (1950) shows that the solution point is the point that is the solution of the problem max ður pr Þðus ps Þ:

(53)

ður ;us Þ2S

Since the solution has to be Pareto-optimal, (53) is equivalent to max ður pr Þðus ps Þ:

(54)

ður ;us Þ2F

Pðs Þ : Then, Now we solve the problem (54). Let EðuÞ represent E exp lrllrsþl s ður pr Þðus ps Þ ¼ ð1 pr Þð1 ps Þ þ ½EðuÞ2 " # ls lr as ls lr þls as ls lr þls ð 1 pr Þ EðuÞ þ ð1 ps Þ EðuÞ : ar lr ar l r Thus, problem (54) is equivalent to min ð1 pr Þ a1 ;a2

as ls ar lr

llþls r

s

EðuÞ þ ð1 ps Þ

as ls ar lr

l lþlr r

s

EðuÞ:

(55)

Thus, when as ls ð1 pr Þls ¼ ; ar lr ð1 ps Þlr

(56)

26

X. Gan et al.

ður pr Þðus ps Þ is maximized. Therefore, the Nash Bargaining solution is represented by the payoffs given in (50) and (51) with parameters ar; as that satisfy ar >0; as >0, ar þ as ¼ 1; and (56). From (56), we can see that as ls =ar lr decreases in the retailer’s reservation payoff pr and increases in the supplier’s reservation payoff ps . Thus, the retailer’s payoff increases in pr and decreases in ps . The same property holds for the supplier. These properties imply that in the Nash Bargaining solution, each agent’s payoff increases with his own reservation payoff and decreases with the other agent’s reservation payoff. When pr ¼ ps , l lnðas ls =ar lr Þ ¼ 0, the side payment disappears. Finally, we can see that the Nash bargaining solution assigns a player a higher payoff when the other player becomes more risk averse.

5.3

Designing a Coordinating Contract

For the special supply chain considered in Sect. 4.2, we can use either a buy-back or a revenue-sharing contract to allocate the channel profit. Under either of these contracts, the retailer’s problem is: lr ls PðsÞ max 1 E exp : s2S lr þ ls This problem is equivalent to problem (43), which implies that the retailer would voluntarily choose the optimal external action s . So we have the following two results. Theorem 7. If the parameters of a buy-back contract satisfy b¼ w¼p

lr ðp vÞ; lr þ ls

lr ls þ c; lr þ ls lr þ ls

then the buy-back contract along with the side payment l lnðas ls =ar lr Þ to the supplier coordinates the supply chain. The profit allocation is given in (45)–(46). Theorem 8. If the parameters of a revenue-sharing contract satisfy w¼

ls c; lr þ l s

then the revenue-sharing contract along with the side payment l lnðas ls =ar lr Þ to the supplier coordinates the supply chain. The profit allocation is given in (45)–(46).

Coordination of Supply Chains with Risk-Averse Agents

27

Note that if ar and as satisfy condition (56), then both the revenue-sharing and the buy-back contracts achieve the Nash Bargaining solution. By adjusting the bargaining coefficients ar and as , one can attain any point in F. Thus, both these contracts are flexible. We should note that in general, Pareto-optimal sharing rules are not proportional as in the case with exponential utility functions. Wilson (1968) provides a necessary and sufficient condition for Pareto-optimality of a sharing rule in a channel with N agents. The condition is stated in the following theorem. Theorem 9. Given an external action s, a necessary and sufficient condition for Pareto-optimality of a sharing rule is that there exists nonnegative weights a ¼ ða1 ; a2 ; . . . ; aN Þ and a function m : R1 ! R1 ; such that X

Pi ðs; uðsÞÞ ¼ PðsÞ;

(57)

i

almost surely, and for each i ai g0i ðPi ðs; uðsÞÞÞ ¼ mðPðsÞÞ;

(58)

almost surely. In what follows, we give an example in which the sharing rule is not of the form of (45)–(46). Here we see that the Pareto-optimal sharing rule depends on the realized channel profit, i.e., it depends on the chosen external action as well as the realized random event. Example 6. Let gr ðxÞ ¼ 1 elr x ; gs ðxÞ ¼

1 els x ; xbx0 ; 1 ex0 ls þ l1s ex0 ls ðx x0 Þ; x>x0 :

(59) (60)

In this example, the retailer’s utility function is the same as in (37), but the supplier’s utility is changed in a way that his risk attitude is the same as in (37) at low profit levels and he is risk neutral at higher profit levels. Proposition 6. For a given external action s for the channel under consideration, a sharing rule u is a Pareto-optimal sharing rule if and only if as ls s PðsÞ l ln aars llsr ; PðsÞb lrlþl x l ln 0 ar lr r ; Pr ðs; u ðsÞÞ ¼ lr þls as ls : ls x0 l ln as ls ; PðsÞ> x l ln 0 lr ar lr lr ar lr 8 <

Ps ðs; u ðsÞÞ ¼

8 <

ls lr þls

lr lr þls PðsÞ

þ l ln aars llsr ;

: PðsÞ ls x0 þ l ln as ls ; lr

ar lr

as ls s PðsÞb lrlþl x l ln 0 ar lr r ; lr þls PðsÞ> lr x0 l ln aars llsr

(61)

(62)

28

X. Gan et al.

Fig. 1 Example of a Paretooptimal frontier

u2 Pareto-optimal Frontier

Ψs5 Ψs4

Ψs3

r2

Ψs1

Ψs2

r1

u1

1 almost surely, where ar ; as >0, ar þ as ¼ 1, and l ¼ lr þl . s Proof. Let

8 lr þls as ls < ðar lr Þlls ðas ls Þllr exp lr ls t ; t b x l ln 0 lr ar lr ; h lr þl s i mðtÞ ¼ : ðar lr Þlls ðas ls Þllr exp ls x0 l ln as ls ; t> lr þls x0 l ln as ls : ar lr lr ar lr Then, according to Theorem 9, (61) and (62) are Pareto-optimal since conditions (57) and (58) are satisfied. □ We can see that under the Pareto-optimal sharing rule, the retailer’s profit increases linearly with the channel’s realized profit when the latter is below a certain level, and remains unchanged thereafter. This is not a proportional sharing rule, and consequently, neither a buy-back nor a revenue-sharing contract along with side payments would coordinate the channel. It appears that new contract forms need to be designed to achieve coordination in such cases. In order to obtain s , we outline the following procedure. First, we compute Cs for each S s 2 S according to (36), and then we find the Pareto-optimal frontier of the set s2S Cs . Any action pair ðs ; u ðs ÞÞ that leads to a point on this frontier is Pareto-optimal. Note that s may not be unique. To illustrate this procedure, let us assume S ¼ fs1 ; s2 ; s3 ; s4 ; s5 g for convenience in exposition. Suppose that the sets Cs for s 2 S are as shown in Fig. 1. Then the frontier consisting of Pareto-optimal solutions is shown as the bold-faced boundary in the figure. Construction of such a frontier in general would require development of numerical procedures. This is not the focus of this chapter, and it is a topic for future research.

6 Discussion One of our main findings is that in any Pareto-optimal joint action, the retailer and the supplier must share the risk appropriately. Specifically, the less risk averse an agent is, the more risk he assumes by taking a larger portion of the channel’s

Coordination of Supply Chains with Risk-Averse Agents

29

random profit. The agents’ final payoffs can be adjusted by a side payment depending on their respective bargaining powers. In the extreme case when one of the agents is risk neutral, then that agent may assume all of the risk. Owing to the risk-sharing effect, the supply chain, when considered as a single entity, is less risk averse than either the risk-averse retailer or the risk-averse supplier if considered as the single owner of the whole channel. For example, in Case 2, the channel’s problem according to (25) is equivalent to solving the problem max ½EðPðsÞÞ lVðPðsÞÞ: s2S

If the retailer or the supplier were to own the channel, he would solve the problem max ½EPðsÞ lr VðPðsÞÞ or max ½EPðsÞ ls VðPðsÞÞ: s2S

s2S

Since l

N X i¼1

ai ui ðPðsÞÞ; ai 2 ½0; 1;

N X

ai ¼ 1;

i¼1

where the agent i’s utility function is denoted as ui ðPðsÞÞ; i ¼ 1; 2; . . . N. In this case, each agent’s identity is not preserved and the critical issue is how to determine the weights ai ; i ¼ 1; 2; . . . N. Once the optimal action for the supply chain is obtained by maximizing uðPðsÞÞ; the profit could be allocated according to some weighting scheme, possibly different from ai ; i ¼ 1; 2; . . . N: It is clear that this method generalizes the risk-neutral case. However, we do not follow this method because it does not guarantee Pareto-optimality of the final outcome.

30

X. Gan et al.

7 Conclusion and Further Research We have proposed a definition of coordination of a supply chain consisting of riskaverse agents. We show that to coordinate such a chain, the first step is to characterize the set of Pareto-optimal solutions and select a solution from this set based on the agents’ respective bargaining powers. The second step is to design a contract to achieve the selected solution. In the risk-neutral case, it is easy to see that an action pair is Pareto-optimal if and only if the supply chain’s expected profit is maximized. But in the risk-averse case, it is more difficult to find Pareto-optimal actions. We characterize Paretooptimal solutions in three specific cases of a supply chain involving one supplier and one retailer, and in each case we design a flexible contract to coordinate the channel. Furthermore, we discuss the bargaining issue in one of the cases. We provide answers to the following questions: What is the optimal external action of the supply chain and what is the optimal sharing rule? In the specific cases that we have considered, we are able to obtain Paretooptimal actions by first obtaining a Pareto-optimal sharing rule that can be used with any external action. This property allows us to obtain an objective function for the supply chain, whose optimization yields an external action, which together with the sharing rule provides us with Pareto-optimal solutions. In more general cases, however, we do not have the above property, and therefore, the sequential procedure used in the special cases does not work. In such cases, we show by a specially constructed example, that obtaining Paretooptimal solutions requires finding first the Pareto-optimal sets corresponding to external actions, and then identifying the Pareto-optimal frontier of the union of these sets. Moreover, the standard contract forms that work for risk-neutral cases do no longer coordinate, and research is required to find new coordinating contract forms.

References Agrawal V, Seshadri S (2000a) Impact of uncertainty and risk aversion on price and order quantity in the newsvendor problem. Manuf Serv Oper Manage 4:410–423 Agrawal V, Seshadri S (2000b) Risk intermediation in supply chains. IIE Trans 32:819–831 Arrow KJ (1951) Social choice and individual values. Wiley, New York Bouakiz M, Sobel MJ (1992) Inventory control with an expected utility criterion. Oper Res 40:603–608 Buzacott J, Yan H, Zhang H (2002) Optimality criteria and risk analysis in inventory models with demand forecast updating. Working paper, The Chinese University of Hong Kong, Shatin Cachon GP (2003) Supply coordination with contracts. In: Kok T, Graves S (eds) Handbooks in operations research and management science. North-Holland, Amsterdam Cachon GP, Lariviere M (2005) Supply chain coordination with revenue-sharing contracts: strengths and limitations. Manage Sci 51:30–44

Coordination of Supply Chains with Risk-Averse Agents

31

Chen F, Federgruen A (2000) Mean-variance analysis of basic inventory models. Working paper, Columbia University, New York Chen X, Sim M, Simchi-Levi D, Sun P (2007) Risk aversion in inventory management. Oper Res 55:828–842 Chopra S, Meindl P (2001) Supply chain management. Prentice-Hall, New Jersey, NJ Eeckhoudt L, Gollier C, Schlesinger H (1995) The risk averse (and prudent) newsboy. Manage Sci 41:786–794 Eliashberg J, Winkler RL (1981) Risk sharing and group decision making. Manage Sci 27:1221–1535 Gan X, Sethi SP, Yan H (2005) Coordination of a supply chain with a risk-averse retailer and a risk-neutral supplier. Prod Oper Manage 14:80–89 Gaur V, Seshadri S (2005) Hedging inventory risk through market instruments. Manuf Serv Oper Manage 7(2):103–120 Harsanyi JC (1955) Cardinal welfare, individualistic ethics, and interpersonal comparisons of utility. J Polit Econ 63:309–321 Jorion P (2006) Value at risk. McGraw-Hill, New York, NY Lau HS (1980) The newsboy problem under alternative optimization objectives. J Oper Res Soc 31:393–403 Lau HS, Lau AHL (1999) Manufacturer’s pricing strategy and return policy for a single-period commodity. Eur J Oper Res 116:291–304 Lavalle IH (1978) Fundamentals of decision analysis. Holt, Rinehart, and Winston, New York Levi H, Markowitz H (1979) Approximating expected utility by a function of mean and variance. Am Econ Rev 69:308–317 Markowitz H (1959) Portfolio selection: efficient diversification of investment. Cowles foundation monograph 16, Yale University Press, New Haven, CT Nagarajan M, Bassok Y (2008) A bargaining framework in supply chains: the assembly problem. Manage Sci 54:1482–1496 Nash JF (1950) The bargaining problem. Econometrica 18:155–162 Pasternack BA (1985) Optimal pricing and returns policies for perishable commodities. Mark Sci 4:166–176 Pratt WJ (1964) Risk aversion in the small and in the large. Econometrica 32:122–136 Raiffa H (1970) Decision analysis. Addison-Wesley, Reading, MA Schweitzer ME, Cachon GP (2000) Decision bias in the newsvendor problem with a known demand distribution: experimental evidence. Manage Sci 46:404–420 Szeg€o G (ed) (2004) New risk measures for the 21st century. Wiley, West Sussex Tayur S, Ganeshan R, Magazine M (eds) (1999) Quantitative models for supply chain management. Kluwer, Boston, MA Tsay A (2002) Risk sensitivity in distribution channel partnerships: implications for manufacturer return policies. J Retailing 78:147–160 Van Mieghem JA (2003) Capacity management, investment, and hedging: review and recent developments. Manuf Serv Oper Manage 5:269–302 Van Neumann J, Morgenstern O (1944) Theory of games and economic behavior. Princeton University Press, Princeton, NJ Wilson R (1968) The theory of syndicates. Econometrica 18:155–162

.

Addendum to “Coordination of Supply Chains with Risk-Averse Agents” by Gan, Sethi, and Yan (2004) Xianghua Gan, Suresh P. Sethi, and Houmin Yan

Abstract In “Coordination of Supply Chains with Risk-Averse Agents” (POMS, Volume 13, 2004), we study the issue of coordination in supply chains involving risk-averse agents, and define a coordinating contract as one that results in a Paretooptimal solution acceptable to each agent. We then develop coordinating contracts in various cases. In the case where the supplier and the retailer each maximizes his own expected utility, we also show that our contract yields the Nash Bargaining solution. In this addendum, we first review some related works that have appeared since the publication of the paper, and then discuss directions for future research. Keywords Supply chain management • Risk aversion • Pareto-optimality • Coordination • Nash bargaining

1 Introduction In Gan et al. (2004), we study the issue of coordination in supply chains involving risk-averse agents, and define a coordinating contract as one that results in a Paretooptimal solution acceptable to each agent. We develop coordinating contracts in three specific cases (1) the supplier is risk neutral and the retailer maximizes his expected profit subject to a downside risk constraint, (2) the supplier and the retailer

X. Gan (*) Department of Logistics and Maritime Studies, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong S.P. Sethi School of Management, SM30, The University of Texas at Dallas, 800W. Campbell Road, Richardson, TX 75080-3021, USA H. Yan Department of Systems Engineering and Engineering Management, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong T.-M. Choi and T.C. Edwin Cheng (eds.), Supply Chain Coordination under Uncertainty, International Handbooks on Information Systems, DOI 10.1007/978-3-642-19257-9_2, # Springer-Verlag Berlin Heidelberg 2011

33

34

X. Gan et al.

each maximizes his own mean-variance trade-off, and (3) the supplier and the retailer each maximizes his own expected utility. Moreover, in case (3) we show that our contract yields the Nash Bargaining solution. In each case, we demonstrate how to find the set of Pareto-optimal solutions, and then design a contract to achieve the solutions. We also exhibit a case in which we obtain Pareto-optimal sharing rules explicitly, and outline a procedure to obtain Pareto-optimal solutions. Gan et al. (2004) was published in Productions and Operations Management and won the Wickham-Skinner Best Paper Award at the Production and Operations Management Society Cancun meeting in 2004. This paper and Gan et al. (2005) are identified as part of a knowledge cluster by means of factor analysis in the paper entitled “The evolution of the intellectual structure of operations management1980–2006: A citation/co-citation analysis” by Pilkington and Meredith (2009). Specifically, the clusters in the general knowledge structure in the 2000s appear on Fig 7 on p.196 of their paper.

2 Related Literature since the Publication of Gan et al. (2004) Since the publication of Gan et al. (2004), there has been a considerable amount of literature devoted to contracts that coordinate supply chains involving risk-averse agents. Some papers study coordination of supply chains with agents who maximize their own mean-variance objectives, for example, Choi et al. (2008) and Wei and Choi (2010). Some papers study management of disruption risk faced by supply chains, for example, Tomlin (2006) and Bakshi and Kleindorfer (2009). Chen and Seshadri (2006) use optimal control theory to show that the contract menu proposed in Agrawal and Seshadri (2000) is optimal. Shi and Chen (2007) focus on Pareto€ u et al. (2007) study optimal contracts for supply chains with risk-averse agents. Ulk€ the optimal allocation of risk between two agents in a supply chain. Sobel and Turcic (2008) employ the Nash bargaining solution to determine the value of an opportunity to negotiate a contract in an archetypal supply chain game. Another stream of literature focuses on inventory management for risk-averse agents. Some papers incorporate financial instruments into inventory management, for example, Gaur and Seshadri (2005), Martı´nez-de-Albe´niz and Simchi-Levi (2005), and Chen et al. (2008). Choi et al. (2009) study a multi-product newsvendor under lawinvariant coherent risk measures. In the following, we limit our review to papers that focus on supply chain coordination. Gan et al. (2005) examine coordinating contracts for a supply chain consisting of one risk-neutral supplier and one risk-averse retailer. They design an easy-toimplement risk-sharing contract that accomplishes the coordination as defined in Gan et al. (2004). Martı´nez-de-Albe´niz and Simchi-Levi (2005) develop a framework that provides buyers with the ability to select multiple contracts at the same time in order to optimize their expected profit. For this purpose, they define a new type of contract, called a portfolio contract, which is in fact a combination of many

Addendum to “Coordination of Supply Chains with Risk-Averse Agents”

35

traditional contracts, such as long-term, option, and flexibility contracts. Then they specialize the model to the case of a portfolio consisting of option contracts, and show that a modified base-stock policy is optimal. Chen and Seshadri (2006) revisit the problem in Agrawal and Seshadri (2000), and use optimal control theory to prove that the contract menu proposed in that paper is optimal. They also show that the menu is optimal among nearly all contracts. Shi and Chen (2007) study a supply chain with two agents, each maximizing his own probability of achieving a predetermined target profit. They investigate two types of contracts: linear tariff contracts and buy-back contracts. They first identify the Pareto-optimal contract(s) for each type, and then evaluate the performance of those contracts. They also show that in some cases, a wholesale price contract can coordinate the supply chain whereas a buy-back contract cannot. € u et al. (2007) consider a supply chain with a contract manufacturer (CM) Ulk€ and a number of original equipment manufacturers (OEMs). Since investment into productive resources is made before demand realization, the supply chain faces the € u et al. (2007) investigate two scenarios in risk of under- or over-investment. Ulk€ which this risk is borne by the OEM and CM, respectively. They show that premium-based schemes are effective in inducing the best party to bear the risk, and conclude that they function well despite information asymmetry when double marginalization is not very high. Note that premium-based schemes are similar to the side payment scheme in Gan et al. (2004). Choi et al. (2008) study channel coordination in a supply chain consisting of a retailer and a supplier, each maximizing his mean-variance trade-off. They find that incorporation of risk aversion substantially affects the achievability of channel coordination. They also show that channel coordination depends on the difference between the risk preferences of the retailer and the supplier. Sobel and Turcic (2008) study a supply chain that faces a newsvendor problem. They characterize the Nash-optimal contracts, and contrast the risk-neutral case with the risk-averse one. They show that risk aversion has a significant impact on the contract terms and one firm’s risk aversion may be advantageous to the other. They suggest that a firm should consider its own and its prospective partner’s sensitivities to risk when it looks for a supply chain partner. Bakshi and Kleindorfer (2009) consider disruption risk management in a global supply chain with two participants who face interdependent losses resulting from supply chain disruptions. They use the Harsanyi-Selten-Nash Bargaining framework to model the supply chain participants’ choice of risk mitigation investments. The bargaining approach allows a framing of both joint financing of mitigation activities before a disruption and loss-sharing after the disruption. Wei and Choi (2010) study coordination of a supply chain with a wholesale pricing and a profit sharing scheme (WPPS) under the mean-variance decision framework. They show that there exists a unique equilibrium of the Stackelberg game with WPPS in the decentralized case. They also show that the manufacturer may benefit from pretending to be more risk-averse than he actually is. Finally, they propose a method to prevent this from happening.

36

X. Gan et al.

3 Future Research The issue of supply chain coordination with risk-averse agents has drawn much attention in recent years and has been studied in a number of research papers. Future research on this issue might be conducted in the following directions. First, optimization problems for a single firm can be revisited from the supply chain’s point of view. In the literature, numerous papers study a single risk-averse firm’s optimal decisions regarding order/production quantities, item prices, etc. Usually, those papers assume that the wholesale price, the amount of a side payment if any, refund on the returned units, etc., are exogenously given. More often than not, the single firm’s optimal decisions do not lead to a Pareto-optimal solution for the whole supply chain. In this case, it is worthwhile to study those problems from a system-wide view. As suggested in Gan et al. (2004), all of the agents in a supply chain can negotiate wholesale price, side payment, and refund policy, and come up with Pareto-optimal decisions on order/production quantities, item prices, etc. Recently, asymmetric information problems have been widely studied in the OR/MS literature. In these studies, the agents are usually assumed to be riskneutral. When those agents are risk-averse, the results in risk-neutral case may not apply and future research is warranted. For example, when the production cost of the supplier is asymmetric, C¸akanyldrm et al. (2010) show that a Pareto-optimal solution is achievable. However, when the agents in a supply chain are risk-averse, the achievability of a Pareto-optimal solution is not clear. The decisions of the agents in a supply chain depend crucially on customer demand. However, in the OR/MS literature, customer demand is usually assumed, for simplification, to be a function of the selling price, and the issue of customers’ risk-aversion is ignored. When the customer demand depends on risk attitudes of the customers, it might be interesting to investigate how the agents in supply chain share risk with customers.

References Agrawal V, Seshadri S (2000) Risk intermediation in supply chains. IIE Trans 32:819–831 Bakshi N, Kleindorfer P (2009) Co-opetition and investment for supply-chain resilience. Prod Oper Manage 2009:583–603 C¸akanyldrm M, Feng Q, Gan X, Sethi SP (2010) Contracting and coordination under asymmetric production cost information. Working paper, University of Texas at Dallas, TX Chen YJ, Seshadri S (2006) Supply chain structure and demand risk. Automatica 2006:1291–1299 Chen F, Gao F, Chao X (2008) Joint optimal ordering and weather hedging decisions: a newsvendor model. Working paper, Chinese University of Hong Kong, Hong Kong Choi S, Ruszczynski AR, Zhao Y (2009) A multi-product risk-averse newsvendor with law invariant coherent measures of risk. Working paper, Rutgers University, New Jersey Choi TM, Li D, Yan H, Chiu CH (2008) Channel coordination in supply chains with agents having mean-variance objectives. Omega 36:565–576

Addendum to “Coordination of Supply Chains with Risk-Averse Agents”

37

Gan X, Sethi SP, Yan H (2004) Coordination of supply chains with risk-averse agents. Prod Oper Manage 13:135–149 Gan X, Sethi SP, Yan H (2005) Coordination of a supply chain with a risk-averse retailer and a risk-neutral supplier. Prod Oper Manage 14:80–89 Gaur V, Seshadri S (2005) Hedging inventory risk through market instruments. Manuf Serv Oper Manage 7:103–120 Martı´nez-de-Albe´niz V, Simchi-Levi D (2005) A portfolio approach to procurement contracts. Prod Oper Manage 14:90–114 Pilkington A, Meredith J (2009) The evolution of the intellectual structure of operations management-1980–2006: a citation/co-citation analysis. J Oper Manage 27:185–202 Shi C, Chen B (2007) Pareto-optimal contracts for a supply chain with satisfying objectives. J Oper Res 58:751–759 Sobel M, Turcic D (2008) Risk aversion and supply chain contract negotiation. Working paper, Case Western Reserve University, Cleveland, OH Tomlin B (2006) On the value of mitigation and contingency strategies for managing supply chain disruption risks. Manage Sci 52:637–659 € u S, Toktay LB, Y€ Ulk€ ucesan E (2007) Risk ownership in contract manufacturing. Manuf Serv Oper Manage 9:225–241 Wei Y, Choi TM (2010) Mean-variance analysis of supply chains under wholesale pricing and profit sharing schemes. Eur J Oper Res 204:255–262

.

A Review on Supply Chain Coordination: Coordination Mechanisms, Managing Uncertainty and Research Directions Kaur Arshinder, Arun Kanda, and S.G. Deshmukh

Abstract The Supply Chain (SC) members are dependent on each other for resources and information, and this dependency has been increasing in recent times due to outsourcing, globalization and rapid innovations in information technologies. This increase in dependency brings some extent of risk and uncertainty too along with benefits. To meet these challenges, SC members must work towards a unified system and coordinate with each other. There is a need to identify the coordination mechanisms which helps in addressing the uncertainty in supply chain and achieving supply chain coordination. A systematic literature review is presented in this paper to throw light on the importance of SC coordination. The objectives of this paper are to: Report and review various perspectives on SC coordination issues, understand and appreciate various mechanisms available for coordination and managing SC uncertainty and identify the gaps existing in the literature. Perspectives on various surrogate measures of supply chain coordination have been discussed followed by the scope for further research. Keywords Coordination mechanisms • Supply chain coordination • Supply chain coordination index • Supply chain uncertainty

This paper is based on earlier version of the following paper: Arshinder K, Kanda A, Deshmukh SG (2008) Supply chain coordination: perspectives, empirical studies and research directions. Int J Prod Econ 115(2):316–335. This paper is also based on the doctoral research work done by Arshinder (2008) at Indian Institute of Technology Delhi, India. K. Arshinder (*) Department of Management Studies, Indian Institute of Technology Madras, Chennai 600036, India e-mail: [email protected] A. Kanda • S.G. Deshmukh Department of Mechanical Engineering, Indian Institute of Technology Delhi, New Delhi 110016, India e-mail: [email protected]; [email protected] T.-M. Choi and T.C. Edwin Cheng (eds.), Supply Chain Coordination under Uncertainty, International Handbooks on Information Systems, DOI 10.1007/978-3-642-19257-9_3, # Springer-Verlag Berlin Heidelberg 2011

39

40

K. Arshinder et al.

1 Introduction Supply chain has evolved very rapidly since 1990s showing an exponential growth in papers in different journals of interest to academics and practitioners (Burgess et al. 2006). The rise in papers on supply chain (SC) as well as the case studies in different areas in different industries motivates to study SC issues further. Supply chains are generally complex with numerous activities (logistics, inventory, purchasing and procurement, production planning, intra-and inter-organizational relationships and performance measures) usually spread over multiple functions or organizations and sometimes over lengthy time horizons. Supply chains tend to increase in complexity and the involvement of numerous suppliers, service providers, and end consumers in a network of relationships causes risks and vulnerability for everyone (Pfohl et al. 2010). The continuous evolving dynamic structure of the supply chain poses many interesting challenges for effective system coordination. Supply chain members cannot compete as independent members. The product used by the end customer passes through a number of entities contributed in the value addition of the product before its consumption. Also, the practices like globalization, outsourcing and reduction in supply base have exacerbated the uncertainty and risk exposure as well as more prone to supply chain disruption. Earlier literature considers risks in relation to supply lead time reliability, price uncertainty, and demand volatility which lead to the need for safety stock, inventory pooling strategy, order split to suppliers, and various contract and hedging strategies (Tang 2006). But today’s supply networks have become very complex and vulnerable to various supply chain risks hence these issues have pulled attention of various academics and practitioners for the last few years (Oke and Gopalakrishnan 2009). Uncertainty relates to the situation in which there is a total absence of information or awareness of a potential event occurrence, irrespective of whether the outcome is positive or negative. The terms risk and uncertainty are frequently used interchangeably (Ritchie and Brindley 2007). As firms move to leaner operating models and increasingly leverage global sourcing models, uncertainty in both supply and demand is growing along with supply chain complexity. To improve the overall performance of supply chain, the members of supply chain may behave as a part of a unified system and coordinate with each other. Thus “coordination” comes into focus. There seems to be a general lack of managerial ability to integrate and coordinate the intricate network of business relationships among supply chain members (Lambert and Cooper 2000). Stank et al. (1999) studied inter-firm coordination processes characterized by effective communication, information exchange, partnering, and performance monitoring. Lee (2000) proposes supply chain coordination as a vehicle to redesign decision rights, workflow, and resources between chain members to leverage better performance such as higher profit margins, improved customer service performance, and faster response time.

A Review on Supply Chain Coordination

41

Though, there are efforts in literature regarding coordination of different functions of the supply chain, the study of coordinating functions in isolation may not help to coordinate the whole supply chain. It appears that the study of supply chain coordination (SCC) is still in its infancy. Though, the need for coordination is realized, a little effort has been reported in the literature to develop a holistic view of coordination. It is interesting to note the following perspectives on supply chain coordination as reported in the literature: • Collaborative working for joint planning, joint product development, mutual exchange information and integrated information systems, cross coordination on several levels in the companies on the network, long term cooperation and fair sharing of risks and benefits (Larsen 2000). • A collaborative supply chain simply means that two or more independent companies work jointly to plan to execute supply chain operations with greater success than when acting in isolation (Simatupang and Sridharan 2002). • Kleindorfer and Saad (2005) asserted that continuous coordination, cooperation, and coordination among supply chain partners are imperative for risk avoidance, reduction, management and mitigation such that the value and benefits created are maximized and shared fairly. • Supply chain coordination is a strategic response to the challenges that arise from the dependencies supply chain members (Xu and Beamon 2006). • Supply chain coordination can be defined as identifying interdependent supply chain activities between supply chain members and devise mechanisms for manage those interdependencies. It is the measure of extent of implementation of such aggregated coordination mechanisms, which helps in improving the performance of supply chain in the best interests of participating members (Arshinder 2008). Various perspectives have been presented in the literature for coordinating supply chain (discussed in Sect. 2). These perspectives and classification of coordination literature has been adopted from the review paper by Arshinder et al. (2008a), however, the authors are motivated to revise the paper with view of incorporating uncertainty in SCC and up gradation of coordination mechanisms. The following developments have motivated the authors to upgrade the current review paper. • Growth in reporting of coordination mechanisms in supply chain. • Managing uncertainty has become more and more challenging, which can be tackled with SCC. • Information technology has been evolving and playing an important role in making global supply chain seamless. To develop a better understanding of the coordination issues in supply chain, a systematic literature review is required to throw light on the importance of SCC and specifically to address the objectives as: to understand and appreciate SCC in different processes of supply chain, to explore various coordination mechanisms to coordinate the supply chain, to understand the role of SCC in managing SC uncertainty and to relate surrogate measures of SCC with supply

42

K. Arshinder et al.

chain performance. The last objective is to identify the gaps existing in the literature followed by few research directions. The terms like integration, collaboration, cooperation and coordination are at times complementary and at times contradictory to each other and when used in the context of supply chain can easily be considered as a part of SCC. This assumption can be followed without loss of generality as the elements like integration (combining to an integral whole); collaboration (working jointly) and cooperation (joint operation) are the elements of coordination.

2 Supply Chain Coordination Literature Classification and Observations The papers related to supply chain coordination were searched using library databases covering a broad range of journals (Appendix). The papers were selected based on the issues addressed by these papers: How to define supply chain coordination and the imperatives of SCC? How to achieve supply chain coordination? Will coordinated supply chain be beneficial to all the individual members of the supply chain? What is the impact of SCC on the performance of various activities and processes of a supply chain? How SCC can help in mitigating supply chain uncertainties? The papers in response to the above mentioned questions were gathered and classified in categories presented in the following sections. To capture each and every aspect of SCC an attempt has been made to classify the literature on SCC as follows: • Perspectives and conceptual models on supply chain coordination. • Joint consideration of functions or processes by supply chain members at different levels to coordinate the supply chain. • Various supply chain coordination mechanisms adopted in the supply chain. • Supply chain coordination to manage uncertainties in the supply chain. • Empirical case studies in supply chain coordination. A schematic overview of hierarchical classification of literature is shown in Fig. 1 which shows that how the different categories of coordination will help in understanding the importance of SCC, utility of coordination mechanisms and the application of SCC on real life problems.

2.1

2.1.1

Perspectives and Conceptual Models on Supply Chain Coordination Challenges in Coordinating the Supply Chain

In any system, the smooth functioning of entities is the result of well-coordinated entities. It may be very difficult to define “coordination” precisely, but the lack of

A Review on Supply Chain Coordination

43

Supply Chain Coordination

Perspectives and conceptual models on SCC

Joint consideration of functions/ processes by various SC members

Coordination across functions of the supply chain

Supply chain coordination mechanisms

Integrated ProcurementProductionDistribution processes

Contracts

Supply uncertainty

Information technology and Information sharing

Supply chain coordination to manage uncertainty

Production disruptions

Demand uncertainty

Other collaborative initiatives

Fig. 1 Overview of the literature classification scheme

coordination can be easily articulated through a variety of surrogate measures. The supply chain members have conflicting goals or objectives and disagreements over domain of supply chain decisions and actions. It must be noted that a typical supply chain also deals with human systems, and hence, which may pose following challenges and difficulties in coordinating supply chain members. • The individual interest, local perspective and opportunistic behavior of supply chain members results in mismatch of supply and demand (Fisher et al. 1994). • The traditional performance measures based on the individual performance may be irrelevant to the maximization of supply chain profit in a coordinated manner. Similarly, the traditional policies, particularly rules and procedures, may not be relevant to the new conditions of inter organizational relationship. There has been over reliance on technology in trying to implement IT (Lee et al. 1997; McCarthy and Golocic 2002). • According to Piplani and Fu (2005), supply chain “plug and play” misalignment is associated with the difficulties involved in dynamically interchanging products (with short life cycle) and partners in the fast changing business environment. • The organizations want to reach to the best suppliers regardless their location globally, which brings many risks and uncertainties in managing cross border supply chains.

44

K. Arshinder et al.

• The benefits accrued by the whole supply chain after joint determination of supply chain performance indicators by supply chain members has no value in the absence of fair share mechanisms. There are multiple benefits accruing from effective SCC. Some of these include: elimination of excess inventory, reduction of lead times, increased sales, improved customer service, efficient product developments efforts, low manufacturing costs, increased flexibility to cope with high demand uncertainty, increased customer retention, and revenue enhancements (Fisher et al. 1994; Lee et al. 1997). 2.1.2

Various Perspectives and Conceptual Models on SCC

The literature reviewed by Burgess et al. (2006) showed that there is relative paucity of strong multi-theoretic approaches in supply chain. By looking at the problems of managing relationships between supply chain members, a need arises to tackle this problem using coordination theory. The most commonly accepted definition of coordination in the literature is “the act of managing dependencies between entities and the joint effort of entities working together towards mutually defined goals” (Malone and Crowston 1994). Coordination is perceived as a prerequisite to integrate operations of supply chain entities to achieve common goals. Various perspectives are reported in the literature regarding SCC. The researchers have described SCC either in the context of the application of coordination in different activities of supply chain or they are derived from other disciplines, summarized in Table 1. Several coordination strategies have been developed to align supply chain processes and activities to ensure better supply chain performance. The papers addressing various forms of coordination are Buyer–Vendor coordination by coordinating Procurement–Inventory–Production–Distribution processes (Goyal and Deshmukh 1992; Thomas and Griffin 1996; Sarmiento and Nagi 1999; Sarmah et al. 2006). Hoyt and Huq (2000) presented a literature review on the buyer-supplier relationship from the perspective of transaction cost theory, strategy structure theory and resource-based theory of the firm. There is abundant literature on conceptual based supply chain partnership but the testing of these concepts is required by utilization of operations research in supply chain (Maloni and Benton 1997). Various models have been discussed presenting various form of coordination such as price changes, quantity discounts (Sharafali and Co 2000), and partial deliveries and establishing their joint policies in context of manufacturing firms (Sarmah et al. 2007), information sharing and decision-making coordination (Sahin and Robinson 2002). Some of the coordination forms can be seen in Table 2. Power (2005) reviewed three principal elements of supply chain integration: information systems, inventory management and supply chain relationships aiming at reducing costs and improving customer service levels. The emerging area of supply chain coordination is outsourcing practices in case of insufficient production capacity of suppliers (Sinha and Sarmah 2007).

A Review on Supply Chain Coordination Table 1 Various perspectives on supply chain coordination Author (year) Perspective Narus and Anderson Cooperation among independent but (1995) related firms to share resources and capabilities to meet their customers’ most extraordinary needs Lambert et al. (1999) A particular degree of relationship among chain members as a means to share risks and rewards that result in higher business performance than would be achieved by the firms individually Larsen (2000) Collaborative working for joint planning, joint product development, mutual exchange information and integrated information systems, cross coordination on several levels in the companies on the network, long term cooperation and fair sharing of risks and benefits Lee (2000) Supply chain coordination as vehicle for redesigning decision rights, workflow, and resources between chain members to leverage better performance Simatupang et al. Given the nature of the interdependencies (2002) between units, coordination is necessary prerequisite to integrate their operations to achieve the mutual goal of the supply chain as a whole as well as those of these units Larsen et al. (2003) Where two or more parties in the supply chain jointly plan a number of promotional activities and work out synchronized forecasts, on the basis of which the production and replenishment processes are determined Hill and Omar Coordination can be achieved when the (2006) supply chain members jointly minimize the operating costs and share the benefits after jointly planning the production and scheduling policies Arshinder (2008) Identifying interdependent supply chain activities between SC members and devise mechanisms for manage those interdependencies. It is the measure of extent of implementation of such aggregated coordination mechanisms, which helps in improving the performance of supply chain in the best interests of participating members

45

Context Resource sharing

Risk and reward sharing

Holistic view of coordination

Workflow/resource dependency

Mutuality

Joint promotional activities, forecasting

Joint decision-making, benefit sharing

Linking coordination mechanisms with SC performance

46

K. Arshinder et al.

Table 2 Different forms of coordination viewed in supply chain S. No. Coordination Author (year) perspectives 1 Coordination of Goyal and Deshmukh (1992), functions or process Thomas and Griffin across SC members (1996) Sarmiento and Nagi (1999) 2 Coordination by Hoyt and Huq (2000), Sahin information sharing and Robinson (2002), Huang et al. (2003), Simatupang et al. (2002) 3 Supply chain Power (2005) partnerships 4

5

Coordination mechanisms and performance Problems in coordinating SC

6

Coordination by IT

7

Implementation issues in coordination

Issues in coordination Integrated procurement, production, distribution and inventory systems Value of information sharing and sharing modes, incentive alignment

Communication, Inventory management and supply chain partnerships Lee et al. (1997) Channel coordination, operational efficiency and information sharing Fawcett and Magnan (2002), Lack of information transparency, incentive Simatupang and Sridharan misalignment (2002) Li et al. (2002), Mc Laren Internet based integration of et al. (2002) complex supply chain processes, cost and benefits of different information systems coordinating supply chain Barratt (2004) Cultural, strategic and implementation elements of supply chain coordination

The other pragmatic initiatives such as Collaborative Planning, Forecasting and Replenishment (CPFR) (Larsen et al. 2003) and Supply Chain Operations Reference (SCOR) (Huan et al. 2004) may have relevance from practitioner’s point of view. Even though coordination improves the performance of the supply chain, it may not always be beneficial to coordinate the supply chain members. The high adoption costs of joining inter-organizational information systems and information sharing under different operational conditions of organizations may hurt some supply chain members (Zhao and Wang 2002). Therefore, it is essential to investigate the conditions under which supply chain coordination is beneficial, so that it should not result in higher supply chain costs and imprecise information. Observations and Gaps Regarding Various Perspectives and Conceptual Models on SCC (a) There seems to be no standard definition of SCC. Various perspectives on SCC as reported in the literature are testimony to this. The differences in perceptions

A Review on Supply Chain Coordination

47

are there because of the different expectations of the various stakeholders and the respective problem domain. Some of these perspectives present the inherent capability or intangibles required to coordinate like responsibility, mutuality, cooperation and trust. The other perspectives can be visualized, based on the coordination effort required in achieving common goals in different activities of supply chain. Since the activities are different, the coordination requirements also vary with the complexity of the activity. The most challenging coordination perspective is to extend the concept of coordination from within an organization to coordination between organizations. (b) By looking at these different perspectives, the SCC can be viewed as a set of following steps: 1. Identify why supply chain members want to coordinate and for which activity/process they are interdependent? Different interdependencies among supply chain members can be: ordering, procurement, inventory management, production, design and development, replenishment, forecasting and distribution. 2. Identify which activity or a set of activities needs to be coordinated, complexities in the activity (activities) and degree of coordination required. 3. Identify the reason to coordinate. Is it the demand uncertainty and/or supply uncertainty, double marginalization or other external risk in the supply chain, which can be addressed by coordination? 4. Identify whether a single or a combination of coordination mechanism are required to tackle the complexities in managing the interdependencies like resource sharing, knowledge sharing, information sharing, joint working, joint decision making, joint design and development of product, joint promotions, implementing information systems, designing risk sharing contracts. (c) Though there are attempts to focus on coordinating the different processes of supply chain, most of the papers reviewed have discussed the work done on analytical models with joint decision making of different process. The literature seems to be lacking in developing empirical relationship between coordination means and mechanisms (Information sharing, trust and IT) and SCC. (d) There is a need to embrace a variety of perspectives on supply chain coordination, various coordination issues and the means and mechanisms to achieve coordination in a holistic manner. (e) Various coordination mechanisms suggested in these models help in improving the various performance measures of the supply chain. These mechanisms include: joint decision-making, information sharing, resource sharing, implementing information technology, joint promotional activities, etc. The other motivation seems to be the ability of supply chain members to share the risks and subsequently share the benefits. (f) There is a need to monitor coordination in supply chain because of the adverse effects of lack of coordination on supply chain performance. There seems to be no measure to quantify coordination. Some models can be proposed to

48

K. Arshinder et al.

quantify and assess the strength of coordination on the basis of coordination mechanisms. (g) More empirical studies are required regarding the proper implementation of coordination mechanisms, so that combinations of different feasible coordination mechanisms can capture the impact of coordination on various supply chain performance measures. The above conceptual models on supply chain coordination have been presented in a fragmented manner. It is important to understand various SC functions to be coordinated. The complexity in coordinating various SC members may also depend on the interface to which two supply chain members belong. The following section presents the importance of SC coordination in various SC functions as well as in different SC processes at various supply chain interfaces.

3 Joint Consideration of Functions or Processes by Supply Chain Members at Different Levels to Coordinate Supply Chain Coordination can be visualized in different functions such as logistics, inventory management, forecasting, transportation, etc. Similarly, various interface such as supplier-manufacturer; manufacturer-retailer, etc. can be effectively managed using coordination.

3.1

Coordinating Functions Across Supply Chain Members

The supply chain members perform different functions or activities like logistics, inventory management, ordering, forecasting and product design involved in management of flow of goods, information and money. In traditional supply chain individual members of supply chain have been performing these activities independently. The supply chain members may earn benefits by coordinating various activities as discussed in following subsections. Logistics has traditionally been defined as the process of planning, implementing, and controlling the efficient flow and storage of goods, services, and related information as they travel from point of origin to point of consumption. The uncertainty and complexity of decision making regarding logistics operations: diversified customers and their different requirements, different resources required, increasing rate of unanticipated change and level of goal difficulty among logistics provider and the customer (supplier, manufacturer, distributor and retailer), geographically dispersed networks of multiple manufacturing sites lead to the need of coordination in this process (Huiskonen and Pirttila 2002). The challenges lie in managing the network complexities to collectively create value to the end customer

A Review on Supply Chain Coordination

49

(Stank et al. 1999; Stock et al. 2000) and integrating the logistics with whole supply chain with the help of electronic communication. The major decisions regarding inventory management include: determination of the order quantity, the timing of order, reorder point and the replenishment of inventory. The factors which are considered while deciding the inventory policy are customer demand (deterministic and random), number of members in supply chain, replenishment lead time, number of different products stored, length of the planning horizon, service level requirements and costs comprised of cost of production, transportation, taxes and insurance, maintenance, obsolescence opportunity cost, stock out, etc. The changes even in one of the above factors affect the decisions regarding inventory policy. The factors related to inventory policy are highly dynamic because of changing market condition, supply uncertainty; different and conflicting inventory policies among supply chain members, and unavailability of inventory information of other members. To face the dynamic situation, the members of supply chain have realized the importance of coordination in inventory management. The supply chain members may coordinate by joint consideration of the system wide costs (Huq et al. 2006; Wu and Ouyang 2003; Gurnani 2001; Barron 2007), sharing cost and price information (Boyaci and Gallego 2002; Piplani and Fu 2005), synchronizing order processing time (Zou et al. 2004; Lu 1995; Yao and Chiou 2004; Barron 2007) and networked inventory management information systems (Verwijmeren et al. 1996). These policies may sometime hurt one of the supply chain members. To compensate losses, different mechanisms have been proposed as quantity discounts, revenue sharing contracts and incentive alignment policies (Li et al. 1996; Moses and Seshadri 2000; Chen and Chen 2005). The different models results in reduction in ordering cost, holding cost, purchasing cost, and supply chain system wide costs and improvement in customer service level and product availability and product variety. The organization has perceived the need of reviving the traditional purchasing function in view of degree of participation and expertise of suppliers to a new evolving function called “strategic sourcing”(Gottfredson et al. 2005). The suppliers can form strategic partnerships by having common goals and sharing forecast information to have updated single forecasting process, which results in substantial cost reduction in whole supply chain (Zsidisin and Ellram 2001; Aviv 2001). The increasing rate of changing technologies, innovation, customer expectations, competition, and risk involved with new product entry and at the same time keeping the product design process cost efficient, is a challenging job. Kim and Oh (2005) presented systems dynamics approach to coordinate supplier and manufacturer decisions regarding improvement in quality and the new product development. Petersen et al. (2005) presented the findings from an empirical survey about the capabilities of suppliers required in coordinating the product design process with supplier. The coordination at design stage may result in better design and improved financial performance if the supplier has sufficient knowledge required to design the product.

50

3.2

K. Arshinder et al.

Coordinating Different Processes of the Supply Chain

A supply chain process consists of a set of activities taken together. Various processes in supply chain are procurement, production and distribution. These processes can be accomplished when some activities are performed like procurement process comprised supplier management, ordering, acquisition, replenishment, inspection activities, etc. Integration of different processes into a single optimization model to simultaneously optimize decision variables of different processes that have traditionally been optimized sequentially helps in improving the performance of SC (Park 2005). These processes sometimes face conflicting issues which are presented in Table 3. Isolated decision making in functionally related supply chain processes might weaken the supply chain system wide competitiveness. The different supply chain processes can be coordinated by implementing joint production delivery policies, common cycle approach, identical replenishment cycle (Yang and Wee 2002) and joint lot scheduling models (Kim et al. 2006). The coordination problems and the related issues at the interfaces of supply chain are presented in Table 4.

3.2.1

Production and Distribution Coordination

Integration of production and distribution processes may lead to a substantial saving in global costs and to an improvement in relevant service by exploiting scale economies of production and transportation, balancing production lots and vehicle loads, and reducing total inventory and stockout. Chikan (2001) gave a theoretical background of integrated production/logistics systems on the basis Table 3 Conflicting issues in supply chain processes SC processes Conflicting issues in supply chain processes Production and distribution The difference in performance metrics such as improvement in coordination quality of production, reduction in cost and improvement in service levels for distribution may also give rise to conflict Production sub functions are usually concentrated in the organization, while distribution sub functions are spread over (Chikan 2001) Production function is obsessed with low cost production, with large batch sizes and efficient and smooth production schedules (Pyke and Cohen 1993) and the distribution function is concerned with customer service as first priority, small batch sizes and frequent changeovers (Pyke and Cohen 1993) Procurement and production Suppliers typically want manufacturers to commit themselves coordination to purchasing large quantities in stable volumes with flexible delivery dates Manufacturers require just-in-time (JIT) supply in small batches from their suppliers due to changing demand and their unwillingness to hold inventories

Information sharing and IT

Coordination mechanism

Logistics provider and clientsa

Structure of supply chain

Joint decision making and benefit sharing Quantity discounts

Manufacturer–retailer

Single-supplier–multibuyers

Joint system cost consideration, quantity discounts

Supply chain network

Single-supplier–multibuyer Seller–buyer

Joint decision making and quantity discounts

Need of system cooperation Independent management IT and mutual benefits of inventories

Different order intervals

Mismatch goals between Information sharing, aligning Logistics provider and shipper and goals, EDI, contracts clientsa transportation provider Lack of integration EDI Logistics provider and between logistics and clientsa supply chain Need of relation Information sharing, IT, Logistics provider and improvement integrating role manufacturer between logistics and client

Mismatch goals between shipper and transportation provider

Coordination problem

Moses and Seshadri (2000) Need of risk sharing, mismatch in stock level and review period Gurnani (2001) Mismatch in timing of order

Verwijmeren et al. (1996)

Inventory Lu (1995) and Yao and Chiou (2004) Li et al. (1996)

Huiskonen and Pirttila (2002)

Stock et al. (2000)

Stank and Goldsby (2000)

Logistics Stank et al. (1999)

Author

Table 4 Coordination in various activities and interfaces of supply chain

Analytical model

Optimization

Network solution

Game theoretic model

Analytical model

Conceptual survey

Empirical survey

Conceptual framework

Empirical survey

Methodology used

Minimize cost

(continued)

Improving customer service level, increasing product variety, and lower supply chain system wide costs Minimize cost

Minimize costs (ordering + holding + purchasing) Maximize profits

Good relationship

Operational performance and financial performance

Inventory level, transportation costs, warehousing costs, ordering costs, order cycle variance, on time deliveries and unacceptable deliveries Channel cycle time and inventory level

Performance measure

A Review on Supply Chain Coordination 51

Misaligned inventory decisions

Independent cost consideration Different cycle times

Piplani and Fu (2005)

Huq et al. (2006)

Inventory-distribution Haq and Kannan (2006)

Forecasting Aviv (2001)

Barron (2007)

Chen and Chen (2005)

Zou et al. (2004)

Wu and Ouyang (2003)

Potential lie in reducing costs by considering all costs jointly

Multi-supplier–singleassembler

Single-manufacturer– multi-retailer Vendor–buyer

Single-wholesaler– multi-retailer

Structure of supply chain

Joint decision making

Joint consideration of cost

Joint consideration of costs at Multi-echelon each level

Manufacturer–retailer

Multi-warehouse– multi-retailer Serial supply chain (multi-echelon)

Joint consideration of cost, Manufacturer–retailer savings sharing, quantity discounts Cost sharing and service level Multi-echelon contracts

Jointly plan pricing and inventory replenishment policies Order coordination, information sharing Joint cost consideration with shortages Information sharing and revenue sharing contracts

Coordination mechanism

Independent decision Joint decision making and making of forecasting demand information sharing

Lack of coordination in lot sizing decisions and pricing Mismatch in timing of order Independent cost calculation Different order processing times of suppliers and incentive conflicts Need of risk sharing

Boyaci and Gallego (2002)

Zhao et al. (2002)

Coordination problem

Author

Table 4 (continued) Performance measure

Pareto improvement

Maximize profits, minimize costs (holding + shortage)

Minimize cost and improve service level Minimize cost

Minimize cost

Fuzzy AHP and genetic Minimize costs (inventory algorithm carrying + production + transportation)

Analytical model

Multi agent technology Minimize inventory holding cost and genetic algorithm Mathematical model Minimize distribution cost and lead and simulation time Analytical model Minimize costs (ordering + holding)

Mathematical

Analytical model (extension to newsboy model)

Analytical model

Simulation

Analytical optimization Maximize channel profits (wholesale problem price-inventory related costs)

Methodology used

52 K. Arshinder et al.

a

Multi-echelon

Joint production distribution cost minimization Joint cost minimization with global BOM

Coordinate production scheduling and vehicle routing

Multi-echelon

Multi-echelon

Multi-echelon

Near optimal cost and service Single-manufacturer– level, plan jointly single-distributor– single-retailer

Joint production distribution cost minimization

Conflict in finding Joint decision making and Single-supplier–multinumber of deliveries quantity discounts buyer of an order by vendor and buyer Holding costs increases For different holding costs of Single-supplier–singleas goods move members, find order buyer downstream in supply quantity and share chain benefits Managing complexities Synchronizing production Multi-echelon (5 cycles and risk pooling levels) effects

Lack of integration in different processes of supply chain Conflict between large batch size (production) and small batch size (distribution) Costs of carrying inventory at multi location, results more inventory level in whole supply chain Lack of integration in different processes Need for coordinating production and distribution

Clients can be supplier, manufacturer, distributor and retailer

Hwarng et al. (2005)

Hill and Omar (2006)

Production-inventory Yang and Wee (2002)

Jang et al. (2002)

Ganeshan (1999)

Chandra and Fisher (1994)

Pyke and Cohen (1993)

Production-distribution Jayaraman and Pirkul (2001)

Simulation

Mathematical model

Mathematical model

Mathematical and simulation Lagrangian heuristics and genetic algorithm

Local improvement heuristics

Constrained optimization problem

Lagrangian relaxation scheme

Average stock level, average backlog and average total cost

Minimize costs (production + shipping + holding)

Minimize costs (holding + ordering)

Minimize costs (purchasing + production + distribution) Minimize costs (production + distribution)

Minimize costs (fixed cost of facilities + holding + distribution)

Production cost and service level

Minimize costs (purchasing + production + distribution)

A Review on Supply Chain Coordination 53

54

K. Arshinder et al.

of institutional economics, discussed business issues regarding integration of these two functions and how this connection is handled in education. Jayaraman and Pirkul (2001) developed an integrated production distribution model comprised fixed cost, purchasing cost, production cost and distribution cost, taken simultaneously. Pyke and Cohen (1993) presented an integrated production distribution model and examined its performance characteristics (production cost and service level). Hill (1997) determined the production and shipment schedule for an integrated system to minimize average total cost per unit time. Kim et al. (2006) developed a mathematical optimization problem in multiple plants in parallel and single retailer supply chain system. The joint optimization of costs was carried out to determine the production cycle length, ordering quantity and frequency, and production allocation ratios for multiple plants. Dotoli et al. (2005) proposed a three-level hierarchical methodology for a supply chain network design at the planning management level. The network is so designed where the members are selected based on the performance followed by optimizing the communication and transportation links of supply chain. The performance measures used were operating costs, cycle time, energy saving, product quality and environmental impact.

3.2.2

Procurement and Production Coordination

Goyal and Deshmukh (1992) reviewed the literature on Integrated ProcurementProduction (IPP) systems. The different models of IPP were classified into the categories based on number of products, planning horizon, solution method employed, joint replenishment orders, and algorithmic issues in their study. Munson and Rosenblatt (2001) presented a purchasing-production integrated model and compared the cases of centralized SC and decentralized SC. It was found that decentralized SC gives same results as that of centralized supply chain if quantity discounts are considered at both upstream and downstream interfaces.

3.2.3

Production and Inventory Coordination

Lu (1995) considered heuristics approach for single vendor multi-buyer problem based on equal sized shipments. With the coordination of the replenishments of different items, the vendor can reduce his total annual cost by 30%. The buyers also benefit from the multi-buyer model by reducing their costs. Hoque and Goyal (2000) developed an optimal solution procedure for optimal production quantity in single vendor single buyer production inventory system with unequal and equal sized shipments from the vendor to the buyer and under the capacity constraint of the transport equipment by using simple interval search approach. Arreola-Risa (1996) considered the situation of multi-item production–inventory system with stochastic demands and capacitated production under deterministic or exponentially distributed unit manufacturing times. The observed results are that variation

A Review on Supply Chain Coordination

55

in the production environment increases the optimal inventory levels. The impact of capacity utilization in optimal base stock level is non-linear function of demand rate. Grubbstrom and Wang (2003) developed a multi-level capacity constrained model with stochastic demand. The Laplace transform was used as tool to construct the model and dynamic programming was used to solve and to find out the net present value (NPV) as an objective function. It was observed that for higher levels of capacity, the stochastic solution continues to improve performance of the system, albeit at a very slow rate and then takes advantage of increasing availability of the capacity resources. Kim et al. (2006) considered common production cycle length, delivery frequency and quantity in three level supply chain in joint economic procurement, production and delivery policy.

3.2.4

Distribution and Inventory Coordination

Jayaraman (1998) developed an integrated mathematical programming mixedinteger model for minimization of the total distribution cost associated with all three decision components i.e. facility locations, inventory parameters and transportation alternative selection, all investigated jointly. The integrated model permits a more comprehensive evaluation of the different trade-off that exists among the three strategic issues. Yokoyama (2002) developed an integrated optimization model of inventory-distribution system in which any consumer point can be supplied by multiple distribution centers. The order-up-to-R, periodic review inventory policies and transportation problem are considered simultaneously. Simulation and linear programming was used to calculate the expected costs and a random local search method was developed to determine optimum target inventory, which was then compared with genetic algorithm. Haq et al. (1991) formulated a mixed integer programming for integrated production–inventory-distribution model. The objective of the model was to determine optimal production and distribution quantities through various channels, optimal levels of inventory at various production stages and at warehouses over 6-month planning periods considering set up time cost, lead time, production losses and recycling of losses with backlogging. Observations and Gaps in Different Activities and at the Interfaces of Supply Chain (a) In the literature, different problems in coordinating the activities with various approaches have been discussed. The main objective considered in coordinating different problems in some activity is either minimizing the costs or maximizing profits. The coordination of same activities at different levels of supply chain reduces the supply chain costs. (b) The common problems addressed in literature are the joint consideration of different costs in an activity. These costs are associated with the supply chain coordination problems of joint ordering by buyers to some supplier, jointly plan order quantity between supplier and buyer, jointly order delivery to the buyers and joint replenishment activities in terms of coordinated lead times.

56

K. Arshinder et al.

The coordination problems have also been extended for coordinating different processes to collectively consider the costs of different processes to minimize the overall cost of supply chain. (c) The methodologies adopted to tackle the problem include: analytical, mathematical and optimization tools. Most of the studies regarding SCC are conducted on a two-level dimension because of the simple supply chain structure (Ganeshan 1999; Hill 1997) and discussed production delivery policies and joint stocking with discounts (Weng and Parlar 1999) at two level supply chain. To effectively allocate the production requirement and capture supply chain dynamics, various models have been dealt with joint purchasing policies in multiple supplier environment (Zou et al. 2004) and considering total cost of logistics. The investigations are required in supply chain encompassing multiple levels that consider the complex interactions between the upstream and downstream sites and gives a more real picture of supply chain. (d) The following are some gaps, which if considered, may further enhance coordination and performance of supply chain: • The whole supply chain is required to coordinate, so models can be extended to consider more than one activity. • The only coordination mechanism used by most of the authors is joint consideration of costs. From the literature regarding coordination models it can be observed that a number of coordination mechanisms (information sharing, roles integration, information technology) are possible to solve the coordination problem. There can be situations where two mechanisms are required to reduce the supply chain costs for example information sharing and quantity discounts. • The consideration of one performance measure may not justify the value of coordination. So, a number of performance measures are required to capture the impact of coordination in a holistic manner. Along with the measures like costs and profits, the benefits of coordination may also be indicated with the help of performance measures like: improving responsiveness by timely information sharing in whole supply chain, reducing inventory delays and information lead time by implementing good information systems and evaluating risks and rewards due to coordination. • The analytical and mathematical approaches used to coordinate activities and processes of supply chain may not tackle the dynamics of supply chain. Hence, simulation approach may be a good choice to view the overall coordination scenario of the whole supply chain. • Most of the studies on coordination are done for two level supply chains. This assumption may restrict the usage of models, as these models may not handle the ever-changing variables of supply chain. • The assumption of integrated different functions and processes leads to cost reduction, but models are required to evaluate or measure the degree of coordination (which leads to improvement in the supply chain performance).

A Review on Supply Chain Coordination

57

• The recent trend of outsourcing the logistics operations to third party logistics provider (3PL) has reduced many discrepancies related to replenishment of goods (Jayaram and Tan 2010). The studies are required how to 3PLs can be an information source to coordinate suppliers and buyers. The knowledge and expertise of 3PLs on routes, fleet size and fleet type can be leveraged in optimizing the procurement-production-distribution problems and integrating with 3PLs. • To gain the advantage of common logistics provider and information systems, the supply chain members at same level may coordinate horizontally. Very few papers have discussed horizontal collaboration (Arshinder et al. 2006; Bahinipati et al. 2009) by using multi-criteria decision making models. Some quantitative models can be proposed to quantify such kind of coordination also. In this section we can observe that how supply chain coordination is required in each SC process. Various processes have been coordinated by adopting different means mechanisms of coordination. By looking at the need of coordination in SC, the researchers may like to know various existing coordination mechanisms, which can be adopted to coordinate supply chain across different industries. The next section presents various coordination mechanisms, which can be adopted as per the suitable supply chain environment.

4 Various Supply Chain Coordination Mechanisms Adopted in the Supply Chain The dependencies between supply chain members can be managed by some means and mechanisms of coordination. By utilizing coordination mechanisms, the performance of supply chain may improve. There are different types of coordination mechanisms as discussed in the following subsection.

4.1

Supply Chain Contracts

Supply chain members coordinate by using contracts for better management of supplier buyer relationship and risk management. The contracts specify the parameters (like quantity, price, time, and quality) within which a buyer places orders and a supplier fulfills them. The objectives of supply chain contracts are: to increase the total supply chain profit, to reduce overstock/understock costs and to share the risks among the supply chain partners (Tsay 1999). The contracts counter double marginalization that is by decreasing the costs of all supply chain members and total supply chain costs when they coordinate as against the costs incurred

58

K. Arshinder et al.

when the SC members act independently. The problem of double marginalization and risks like overstock and understock has been widely been observed single period inventory models with less shelf life of product. Most of the contracts have been proposed as single period models. Various contracts are defined in Table 5. Buyback contracts or returns policy has been widely used coordination contract in textile and fashion industry. In buyback contracts a manufacturer offers retailer either full credit for a partial return of goods a partial credit for all unsold goods. In case of retail competition the manufacturer will be benefited from the returns policy when the production costs are sufficiently low and demand uncertainty is not too great (Padmanabhan and Png 1997). Krishnan et al. (2004) have analyzed that Table 5 Definitions of supply chain contracts S. Supply chain Definition No. contract 1 Buy back The manufacturer (seller) agrees to buy back the unsold units from the retailer (buyer) for agreed upon prices at the end of the selling season 2 Revenue In a revenue sharing contract, the sharing buyer shares some of his revenues with the seller, in return for a discount on the whole sale price 3 Sales rebate The sales rebate provides a direct incentive to the retailer to increase sales by means of a rebate paid by the supplier for any item sold above a certain quantity 4 Quantity It couples the customer’s flexibility commitment to purchase no less than a certain percentage below the forecast with the supplier’s guarantee to deliver up to a certain percentage above 6 Trade policy This policy deals with how the total profit is shared among supply chain entities 7 Reservation This policy offers discounts to the policy products reserved and the products which are not reserved are sold at retail price After the selling season, the unsold 8 Markdown money units are sold at discounted price (price discount) 9 Quantity During the selling period, the seller discount offers discounts based on quantity of goods purchased

Author (year)

Remarks

Mantrala and Improves the coordination, Raman increases (1999), Hau and Li (2008) sales, risk sharing Yao et al. (2008), More flexible in terms in terms Zhou and of whole sale Wang (2009) price Wong et al. Provides direct (2009) incentives for retailers to increase sales Tsay (1999)

Gives more flexibility in order quantity

Ding and Chen (2008)

Offers better profit sharing

Chen and Chen (2009)

Reduces the uncertainty in demand

Lee (2001), Pan et al. (2009)

Improves profit of the channel

Weng (2004)

Improves the sales

A Review on Supply Chain Coordination

59

buyback contract coupled with promotional cost sharing agreements between manufacturer and retailer result in supply chain coordination. The other consideration in buyback contract is the case of information sharing and asymmetrical information between the supply chain members (Yao et al. 2005; Yue and Raghunathan 2007). Bose and Anand (2007) proposed that by assuming transfer price exogenous the buyback contract is Pareto efficient. Yao et al. (2008) proposed an analytical model to analyse the impact of stochastic and price dependent demand on returns policy between manufacturer and retailer. The other variants of buyback contracts discussed in literature are: stochastic salvage capacity in fashion industry (Lee and Rhee 2007); two period contract model in case of decentralized assembly system (Zou et al. 2008); in case of updating of information in supply chain (Chen et al. 2006) and by including the risk preferences of the SC members (He et al. 2006). In case of quantity flexibility contract, the buyer is allowed to modify the order within limits agreed to the supplier as demand visibility increases closer to the point of sale. The buyer modifies the order as he gains better idea of actual market demand over time. Tsay and Lovejoy (1999) proposed quantity flexibility contracts for two independent members of the supply chain model to design incentives for the two parties to determine system wide optimal outcome. The efficiency can be improved when buyer is ready to pay more to the supplier for increased flexibility. Tsay and Lovejoy (1999) proposed a framework for the design of quantity flexibility in three level supply chains, behavioural models in response to quantity flexibility contracts and the impact on the supply chain performance measures: inventory levels and order variability. More output flexibility comes at the expense of greater inventory cost, so inventory management has been viewed as the management of process flexibilities. It is observed that the quantity flexibility contracts can dampen the transmission of order variability throughout the supply chain. Milner and Rosenblatt (2002) analysed two period quantity flexibility contract in which the buyer is allowed to adjust second order paying a per unit order adjustment penalty. This contract can reduce the potentially negative effect of correlation of demand between two periods, but the order quantity flexibility reduces the profits of the buyer. Barnes-Schuster et al. (2002) proposed two period options contracts where buyer has flexibility to respond to market changes in second period and coordinate the supply chain channel. Sethi et al. (2004) developed a model to analyze a quantity flexibility contract involving multiple periods, rolling horizon demand and forecast updates including demand and price information updates. In revenue sharing contract, the supplier charges the buyer a low wholesale price and shares a fraction of the revenues generated by the buyer (Giannoccaro and Pontrandolfo 2004; Cachon and Lariviere 2005; Koulamas 2006). The SC members can design contracts based on discounts: lot size based or volume based. Yao et al. (2008) developed a revenue sharing model in the case of retail competition by considering price sensitivity. vander Rhee et al. (2010) has considered multi echelon (more than two) supply chain members and simultaneously installed revenue sharing contracts between all pairs of adjacent supply chain members to coordinate the supply chain.

60

K. Arshinder et al.

A discount is lot size based if the pricing schedule offers discounts based on the quantity ordered in a single lot. A discount is volume based if the discount is based on the total quantity purchased over a given period regardless of the number of lots purchased over that period (Rubin and Benton 2003; Weng 2004). Chauhan and Proth (2005) proposed a profit sharing model under price dependent demand proportional to their risks based on expected customer demand.

4.2

Role of Information Sharing and Information Technology

IT is used to improve inter-organizational coordination (McAfee 2002; Sanders 2008) and in turn, inter-organizational coordination has been shown to have a positive impact on select firm performance measures, such as customer service, lead-time, and production costs (Vickery et al. 2003). Information technology helps to link the point of production seamlessly with the point of delivery or purchase. It allows planning, tracking and estimating the lead times based on the real time data. Advances in Information Technology [e.g. internet, EDI (electronic data interchange), ERP (enterprise resource planning), e-business and many more] enable firms to rapidly exchange products, information, and funds and utilize collaborative methods to optimize supply chain operations. Internet and web can enhance effective communication, which helps members of supply chain review past performance, monitor current performance and predict when and how much of certain products need to be produced and to manage workflow system (Liu et al. 2005). Fin (2006) investigated the relation between EDI in apparel industry and three performance levels: operational, financial and strategic. This helped in reduction of lead time from several weeks to 3 days. According to Soliman and Youssef (2001), e-business strategy refers to the way internet tools are selected and used in relation to the needs of integration and coherent with other organizational and managerial tools: e-commerce (Swaminathan and Tayur 2003) can be used to support processes such as sales, distribution and customer service processes, support to sourcing, procurement, tendering, and order fulfillment processes, and e-manufacturing (Kehoe and Boughton 2001). Devaraj et al. (2007) analyzed the relationship between supplier integration and customer integration with supply chain performance when supported by e-business technologies. E-business capability supporting supply chain technologies such as customer orders, purchasing and collaboration between suppliers and customer enhances the production information integration intensity, which in turn improves the supply chain performance. Skipper et al. (2008) proposed a conceptual model to link level of interdependence among supply chain with supply chain performance moderated by different types of IT needed to achieve different levels of coordination. The framework is supported by interdependence theory and coordination theory. The coordination processes between globally dispersed and mobile supply chain members is becoming more and more information intensive. The recent trends in intelligent wireless

A Review on Supply Chain Coordination

61

web services have proved enhancement in the mobile real time supply chain coordination (Saroor et al. 2009). The various coordination problems handled by information systems are: little value to the supplier because of competitive bidding, forced implementation of IT, incompatible information system at different levels of supply chain, greater lead times, inefficient purchase order and misaligned e-business strategies and coordination mechanisms (Porter 2001). Stank et al. (1999) report that the food firms benefit from more accurate and timely information and IT or EDI improves inventory management and helps in comprehension of the order cycle. Yusuf et al. (2004) examined key dimensions of implementation of ERP system in Rolls Royce. The implementation of latest information system only may not be sufficient to integrate supply chain members, since at times; faulty implementation may result in the poor performance of supply chain. Li et al. (2009) carried out an empirical study to explore relationship between IT, supply chain integration and supply chain performance of Chinese manufacturing organization. Supply chain integration mediates the relationship between IT implementation and supply chain performance. Hence, IT can be a good enabler to integrate supply chain. But it is important to take into account the justification of IT in changing business environment. It must take into account the appropriate usage, investment justification and align with business environment to achieve competitive advantage (Gunasekaran et al. 2006). The supply chain members coordinate by sharing information regarding demand, orders, inventory, shipment quantity, POS data, etc. Timely demand information or advanced commitments from downstream customers helps in reducing the inventory costs by offering price discounts and this information can be a substitute for lead time and inventory (Reddy and Rajendran 2005). The value of information sharing increases as the service level at the supplier, supplier-holding costs, demand variability and offset time increase, and as the length of the order cycle decrease (Bourland et al. 1996; Chen et al. 2000). The higher the level of information sharing, the more important the effective supply chain practice is to achieve superior performance (Zhou and Benton 2007). Some comparative studies have done in which no information sharing policy is compared with full information sharing policy. Information sharing policy results in inventory reductions and cost savings (Yu et al. 2001). Cachon and Fisher (2000) presented a simulation-based comparative study, where the supply chain costs are 2.2% lower on average with full information sharing policy than with traditional information policy and the maximum difference is 12.1%. Also, this results in faster and cheaper order processing that leads to shorter lead times. The point of sales (POS) data helps the supplier to better anticipate future orders of the retailers and reduces the bullwhip effect (Dejonckheere et al. 2004). The supplier may take advantage of the retailers’ inventory information in allocating the stock to retailers optimally (Moinzadeh 2002). Ding et al. (2011) has investigated the mechanism of providing incentive to retailer by upstream partner for implementing demand information sharing in the context of three-echelon supply chain system. A cooperative game approach is

62

K. Arshinder et al.

proposed to address the problem of profit allotment between partners to effectively motivate the partners to be cooperative with each other.

4.3

Other Collaborative Initiatives

Joint consideration of replenishment (Yao and Chiou 2004; Chen and Chen 2005), inventory holding costs with dynamic demand (Boctor et al. 2004), collaborative planning (Aviv 2001), costs of different processes (Haq and Kannan 2006; Jayaraman and Pirkul 2001; Ganeshan 1999), frequency of orders (Yang and Wee 2002; Barron 2007), batch size (Pyke and Cohen 1993; Boyaci and Gallego 2002), product development (Kim and Oh 2005) to improve the performance of supply chain. A supply chain member may design a scheme to share profits at the end of period. The supply chain members share profit by determining optimal order quantity of single supplier and multi-buyer supply chain and achieve coordination (Jain et al. 2006). A coherent decision-making helps in resolving conflicts among supply chain members and in exceptions handling in case of any future uncertainty. There are many factors involved in achieving coordination like human, technology, strategies, relationship, rewards, sharing of knowledge, sharing benefits, aligning goals, scheduling of frequent meetings of stakeholders for conflict resolution, understanding of nature of intermediates and knowledge of supply chain concepts, status or power difference and resistance in following the instructions of other organizations (Lu 1995; Gittell and Weiss 2004). Simatupang et al. (2004) explored a fashion firm to see how coordination is driven by its responsibility interdependence, uncertainty, and inter-functional conflict. By properly identifying different points of coordination, the performance improvement was effected. Vendor Managed Inventory (VMI) is a supply chain initiative whereby a supplier assumes responsibility for maintaining inventory levels and determining order quantities for its customers. A number of benefits from VMI adoption have been reported in literature: reduction in inventories, shorter order intervals and more frequent deliveries. A VMI program typically involves the use of a software platform, the sharing of demand forecasts and/or cost information, timely communications, set liability levels, and risk-sharing parameters and common goal sharing between the buyer and the supplier. VMI can be particularly beneficial in the products with high demand variance and high outsourcing costs (Cheung and Lee 2002). Collaborative Planning, Forecasting and Replenishment (CPFR) is a collaboration initiative where two or more parties in the supply chain jointly plan a number of promotional activities and work out synchronized forecasts, on the basis of which the production and replenishment processes are determined (Larsen et al. 2003). Some of the benefits of CPFR are increased sales, higher service levels, faster order response time, lower product inventories, faster cycle times, reduced capacity requirements, reduced number of stocking points, improved forecast accuracy and lower system expenses. Danese et al. (2004) explored the relationship between the

A Review on Supply Chain Coordination

63

types of interdependencies (one way and two way communications) among the units involved in the CPFR processes and the activated coordination mechanisms (Liaison positions, meetings, task forces, standing committees and integrating managers) in three case studies for all the steps of implementation of CPFR. The case studies were considered from different industries: pharmaceutical, automotive and mechanical. This relationship may help managers in the decision making process to select the most appropriate action to perform to implement CPFR. Quick response (QR) is another inventory management initiative which can be undertaken to coordinate supply chain members by responding quickly to market changes with reduced lead time. The response time is reduced as a retailer sends POS data to its supplier. The supplier makes use of this information to improve the demand forecast and production/distribution schedules (Iyer and Bergen 1997; Simchi-Levi et al. 2007). Choi and Sethi (2010) have reviewed QR supply chains from both supply and demand perspectives and classified the literature as supply information management, demand information management and supporting technologies. It is concluded that there are challenges to implement QR in multiple decision points, which needs to be met by continuously innovating new technologies like Radio Frequency Identification Devices (RFID). The Supply Chain Operations Reference (SCOR) model helps in evaluating and improving enterprise wide supply chain performance and management. SCOR is structured on four levels: plan, source, make and deliver. It brings order to the diverse activities that make up the supply chain, and provides common terminology and standard process descriptions. The model allows companies to: evaluate their own processes effectively, compare their performance with other, companies both within and outside their industry segment, pursue specific competitive advantages, use benchmarking and best practice information to prioritize their activities, quantify the benefits of implementing change and identify software tools best suited to their specific process requirements (Huan et al. 2004). Observations and Gaps in Coordination Mechanisms (a) The supply chain contracts can be a useful mechanism to resolve the conflict and risk related problems. The use of information technology in handling transactions online between supply chain members reduces the response time. The members can plan their operational activities by sharing or retrieving the data from each other. It helps in streamlining the processes and reduces supply chain costs. (b) The members might have different technologies, skill and different type of knowledge about market. To handle any future exceptions or uncertainties, the members may jointly plan supply chain activities like ordering, replenishment, and forecasting and product design. (c) The following gaps regarding coordination mechanisms need attention to enhance coordination: • Since the role and utility of all coordination mechanisms is handling different phases of supply chain. To coordinate supply chain as a whole,

64

K. Arshinder et al.

the consideration of all coordination mechanisms may give very good performance. • Most of the models describing coordination mechanisms are dealt in two level supply chain, which can be extended to multi-level supply chain. The relation between different coordination mechanisms and the performance measures of supply chain need to be developed. The models handling the problems of coordination have emphasized on single performance measures. The supply chain dynamics may be captured by considering a number of performance measures of supply chain. • Supply chain contracts are designed to motivate the downstream member to order more than his/her optimal order quantity. The downstream member always faces uncertainty of overstock or under stock. The upstream member always faces uncertainty that whether the downstream member will send the order matching the upstream member’s capacity. The contracts like buyback and revenue sharing contracts can enhance expected sales and reduces stock outs. Quantity flexibility contracts can reduce the overstock problems of downstream members. These performance indicators are equally important, which needs more research attention. • The contract decision variables at different interfaces of the supply chain in multi echelon environment interact with each other. For example the contract adopted by supplier and manufacturer is sometimes dependent on the contract adopted by same manufacturer with his/her distributor in a same supply chain. There is a need to explore such relationship and to explore different combinations of contracts at different interfaces of supply chain. The major driver of SCC is the conflict or uncertainty, which needs to be addressed by selecting suitable coordination mechanism. But, it is important to understand at the same time, to what extent SCC can help in mitigating supply chain uncertainty (presented in the next section).

5 Supply Chain Coordination to Manage Uncertainty in the Supply Chain Supply chain uncertainty has been captured in various forms like supply uncertainty, production or operational uncertainty and demand uncertainty. In the supply chain coordination literature, various coordination mechanisms have been adopted to manage supply chain uncertainty like uncertainty in capacity, demand, lead time, quantity and production and supply disruptions (Tang and Musa 2011) as shown in Fig. 2. Many papers have emphasized on supply chain contracts and information exchange/sharing to manage supply chain uncertainties. Whereas, the other set of papers discussed the joint consideration of costs and profits of all supply chain members while taking decisions regarding ordering and replenishment. This joint

A Review on Supply Chain Coordination Supply side uncertainty

Supplier Production/ Operational uncertainty

65 Supply side uncertainty

Buyer Production/ Operational uncertainty

Buyer’s side uncertainty

Supplier

Coordination mechanisms

Demand uncertainty

Buyer

Coordinate as SC members are part of one system to manage uncertainty and to share risks and rewards

Supply Chain Performance Improvement Fig. 2 Managing supply chain uncertainty with supply chain coordination

consideration of costs or profits (centralized system) helps to improve the performance of supply chain over a decentralized case (independent decision making). Due to the increased technological innovations, the products’ lifecycle has largely shortened. Seasonal and perishable goods can be attributed to this kind. Such products have longer production and delivery lead time than their selling season (Mantrala and Raman 1999). So the orders should be placed before the selling season starts. Some of the important challenges in integrating the supply chain are tackling issues such as managing complex supply chain structures, demand uncertainty and leftover units after selling season. In a single period inventory model, better coordination can be achieved by inducing the retailer/ buyer to order more in order to avoid their risk of under stocking through some negotiations with the manufacturers/seller. The manufacturer offers integrated decision making policies like returns policy, sales rebate policy, price discount/ volume discount policy, etc. to raise the order quantity and improves sales (Yao et al. 2008). Past research has proved that introduction of various contracts improve the performance of the supply chain as well of each entity in supply chain.

66

K. Arshinder et al.

The contracts have been discussed for single period inventory models with either deterministic demand or uncertain demand or price dependent demand. Apart from contracts there can be some incentive function to achieve flexible cost allocation between supplier and buyer to coordinate the supply chain and to manage uncertainty in supply (Zimmer 2004). Hou et al. (2010) have proposed a model considering one manufacturer and two suppliers: the main supplier is cheaper but prone to disruption risk and backup supplier is more reliable but an expensive. The authors have developed a non-linear optimization model to determine the optimal values of buyer’s order quantity and optimal buyback price under both supply and demand uncertainty. Early supply involvement reduces the likelihood of supply disruptions and negative supply events (Zsidsin and Smith 2005). The contracts like advanced purchase commitments can help mitigating supply uncertainty, where unsatisfied demand can be backordered from risky supplier (Serel 2007). The other kind of uncertainty due to disruption can be observed as disruptions in the production process at manufacturer’s facility. Qi et al. (2004) proposed a model for short life cycle product with demand as decreasing function of retail price considering disruptions. The model considered the two periods wherein the second period demand change can lead to the disruption, which may affect the production plan of supplier. The wholesale quantity discounts may coordinate the supply chain in this scenario of disruption. The similar kind of disruption can be seen in terms of production costs. Xiao and Qi (2008) developed two-period model for onemanufacturer and two competing retailers supply chain under production costs disruption. The authors have analyzed two mechanisms; an all unit quantity discounts and incremental quantity discounts the under production disruptions for possible coordination scenarios. A risk sharing contract is proposed where at the end of period the retailer compensates manufacturer’s losses due to overproduction or manufacturer compensates retailer’s losses due to over stock in case of supply chain with two stage demand information updating (Chen et al. 2006). There can be several benefits of splitting the single period order into multiple ordering to update the demand information and revise the order in the subsequent orderings. It has impact on production costs of the manufacturer due to slow production and fast production as against the multiple different orders (Liu et al. 2004). The other effect of multiple ordering can be seen on holding cost, lead time, backorders, varying wholesale and retail price and consideration of demand for multiple periods. The methodology adopted for handling multiple ordering ranges from newsboy problem to analytical models with simulation to the dynamic programming. The decision variables have been the order quantities and/or the varying wholesale prices, retail prices and buyback prices in multi-period situation (Lee 2007; Zhou and Wang 2009; Pan et al. 2009). Other aspect of capturing demand uncertainty is by using fuzzy demand. The expected profits of coordinated supply chain outperform the expected profits in the case of no coordination under fuzzy demand (Xu and Zhai 2010). Barbarosoglu (2000) has proposed a decision support model for improving supplier–buyer coordination by using supply contracts where the buyer’s commitment is considered as a

A Review on Supply Chain Coordination

67

function of time at the contract renewal time to reduce the supply chain nervousness. A pricing model is formulated to address partnership expectations for a fair sharing of savings of the supply chain members. Observations and Gaps in Uncertainty and Supply Chain Coordination (a) Most of the studies are restricted to two level serial supply chains. In reality, supply chain can have divergent and convergent multi-echelon structures. The literature seems lacking to address the uncertainty concerns in such structures. (b) The literature has emphasized more on demand uncertainty, whereas, supply uncertainty can be of equal concern in the era of globalization and outsourcing. Moreover, the quantitative models can be proposed to explore the impact of supply uncertainty on supply chain performance. (c) There are very few studies on splitting the single period order into multiple orders. The supply chain members can take advantage of more accurate information over a period of selling season and hence resolve supply chain inefficiencies. (d) The buyback contract is the only contract which has been discussed in multi ordering models to manage the risk. There is a scope to explore combination of other contracts in multiple ordering over single season.

6 Discussion A number of difficulties in SCC are identified based on the literature. These difficulties have been identified from different activities, interfaces and the number of levels in the supply chain. It has been realized that the difficulties in SCC and independent working of supply chain members lead to poor performance. The coordination problems are solved by implementing some coordination mechanisms in supply chain activities, which may result in the improvement of some performance measures. The SC activities have been considered in isolation to solve their respective coordination problem. The coordination problems may not be same in all activities of supply chain. The requirements of coordinating whole SC may vary with SC activity, with some interface of SC, with number of echelons in SC and with process of SC. There are different activities and different coordination problems in whole supply chain. Coordinating one activity may not help to improve supply chain system wide performance.

6.1

Existing Models of Coordination and the Gaps in These Models

There are some initiatives and models (such as CPFR and SCOR) which may help in collaboration along the supply chain. These models consist of so many steps and

68

K. Arshinder et al.

the implementation of such processes takes time. Various guidelines are required to implement these models in practice. It is difficult to link the guidelines directly to the performance of supply chain. It may take a number of years to know the performance improvement by implementing these models, as there is no set measure to quantify coordination which can be linked with practice (or which may result due to implementation of these models) of these models. It is difficult to get a quantitative measure after implementing models like CPFR and SCOR, which may indicate about whether SC is coordinated or not. The coordination models discussed have different performance measures at single level and at interface of supply chain, which are not aligned with the whole supply chain. To monitor coordination in supply chain, same performance measures throughout will help in evaluating the value of coordination. There are different mechanisms, which when applied, result in different trade-offs of performance measures of coordinated supply chain because of different characteristics of performance measures. The complexity of considering whole supply chain and the performance trade-offs cannot be handled with the models discussed in the literature. These difficulties can be easily tackled by approaches like fuzzy logic (Ross 1997) and multi-objective genetic algorithms (Deb 2002). Fuzzy logic is applied in the situation where understanding is quite judgmental and the processes where human reasoning and human decision making is involved like the complexities in supply chain. The optimum values of decision variables in multi objective environment can be easily determined with the help of tools like Genetic Algorithm.

6.2

Proposed Framework to Quantify Coordination

The controlling parameter of achieving coordination is the impact of application of coordination mechanisms (CMs) on the performance measure. It can be observed from the Decision-coordination mechanism matrix given in Table 6 that how different coordination mechanisms can be used for various supply chain decisions. The proper implementation and usage of coordination mechanisms improve the performance of the supply chain (Arshinder 2008). It can be observed that the problems and conflicts in coordinating the supply chain members can be resolved through coordination mechanisms. The importance of coordination mechanism may help in determining the value of coordination in supply chain.

6.2.1

Framework Using Various Coordination Mechanisms

A framework has been proposed based on the usage of coordination mechanisms and their importance in managing uncertainty and resolving various kinds of conflicting problems in coordination. The coordination mechanisms can be classified as:

Coordination mechanism

Supply chain network Integrated procurementproduction

Integrated production-distribution Joint consideration of cost

X

X

Coordinated timing of the order

Coordinated timing of replenishment Inventory management in a network Forecasting

X

X

X

X

X

X

Supply Information chain technology contracts

Inventory Coordinated order quantity

Logistics Coordination issues in 3PL provider and customer Integrating the logistics activity geographically dispersed network/supply chain

Supply chain decision

Table 6 Decision-coordination mechanism matrix

X

X X

X

Jayaraman and Pirkul (2001), Pyke and Cohen (1993), Kim et al. (2006) Dotoli et al. (2005) Goyal and Deshmukh (1992), Munson and Rosenblatt (2001)

Aviv (2001)

X

X

Speed of delivery, status of order, accuracy of information, invoicing on delivery, cash-flow improvements, accurate invoicing, transportation costs, warehousing costs, inventory levels, ordering costs, stock-outs, order cycle time, order cycle variance, on time deliveries

Performance measures

Supply chain system wide cost

Supply chain system wide cost

Bullwhip effect, holding cost, system wide cost

Li et al. (1996), Boyaci and Gallego (2002), Inventory levels, ordering costs, customer Piplani and Fu (2005), Zou et al. (2004), service level, holding costs, product Wu and Ouyang (2003) variety, purchasing costs, product availability, unacceptable delivery, Lu (1995), Moses and Seshadri (2000), system wide costs Gurnani (2001), Zhao et al. (2002) Huq et al. (2006), Barron (2007)

Stock et al. (2000)

Huiskonen and Pirttila (2002)

Authors

Verwijmeren et al. (1996)

X

X

X

X

Joint decision making

X

X

X

X

Information sharing

A Review on Supply Chain Coordination 69

70

• • • •

K. Arshinder et al.

Supply chain contracts (M1) Information Technology (M2) Information sharing (M3) Joint decision making (M4)

This is not an exhaustive list of coordination mechanisms. These coordination mechanisms can be different in number as per the requirements of supply chain for example dependent on the type of industry and type of interdependencies between SC members. In the present framework four coordination mechanisms are considered because of their extensive discussion in literature. It can be assumed without any loss of generality, that if the coordination mechanism is applied properly, it will help in achieving SCC. Since, supply chain involves certain members who are human beings and the human system is the most complex system to be managed in organization study. There is bound to be conflicts and problems in the traditional supply chain, which call for an urgent need to implement coordination mechanisms in supply chain. The coordination mechanisms are from different domains, require different conditions and can operate in different situations. But, one thing common in all mechanisms is that all mechanisms are implemented to improve the performance of supply chain and to resolve confusion and uncertainty among SC members due to independent decision making. To know more about the importance of coordination mechanism, one way is to study all the activities in some process, identify the dependent activities in that process and select the coordination mechanism to coordinate all activities of a process (Arshinder et al. 2006). Since, whole supply chain needs to be coordinated; the usage of all four coordination mechanisms and performance improvement achieved by these mechanisms will help in evaluating SCC. A better way to find some quantitative index of supply chain coordination is by incorporating the strength of coordination mechanisms by following steps shown in Fig. 3. The quantitative index can be represented as Supply Chain Coordination Index (SCCI) can be viewed as a function of implementation of coordination mechanisms. SCCI for four coordination mechanisms can be represented as: SCCI ¼ f ðM1; M2; M3; M4Þ The above function is to be formulated in such a way that the combined impact of performance improvement by using all mechanisms is considered. This formulation poses two challenges: 1. It is required to represent all coordination mechanisms with a unique scale. 2. It is required to evaluate improvement in performance measures qualitatively or quantitatively by using coordination mechanisms. The methodologies like AHP and Fuzzy logic may help to represent coordination mechanisms with a unique scale. The performance improvement can be captured either empirically with the help of judgments given by managers or

A Review on Supply Chain Coordination

71

Define the structure of supply chain

Set performance measures for whole supply chain

Choose input variables

Run the simulation for The case when the members are working independently Observe the impact on performance measures and set as PM(w/c) (without coordination)

Select the coordination mechanisms and run the simulation

Supply chain contracts (M1) Buyback Revenue sharing Quantity flexibility Quantity discounts

Determine performance measure (PMM1C)

Information Technology (M2) Email Internet EDI ERP POS

Determine performance measure (PMM2C)

Information sharing (M3) Demand Inventory Lead time Production schedule Capacity Cost

Determine performance measure (PMM3C)

Joint decisionmaking (M4) Cost consideration Replenishment Forecasting Ordering

Determine performance measure (PMM4C)

Determine the percentage improvement in performance measures with respect to case of without coordination PMMi = (PMMiC – PMMi (w/c))/ PMMi (w/c) for i=1,2,3,4

Assign weights to different coordination mechanisms (WMi, for i= 1,2,3,4) based on relative improvement of percentage of all CMs by devising some scale (AHP).

WM1

WM2

WM3

WM4

SCCI = WM1PMM1+ WM2PMM2 + WM3 PMM3 + WM4 PMM4

Fig. 3 The proposed model to quantify supply chain coordination index (SCCI)

with the help of simulating the scenarios of using these coordination mechanisms to obtain same performance measures. The improvement in performance measures will motivate supply chain members to implement coordination mechanisms.

72

K. Arshinder et al.

One of the efforts has been proposed based on the implementation of all coordination mechanisms with the help of graph theoretic approach (Kaur et al. 2006). This methodology is based on allocation of relative importance of these coordination mechanisms given by the judgments of managers. These judgments are based on the implementation of mechanisms and the importance of mechanisms based on the performance improvement by these mechanisms.

6.2.2

Relation Between Coordination Mechanisms and Performance Measures with Simulation

A simulation approach can also be a useful tool in capturing the different scenarios of coordination mechanisms and their impact on selected performance measures. Certain assumptions can be considered regarding the levels of supply chain, one period or multiple period model and various operational variables like order quantity, holding and shortage costs, etc. The implementation of various coordination mechanisms can be simulated to analyze same performance measures with same assumptions. Some constraints can be included in the model which takes care of the fact that none of the supply chain member will face losses by implementing coordination mechanisms. The improvement in performance measures will give an idea about the capability of an organization to achieve coordination. The model proposed in Fig. 3, helps in evaluating SCCI. The first few steps can be used in simulation to determine the performance measures. Some input variables may be selected like different costs, price, inventory policies, lead time, capacity and type of coordination mechanisms at all levels of supply chain in a pre defined structure of supply chain. The assumptions for demand (uncertain and price dependent), lead time and time horizon can be set for the simulation and run the simulation to obtain certain performance measures. The performance measures are function of input variables. The problem may be multi objective based on the selected performance measures of supply chain. The results of simulation that is improvement in the performance measures by applying different coordination mechanisms can be combined using again some hybrid frameworks like: AHP, Fuzzy logic and/or Graph theoretic approach to determine SCCI.

6.2.3

Hybrid Framework Using Various Coordination Mechanisms and Simulation

The coordination mechanisms (M1, M2, M3 and M4) have different characteristics and their impact on the performance measures may also be different. The simulation can be carried out without implementing coordination mechanisms and then the results are compared with the situation with considering the coordination

A Review on Supply Chain Coordination

73

mechanisms as shown in Fig. 3. A framework is required which can capture and combine the values of performance improvements by coordination mechanisms and their relative importance. To make the results consistent, the performance improvements can be normalized in terms of percentages. This framework may have capability to find the relative importance of respective coordination mechanisms by using AHP and/or Fuzzy logic (Arshinder et al. 2007). A scale can be devised based on the difference in the percentage improvements by CMs. This scale may help in determining the relative importance or weights of CMs. The linear equation of SCCI can be derived from the proposed model to determine the value of SCCI.

6.3

Insights Gained from Proposed Framework

The proposed framework helps in defining and measuring SCC. Supply chain coordination can be used to enhance system wide performance enabled due to the implementation of coordination mechanisms selected based on the type of industry and the interdependencies between supply chain members keeping in view mutual interests of all SC members. Supply chains can be coordinated by identifying interdependent activities between supply chain members required to accomplish SC objectives. Once interdependencies are identified, some means of mechanism(s) are devised to manage the decision variables pertaining to interdependent activity. The independent evaluation of decision variables of interdependent activities by SC members represents the case of uncoordinated supply chains. Once, coordination mechanism is selected to manage interdependencies, SC members can simulate and compare the scenarios: one with using CM and other without coordination mechanisms. The expected values of improvement in certain performance measures may help to realize the value of coordination. Same steps can be used for all processes of supply chain. Various functions can be explored for SCCI depending on the number and implementation of CMs. Suitable techniques can be used such as Multi-CriteriaDecision-Making (MCDM) models to quantify SCCI as a function of various CMs.

6.4

Surrogate Measures of Supply Chain Coordination

To innovate continuously is the base line for all the organizations, which makes the supply chain more dynamic in nature. It is important to capture the performance of supply chain. The highly uncertain environment in supply chain brings in the challenges to have fix kind of performance measures. Gunasekaran et al. (2001) developed a framework for measuring supply chain performance for each activity

74

K. Arshinder et al.

of plan, source, make and deliver under strategic, tactical and operational decisions. The literature on supply chain performance measures is lacking in presenting standard performance metrics. The problem manifolds when the question comes to measure supply chain coordination. There is scarcity of studies to evaluate coordination in supply chain. The following performance measures can be good indicators of supply chain coordination. (a) Supply chain profitability. Joint consideration of order quantity, costs or profits may lead to improvement in supply chain performance. Regardless of the number of entities in supply chain, the joint consideration of order quantity in supply chain for single period model improves the profitability of whole supply chain (Arshinder et al. 2009a). Most of the contracts reported in literature have expected profits as a performance indicator too. (b) Supply chain flexibility. When the supply chain members coordinate with each other by using contracts, it gives more flexibility to supply chain members to change order quantity, price, cost and lead time. The lower and upper bound can be set for decision variables of contracts (coordination mechanisms) to ensure that the performance of each SC member in a centralized case (consideration of all SC members to be a part of one system) with appropriate coordination mechanism is better than decentralized case (individual supply chain member). Various supply chain contracts present different kinds of supply chain flexibility (Arshinder et al. 2008a). (c) Mitigating uncertainty or risk sharing. The recent issue in supply chain coordination is “How to allocate the total gain in the supply chain achieved due to coordination after mitigating risk?” Many studies have recently developed game theoretic models to fairly share the rewards among supply chain members. The risk mitigation in the form of gain in whole supply chain can be a surrogate measure of SCC. In similar way the extra share of profit allocated out of total gain in SC due to coordination can also reflect the coordinated supply chain. It has also been observed that as the demand variance is increased, the coordinated supply chain due to contracts outperform the independent case of supply chain (Arshinder et al. 2008b). The SC members can devise the contracts in which supplier gives assurance to the buyer to supply emergency orders in case of sudden surge in demand to share risk of losing a customer. Whereas, the buyer can share the extra cost incurred by the supplier in producing emergency orders in view of uncertainty in demand. How well such kind of contracts is designed can be a good indicator of coordination to share risks due to uncertainty in supply chain (Serel 2007). (d) Supply chain coordination index. As it has been discussed that various combination of coordination mechanisms can improve the performance of supply chain. Many situations in supply chain need more than one coordination mechanisms like VMI with quantity discounts, supply chain contracts with information sharing, supply chain contracts with joint decision making (joint consideration of costs). Such kind of index has been developed in Arshinder et al. (2009b) (also mentioned in the proposed framework).

A Review on Supply Chain Coordination

75

7 Major Challenges and Future Research Directions Coordinating the supply chain across organizational boundaries may be one of the most difficult aspects of supply chain management. Many firms simply are unaware of the fundamental dynamics of supply chains, but even those firms that are enlightened enough to understand these dynamics are often unable to realize inter-organizational coordination. Often the most effective supply chains have a dominating organization that sees the benefits of SCC and forces the rest of the supply chain to comply (i.e., global leader in retailing such as Wal-Mart). Many supply chains, however, either do not have a dominant organization, or the dominating organization is unenlightened. In these instances, coordinating the supply chain is most difficult. Typically, it is observed that the SCC problems could be due to the conflicting objectives that leads to a short time relationships with SC members, hence the environment and expectations changes frequently with dealing with new members. On this background, it is essential that the SC members need to appreciate the importance of coordination. This paper has attempted to deliberate on various theoretical perspectives on SCC. The objective to achieve coordination is limited only to the individual functions, to the single coordination mechanism at interfaces of supply chain and to achieve restricted performance measures. A holistic approach towards coordination in whole supply chain is a big challenge, which motivated to propose the issues of SCC in this paper. The mechanisms for coordination need to be studied in detail. The coordination mechanisms can further be of different sub types. To coordinate the whole supply chain, the aggregation of the impact of all coordination mechanisms on the performance of supply chain is required. Various combinations may be explored with the help of simulation. Supply chain contracts have proved to coordinate single period supply chains. The research is required to explore the utility of contracts in multi-period cases. In multi period model, the supply chain members are more expose to the uncertainty as they are dealing with supply chain members frequently. How various coordination mechanisms can be allied in multi period problems as well as can we evaluate coordination in such case? Very few studies have been reported to quantify risk or uncertainty in supply chain. The Bullwhip effect has extensively been discussed in the literature. Actually, there can be many variations seen in supply chain like supply uncertainty, delay in delivery having cascading effect as we go downwards in the supply chain, which is similar to the order variation in Bullwhip effect. How SCC can help in mitigating such uncertainties is one of the important research issues? Acknowledgements The authors are grateful to the reviewers and the whole editorial team of International Journal of Production Economics, Elsevier for constructive suggestions and for considering our paper for publication. The authors are also thankful to the referees of Springer’s Research Handbook Series on “Innovative Schemes for Supply Chain Coordination and Uncertainty” for the comments and suggestions to improve the quality of our paper.

76

K. Arshinder et al.

Appendix List of Journals Refereed in Review Paper 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16.

Computers & Industrial Engineering Computers & Operations research European Journal of Operational Research IIE Transactions International Journal of Logistics and System Management International Journal of Logistics Management International Journal of Operations and Production Management International Journal of Physical Distribution & Logistics Management International Journal of Production Economics International Journal or Production Research Journal of Operations Management Management Science Omega Supply Chain Management: An International Journal Transportation Research (Part E) Other Journals from Emerald, Inderscience and Sciencedirect portal

References Arreola-Risa A (1996) Integrated multi-item production-inventory systems. Eur J Oper Res 89(2):326–340 Arshinder K (2008) An integrative framework for supply chain coordination. Unpublished doctoral thesis, Indian Institute of Technology Delhi, New Delhi, Arshinder K, Kanda A, Deshmukh SG (2006) A coordination based perspective on the procurement process in supply chain. Int J Value Chain Manage 1(2):117–138 Arshinder K, Kanda A, Deshmukh SG (2007) Coordination in supply chains: an evaluation using fuzzy logic. Prod Plann Control 18(5):420–435 Arshinder K, Kanda A, Deshmukh SG (2008a) Supply chain coordination: perspectives, empirical studies and research directions. Int J Prod Econ 115(2):316–335 Arshinder K, Kanda A, Deshmukh SG (2008b) Development of a decision support tool for supply chain coordination using contracts. J Adv Manage Res 5(2):20–41 Arshinder K, Kanda A, Deshmukh SG (2009a) A coordination theoretic model for three level supply chains using contracts. Sadhana Acad Proc Eng Sci 35(5):767–798 Arshinder K, Kanda A, Deshmukh SG (2009b) A framework for evaluation of coordination by contracts: a case of two-level supply chains. Comput Ind Eng 56:1177–1191 Aviv Y (2001) The effect of collaborative forecasting on supply chain performance. Manage Sci 47(10):1326–1343 Bahinipati BK, Kanda A, Deshmukh SG (2009) Horizontal collaboration in semiconductor manufacturing industry supply chain: an evaluation of collaboration intensity index. Comput Ind Eng 57(3):880–895

A Review on Supply Chain Coordination

77

Barbarosoglu G (2000) An integrated supplier-buyer model for improving supply chain coordination. Prod Plann Control 11(8):732–741 Barnes-Schuster D, Bassok Y, Anupindi R (2002) Coordination and flexibility in supply contracts with options. Manuf Serv Oper Manage 4(3):171–207 Barratt M (2004) Understanding the meaning of collaboration in the supply chain. Supply Chain Manage Int J 9(1):30–42 Barron CLE (2007) Optimizing inventory decisions in a multi stage multi customer supply chain: a note. Transp Res E 43(5):647–654 Boctor FF, Laporte G, Renand J (2004) Models and algorithms for the dynamic joint replenishment problem. Int J Prod Res 42(13):2667–2678 Bose I, Anand P (2007) On returns policies with exogenous price. Eur J Oper Res 178(3): 782–788 Bourland KE, Powell SG, Pyke DF (1996) Exploiting timely demand information to reduce inventories. Eur J Oper Res 92(2):239–253 Boyaci T, Gallego G (2002) Coordinating pricing and inventory replenishment policies for one wholesaler and one or more geographically dispersed retailers. Int J Prod Econ 77(2):95–111 Burgess K, Singh PJ, Koroglu R (2006) Supply chain management: a structured review and implications for future research. Int J Oper Prod Manage 26(7):703–729 Cachon GP, Fisher M (2000) Supply chain Inventory management and the value of shared information. Manage Sci 46(8):1032–1048 Cachon GP, Lariviere MA (2005) Supply chain coordination with revenue sharing contracts: strengths and limitations. Manage Sci 51(1):30–44 Chandra P, Fisher ML (1994) Coordination of production and distribution planning. Eur J Oper Res 72(3):503–517 Chauhan SS, Proth JM (2005) Analysis of a supply chain partnership with revenue sharing. Int J Prod Econ 97(1):44–51 Chen TH, Chen JM (2005) Optimizing supply chain collaboration based on joint replenishment and channel coordination. Transp Res E 41(4):261–285 Chen LH, Chen YC (2009) A newsboy problem with simple reservation arrangement. J Comput Ind Eng 56(1):157–160 Chen F, Drezner Z, Ryan JK, Levi DS (2000) Quantifying the bullwhip effect in a simple supply chain: the impact of forecasting, lead times, and information. Manage Sci 46(3):436–443 Chen H, Chen J, Chen YF (2006) A coordination mechanism for a supply chain with demand information updating. Int J Prod Econ 103(1):347–361 Cheung KL, Lee HL (2002) The inventory benefit of shipment coordination and stock rebalancing in a supply chain. Manage Sci 48(2):300–306 Chikan A (2001) Integration of production and logistics – in principle, in practice and in education. Int J Prod Econ 69(2):129–140 Choi TM, Sethi S (2010) Innovative quick response programs: a review. Int J Prod Econ 114:456–475 Danese P, Romano P, Vinelli A (2004) Managing business processes across supply networks: the role of coordination mechanisms. J Purch Supply Manage 10(5):165–177 Deb K (2002) Multi-objective optimization using evolutionary algorithms. Wiley, Chichester Dejonckheere J, Disney SM, Lambrecht MR, Towill DR (2004) The impact of information enrichment on the Bullwhip effect in supply chains: a control engineering perspective. Eur J Oper Res 153(3):727–750 Devaraj S, Krajewski L, Wei JC (2007) Impact of eBusiness technologies on operational performance: the role of production information integration in the supply chain. J Oper Manage 25(6):1199–1216 Ding D, Chen J (2008) Coordinating three level supply chain with flexible returns policies. Omega 36(4):865–876 Ding H, Guo B, Liu Z (2011) Information sharing and profit allotment based on supply chain coordination. Int J Prod Econ 133(1):70–79

78

K. Arshinder et al.

Dotoli M, Fanti MP, Meloni C, Zhou MC (2005) A multi-level approach for network design of integrated supply chain. Int J Prod Res 43(20):4267–4287 Fawcett SF, Magnan GM (2002) The rhetoric and reality of supply chain integration. Int J Phys Distrib Logistics Manage 32(5):339–361 Fin B (2006) Performance implications of information technology implementation in an apparel supply chain. Supply Chain Manage Int J 11(4):309–316 Fisher ML, Raman A, McClelland AS (1994) Rocket science retailing is almost here: are you ready? Harv Bus Rev 72(3):83–93 Ganeshan R (1999) Managing supply chain inventories: a multiple retailer, one warehouse, multiple supplier model. Int J Prod Econ 59(1–3):341–354 Giannoccaro I, Pontrandolfo P (2004) Supply chain coordination by revenue sharing contracts. Int J Prod Econ 89(2):131–139 Gittell JH, Weiss L (2004) Coordination networks within and across organizations: a multi-level framework. J Manage Stud 41(1):127–153 Gottfredson M, Puryear R, Phillips S (2005) Strategic sourcing: from periphery to the core. Harv Bus Rev 83(2):132–139 Goyal SK, Deshmukh SG (1992) Integrated procurement-production systems: a review. Eur J Oper Res 62(1):1–10 Grubbstrom RW, Wang Z (2003) A stochastic model of multi-level/multi-stage capacityconstrained production-inventory systems. Int J Prod Econ 81–82(1):483–494 Gunasekaran A, Patel C, Tirtiroglu E (2001) Performance measures and metrics in a supply chain environment. Int J Oper Prod Manage 21(1–2):71–87 Gunasekaran A, Ngai EWT, McGaughey RE (2006) Information technology and systems justification: a review for research and applications. Eur J Oper Res 173(3):957–983 Gurnani H (2001) A study of quantity discount pricing models with different ordering structures: order coordination, order consolidation, and multi-tier ordering hierarchy. Int J Prod Econ 72(3):203–225 Haq AN, Kannan G (2006) Design of an integrated supplier selection and multi echelon distribution inventory model in a built-to-order supply chain environment. Int J Prod Res 44(10):1963–1985 Haq AN, Vrat PA, Kanda A (1991) An integrated production-inventory-distribution model for manufacture of urea: a case. Int J Prod Econ 25(1–3):39–49 Hau Z, Li S (2008) Impacts of demand uncertainty on retailer’s dominance and manufacturerretailer supply chain coordination. Omega 36(5):697–714 He J, Chin KS, Yang JB, Zhu DL (2006) Return policy model of supply chain management for single-period products. J Optim Theory Appl 129(2):293–308 Hill RM (1997) The single-manufacturer single retailer integrated production-inventory model with a generalized policy. Eur J Oper Res 97(3):493–499 Hill RM, Omar M (2006) Another look at the single-vendor single-buyer integrated productioninventory problem. Int J Prod Res 44(4):791–800 Hoque MA, Goyal SK (2000) An optimal policy for a single-vendor single-buyer integrated production-inventory system with capacity constraint of the transport equipment. Int J Prod Econ 65(3):305–315 Hou J, Zeng AZ, Zhao L (2010) Coordination with a backup supplier through buy-back contract under supply disruption. Transp Res E 46:881–895 Hoyt J, Huq F (2000) From arms-length to collaborative relationships in the supply chain: an evolutionary process. Int J Phys Distrib Logistics Manage 30(9):750–764 Huan SH, Sheoran SK, Wang G (2004) A review and analysis of supply chain operations reference (SCOR) model. Supply Chain Manage Int J 9(1):23–29 Huang G, Lau J, Mak K (2003) The impacts of sharing production information on supply chain dynamics: a review of the literature. Int J Prod Res 41(7):1483–1517 Huiskonen J, Pirttila T (2002) Lateral coordination in a logistics outsourcing relationship. Int J Prod Econ 78(2):177–185 Huq F, Cutright K, Jones V, Hensler DA (2006) Simulation study of a two-level warehouse inventory replenishment system. Int J Phys Distrib Logistics Manage 36(1):51–65

A Review on Supply Chain Coordination

79

Hwarng HB, Chong CSP, Xie N, Burgess TF (2005) Modelling a complex supply chain: understanding the effect of simplified assumptions. Int J Prod Res 43(13):2829–2872 Iyer AV, Bergen ME (1997) Quick response in manufacturer-retailer channels. Manage Sci 43:559–570 Jain K, Nagar L, Srivastava V (2006) Benefit sharing in inter-organizational coordination. Supply Chain Manage Int J 11(5):400–406 Jang YJ, Jang SY, Chang BM, Park J (2002) A combined model of network design and production/ distribution planning for a supply network. Comput Ind Eng 43(1–2):263–281 Jayaram J, Tan K (2010) Supply chain integration with third-party logistics providers. Int J Prod Econ 125(2):262–271 Jayaraman V (1998) Transportation facility location and inventory issues in distribution network design. Int J Oper Prod Manage 18(5):471–494 Jayaraman V, Pirkul H (2001) Planning and coordination of production and distribution facilities for multiple commodities. Eur J Oper Res 133(2):394–408 Kaur A, Kanda A, Deshmukh SG (2006) A graph theoretic approach to evaluate supply chain coordination. Int J Logistics Syst Manage 2(4):329–341 Kehoe DF, Boughton NJ (2001) New paradigms in planning and control across manufacturing supply chains: the utilization of internet technologies. Int J Oper Prod Manage 21(5–6):582–593 Kim T, Hong Y, Lee J (2006) Joint economic production allocation and ordering policies in a supply chain consisting of multiple plants and a single retailer. Int J Prod Res 43(17): 3619–3632 Kim B, Oh H (2005) The impact of decision making sharing between supplier and manufacturer on their collaboration performance, Supply Chain Management: An International Journal 10 (3-4):223–236 Kleindorfer PR, Saad GH (2005) Managing disruptions risks in supply chains. Prod Oper Manage 14(1):53–68 Koulamas C (2006) A newsvendor problem with revenue sharing and channel coordination, Decision Sciences 37(1):91–100 Krishnan H, Kapuscinski R, Butz DA (2004) Coordinating contracts for decentralized supply chains with retailer promotional effort. Manage Sci 50(1):48–63 Lambert DM, Cooper MC (2000) Issues in supply chain management. Ind Mark Manage 29(1): 65–83 Lambert DM, Emmelhainz MA, Gardner JT (1999) Building successful partnerships, Journal of Business Logistics 20(1):165–181 Larsen ST (2000) European logistics beyond 2000. Int J Phys Distrib Logistics Manage 30(6): 377–387 Larsen TS, Thernoe C, Anderson C (2003) Supply chain collaboration theoretical perspective and empirical evidence. Int J Phys Distrib Logistics Manage 33(6):531–549 Lee HL (2000) Creating value through supply chain integration. Supply Chain Manage Rev 4(4): 30–36 Lee HC (2001) Coordinated stocking, clearance sales, and return policies for a supply chain. Eur J Oper Res 131(3):491–513 Lee HC (2007) Coordination on stocking and progressive pricing policies for a supply chain. Int J Prod Econ 106(1):307–319 Lee CH, Rhee BD (2007) Channel coordination using product returns for a supply chain with stochastic salvage capacity. Eur J Oper Res 177(1):214–238 Lee HL, Padmanabhan V, Whang S (1997) Information distortion in supply chain: the bullwhip effect. Manage Sci 43(4):546–558 Li SX, Huang Z, Ashley A (1996) Improving buyer-seller system cooperation through inventory control. Int J Prod Econ 43(1):37–46 Li Z, Kumar A, Lim YG (2002) Supply chain modeling: a coordination approach. Integr Manuf Syst 13(8):551–561 Li G, Yang H, Sun L, Sohal AS (2009) The impact of IT implementation on supply chain integration and performance. Int J Prod Econ 120(1):125–138

80

K. Arshinder et al.

Liu K, Li JA, Lai KK (2004) Single period, single product news vendor model with random supply shock. Eur J Oper Res 158(3):609–625 Liu J, Zhang S, Hu J (2005) A case study of an inter-enterprise workflow-supported supply chain management system. Inf Manage 42(3):441–454 Lu L (1995) A one-vendor multi-buyer integrated model. Eur J Oper Res 81(2):312–323 Malone T, Crowston K (1994) The interdisciplinary study of coordination. ACM Comput Surv 26(1):87–119 Maloni MJ, Benton WC (1997) Supply chain partnerships: opportunities for operations research. Eur J Oper Res 101(3):419–429 Mantrala MK, Raman K (1999) Demand uncertainty and supplier’s returns policies for a multistore style-good retailer. Eur J Oper Res 115(2):270–284 McAfee A (2002) The impact of enterprise information technology adoption on operational performance: an empirical investigation. Prod Oper Manage 11(1):33–53 McCarthy S, Golocic S (2002) Implementing collaborative planning to improve supply chain performance. Int J Phys Distrib Logistics Manage 32(6):431–454 Mc Laren T, Head M, Yuan Y (2002) Supply chain collaboration alternatives: understanding the expected costs and benefits, Internet Research 12(4):348–364 Milner JM, Rosenblatt MJ (2002) Flexible supply contracts for short life cycle goods: the buyer’s perspective. Nav Res Logist 49(1):25–45 Moinzadeh K (2002) A multi-echelon inventory system with information exchange. Manage Sci 48(3):414–426 Moses M, Seshadri S (2000) Policy mechanisms for supply chain coordination. IIE Trans 32(3):254–262 Munson CL, Rosenblatt MJ (2001) Coordinating a three-level supply chain with quantity discounts. IIE Trans 33(5):371–384 Narus JA, Anderson JC (1995) Rethinking distribution: adaptive channels. Harv Bus Rev 74(4):112–120 Oke A, Gopalakrishnan M (2009) Managing disruptions in supply chains: a case study of a retail supply chain. Int J Prod Econ 118(1):168–174 Padmanabhan V, Png IPL (1997) Manufacturer’s returns policies and retailer’s competition. Mark Sci 16(1):81–94 Pan K, Lai KK, Liang L, Leung SCH (2009) Two period pricing and ordering policy for the dominant retailer in a two echelon supply chain with demand uncertainty. Omega 37(4):919–929 Park YB (2005) An integrated approach for production and distribution planning in supply chain management. Int J Prod Res 43(6):1205–1224 Petersen KJ, Handfield RB, Ragatz GL (2005) Supplier integration into new product development: coordinating product, process and supply chain design. J Oper Manage 23(3–4):371–388 Pfohl H, Kohler H, Thomas D (2010) State of the art in supply chain risk management research: empirical and conceptual findings and a roadmap for the implementation in practice. Logistics Res 2(1):33–44 Piplani R, Fu Y (2005) A coordination framework for supply chain inventory alignment. J Manuf Technol Manage 16(6):598–614 Porter ME (2001) Strategy and the internet. Harv Bus Rev 79(2):63–78 Power D (2005) Supply chain management integration and implementation: a literature review. Supply Chain Manage Int J 10(4):252–263 Pyke DF, Cohen MA (1993) Performance characteristics of stochastic integrated production distribution systems. Eur J Oper Res 68(1):23–48 Qi X, Bard JF, Yu G (2004) Supply chain coordination with demand disruptions. Omega 32:301–312 Reddy AM, Rajendran C (2005) A simulation study of dynamic order-up-to policies in a supply chain with non-stationary customer demand and information sharing. Int J Adv Manuf Technol 25(9–10):1029–1045 Ritchie B, Brindley C (2007) Supply chain risk management and performance: a guiding framework for future development. Int J Oper Prod Manage 27(3):303–322 Ross TJ (1997) Fuzzy logic with engineering applications. McGraw Hill, Singapore

A Review on Supply Chain Coordination

81

Rubin PA, Benton WC (2003) A generalized framework for quantity discount pricing schedules. Decis Sci 34(1):173–188 Sahin F, Robinson P (2002) Flow coordination and information sharing in supply chains: review, implications and directions for future research. Decis Sci 33(4):505–536 Sanders NR (2008) Pattern of information technology use: the impact on buyer–suppler coordination and performance. J Oper Manage 26(3):349–367 Sarmah SP, Acharya D, Goyal SK (2006) Buyer–vendor coordination models in supply chain management. Eur J Oper Res 175(1):1–15 Sarmah SP, Acharya D, Goyal SK (2007) Coordination and profit sharing between a manufacturer and a buyer with target profit under credit option. Eur J Oper Res 182(3):1469–1478 Sarmiento AM, Nagi R (1999) A review of integrated analysis of production–distribution systems. IIE Trans 31(11):1061–1074 Saroor J, Tarokh MJ, Shemshadi A (2009) Initiating a state of the art system for real-time supply chain coordination. Eur J Oper Res 196(2):635–650 Serel DA (2007) Capacity reservation under supply uncertainty. Comput Oper Res 34(4): 1192–1220 Sethi SP, Yan H, Zhang H (2004) Quantity flexibility contracts: optimal decisions with information updates. Decis Sci 35(4):691–712 Sharafali M, Co HC (2000) Some models for understanding the cooperation between supplier and the buyer. Int J Prod Res 38(15):3425–3449 Simatupang TM, Sridharan R (2002) The collaborative supply chain. Int J Logistics Manage 13(1):15–30 Simatupang TM, Wright AC, Sridharan R (2002) The knowledge of coordination for supply chain integration. Bus Process Manage J 8(3):289–308 Simatupang TM, Sandroto IV, Lubis SBH (2004) Supply chain coordination in a fashion firm. Supply Chain Manage Int J 9(3):256–268 Simchi-Levi D, Kaminsky P, Simchi-Levi E (2007) Designing and managing the supply chain, 3rd edn. McGraw Hill, New York Sinha S, Sarmah SP (2007) Supply chain coordination model with insufficient production capacity and option for outsourcing. Math Comput Modell 46:1442–1452 Skipper JB, Craighead CW, Byrd TA, Rainer RK (2008) Towards a theoretical foundation of supply network interdependence and technology-enabled coordination strategies. Int J Phys Distrib Logistics Manage 38(1):39–56 Soliman F, Youssef M (2001) The impact of some recent developments in e-business on the management of next generation manufacturing. Int J Oper Prod Manage 21(5–6):538–564 Stank TP, Crum MR, Arango M (1999) Benefits of interfirm coordination in food industry in supply chains. J Bus Logistics 20(2):21–41 Stank TP and Goldsby TJ (2000) A framework for transportation decision making in an integrated supply chain, Supply Chain Management: An International Journal 5(2):71–77 Stock GN, Greis NP, Kasarda JD (2000) Enterprise logistics and supply chain structure: the role of fit. J Oper Manage 18(5):531–547 Swaminathan JM, Tayur SR (2003) Models for supply chains in E-business. Manage Sci 49(10):1387–1406 Tang CS (2006) Perspectives in supply chain risk management. Int J Prod Econ 103(2):451–488 Tang O, Musa SN (2011) Identifying risk issues and research advancements in supply chain risk management, International Journal of Production Economics 133(1):25–34 Thomas DJ, Griffin PM (1996) Coordinated supply chain management. Eur J Oper Res 94(1): 1–15 Tsay A (1999) The quantity flexibility contract and supplier–customer incentives. Manage Sci 45 (10):1339–1358 Tsay AA, Lovejoy WS (1999) Quantity flexibility contracts and supply chain performance. Manuf Serv Oper Manage 1:89–111 vander Rhee B, vander Veen JAA, Venugopal V, Nall VN (2010) A new revenue sharing mechanism for coordinating multi-echelon supply chains. Oper Res Lett 38(4):296–301

82

K. Arshinder et al.

Verwijmeren M, Vlist PV, Donelaar KV (1996) Networked inventory management information systems: materializing supply chain management. Int J Phys Distrib Logistics Manage 26(6):16–31 Vickery SK, Jayaram J, Droge C, Calantone R (2003) The effects of an integrative supply chain strategy on customer service and financial performance: an analysis of direct versus indirect relationships. J Oper Manage 21(5):523–539 Weng ZK (2004) Coordinating order quantities between the manufacturer and the buyer: a generalized newsvendor model. Eur J Oper Res 156(1):148–161 Weng ZK, Parlar M (1999) Integrating early sales with production decisions: analysis and insights, IIE Transactions, 31(11): 1051–1060 Wong WK, Qi J, Leung SYS (2009) Coordinating supply chains with sales rebate contracts and vendor-managed inventory. Int J Prod Econ 120(1):151–161 Wu K, Ouyang L (2003) An integrated single-vendor single-buyer inventory system with shortage derived algebraically. Prod Plann Control 14(6):555–561 Xiao T, Qi X (2008) Price competition, cost and demand disruptions and coordination of a supply chain with one manufacturer and two competing retailers. Omega 36:741–753 Xu L and Beamon B (2006) Supply Chain Coordination and Cooperation Mechanisms: An Attribute-Based Approach, The Journal of Supply Chain Management 42(1):4–12 Xu R, Zhai X (2010) Analysis of supply chain coordination under fuzzy demand in a two-stage supply chain. Appl Math Model 34:129–139 Yang PC, Wee HM (2002) A single-vendor and multiple buyers production-inventory policy for deteriorating item. Eur J Oper Res 143(3):570–581 Yao MJ, Chiou CC (2004) On a replenishment coordination model in an integrated supply chain with one vendor and multiple buyers. Eur J Oper Res 159(2):406–419 Yao D, Yue X, Wang X, Liu JJ (2005) The impact of information sharing on a returns policy with the addition of a direct channel. Int J Prod Econ 97(2):196–209 Yao Z, Leung SCH, Lai KK (2008) Analysis of the impact of price sensitivity factors on the returns policy in coordinating supply chain. Eur J Oper Res 187(1):275–282 Yokoyama M (2002) Integrated optimisation of inventory-distribution systems by random local search and a genetic algorithm. Comput Ind Eng 42(2–4):175–188 Yu Z, Yan H and Cheng TCE (2001) Benefits of information sharing with supply chain partnerships, Industrial Management and Data Systems 101(3):114–119 Yue X, Raghunathan S (2007) The impacts of the full returns policy on a supply chain with information asymmetry. Eur J Oper Res 180(2):630–647 Yusuf Y, Gunasekaran A, Abthorpe MS (2004) Enterprise information systems project implementation: a case study of ERP in Rolls-Royce. Int J Prod Econ 87(3):251–266 Zhao W, Wang Y (2002) Coordination of joint pricing-production decisions in a supply chain. IIE Trans 34(8):701–715 Zhao X, Xie J, Zhang WJ (2002) The impact of information sharing and ordering coordination on supply chain performance. Supply Chain Manage Int J 7(1):24–40 Zhou H, Benton WC Jr (2007) Supply chain practice and information sharing. J Oper Manage 25(6):1348–1365 Zhou YW, Wang SD (2009) Manufacturer-buyer coordination for newsvendor products with two ordering opportunities and partial backorders. Eur J Oper Res 198(3):958–974 Zimmer K (2004) Supply chain coordination with uncertain just-in-time delivery. Int J Prod Econ 77:1–15 Zou X, Pokharel S, Piplani R (2004) Channel coordination in an assembly system facing uncertain demand with synchronized processing time and delivery quantity. Int J Prod Res 42(22):4673–4689 Zou X, Pokharel S, Piplani R (2008) A two-period supply contract model for a decentralized assembly system. Eur J Oper Res 187(1):257–274 Zsidisin GA, Ellram LM (2001) Activities related to purchasing and supply management involvement in supplier alliances. Int J Phys Distrib Logistics Manage 31(9):629–646 Zsidsin GA, Smith ME (2005) Managing supply risk with early supplier involvement: a case study and research propositions. J Supply Chain Manage 41(4):44–57

Control Policies for Multi-echelon Inventory Systems with Stochastic Demand Qinan Wang

Abstract Although supply chain management has become an important management paradigm, the optimal control of a stochastic multi-echelon supply chain inventory system is still largely an open issue. An inventory control policy for such a system has to consider at least three aspects: order coordination, information sharing, and stock or risk pooling. Each aspect can generate significant benefits. Nevertheless, an inventory control policy that can fully optimize system performance on all dimensions, even if it exists, would be very difficult to determine. In this paper, we provide a literature review of inventory control policies for multiechelon supply chain systems with uncertain demand. We first review generic policies developed by extending the basic inventory control policies at a single location. Subsequently, we discuss the themes of coordinated replenishment and information sharing for multi-echelon inventory systems and review the relevant literature. This framework highlights the key factors that drive the performance of multi-echelon inventory systems and shed lights on directions of future research in this area. Keywords Multi-echelon inventory control policy • Review • Supply chain management

1 Introduction A supply chain consists of all organizations involved directly or indirectly in the provision of a product and/or service required by end customers. Inventory management in a supply chain spans all movement and storage of raw materials, work-in-process inventory, and finished goods from point of origin to point of consumption. Traditionally, inventory decisions are made locally at each stocking Q. Wang (*) Nanyang Business School, Nanyang Technological University, Singapore, Singapore 639798 e-mail: [email protected] T.-M. Choi and T.C. Edwin Cheng (eds.), Supply Chain Coordination under Uncertainty, International Handbooks on Information Systems, DOI 10.1007/978-3-642-19257-9_4, # Springer-Verlag Berlin Heidelberg 2011

83

84

Q. Wang

point. However, companies have long realized that they can achieve tremendous benefits by integrating their inventory operations with their business partners. Contemporary supply chain management calls for the application of a total systems approach to supply chain inventory management (Jacobs et al. 2009). There has been a great deal written on the control and management of supply chain inventory systems. The literature on supply chain inventory systems has been reviewed from different perspectives (see, e.g., Clark 1972; Federgruen 1993; Axs€ater 1993a, 2003b; Tsay et al. 1998; Ganeshan et al. 1998; Tan 2001; Sahin and Robinson 2002; Li and Wang 2007). In particular, Federgruen (1993) discussed discrete-time centralized planning models, and Axs€ater (1993a, 2003b) discussed continuous-review centralized planning models for multi-level inventory systems with stochastic demand. Tan (2001) reviewed the evolution of the supply chain management philosophy. More recently, Li and Wang (2007) reviewed the literature on coordination mechanisms of supply chain systems. We review the recent developments on centralized control policies for multiechelon inventory systems with stochastic demand in this paper. Except for some special cases, the optimal control of a supply chain inventory system with stochastic demand is still largely an open problem. We attempt to review the literature to highlight the deficiencies of existing multi-echelon inventory control policies in their capacity to optimize system performance. For this purpose, we adopt a framework to review the literature based on the basic inventory control policies at a single facility. As multi-echelon inventory systems are comprised of individual facilities, their control has been treated as an extension of the inventory control problem at a single facility. Consequently, the multi-echelon inventory control literature has been built based on inventory control policies at a single facility. We first review generic inventory control models for multi-echelon inventory systems that are developed by applying inventory control policies at a single facility. This approach helps us understand the evolution, and highlights the fundamental issues and methodologies for the control of multi-echelon inventory systems. However, a multi-echelon inventory control system is not a simple and straightforward extension of inventory control at individual facilities. The control and management of a multi-echelon inventory system brings about completely new issues and challenges. We subsequently discuss these challenges and review the relevant literature. This discussion highlights the deficiencies of the current literature and enhances our understanding about the key factors that drive the performance of multi-echelon inventory systems. The review also points out new challenges and directions for future research on multi-echelon inventory control systems. The rest of the paper is organized as follows. In Sect. 2, we discuss three basic inventory control policies at a single location, and the structures and issues of multiechelon inventory control systems. Subsequently, in Sects. 3–5, we review multiechelon inventory control systems that are built respectively on the three basic installation inventory control policies. In Sect. 6, we discuss the key factors that drive the performance of multi-echelon inventory systems and review the relevant

Control Policies for Multi-echelon Inventory Systems with Stochastic Demand

85

literature. Finally, in Sect. 7, we conclude the study and summarize possible directions for future research.

2 Multi-echelon Inventory Systems and Control Policies There are various supply chain network structures. The simplest is a serial system that has a single facility at each stage. The next simplest structure is an assembly system in which multiple parts are assembled into a single component and, therefore, a single facility can have multiple suppliers. A distribution system supplies a product to many customers and, therefore, can have multiple customers for a supplier. A tree system combines assembly systems and distribution systems, and a general system can include any of the above as part of the system (Zipkin 2000). A prototype network structure for previous studies on multi-echelon inventory systems is a two-level distribution system whereby a central warehouse supplies a product to a group of retailers. This structure includes a serial system as a special case. An assembly system can also be considered as a special case as it can be decomposed into multiple serial systems under certain conditions (Rosling 1989). More importantly, this structure includes all the fundamental issues for a multiechelon inventory control system and yet does not have the complexity of a general system. Unless otherwise stated, we use the distribution system that is depicted in Fig. 1 for the discussion. Furthermore, we assume that the warehouse obtains supply from a perfectly reliable external source that can fill an order without delay. As illustrated in Fig. 1, an installation is a single facility, and an echelon comprises the facility at the current stage and all facilities at the downstream stages. The following assumptions are typical (1) demand generates randomly and independently at the retailers, (2) all shortages are backordered, (3) all lead times are constant and deterministic, and (4) inventory holding costs and backorder costs are linear. In addition, demand processes are assumed to have stationary and independent increments. This condition holds true for the commonly used compound Poisson demand process and the normal demand model (Rao 2003).

2.1

Basic Inventory Control Policies for a Single Facility

Assume that the central warehouse has unlimited capacity and is perfectly reliable in fulfilling customer orders. Under this condition, inventory control decisions at a retailer can be made independently of inventory control decisions at other facilities in the system. An inventory control policy for the retailer in this case is referred to as an installation inventory control policy. Inventory management at a single facility or installation consists of two fundamental decisions: how much to order and when to order (Zipkin 2000). The objective is to minimize long-run expected inventory related costs including

86

Q. Wang

Echelon Inventory System

External Supplier

Warehouse

Retailer 1

Customer

. . .

Retailer N

Customer

Fig. 1 A two-echelon distribution system

ordering costs, inventory holding costs, and stockout costs. Fundamental trade-offs exist for safety stock and order quantity. Namely, a higher safety stock lowers stockout costs but raises inventory levels or inventory holding costs, and a smaller order quantity (or more frequent ordering) lowers inventory levels or inventory holding costs but increases ordering costs. We desire an inventory control policy that can balance the cost components and minimize their total. There are three basic installation inventory control policies. If inventory positions can be monitored continuously, there are two approaches to decide when to order. We can adopt a stock-based approach to place an order when the inventory position (inventory on hand plus outstanding orders minus backorders) reaches a reorder point or a time-based approach to order periodically in a fixed replenishment interval. The stock-based approach leads to the well-known continuous-review batch-ordering (R, Q) policy, and the time-based approach leads to the fixed-interval order-up-to (S, T) policy. On the other hand, if inventory positions can be monitored only periodically, a base-stock policy is adopted.

2.1.1

The Continuous-Review Batch-Ordering (R, Q) Policy

Under this policy, a batch of Q units is ordered from the supplier when the inventory position drops to a pre-determined reorder point R. The structural properties and optimization procedures for this policy have been well discussed (Silver and Peterson 1998; Zipkin 2000). The two decision variables of this policy, i.e. the reorder point R and order quantity Q, cannot be determined separately. According to Zheng (1992), if the order quantity is determined by the deterministic economic order quantity model,

Control Policies for Multi-echelon Inventory Systems with Stochastic Demand

87

and the reorder point is determined optimally given the order quantity, the maximum cost deviation as compared to the optimal (R, Q) policy is 12.5%.

2.1.2

The Fixed-Interval Order-up-to (S, T) Policy

Under this policy, the inventory position is reviewed in a fixed time interval of T and inventory is ordered, if necessary, at a review point to raise the inventory position to S. Similar to R and Q for the (R, Q) policy, S and T are decision variables for this policy. This policy provides a time-based alternative to the stockbased (R, Q) policy. Although time-based control that uses fixed replenishment schedules to coordinate production and inventory activities has long been a common practice (Graves 1996), the (S, T) policy has not been widely applied in the literature. One reason is that it is dominated by the (R, Q) policy. The structural properties of this policy have been analyzed recently by Rao (2003). It is shown that the long-run inventory related cost under this policy can be much higher than that under the (R, Q) policy. The worst case scenario identified by Rao (2003) is a 43% cost increase. The superiority of the (R, Q) policy comes from the use of continuous review or real time stock information to optimize replenishment decisions.

2.1.3

The Periodic-Review Base-Stock (s, S) Policy

If inventory positions can be monitored only periodically, for example, due to restrictions of physical conditions such as 1 day, 1 week, etc., time is typically modeled as a sequence of discrete time points. Time periods are defined as intervals between time points. Assume that all significant events occur at the time points. The decision problem is then to determine the order quantity at each review point. The problem was first formulated as a dynamic programming model under the assumption that demands at different time points (or equivalently in different time periods) are independent (Arrow et al. 1951; Dvoretzky et al. 1952). Since then, a massive literature on this topic has accumulated. Porteus (1990) provided a thorough review. Assume that the planning horizon is finite or otherwise demand and costs are stationary. The optimal inventory control policy can be fully characterized for this problem. Namely, if there are no economies of scale in ordering, the optimal inventory control policy is to order at each time point to raise the inventory position to a fixed base-stock level S. This policy is referred to as the base-stock policy. If there are economies of scale in ordering, the optimal inventory control policy is a generalization of the single-parameter base-stock policy: order to raise the inventory position to S if the current inventory is less than s (and order nothing otherwise). This modified base-stock policy is referred to as a (s, S) policy. Various algorithms have been developed to compute optimal (s, S) policies (see, e.g., Veinott and Wagner 1965; Zheng and Federgruen 1991).

88

2.2

Q. Wang

Multi-echelon Inventory Control Policies

Regardless of the source of supply, the central warehouse is usually not perfectly reliable and, even if it is, this may not be economically desirable. As such, an order from a retailer will be filled depending on availability of stock at the warehouse. Consequently, the inventory decisions at the retailers and the warehouse cannot be made independently. In other words, inventory control decisions at the warehouse and retailers should be considered together and integratively to optimize system performance. This leads to a multi-echelon inventory control problem. Obviously, each facility in a multi-echelon inventory control system faces the same fundamental decisions and issues for an installation inventory control problem. Therefore, a multi-echelon inventory control policy can be developed by applying an installation inventory control policy at each facility. Indeed most multi-echelon inventory control policies have been developed in this way. These policies are generic multi-echelon inventory control policies. This development provides a framework to review and discuss the multi-echelon inventory control literature. We will review the multi-echelon inventory control models that are developed based on the three basic installation inventory control policies separately in the next three sections. A multi-echelon inventory control system, however, is not a simple and straightforward addition of separate inventory control problems at individual facilities. The control of a multi-echelon inventory system brings about completely new issues and challenges. First, inventory decisions and activities at the warehouse and the retailers must be carefully planned and coordinated. Control policies as a result of local optimization are usually not able to optimize system performance. This issue is particularly important for decentralized supply chains in which inventory control decisions are usually made locally (Li and Wang 2007). Second, information sharing becomes a key driving factor for system performance. Accurate, timely and easily accessible information on demand and stock or advanced commitment from downstream customers can significantly improve system performance. Cachon and Fisher (2000) demonstrated that information sharing can lower system costs by an average of 2.2%, with a maximum of 12.1%. The benefits of demand information may be even higher (Lee et al. 2000). Finally, pooling lead time demand at retailers can also lead to significant reductions in system inventories and/or stockouts. Because of these issues, conventional relationships between installation inventory control policies may no longer be applicable. For example, although the basic (R, Q) policy dominates the basic (S, T) policy at a single facility, a fixed-interval order-up-to policy may perform significantly better than a stock-based batch-ordering policy for a distribution system with multiple retailers. As shown by Wang and Axs€ater (2010) and Wang (2010), although a stock-based batch-ordering policy still has the advantage of using real time stock information, a time-based base-stock policy can provide a better mechanism to coordinate replenishments and pool lead time demand for retailers. When the benefits of order coordination and stock pooling are more significant than the value of stock

Control Policies for Multi-echelon Inventory Systems with Stochastic Demand

89

information, the latter outperforms the former. We discuss these issues and review the relevant literature in Sect. 6. This discussion will highlight the key factors that drive the performance of a multi-echelon inventory control system.

3 Stock-Based Batch-Ordering Policies A typical stock-based batch-ordering policy for a multi-echelon inventory system is to apply a continuous-review ðR; QÞ policy at each facility. This means that a batch of size Q is ordered when the inventory position drops to or below the reorder point R. The batch size and reorder point may vary for different facilities. In addition, the reorder point can be determined based on installation stock or echelon stock. For convenience of discussion, we define the following conditions and parameters (1) there are N retailers, (2) the demand at retailer i follows a simple N P Poisson process with mean li and l ¼ li , and (3) the lead time is Li for retailer i¼1

i and L0 for the warehouse.

3.1

One-for-One Replenishments

Early applications of the (R, Q) policy started with one-for-one replenishments at all facilities. Under this assumption, facility i sets up an order-up-to level Si and orders one unit from the supplier when its inventory position drops to or below Si , or equivalently facility i adopts a continuous-review (R, Q) policy with R ¼ Si 1 and Q ¼ 1. Consider the operations at the warehouse when the warehouse and retailers all adopt one-for-one replenishments. A retailer order at time t is filled immediately if there is stock at the warehouse or otherwise until stock is available. The maximum delay occurs when an order has to be met with a unit ordered from the external supplier at time t. Consequently, all retailer orders placed prior to time t have been filled by time t þ L0 . Therefore, the outstanding orders at time t þ L0 , i.e. units ordered from the external supplier but not yet delivered to the warehouse, are the retailer orders or demands that have occurred between t and t þ L0 . When demand is Poisson, the number of outstanding orders at the warehouse follows a Poisson distribution with mean l L0 . Backorders occur at the warehouse as soon as the number of outstanding orders at the warehouse exceeds the inventory position S0 . Let y denote the number of outstanding orders and EðB0 Þ denote the average number of backorders. We have EðB0 Þ ¼ E½ðy S0 Þþ , where xþ ¼ maxf0; xg. By Little’s law, the average delay at

90

Q. Wang

the warehouse is then equal to EðB0 Þ= l0 . This result holds even if the lead time at the warehouse is stochastic (Axs€ater 1993a). The actual lead time at a retailer is equal to the constant lead time (i.e. Li ) plus a random delay at the warehouse. According to the discussion above, the delay at the warehouse is a random variable with an expected value of EðB0 Þ= l0 . The average lead time at retailer i, denoted by Li , is then equal to Li ¼ Li þ EðB0 Þ= l0 . Sherbrooke (1968) first developed the METRIC approximation by assuming that the lead time at retailer i is constant at Li . With this assumption, expected inventory levels and backorders at the retailers can be easily evaluated. However, this approximation often leads to significant errors. Various subsequent studies have attempted to improve and extend the METRIC approach to other situations (Muckstadt 1973, 1979; Graves 1985; Sherbrooke 1986; Lee and Moinzadeh 1987a). Shortly after the METRIC approximation was introduced, Simon (1971) developed an exact solution for the problem. Exact solution procedures that are more efficient and/or under more general conditions followed (Shanker 1981; Svoronos and Zipkin 1991). In particular, Axs€ater (1990a, 1993c) provided a very efficient recursive procedure to evaluate one-for-one replenishment policies when retailers are identical. Forsberg (1995) provided a solution when demand follows a compound Poisson process. The exact solution has also been extended to the case in which the warehouse adopts a general batch-ordering policy. Let CðSw ; Sr Þ denote the total inventory holding and shortage costs per time unit when applying one-for-one replenishments with inventory positions Sw and Sr at the warehouse and a retailer, respectively. When retailers use one-for-one replenishments and the warehouse adopts a general batch-ordering policy ðRw ; Qw Þ, the total inventory holding and þ Qw RwP shortage costs per time unit is given by ð1= Qw Þ Cðj; Sr Þ (Axs€ater 1993a). j¼Rw þ1

3.2

General Batch-Ordering Policies

Apparently, one-for-one replenishments are applicable only when there are no setup costs for orders. When there are economies of scale in ordering, a general batchordering policy is more appropriate. However, a general batch-ordering policy is usually more difficult to evaluate. Let Qi denote the order quantity of retailer i. Then retailer i places orders to the warehouse according to an Erlang renewal process with Qi stages. As a result, the demand process at the warehouse is a superposition of N such processes. The problem is still tractable for serial systems. For a two-level distribution system with a single retailer or N ¼ 1, let the retailer use a batch-ordering policy ðRr ; Qr Þ and the warehouse adopt a batch-ordering policy ðRw ; Qw Þ (both as batches of Qr ). Using the results for one-for-one replenishments, the total inventory holding and shortage costs per time unit is given by þ Qw RrP þ Qr RwP ð1= Qw Qr Þ Cðj Qr ; kÞ (Axs€ater 1993c). j¼Rw þ1 k¼Rr þ1

Control Policies for Multi-echelon Inventory Systems with Stochastic Demand

91

Many studies have developed batch-ordering inventory control policies for general serial systems. The following are a few notable examples. De Bodt and Graves (1985) developed an approximate solution using a nested policy in which an order is delivered to the retailer upon the arrival of a shipment from the external supplier at the warehouse. Badinelli (1992) put forward a model of the steady-state values of on-hand inventory and backorders at each facility when each member adopts an installation-stock (R, Q) policy. More recently, Chen and Zheng (1994a) first evaluated echelon-stock (R, nQ) policies and developed a recursive procedure to compute the steady-state inventory levels of the system when demand follows a simple Poisson process. Subsequently, they extended the analysis to situations in which demand follows a compound Poisson process and developed near-optimal echelon-stock (R, nQ) policies (Chen and Zheng 1998) and to situations in which materials flow in fixed batch sizes (Chen 2000). In particular, Chen (1999a) proved that, for a two-stage serial system with zero lead time at the warehouse, a nested stock-based batch-ordering policy can achieve at least 94% system optimality. This is probably the only performance evaluation for a multi-echelon batch-ordering policy. When there are multiple retailers, decisions at the retailers must be coordinated and integrated in order to optimize system performance. However, members in a supply chain traditionally have little communication about their demand and inventory activities. As a result, early developments of stock-based batch-ordering policies considered installation policies in which each member makes its inventory decision separately using only local stock and demand information. Deuermeyer and Schwarz (1981) considered a distribution system consisting of one supplier and multiple identical retailers, and developed an installation-stock policy. They approximated each facility as a single-location inventory system and used decomposition as an adaptation of the METRIC technique. The problem has been examined by many others and more accurate approximate solutions have been developed since then (Moinzadeh and Lee 1986; Lee and Moinzadeh 1987a, b; Svoronos and Zipkin 1988). Exact evaluations of on-hand inventories and backorders for the system have been developed by Axs€ater (1993c) when retailers are identical and face independent simple Poisson demand processes. Forsberg (1996) and Axs€ater (2000) then generalized the results to the case of non-identical retailers under simple and compound Poisson demand. Cheung and Hausman (2000) provided exact performance evaluations for the supplier when retailers order in batches of a basic quantity. Other approximate optimizations have been developed by Axs€ater et al. (2002), Axs€ater (2003a) and Gallego et al. (2007). Instead of using local or installation inventory position, a stock-based backordering policy can be developed based on echelon inventory position, or the sum of the local inventory position and the inventory positions at all its downstream members. Since an echelon-stock policy incorporates stock information at downstream facilities for inventory control, it is superior to an installation-stock policy. As shown by Axs€ater and Rosling (1993) and Axs€ater and Juntti (1996), although the relative performance of the two policies seldom deviates by more than 5%, it is

92

Q. Wang

clear that the echelon-stock policy outperforms the installation policy for serial and assembly systems. However, there are two barriers for the wide application of the echelon-stock policy. First, its implementation requires a centralized information system that enables a supply chain member to have access to the demand and inventory information at all its downstream members in order to continuously monitor its echelon stock. Second, echelon-stock policies are usually more difficult to evaluate than installation-stock policies. Axs€ater (1997) and Chen and Zheng (1997) provided exact evaluations of echelon-stock batch-ordering policies for twolevel distribution systems. When there are multiple retailers, neither an installation-stock policy nor an echelon-stock policy fully utilizes the inventory state of a multi-echelon inventory system to optimize system performance. This is obvious for an installation policy since the inventory control at the supplier utilizes no inventory information from the retailers. For an echelon-stock policy, the inventory control decision at the supplier utilizes the echelon-stock (i.e. the total inventory position at all retailers) information rather than the inventory status at the individual retailers. In this sense, neither policy may be able to optimize system performance. Stock-based batch-ordering policies have also been widely used to analyze multi-echelon inventory systems with multiple supply modes (see, e.g., Lee 1987; Moinzadeh and Nahmias 1988; Axs€ater 1990b; Moinzadeh and Schmidt 1991; Johansen and Thorstenson 1998) and information sharing (see, e.g., Bourland et al. 1996; Chen 1998; Gavieneni 2002; Gurbuz et al. 2007). The issue of information sharing will be discussed further in Sect. 6.

4 Fixed-Interval Order-up-to Policies When inventory position can be monitored continuously, most previous research work on multi-echelon inventory systems has been confined to stock-based batchordering policies. Applications of the fixed-interval order-up-to (S, T) policy to build multi-echelon inventory control systems are relatively limited. One reason is that the (S, T) policy is dominated by the (R, Q) policy for a single location. This dominance, however, does not extend to multi-echelon inventory systems. Wang and Axs€ater (2010) and Wang (2010) demonstrated recently that time-based control can perform significantly better than stock-based control under certain conditions for distribution systems with multiple retailers. Previous studies have considered control policies for multi-echelon inventory systems with fixed replenishment intervals. These studies assume that replenishment intervals are determined exogenously but allow the flexibility of coordinating replenishments. A notable feature is the concept of nesting (Graves 1996; van Houtum et al. 2007). This condition is achieved under two constraints: the integer-ratio constraint requires that the replenishment interval at a stage be an integer multiple of the replenishment interval at the next downstream stage; and the synchronization constraint requires that a shipment from an upstream can be

Control Policies for Multi-echelon Inventory Systems with Stochastic Demand

93

forwarded immediately to the next downstream stage if so desired. These policies can be considered as applications of the fixed-interval order-up-to (S, T) with a given interval T. On the other hand, for a given interval T, the decision problem is similar to a periodic-review inventory control problem with constraints on the replenishments at certain time points. Therefore, base-stock policies are adopted at replenishment points. Eppen and Schrage (1981) first considered a fixed replenishment interval policy for retailers to synchronize their replenishments. The retailers are identical, face independent stochastic demand, and order together periodically from an outside supplier through a depot. The replenishment interval is given, and stock is allocated to the retailers entirely and immediately upon its arrival at the depot. In this way, the depot does not carry inventory and acts only as a cross docking point. By assuming large incoming orders that will always result in an equal probability of stockout at each retailer, they derived approximately optimal base- stock policies at the depot. Subsequently, Schwarz (1989) and Kumar et al. (1995) adopted fixed replenishment interval policies to study the benefits of stock pooling. McGavin et al. (1993, 1997) used this structure to analyze warehouse inventory allocation policies to minimize system lost sales. Atkins and Iyogun (1988), Viswanathan (1997), and Eynan and Kropp (1998), among others, studied joint replenishment policies, and Cetinkaya and Lee (2000) analyzed time-based consolidation policies for vendor-managed inventory systems. In addition, a number of studies have been devoted to optimal control policies for multi-echelon inventory systems when replenishment intervals at all facilities are given (Yano and Carson 1988; Jackson 1988; Graves 1996; Axs€ater 1993b; van Houtum et al. 2007). A few studies have considered full decision models to optimize system performance on both replenishment interval and inventory control policies recently. Feng and Rao (2007) applied the ðS; TÞ policy to a two-stage serial system. They considered the “fixed reorder interval, T, order-up-to base-stock level, R” policy and refer to this policy as the echelon-stock (R, nT) policy. The decision problem is formulated as a mixed-integer nonlinear programming model, in which the cost under each decision alternative is obtained by simulation. They compared the (R, nT) policy to the (R, nQ) policy (Chen and Zheng 1994a, b) numerically and showed that, although the latter dominates the former, the cost differences are often not significant. More recently, Wang and Axs€ater (2010) developed a fixed-interval order-up-to policy for a distribution system with multiple retailers. Let the warehouse set up a basic replenishment period T. Retailers are required to replenish through the warehouse in intervals that are integer multiples of the basic replenishment period. No inventory is carried at the warehouse. They compared this policy to the stockstock batch-ordering policy (Axs€ater 1993c) and showed that the dominance of the latter at a single location does not extend to distribution systems with multiple retailers. The time-based policy can perform significantly better than the stockbased batch-ordering policy in certain situations. However, since the warehouse is restricted to a cross-docking point that does not carry inventory, the efficiency of this policy could be low if the cost of carrying inventory at the warehouse is low.

94

Q. Wang

Subsequently, Wang (2010) generalized this model to allow the warehouse to carry inventory and developed a general integer-ratio policy for a distribution system with multiple and identical retailers. Another motivation for this development is the remarkable achievements of the integer-ratio policy in deterministic settings (Roundy 1985, 1986). Let the warehouse set up a base replenishment interval to replenish inventories from the external supplier. Retailers are required to review their inventory positions and order in fixed time intervals that are integer or integer-ratio multiples of the base replenishment interval at the warehouse. The warehouse and retailers each adopt an echelon-stock order-up-to policy, i.e. order the needed inventories to raise the echelon inventory position to a fixed order-up-to level at each review point. It is shown numerically that, although the stock-based batch-ordering policy generates a lower system cost in many cases, the integer-ratio policy can perform significantly better in certain settings.

5 Periodic-Review Base-Stock Policies When time is discrete, the problem is usually formulated as a dynamic programming or Markov decision model. However, except for some special cases, the large dimension of the inventory state for the exact formulation usually precludes an exact solution. The focus is then shifted to approximate solutions.

5.1

Serial Systems

Clark and Scarf (1960) showed that, for a serial system with M stages, if the planning horizon is finite and there is no setup cost except at the highest installation, the inventory problem for the system can be decomposed into exactly M separate single location problems, one for each echelon. These problems can be specified and solved recursively starting from the highest echelon. The optimal solution at the highest echelon is a ðsM ; SM Þ policy: order the necessary stock from the external supplier to raise the system inventory position to an order-up-to level SM whenever the starting inventory position is at or below a reorder point sM ; the optimal solution at a lower echelon j < M is a modified base stock policy: set up a base stock level Sj for echelon j < M, and ship the necessary stock from the upper level if there is sufficient stock and otherwise whatever available to raise the inventory position at the echelon to Sj . The model and solution were originally suggested by Clark (1958). The seminal work by Clark (1958) and Clark and Scarf (1960, 1962) initiated the research on the optimal control of multi-echelon inventory systems. The basic model and solution have been generalized to various settings since then. Federgruen

Control Policies for Multi-echelon Inventory Systems with Stochastic Demand

95

and Zipkin (1984a) extended the basic model to the infinite-horizon case for both discounted and average costs, and showed that the computations required are much easier than for the finite-horizon case. They also considered the case with multiple locations at the lower echelon and showed that the problem can be approximated by a problem with a single location at the lower echelon. Subsequently, Zipkin (1986a, b) considered the problem with uncertain lead times. Chen (2000) considered optimal control policies for multi-stage serial or assembly inventory systems with batch ordering, and generalized several existing results for the basic model. Recently, van Houtum et al. (2007) generalized the results to serial inventory systems with fixed replenishment intervals. They proved the optimality of the (s, S) policy for a serial system under the integer and synchronization constraints when replenishments intervals are given. Chen and Song (2001) considered a serial system in which demand generates at stage one in each period according to a distribution that is determined by the current state of an exogenous Markov Chain, and showed that the optimal policy for the system is an echelon-stock base-stock policy with state-dependent order-up-to levels. Shang and Song (2003, 2007) and Gallego and Ozer (2003, 2005) developed simple heuristics and bounds and approximations for the optimal policy parameters. Recently, Chao and Zhou (2009) extended the results to serial systems with batch ordering and fixed replenishment schedules. The results for a serial system have been generalized to assembly systems. Schmidt and Nahmias (1985) considered a simple assembly system in which two components are purchased from outside and assembled into a single end item, and characterized the optimal policy. Subsequently, building on Schmidt and Nahmias (1985), Rosling (1989) showed that, when all order and assembly cost functions are linear, an assembly system can be transformed into an equivalent serial system. All carrying, outside order and assembly costs remain linear in the equivalent system. Therefore, an optimal control policy is to apply modified base stock policies in each period for all nodes. The result holds for both finite planning horizon and infinite planning horizon models. Song and Yao (2002) considered the performance and optimization of assembly systems with random lead times. By modeling an assembly system under a given base-stock policy as a M=G=1 queuing system, they derived easy-to-compute performance bounds. When there are setup costs at all stages, it is well known that the optimal control of the inventory system is difficult. Only heuristic solutions and approximations have been developed. Chen and Zheng (1994b) developed lower bounds for multi-echelon stochastic inventory systems including serial, assembly and distribution systems with multiple retailers in which economies of scale in ordering exist at all locations. Recently, Shang and Zhou (2010) considered (r, nQ, T) policies. Under such a policy, each stage reviews its inventory in every T periods and orders according to an echelon-stock (r, nQ) policy. There are two types of fixed costs: one for each order batch Q and another for each inventory review. They developed a method for obtaining heuristic and optimal policy parameters.

96

5.2

Q. Wang

General Systems

Similar to serial systems with setup costs at all stages, an optimal control policy that can fully optimize system performance for a distribution system with multiple retailers is unknown. Previous studies have focused on heuristic solutions and approximations under various restrictions. The reader is referred to Federgruen (1993) for a systematic and detailed discussion on such restrictions and approximations. A number of studies have considered control policies under the restriction that the warehouse does not hold inventory. Under this condition, the warehouse acts as only a cross docking point and a delivery at the warehouse is allocated immediately and entirely to retailers. Eppen and Schrage (1981) started the use of this restriction. Subsequently, Federgruen and Zipkin (1984b) showed that a distribution system with multiple retailers can be approximated by a single-location problem under this condition. Recently, Wang and Axs€ater (2010) adopted this setting to develop a fixedinterval order-up-to policy for a two-level distribution system with multiple retailers. When multiple retailers order from the warehouse at a common point in time, stock allocation becomes an important issue. Federgruen and Zipkin (1984c) provided a systematic and detailed discussion on this problem. They showed that, when stock is allocated based on the inventory positions at individual retailers, the state space of the dynamic programming problem has a very large dimension, which usually precludes an exact solution of the model. They developed a computationally tractable approximate solution with myopic stock allocation. Under a myopic stock allocation policy, an incoming order at the warehouse is allocated to retailers to minimize their expected costs in the very first period in which the allocation has an impact, i.e. in the period where the shipments arrive at their destinations. Since then, different allocation approaches have been considered. A cycle allocation policy allocates an incoming order at the warehouse to retailers to minimize their expected costs in an ordering cycle, i.e. the period from the arrival of the current shipments to the arrival of the shipments from the next order (Federgruen 1993). A general reservation policy reserves a unit of supply at the time of a demand event at a retailer, and ships the reserved supply together to a retailer according to a fixed schedule or when inventory becomes available. This allocation method was also called “virtual allocation” by Graves (1996) and Axs€ater (1993c). To maximize the benefits of stock pooling, stock at the warehouse should be allocated in the lastminute or at the time of shipment (Marklund 2006). Several studies also assumed that inventory positions can be balanced at an allocation point if necessary (see, e.g., Eppen and Schrage 1981; Federgruen and Zipkin 1984c; McGavin et al. 1993; Axs€ater et al. 2002; Sosic 2006). This balance condition means simply that transshipments among retailers or allocation of negative quantities are allowed, although this is not always possible in practice. According to Wang and Axs€ater (2010), allocation policies can have a significant impact on system costs. They showed that system costs may increase by more than 30% under certain conditions when stock at the warehouse is allocated according to a complete reservation allocation policy as compared to a last-minute allocation policy.

Control Policies for Multi-echelon Inventory Systems with Stochastic Demand

97

Despite these progresses, exact evaluations and solutions are rare. Cachon (2001) provided exact evaluations for average inventory, backorders and fill rates for a two-echelon distribution system with multiple retailers and batch-ordering. Chen and Zheng (1994b) established lower bounds for serial, assembly and distribution systems with setup costs at all stages. Although many studies have provided full or partial characterizations of optimal polices for some special cases (see, e.g., Chen 2004a, b; Fox et al. 2006; Chao and Zipkin 2008; Sheopuri et al. 2010), a general optimal solution is hard, if not impossible. The reader may refer to Glasserman and Tayur (1994, 1995), Kapuscinski and Tayur (1998), Gallego and Toktay (2004), Ozer and Wei (2004), Chao and Zipkin (2008), and the references herein for capacitated multi-echelon inventory systems; and Tagaras and Vlachos (2001), Feng et al. (2006), Sheopuri et al. (2010), and the references herein for multi-echelon inventory systems with multiple supply modes.

6 Coordinated Replenishment and Information Sharing Despite the great progresses over the years, we seem to know little about the key factors that drive the performance of a multi-echelon inventory control system (Gallego et al. 2007). This is particularly the case for systems with multiple facilities at a stage (e.g. a distribution system with multiple retailers). Many control policies for multi-echelon inventory systems have been developed by extending installation control policies. However, a multi-echelon inventory control system is more than a simple addition of inventory control problems at individual facilities. It brings about completely new issues and challenges. We discuss two such issues and review the relevant literature in this section.

6.1

Coordinated Replenishment

In general, coordinated replenishment centralizes and synchronizes the ordering decisions for retailers. There are two distinct potential benefits. First, when retailers order together, they may achieve economies of scale in ordering such as quantity discounts from the outside supplier, savings in transportation costs, etc. Second, the system can choose to allocate the stock among the retailers at the time of shipment rather than at the time of ordering. Postponing the allocation allows the system to observe the demands at the retailers in the warehouse lead time, and thus to make a better informed allocation. This can result in significant system cost reductions. Eppen and Schrage (1981) coined the term “statistical economies of scale” for these benefits. This effect is also commonly called the benefits of stock pooling in the literature. Coordinated replenishment for multiple products is commonly referred to as the joint replenishment problem and has been studied extensively (Federgruen et al. 1984;

98

Q. Wang

Khouja and Goyal 2008). The objective is to achieve economies of scale in ordering. Apparently, similar benefits can be achieved by coordinating replenishments for multiple retailers in a supply chain. Axs€ater and Zhang (1999) attempted to develop a joint replenishment policy for a distribution system, but did not find an appropriate policy to improve system performance. A more efficient policy was developed subsequently by Cheung and Lee (2002). They considered a supplier serving multiple retailers and developed the following joint replenishment policy. Let each retailer set up a target inventory position. As demands arrive at the retailers, the inventory positions at the retailers drop below their respective targets. When the cumulative demands over all retailers reach a batch size Q, the supplier orders the quantity from a warehouse to restore the inventory positions at the retailers to their respective target levels. The orders from the warehouse are transported from the external supplier directly to the retailers. The supplier does not carry inventory and acts as only a cross-docking point. They showed that the joint replenishment policy dominates the installation-stock batch-ordering policy (Axs€ater 1993c). The benefits of coordinated replenishments for multiple retailers in a supply chain, however, are beyond the traditional economies of scale in ordering. Cheung and Lee (2002) showed that, if orders from the warehouse are first transported to the supplier and then allocated and shipped to the retailers upon their arrival at the supplier, system costs can be further reduced significantly. This is due to the statistical economies of scale. Eppen and Schrage (1981) first studied this effect by synchronizing ordering decisions for a group of identical retailers. Subsequently, Jackson (1988) extended the basic model to allow the warehouse to carry inventory and demonstrated empirically the benefits of centralizing at least a portion of the total system stock. Schwarz (1989) assessed the value of stock pooling by comparing two systems. In system one, inventories are ordered and shipped directly from an outside supplier to each retailer separately. In system two, this is done through a warehouse, thereby inventories are ordered together and allocated and shipped to the retailers upon their arrival. The effect of stock pooling is measured by the overall reduction in variance of the retailer end-of-cycle net inventory. Kumar et al. (1995) studied this effect along a fixed delivery route using a dynamic inventory control policy. McGavin et al. (1993, 1997) analyzed warehouse inventory allocation policies to minimize system lost sales. Other studies have considered stock pooling in various situations (see, e.g., Eynan and Fouque 2003; Benjaafar et al. 2005; Ve´ricourt et al. 2002; Kukreja et al. 2001; Wee and Dada 2005). In particular, Wang and Axs€ater (2010) and Wang (2010) showed that fixedinterval order-up-to polices are able to provide a more efficient mechanism for supply chain members to coordinate replenishments than stock-based batch-ordering policies. Because of the benefits of coordinated replenishment, particularly the statistical economies of scale, time-based control policies can perform significantly better than stock-based control policies for distribution systems with multiple retailers.

Control Policies for Multi-echelon Inventory Systems with Stochastic Demand

6.2

99

Demand and Stock Information

Iglehart (1964) initiated the research on the use of demand information in inventory control. He considered the inventory control problem for a single location facing a demand distribution with unknown parameters, and developed a Bayesian estimation scheme to update the demand distribution with new information for inventory control. This problem has been studied subsequently by many others (Azoury 1985; Lovejoy 1990; Milner and Kouvelis 2002). Furthermore, Zheng and Zipkin (1990) started to quantify the value of information sharing among multiple products and/or facilities. They considered the scheduling problem for two items competing for a single production facility, and showed that information about outstanding orders of the two products could improve system performance. Zipkin (1995) extended the analysis to a multi-item production facility subsequently. Supply chain management brings about a new issue of sharing information across supply chain members. A massive literature has accumulated on the use of stock and demand information (Gunasekaran and Ngai 2004). Bourland et al. (1996) considered the use of stock information to improve inventory control decisions. They considered a two-level serial system with a supplier and a single retailer, and demonstrated that the supplier could improve its replenishment decision by making use of stock information at the retailer. In the meantime, a number of studies analyzed the use of information sharing to reduce the bullwhip effect in supply chain systems (Lee et al. 1997; Lee and Whang 1999; Chen et al. 2000). The recent interest has focused on the use of advance demand information to improve the efficiency of multi-echelon inventory control systems (see, e.g., Hariharan and Zipkin 1995; Gallego and Ozer 2001; Iyer and Ye 2000; Ozer 2003; Marklund 2006; Axs€ater and Marklund 2008; Wang and Toktay 2008). Many studies have evaluated the value of stock and demand information in supply chain management. The evaluations, however, varied for different studies. Chen (1998) studied the benefits of stock information by comparing the costs for a multi-echelon serial system when using an echelon-stock batch-ordering policy and an installation-stock batch-ordering policy. He observed that echelon-stock information could reduce system cost by an average cost of 1.75%, with a maximum of 9%. Gavieneni et al. (1999) and Gavieneni (2002) considered a similar setting with a capacitated manufacturer and a single retailer, and observed higher values for stock information, ranging from 1 to 35% of system inventory related cost with an average of 14%. In the meantime, Cachon and Fisher (2000) considered a setting with one supplier and N identical retailers. They contrasted the value of information sharing with faster and cheaper order processing, which led to shorter lead times and smaller batch sizes, respectively, and found that implementing information technology to accelerate and smooth the physical flow of goods through a supply chain was significantly more valuable than to expand the flow of information. Furthermore, Gurbuz et al. (2007) extended the joint replenishment policy developed by Cheung and Lee (2002) to use retailer stock information to trigger a joint replenishment order. Following Moinzadeh (2002), they proposed a second-order

100

Q. Wang

trigger rule based on a minimum inventory position at each retailer so that an order is placed to the outside supplier when any retailer’s inventory position drops to a minimum requirement level or the demand at all retailers accumulates to Q units, whichever is earlier. They showed numerically that stock information could improve the joint replenishment policy. However, the improvements were only moderate. Evaluations for the value of demand information varied even greater. Lee et al. (2000) considered a two-level supply chain that consists of a manufacturer and a single retailer who faces a non-stationary auto-correlated demand process, and showed that the manufacturer would experience great savings when demand information was shared by the downstream member. The analysis was subsequently extended by Aviv (2002) to a setting in which companies could observe early market signals to improve their forecasting performance. However, Raghunathan (2001, 2003) pointed out later that the finding of Lee et al. (2000) depends on the critical assumptions that the manufacturer uses only the most recent order information from the retailer to forecast its future orders and that the parameters of the demand process are only accessed by the retailer. If the parameters of the autocorrelated demand process are known to both parties, sharing of demand information is actually of limited value. As the manufacturer can forecast the demand by using the retailer’s order history and the accuracy of forecast increases monotonically with each subsequent time period, the value of information decreases monotonically with each time period and converges to zero. Since the entire retailer order history is available to the manufacturer, the manufacturer is in a position to use the data in its forecasting process. Graves (1999) made similar observations in a slightly different setting.

7 Conclusions and Discussions To summarize the literature review, let us consider the distribution system that is described in Sect. 2. Suppose time is continuous and there are multiple retailers. Several alternative policies have been developed for the inventory control system in the literature. We list these policies in Table 1 below. Previous studies have shown that, if everything else holds constant, (1) Policy 2 dominates Policy 1 because echelon stock provides more accurate information about the system inventory state, (2) the joint replenishment (JR) policies, i.e. Policies 3, 4, and 5, dominate Policy 1, also because joint replenishment utilizes more accurate system stock information, (3) Policy 4 dominates Policy 3 due to the effect of stock pooling, and (4) Policy 5 dominates Policy 3 because more accurate information about the inventory state at retailers is used. Policy 4, Policy 5, and Policy 6 do not dominate each other. These relationships reflect the state of the art for the developments of multiechelon inventory control systems. Optimal control of a multi-echelon inventory system is still largely an open problem. First of all, little is known about the format

Control Policies for Multi-echelon Inventory Systems with Stochastic Demand

101

Table 1 Policies for a distribution system with multiple retailers Policy Policy description References Installation-stock Apply the stock-based (R, Q) policy Axs€ater (1993c, 2000), Forsberg batch-ordering at each facility based on (1997), Cheung and Hausman policy installation stock (2000) Echelon-stock batchApply the stock-based (R, Q) policy Axs€ater (1997), Axs€ater and ordering policy at each facility based on echelon Rosling (1993), Chen and stock Zheng (1997) JR policy without stock Order for all retailers when their Axs€ater and Zhang (1999), pooling demands accumulate to a fixed Cheung and Lee (2002) batch size and ship stock to each retailer directly from the outside supplier JR policy with stock Order for all retailers when their Cheung and Lee (2002) pooling demands accumulate to a fixed batch size and ship stock to a depot where the stock is optimally allocated JR policy using retailer Order for all retailers when their Gurbuz et al. (2007) stock information demands accumulate to a fixed but no stock pooling batch size and/or the inventory position at a retailer reaches a trigger level, and ship stock to each retailer directly from the outside supplier Fixed-interval orderApply the time-based (S, T) policy at Eppen and Schrage (1981), up-to policy each facility based on echelon Jackson (1988), Schwarz stock (1989), Wang and Axs€ater (2010)

of an optimal control policy. Stock-based batch-ordering policies and time-based order-up-to policies have been developed based respectively on the basic (R, Q) policy and (S, T) policy for a single facility. These formats have been adopted mostly because of convenience rather than optimality. For a single facility, the stock-based (R, Q) policy can be optimal and dominates the time-based (S, T) policy. This dominance can carry over to serial systems but not to distribution systems with multiple retailers. Second, little seems to be known about the factors that drive the performance for a multi-echelon inventory system. Previous studies have shown that an optimal policy has to consider at least three aspects (a) careful planning and coordination of inventory activities at the warehouse and retailers, (b) use of accurate and timely stock and demand information, and (c) pooling of stocks. However, the evaluations of these benefits varied significantly for different studies. For example, evaluations for the value of information varied from almost zero to an average system cost reduction of more than 10%. Apparently, the significance of these benefits depends on the setting and the inventory control policy adopted for the study, among other possible confounding factors. Trade-off often exists. For example, a setting or a format of inventory control policy that can fully utilize stock information may not

102

Q. Wang

be able to maximize the benefits of order coordination and stock pooling. Consequently, an inventory control policy that can fully optimize system performance, even if it exists, would be extremely difficult to structure and determine. As a result, the research on multi-echelon inventory control systems has focused on the identification of close-to-optimal policies that have a relatively simple structure and are reasonably easy to determine and implement. While this trend will continue, the question on the efficiency of an inventory control policy to optimize system performance will always be a challenge. Current inventory control policies have focused on some but not all driving factors for system performance. For example, stock-based policies focus more on using stock information to optimize replenishment decisions. In contrast, time-based policies focus more on coordinating system replenishments for both the traditional and statistical economies of scale. In view of these developments, while performance evaluations on multi-echelon inventory control policies to optimize system performance are very desirable, we may develop more efficient multi-echelon inventory control policies by combining the features of different policy formats. Finally, we have focused exclusively on multi-echelon inventory systems where decisions can be centralized by a unique decision maker to optimize system performance. Although many studies have considered decentralized multi-echelon inventory control systems (see, e.g., Cachon and Zipkin 1999; Chen 1999b; Axs€ater 2001, 2005; Caldentey and Wein 2003), the literature on decentralized systems is still limited as compared to the literature on centralized systems. This is inadequate, given the fact that most supply chains consist of members who are separate and independent economic entities. One reason for this inadequacy is that, although a centralized solution can optimize system performance, it is not always in the best interest of every individual member. Therefore, the implementation of a centralized solution or the coordination of a decentralized supply chain that aims to optimize system performance must have a mechanism to align the objectives of individual supply chain members. While a lot of research has been done on this subject for deterministic supply chains (Li and Wang 2007), relatively little seems to have been done on the topic for stochastic supply chain systems. Consequently, the coordination of decentralized multi-echelon inventory systems with stochastic demand and/or lead time represents great opportunities and challenges for future research.

References Arrow K, Harris T, Marschak J (1951) Optimal inventory policy. Econometrica 19:250–272 Atkins D, Iyogun P (1988) Periodic versus ‘can-order’ policies for coordinated multi-item inventory systems. Manage Sci 34:791–796 Aviv Y (2002) Gaining benefits from joint forecasting and replenishment processes: the case of auto-correlated demand. Manuf Serv Oper Manage 4(1):55–74 Axs€ater S (1990a) Simple solution procedures for a class of two-echelon inventory problems. Oper Res 38:64–69 Axs€ater S (1990b) Modeling emergency lateral transshipments in inventory systems. Manage Sci 36:1329–1338

Control Policies for Multi-echelon Inventory Systems with Stochastic Demand

103

Axs€ater S (1993a) Continuous review policies for multi-level inventory systems with stochastic demand. In: Graves S, Rinnooy Kan A, Zipkin P (eds) Handbooks in operations research and management science, vol 4, Logistics of production and inventory. North-Holland, Amsterdam, pp 175–197 Axs€ater S (1993b) Optimization of order-up-to-S policies in two-echelon inventory systems with periodic-review. Nav Res Logist 40:245–253 Axs€ater S (1993c) Exact and approximate evaluation of batch-ordering policies for two-level inventory systems. Oper Res 41(4):777–785 Axs€ater S (1997) Simple evaluation of echelon stock (R, Q) policies for two-level inventory systems. IIE Trans 29:661–669 Axs€ater S (2000) Exact analysis of continuous review (R, Q) policies in two-echelon inventory systems with compound Poisson demand. Oper Res 48(5):686–696 Axs€ater S (2001) A framework for decentralized multi-echelon inventory control. IIE Trans 33:91–97 Axs€ater S (2003a) Approximate optimization of a two-level distribution inventory system. Int J Prod Econ 81–82:545–553 Axs€ater S (2003b) Supply chain operations: serial and distribution inventory systems. In: Graves SC, de Kok T (eds) Supply chain management: design, coordination, and operation. Handbooks in operations research and management science, vol 11, Chap. 10. Elsevier, Amsterdam, pp 525–559 Axs€ater S (2005) A simple decision rule for decentralized two-echelon inventory control. Int J Prod Econ 93–94:53–59 Axs€ater S, Juntti L (1996) Comparison of echelon stock and installation stock policies for twolevel inventory systems. Int J Prod Econ 45:303–310 Axs€ater S, Marklund J (2008) Optimal position-based warehouse ordering in divergent twoechelon inventory systems. Oper Res 56(4):976–991 Axs€ater S, Rosling K (1993) Installation vs. echelon stock policies for multilevel inventory control. Manage Sci 39(10):1274–1280 Axs€ater S, Zhang W (1999) A joint replenishment policy for multi-echelon inventory control. Int J Prod Econ 59:243–250 Axs€ater S, Marklund J, Silver EA (2002) Heuristic methods for centralized control of onewarehouse, N-retailer inventory systems. Manuf Serv Oper Manage 4(1):75–97 Azoury K (1985) Bayes solutions to dynamic inventory models under unknown demand distributions. Manage Sci 31(9):1150–1160 Badinelli R (1992) A model for continuous-review pull policies in serial inventory systems. Oper Res 40(1):142–156 Benjaafar S, Cooper W, Kim J (2005) On the benefits of pooling in production-inventory systems. Manage Sci 51(4):548–565 Bourland K, Powell S, Pyke D (1996) Exploiting timely demand information to reduce inventories. Eur J Oper Res 92:239–253 Cachon G (2001) Exact evaluation of batch-ordering inventory policies in two-echelon supply chains with periodic review. Oper Res 49(1):79–98 Cachon G, Fisher M (2000) Supply chain inventory management and the value of shared information. Manage Sci 46(8):1032–1048 Cachon G, Zipkin P (1999) Competitive and cooperative inventory policies in a two-stage supply chain. Manage Sci 45(7):936–953 Caldentey R, Wein L (2003) Analysis of a decentralized production-inventory system. Manuf Serv Oper Manage 5(1):1–17 Cetinkaya S, Lee C (2000) Stock replenishment and shipment scheduling for vendor managed inventory. Manage Sci 46(2):217–232 Chao X, Zhou S (2009) Optimal policy for a multiechelon inventory system with batch ordering and fixed replenishment intervals. Oper Res 57(2):377–391

104

Q. Wang

Chao X, Zipkin P (2008) Optimal policy for a periodic-review system under a supply capacity contract. Oper Res 56(1):59–70 Chen F (1998) Echelon reorder points, installation reorder points, and the value of centralized demand information. Manage Sci 44(12):S221–S234 Chen F (1999a) 94%-effective policies for a two-stage serial inventory system with stochastic demand. Manage Sci 45(12):1679–1696 Chen F (1999b) Decentralized supply chains subject to information delays. Manage Sci 45 (8):1076–1090 Chen F (2000) Optimal policies for multi-echelon inventory problems with batch ordering. Oper Res 48(3):376–389 Chen S (2004a) The optimality of hedging point policies for stochastic two-product flexible manufacturing systems. Oper Res 52(2):321–323 Chen S (2004b) The infinite horizon periodic review problem with setup costs and capacity constraints: a partial characterization of the optimal policy. Oper Res 52(3):409–421 Chen F, Song J (2001) Optimal policies for multi-echelon inventory problems with Markovmodulated demand. Oper Res 49(2):226–234 Chen F, Zheng Y (1994a) Evaluating echelon stock (R, nQ) policies in serial production/inventory systems with stochastic demand. Manage Sci 40(10):1262–1275 Chen F, Zheng Y (1994b) Lower bounds for multi-echelon stochastic inventory systems. Manage Sci 40(11):1426–1443 Chen F, Zheng Y (1997) One-warehouse multi-retailer systems with centralized stock information. Oper Res 45(2):275–287 Chen F, Zheng Y (1998) Near-optimal echelon-stock (R, nQ) policies in multistage serial system. Oper Res 46(4):592–602 Chen F, Drezner Z, Ryan J, Simchi-Levi D (2000) Quantifying the bullwhip effect in a simple supply chain: the impact of forecasting lead time and information. Manage Sci 46(3):436–443 Cheung K, Hausman W (2000) An exact performance evaluation for the supplier in a two-echelon inventory system. Oper Res 48(4):646–653 Cheung K, Lee H (2002) The inventory benefit of shipment coordination and stock rebalancing in a supply chain. Manage Sci 48(2):300–306 Clark A (1958) A dynamic, single-item, multi-echelon inventory model. Report. Rand Corporation, Santa Monica, CA Clark A (1972) An informal survey of multi-echelon inventory theory. Nav Res Logistics Q 19:621–650 Clark A, Scarf H (1960) Optimal policies for a multi-echelon inventory problem. Manage Sci 6 (4):475–490 Clark A, Scarf H (1962) Approximate solutions to a simple multi-echelon inventory problem. In: Arrow KJ, Karlin S, Scarf H (eds) Studies in applied probability and management science. Stanford University Press, Stanford, CA De Bodt M, Graves S (1985) Continuous review policies for a multi-echelon inventory problem with stochastic demand. Manage Sci 31(10):1286–1299 Deuermeyer B, Schwarz L (1981) A model for the analysis of system service level in warehouse/ retailer distribution systems: the identical retailer case. In: Schwarz LB (ed) Studies in the management sciences, vol 16, Multi-level production/inventory control systems. North-Holland, Amsterdam, pp 163–193 Dvoretzky A, Kiefer J, Wolfowitz J (1952) The inventory problem. Econometrica 20:187–222 Eppen G, Schrage L (1981) Centralized ordering policies in a multiwarehouse system with leadtimes and random demand. In: Schwarz LB (ed) Studies in the management sciences, vol 16, Multi-level production/inventory control systems. North-Holland, Amsterdam, pp 88–110 Eynan A, Fouque T (2003) Capturing the risk-pooling effect through demand reshape. Manage Sci 49(6):704–717

Control Policies for Multi-echelon Inventory Systems with Stochastic Demand

105

Eynan A, Kropp D (1998) Periodic review and joint replenishment in stochastic demand environment. IIE Trans 30:1025–1033 Federgruen A (1993) Centralized planning models for multi-echelon inventory systems under uncertainty. In: Graves S, Rinnooy Kan A, Zipkin P (eds) Handbooks in operations research and management science, vol 4, Logistics of production and inventory. North-Holland, Amsterdam, pp 133–173 Federgruen A, Zipkin P (1984a) Computational issues in an infinite-horizon, multi-echelon inventory model. Oper Res 32(4):818–836 Federgruen A, Zipkin P (1984b) Approximations of dynamic, multilocation production and inventory systems. Manage Sci 30(1):69–84 Federgruen A, Zipkin P (1984c) Allocation policies and cost approximation for multilocation inventory systems. Nav Res Logistics Q 31:97–131 Federgruen A, Groenevelt H, Tijms H (1984) Coordinated replenishments in a multi-item inventory system with compound Poisson demands and constant lead times. Manage Sci 30:344–357 Feng K, Rao U (2007) Echelon-stock (R, nT) control in two-stage serial stochastic inventory systems. Oper Res Lett 35:95–104 Feng Q, Sethi S, Yan H, Zhang H (2006) Are base-stock policies optimal in inventory problems with multi delivery modes? Oper Res 54(4):801–807 Forsberg R (1995) Optimization of order-up-to-S policies for two-level inventory systems with compound Poisson demand. Eur J Oper Res 81:143–153 Forsberg R (1997) Exact evaluation of (R, Q)-policies for two level inventory systems with Poisson demand. Eur J Oper Res 96:130–138 Fox E, Metters R, Semple J (2006) Optimal inventory policy with two suppliers. Oper Res 56 (4):696–707 Gallego G, Ozer O (2001) Integrating replenishment decisions with advance demand information. Manage Sci 47(10):1344–1360 Gallego G, Ozer O (2003) Optimal replenishment policies for multi-echelon inventory systems under advance demand information. Manuf Serv Oper Manage 5:157–175 Gallego G, Ozer O (2005) A new algorithm and a new heuristic for serial supply systems. Oper Res Lett 33:349–362 Gallego G, Toktay L (2004) All-or-nothing ordering under capacity constrain. Oper Res 52 (6):1001–1002 Gallego G, Ozer O, Zipkin P (2007) Bounds, heuristics, and approximations for distribution systems. Oper Res 55(3):503–517 Ganeshan R, Jack E, Magazine M, Stephens P (1998) A taxonomic review of supply chain management research. In: Tayur S, Magazine M, Ganeshan R (eds) Quantitative models for supply chain management, vol 17, International series in operations research and management science. Kluwer, Norwell, MA, pp 839–879 Gavieneni S (2002) Information flows in capacitated supply chains with fixed ordering cost. Manage Sci 48(5):644–651 Gavieneni S, Kapuscinski R, Tayur S (1999) Value of information in capacitated supply chains. Manage Sci 45(1):16–24 Glasserman P, Tayur S (1994) The stability of a capacitated, multi-echelon production-inventory system under a base-stock policy. Oper Res 42(5):913–925 Glasserman P, Tayur S (1995) Sensitivity analysis for base-stock levels in multi-echelon production-inventory systems. Manage Sci 41(2):263–281 Graves S (1985) A multi-echelon inventory model for repairable item with one-for-one replenishment. Manage Sci 31(10):1247–1256 Graves S (1996) A multi-echelon inventory model with fixed replenishment intervals. Manage Sci 42(1):1–18 Graves S (1999) A single-item inventory model for a nonstationary demand process. Manuf Serv Oper Manage 1(1):50–61

106

Q. Wang

Gunasekaran A, Ngai E (2004) Information systems in supply chain integration and management. Eur J Oper Res 159:269–295 Gurbuz M, Moinzadeh K, Zhou Y (2007) Coordinated replenishment strategies in inventory/ distribution systems. Manage Sci 53(2):293–308 Hariharan R, Zipkin P (1995) Customer-order information, leadtimes, and inventories. Manage Sci 41(10):1599–1607 Iglehart D (1964) The dynamic inventory problem with unknown demand distributions. Manage Sci 10(3):429–440 Iyer A, Ye J (2000) Assessing the value of information sharing in a promotional retail environment. Manuf Serv Oper Manage 2(2):128–143 Jackson P (1988) Stock allocation in a two-echelon distribution system or “what to do until your ship comes in”. Manage Sci 34(7):880–895 Jacobs F, Chase R, Aquilano N (2009) Operations and supply management, 12th edn. McGrawHill, New York Johansen S, Thorstenson A (1998) An inventory model with Poisson demands and emergency orders. Int J Prod Econ 56–57:275–289 Kapuscinski R, Tayur S (1998) A capacitated production-inventory model with period demand. Oper Res 46(6):899–911 Khouja M, Goyal S (2008) A review of the joint replenishment problem literature: 1989–2005. Eur J Oper Res 186:1–16 Kukreja A, Schmidt C, Miller D (2001) Stocking decisions for low-usage items in a multilocation inventory system. Manage Sci 47(10):1371–1383 Kumar A, Schwarz L, Ward J (1995) Risk-pooling along a fixed delivery route. Manage Sci 41:344–362 Lee H (1987) A multi-echelon inventory model for repairable items with emergency lateral transshipments. Manage Sci 33:1302–1316 Lee H, Moinzadeh K (1987a) Two-parameter approximations for multi-echelon repairable inventory models with batch ordering policy. IIE Trans 19:140–149 Lee H, Moinzadeh K (1987b) Operating characteristics of a two-echelon inventory system for repairable and consumable items under batch ordering and shipment policy. Nav Res Logistics Q 34:365–380 Lee H, Whang S (1999) Decentralized multi-echelon supply chains: incentives and information. Manage Sci 45(5):633–640 Lee H, Padmanabhan V, Whang S (1997) Information distortion in a supply chain: the bullwhip effect. Manage Sci 43(4):546–558 Lee H, So K, Tang C (2000) The value of information sharing in a two-level supply chain. Manage Sci 46(5):626–643 Li X, Wang Q (2007) Coordination mechanisms of supply chain systems. Eur J Oper Res 179:1–16 Lovejoy W (1990) Myopic policies for some inventory models with uncertain demand distributions. Manage Sci 36(6):724–738 Marklund J (2006) Controlling inventories in divergent supply chains with advance order information. Oper Res 54(5):988–1010 McGavin E, Schwarz L, Ward J (1993) Two-interval inventory-allocation policies in a onewarehouse N-identical retailer distribution system. Manage Sci 39(9):1092–1107 McGavin E, Ward J, Schwarz L (1997) Balancing retailer inventories. Oper Res 45(6):820–830 Milner J, Kouvelis P (2002) On the complementary value of accurate demand information and production and supplier flexibility. Manuf Serv Oper Manage 4(2):99–113 Moinzadeh K (2002) A multi-echelon inventory system with information exchange. Manage Sci 48(3):414–426 Moinzadeh K, Lee H (1986) Batch size and stocking levels in multi-echelon repairable systems. Manage Sci 32:1567–1581 Moinzadeh K, Nahmias S (1988) A continuous review model for an inventory system with two supply modes. Manage Sci 34:761–773

Control Policies for Multi-echelon Inventory Systems with Stochastic Demand

107

Moinzadeh K, Schmidt C (1991) An (S-1, S) inventory system with emergency orders. Oper Res 39:308–321 Muckstadt J (1973) A model for a multi-item, multi-echelon, multi-indenture inventory system. Manage Sci 20:472–481 Muckstadt J (1979) A three-person, multi-item model for recoverable items. Nav Res Logistics Q 26:199–221 Ozer O (2003) Replenishment strategies for distribution systems under advance demand information. Manage Sci 49(3):255–272 Ozer O, Wei W (2004) Inventory control with limited capacity and advance demand information. Oper Res 52(6):988–1001 Porteus E (1990) Stochastic inventory theory. In: Heyman D, Sobel M (eds) Stochastic models, handbooks in operations research and management science, vol 12. Elsevier (North-Holland), Amsterdam Raghunathan S (2001) Information sharing in a supply chain: a note on its value when demand is nonstationary. Manage Sci 47(4):605–610 Raghunathan S (2003) Impact of demand correlation on the value of and incentives for information sharing in a supply chain. Eur J Oper Res 146:634–649 Rao U (2003) Properties of the periodic-review (R, T) inventory control policy for stationary, stochastic demand. Manuf Serv Oper Manage 5(1):37–53 Rosling K (1989) Optimal inventory policies for assembly systems under random demands. Oper Res 37(4):565–579 Roundy R (1985) 98%-effective integer-ratio lot-sizing for one-warehouse multi-retailer systems. Manage Sci 31(11):1416–1429 Roundy R (1986) A 98%-effective lot sizing rule for a multi-product, multi-stage production/ distribution system. Math Oper Res 11(4):699–727 Sahin F, Robinson E (2002) Flow coordination and information sharing in supply chains: review, implications, and directions for future research. Decis Sci 33(4):505–535 Schmidt C, Nahmias S (1985) Optimal policy for a two-stage assembly system under random demand. Oper Res 33:1130–1145 Schwarz L (1989) A model for assessing the value of warehouse risk-pooling: risk-pooling over outside-supplier leadtimes. Manage Sci 35(7):828–842 Shang K, Song J (2003) Newsvendor bounds and heuristic for optimal policies in serial supply chains. Manage Sci 49(5):618–638 Shang K, Song J (2007) Serial supply chains with economies of scale: bounds and approximations. Oper Res 55(5):843–853 Shang K, Zhou S (2010) Optimal and heuristic echelon (r, nQ, T) policies in serial inventory systems with fixed costs. Oper Res 58(2):414–429 Shanker K (1981) Exact analysis of a two-echelon inventory system for recoverable items under batch inspection policy. Nav Res Logistics Q 28:579–601 Sheopuri A, Janakiraman G, Seshdri S (2010) New policies for the stochastic inventory control problem with two supply sources. Oper Res 58(3):734–747 Sherbrooke C (1968) METRIC: a multi-echelon technique for recoverable item control. Oper Res 16:122–141 Sherbrooke C (1986) VARI-METRIC: improved approximation for multi-indenture, multi-echelon availability models. Oper Res 34:311–319 Silver E, Peterson R (1998) Decision systems for inventory management and production. Wiley, New York Simon R (1971) Stationary properties of a two-echelon inventory model for low demand items. Oper Res 19:761–777 Song J, Yao D (2002) Performance analysis and optimization of assemble-to-order systems with random lead times. Oper Res 50(5):889–903 Sosic G (2006) Transshipment of inventories among retailers: myopic vs. farsighted stability. Manage Sci 52(10):1493–1508

108

Q. Wang

Svoronos A, Zipkin P (1988) Estimating the performance of multi-level inventory systems. Oper Res 36(1):57–72 Svoronos A, Zipkin P (1991) Evaluation of one-for-one replenishment policies for multiechelon inventory systems. Manage Sci 37:68–83 Tagaras G, Vlachos D (2001) A periodic review inventory system with emergency replenishments. Manage Sci 47(3):415–429 Tan K (2001) A framework of supply chain management literature. Eur J Purch Supply Manage 7:39–48 Tsay A, Nahmias S, Agrawa N (1998) Modeling supply chain contracts: a review. In: Tayur S, Magazine M, Ganeshan R (eds) Quantitative models for supply chain management, vol 17, International series in operations research and management science. Kluwer, Norwell, MA, pp 299–336 van Houtum G, Scheller-Wolf A, Ji X (2007) Optimal control of serial inventory systems with fixed replenishment intervals. Oper Res 55(4):674–687 Veinott A, Wagner H (1965) Computing optimal (s, S) inventory problems. Manage Sci 11:525–552 Ve´ricourt F, Karaesmen F, Dallery Y (2002) Optimal stock allocation for a capacitated supply system. Manage Sci 48(11):1486–1501 Viswanathan S (1997) Note. Periodic review (s, S) policies for joint replenishment inventory systems. Manage Sci 43:1447–1454 Wang Q (2010) Integer-ratio policies for distribution systems with multiple retailers and stochastic demand. Working paper, Nanyang Business School, Nanyang Technological University, Singapore Wang Q, Axs€ater S (2010) Fixed-interval order-up-to policies for distribution systems with multiple retailers and stochastic demand. Working paper, Nanyang Business School, Nanyang Technological University, Singapore Wang T, Toktay B (2008) Inventory management with advance demand information and flexible delivery. Manage Sci 54(4):716–732 Wee K, Dada M (2005) Optimal policies for transshipping inventory in a retailer network. Manage Sci 51(10):1519–1533 Yano C, Carson C (1988) Safety stocks for assembly systems with fixed production schedules. J Manuf Oper Manage 1:182–201 Zheng Y (1992) On properties of stochastic inventory systems. Manage Sci 38(1):87–103 Zheng Y, Federgruen A (1991) Finding optimal (s, S) policies is about as simple as evaluating a single policy. Oper Res 39(4):654–665 Zheng Y, Zipkin P (1990) A queuing model to analyze the value of centralized inventory information. Oper Res 38(2):296–307 Zipkin P (1986a) Stochastic leadtimes in continuous-time inventory models. Nav Res Logistics Q 33:763–774 Zipkin P (1986b) Inventory service level measures: convexity and approximation. Manage Sci 32 (8):975–981 Zipkin P (1995) Performance analysis of a multi-item production-inventory system under alternative policies. Manage Sci 41(4):690–703 Zipkin P (2000) Foundations of inventory management. McGraw-Hill, New York

Supply Chain Models with Active Acquisition and Remanufacturing Xiang Li and Yongjian Li

Abstract Nowadays economic, marketing, and environment legislation are increasingly driving firms to consider product reuse, thus active acquisition of used products is prevailing in industry. This chapter focuses on the problem regarding the active acquisition and remanufacturing within supply chain scope. We introduce some recent and important research developments, including the centralized system problem with price-sensitive acquisition and demand, the decentralized supply chain problem with vertical channels, and the decentralized supply chain problem with horizontal competition. For each problem, analytical models are presented and main results are elucidated. Finally, further research directions are also pointed out. Keywords Acquisition management • Closed-loop supply chain • Coordination • Remanufacturing

1 Introduction Supply chain management considering return flows has received substantial interests from both industrial and academic worlds. Product returns represent a growing financial concern for firms, with an estimation of $35 billion annually for the USA alone (Meyer 1999). From a traditional perspective, these returns fall into two categories: consumer returns to the retailer during the return period, and product overstocks returned to the upstream manufacturer, both incurring considerable reprocessing cost and revenue loss. This return flow is regarded as “a bad X. Li (*) Research Center of Logistics, College of Economic and Social Development, Nankai University, Tianjin 300071, P.R. China e-mail: [email protected] Y. Li Business School, Nankai University, Tianjin 300071, P.R. China e-mail: [email protected] T.-M. Choi and T.C. Edwin Cheng (eds.), Supply Chain Coordination under Uncertainty, International Handbooks on Information Systems, DOI 10.1007/978-3-642-19257-9_5, # Springer-Verlag Berlin Heidelberg 2011

109

110

X. Li and Y. Li

thing”, and the associated costs are reduced by developing various return policies and incentive strategies (Ferguson et al. 2006). On the other hand, nowadays legislations usually encourage the reuse of product returns. Remanufacturing technology has been developed as an environmentally and economically sound way to deal with the returns after customer usage. As a result, a trend of active acquisition of used product is replacing the traditional passive return, and the remanufacturing industry is being boosted. The relevant activities, however, require effective coordination at multiple locations to retrieve the used product and recover its value, which is usually referred to as reverse supply chain management (Prahinski and Kocabasoglu 2006). The acquisition and remanufacturing management turns out to be a complicated problem, considering the uncertainty of the product returns, and the different, usually conflicting objectives of the supply chain members. This chapter focuses on the supply chain problem with active acquisition and remanufacturing, hoping to provide some recent research models and results. More specifically, there are three streams of researches drawing our attention: the centralized system problem with price-sensitive acquisition and demand, the decentralized supply chain problem with vertical channel, and the decentralized supply chain problem with horizontal competition. First of all, the used product can be actively collected in a market-driven channel by paying a price called acquisition price to end users or core dealers. The related problem is first proposed by Guide and Jayaraman (2000), which delineates the necessity of a careful coordination to balance the returns with the demand. Guide et al. (2003) builds up a quantitative model of a remanufacturing system in which the returns and demand could be controlled by the acquisition price and selling price, respectively, and Bakal and Akcali (2006) extends it into a random remanufacturing yield case. Ray et al. (2005) studies the optimal pricing/trade-in strategies for a durable, remanufacturable product, by characterizing some key factors of such product. Other related papers include Robotis et al. (2005), Qu and Williams (2008), Liang et al. (2009), etc. In sum, all the above papers focus on the centralized control of integrated acquisition and remanufacturing system. As for the decentralized supply chain, the acquisition and remanufacturing problems are usually linked up with game theory and contracting. One stream of the research is on the vertical channel, which is studied under Stackelberg leaderfollower game framework. Savaskan et al. (2004) considers the problem of choosing an appropriate reverse channel for the acquisition of used product. Three channel structures are analyzed and compared, involving the interaction among supply chain members such as the OEM, retailer, and 3PL collector. In the succeeding work, Savaskan and Van Wassenhove (2006) further explores the reverse channel choice problem with one upstream manufacturer and two competing retailers. The reverse supply chain model is also studied in Karakayali et al. (2006) without considering the forward distribution channel. As most papers focus on the policy of commercial returns, e.g., Pasternack(1985), Padmanabhan and Png (1997), Mukhopadhyay and Setoputro (2005), Yue and Raghunathan (2007) and so on, the research on vertical supply chain management with the active acquisition is rather limited.

Supply Chain Models with Active Acquisition and Remanufacturing

111

Furthermore, the study on the acquisition and remanufacturing competition at a horizontal supply chain level is being a hot topic recently. Majumder and Groenevelt (2001a) describes a two-period model under the assumption of indistinguishable quality between remanufactured and new product, but distinguishable brand between the competing OEM and remanufacturer. Majumder and Groenevelt (2001b) extends the model by allowing for competition in the acquisition of used product. Ferrer and Swaminathan (2006) extends Groenevelt and Majumder (2001a) into a multi-period setting where the brand competition is carried on in the second and subsequent periods. Ferguson and Toktay (2006) adopts a similar two-period model focusing on the strategic role of OEM remanufacturing as an entry deterrent to the local remanufacturer. The remanufacturing cost has a quadratic form and the competition is between the sales of new and remanufactured products. Atasu et al. (2008) also analyzes a two-period model to study the effect of timing of the remanufactured product introduction and the use of remanufacturing as a marketing strategy to compete with low-cost OEM competitors. Other related papers include Debo et al. (2005), Heese et al. (2005), Webster and Mitra (2007, 2008), etc. Most of the above models are under Nash game framework and two-period setting. In this chapter, we choose to present six models from the above three research streams, and provide an in-depth discussion on their managerial insights. The purpose is not to give an all-around review or survey on the numerous literature related to product returns, but to offer some connected and evolving works, hoping to reflect new research trends. Also, one of our main focuses is on the collection effort (and price) paid by supply chain members during the active acquisition of the used product, therefore some important researches outside this scope are not involved in the chapter. For example, some papers study the acquisition decision of the used product under a traditional newsvendor framework, where the used product acquisition can be simply regarded as a normal product order, e.g., Ferrer (2003), Vlachos and Dekker (2003), Galbreth and Blackburn (2006), Zikopoulos and Tagaras (2007), Kaya (2010), etc. Some other papers consider the acquisition problem with the location decision of acquisition centers, e.g., Wojanowski et al. (2007), Aras and Aksen (2008), Aras et al. (2008), etc. For a more comprehensive review on these subjects, we refer the reader to Dekker et al. (2004), Guide and Van Wassenhove (2009), Pokharel and Mutha (2009), and Ilgin and Gupta (2010).

2 Centralized System with Price-Sensitive Acquisition and Demand As stated in the last section, in the market-driven channel an acquisition price is paid to end users or core suppliers for the returned used product. This approach is widely adopted as it grants the firm a partial control on the used product returns. Guide and Van Wassenhove (2001) provides a detailed case study of the telephone remanufacturer, ReCellular Inc., buying used phones from a variety of sources and

112

X. Li and Y. Li

and selling remanufactured ones. Similar examples can also be found in the automotive industry, where automotive parts from the end-of-life vehicles are remanufactured and reused (Bakal and Akcali 2006). A common feature of the above cases is that the remanufacturing firm has full pricing power in both used product acquisition and remanufactured product demand, the quantities of which are sensitive to the acquisition price and the selling price, respectively. In this section, we investigate this pricing problem faced by the remanufacturing firm, with the objective of maximizing the profit from remanufacturing. The pricing decision is regarded as a lever to match the demand and supply, and both the deterministic and random yield models can be established.

2.1

Deterministic Model

A remanufacturing firm, ReCellular, collects used phones from the providers grading the phones and selling them in different quality classes. The used phones with quality classification differ in their condition, like the appearance, damage, and age, and the remanufacturing costs for different quality classes are different. On the other hand, all the remanufactured phones have the same quality and the same selling price. The acquisition quantities and demand are deterministic and dependent on the acquisition prices and selling price, respectively. This quantitative model is studied in Guide et al. (2003), and introduced in the following. Suppose that there are N quality classes, 1,2,. . .,N. The remanufacturing cost of class i is ci. The acquisition price of the used product of class i is denoted as fi, and the corresponding return quantity is ri( fi). It is assumed that ri( fi) is a continuous, increasing and twice differentiable function defined on [bi, g ci], where bi is the minimal acquisition price and g is the maximum price at which a remanufactured product can be sold. For convenience, the classes are ordered in such a way that b1 þ c1 < b2 þ c2 < < bN þ cN . On the demand side, let d(p) be the demand of the remanufactured product when the selling price is p, which is a continuous, decreasing and twice differentiable function defined on [b1 + c1, g]. The inverse function of d( p) is denoted by P(d), i.e., P(d(p)) ¼ p. And the inverse function of ri( fi) is denoted by Ai(ri), i.e., Ai(ri( fi)) ¼ fi, for all i. The objective is to determine the acquisition prices fi ; i ¼ 1; . . . ; N, and the selling price p to maximize the profit. The optimal acquisition price is denoted by fi and the resulting acquisition quantity is ri ; i ¼ 1; . . . ; N. As it is a deterministic model, the demand of the remanufactured product should be equal to the quantity of the used product returns to attain the profit maximization, i.e., the prices should satisfy dðpÞ ¼

N X i¼1

ri ðfi Þ:

(1)

Supply Chain Models with Active Acquisition and Remanufacturing

113

Thus, the optimization problem can be formulated as ðP1Þ

max P

f1 ;;fN

N X

! ri ðfi Þ

i¼1

N X

ri ðfi Þ

i¼1

N X

½ri ðfi Þðfi þ ci Þ;

(2)

i¼1

or equivalently, ðP2Þ

N X

max P

r1 ;...;rN

! ri

i¼1

N X i¼1

ri

N X

½ri ðAðri Þ þ ci Þ:

(3)

i¼1

It is natural to ask if the objective function (P1) or (P2) has joint concavity. Examples have shown that the function is not always concave and even not unimodal in some cases. Then the following two questions turn out to be essential: 1. On what conditions the objective function is concave (or unimodal) and in that case, is there any analytical property helpful to solve the problem? 2. Is there any algorithm/heuristic useful for the problem? Guide et al. (2003) provides the following answers. First, the objective (P2) is jointly concave if dP(d) is concave and riA(ri) is convex for any i. Second, under these conditions (called Condition C1 and C2, respectively), there is an M such that fi >bi for all i ¼ 1; . . . ; M and fi ¼ bi for all i ¼ M þ 1; ; N. This property indicates it is optimal to only acquire the used product in class i ¼ 1; ; M for some 1 M N. For the second question, define functions GðpÞ :¼ ddðpÞ and 0 ðpÞ þ p Fi ðfi Þ :¼ rr0iiðfðfiiÞÞ þ ðfi þ ci Þ. Then, the first-order optimality conditions for problem (P1) can be expressed by Fi ðfi Þ ¼ G P

N X

!! ri ðfi Þ

for i ¼ 1; ; M;

(4)

for i ¼ M þ 1; ; N:

(5)

i¼1

and Fi ðfi Þ G P

N X

!! ri ðfi Þ

i¼1

The above properties yield the following algorithm to calculate the optimal acquisition prices under Conditions C1 and C2. Step 1. M: ¼ 1. M ðaM Þ :¼ aM . If M 2, use the equation Fi(ai) ¼ FM(aM) to express ai as Step 2. HM a function of aM, denoted by HiM ðaM Þ, for i ¼ 1; 2; ; M 1.

114

X. Li and Y. Li

P M Step 3. Use the equation dðpÞ ¼ M i¼1 ri Hi ðaM ÞÞ to express p as a function of aM, denoted by KM(aM). Step 4. Obtain the value aM M for aM by equalizing FM(aM) to G(KM(aM)). Step 5. If either M ¼ N or FM ðaM M Þ FMþ1 ðbMþ1 Þ, output the optimal prices: fi ¼ M M M M Hi ðaM Þ for i ¼ 1; ; M, and fi ¼ bi for i ¼ M þ 1; ; N; And p ¼ K ðaM Þ. Otherwise, M :¼ M þ 1 and then go to Step 1. This algorithm shows the procedure to compute the exact optimal prices. However, Steps 4 and 5 have to be carried out numerically if the demand and return functions have complex shapes. Moreover, the algorithm can be only applied when both the conditions C1 and C2 are satisfied. In this regard, a heuristic is derived in Guide et al. (2003) based on the idea of making the prices such that the profit per product is fixed. It is set that fi ¼ max(bi, fnew ci), where fnew is the theoretical acquisition price that the firm is willing to pay for an as-good-as-new returned item, which doesn’t need the remanufacturing. The associated total return P quantity r 0 ðfnew Þ :¼ i;bi þci

2.2

Random Remanufacturing Yield Model

The uncertainty on product returns is regarded as an important characteristic of reverse logistics. In a market-driven collection channel, the quantity uncertainty of the returns is considerably controlled by acquisition price, while the quality uncertainty still exists. This quality uncertainty can be reflected by remanufacturing yield randomness and is considered in this subsection. The problem is motivated by the automotive remanufacturing industry. Typically, the automotive dismantling/remanufacturing firm buys end-of-life vehicles (ELVs) and sells remanufacturable parts, facing both the acquisition pricing and selling pricing problems. However, due to the uncertainty of the returned product quality, there is a random yield in the part remanufacturing process. That is to say, the quality requirement for remanufacturing is not satisfied by all returned parts, and the percentage of remanufacturable ones is uncertain. The random yield is a major concern in this remanufacturing system. Specifically, suppose that the supply of ELVs r(f) is a deterministic and linear function of the acquisition price f, i.e., r(f) ¼ a + bf, where a,b > 0. The demand for the remanufactured part d(p) is a deterministic and linear function of the selling price p, i.e., d(p) ¼ a bp. The random yield is also dependent on the acquisition price and modeled as the product of R and t(f). t(f) is a deterministic, concave and

Supply Chain Models with Active Acquisition and Remanufacturing

115

nondecreasing function of f which converges to 1 as f increases, and R is a random variable denoting the maximum attainable yield rate. The part not remanufacturable or not remanufactured has a salvage value s. It is also supposed that each ELV has a hulk that can be salvaged with a unit price h. The testing cost for each part is c and the remanufacturing cost for remanufacturable part is cr. The objective is to maximize the total expected profit from the remanufactured part sale and the hulk sale. Bakal and Akcali (2006) studies three different models to explore the acquisition pricing and selling pricing decisions of this problem. The first model is the deterministic yield case serving as a benchmark for the random yield cases. The pricing problem is formulated as follows: max f ; p Pðf ; pÞ ¼ ða bpÞðp s cr Þ þ ða þ bf Þðh þ s f cÞ s:t: ða bpÞ Rtðf Þða þ bf Þ

(6)

where the yield R now is a constant. Let K ¼ h + s c and r0 ¼ [a b(s + cr)]/ [t(K/s a/2b)(a + bK)]. It turns out that r0 is a threshold yield rate. If r0 1, then the profit is increasing in R until R ¼ r0 and remains constant thereafter. Otherwise, the profit is strictly increasing in R. The second model is called Postponed Pricing Model (PPM), in which the yield is random and the remanufacturer has the opportunity to set the selling price of the remanufactured part after the realization of random yield. Therefore, a two-step dynamic programme can be formulated for this model as follows. ðP3Þ

max p P1 ðpjr; f Þ ¼ ða bpÞðp s cr Þ þ ða þ bf Þðh þ s f cÞ s:t:ða bpÞ rtðf Þða þ bf Þ (7)

and ðP4Þ

max f Pðf Þ ¼ ER ½P1 ðp ðR; f ÞjR; f Þ

(8)

where p ðr; f Þ denotes the optimal selling price of Problem (P3) given the yield is realized as r and the acquisition price is f. It is generally difficult to explore the property of P(f) in (P3) for arbitrary t(f). Even in the simple case t(f) ¼ (f + m)/ (f + n), the unimodularity of P(f) is hard to prove analytically, though it could be validated by a thorough numerical experiment. In the third model, called Simultaneous Pricing Model (SPM), the acquisition price and selling price are determined simultaneously before the realization of random yield. Then the problem is equal to max f ; p Pðf ; pÞ ¼ ER fðp cr sÞ min½a bp; Rtðf Þða þ bf Þg ðc sÞða þ bf Þ þ hða þ bf Þ;

(9)

116

X. Li and Y. Li

in which v is the unit shortage penalty. The problem is also hard to analyze theoretically and it is conjectured that the objective function is pseudo-concave. The effect of the random yield on this active acquisition and remanufacturing system is explored by a comprehensive computational study. The threshold yield rate r0 turns out to be a critical indicator. The system is referred to as “high margin”/ “low margin” if r0 1/r0 1, which indicates the profit of used product acquisition h + s c f is high/low. In the high margin case, the expected supply always exceeds the demand in both the PPM and SPM scenarios, which indicates the firm creates a buffer inventory. This effect is especially prominent in SPM since the firm has no opportunity to adjust the selling price according to the realization of the yield and the buffer inventory helps to deal with this uncertainty more effectively. Moreover, when the deviation of the maximum attainable yield rate is moderate, both the models generate profits close to the benchmark DPM. In the low margin case, the performance of PPM is also close to DPM, whereas SPM generates lower profit and the firm does not hold buffer inventory any more since all the remanufacturable parts are used to satisfy the demand due to lower profit margin. It is shown that PPM always outperforms SPM by postponing the selling pricing decision after the realization of random yield. However, in the high margin case the difference is not prominent since the high hulk salvage value offsets the loss of delayed yield information to some extent. In fact, the benefit of postponing the selling pricing becomes more significant with lower profit margin, lower yield rate and higher variation. The value of perfect yield information is also studied by the comparison of deterministic maximum yield R ¼ 0.5 and uniformly distributed yield R ~ U(0,1). It can be concluded that the yield information is crucial for the operations of the firm, especially for the lower margin case.

3 Decentralized Supply Chain with Vertical Channels Outsourcing allows a firm to concentrate on its own core competency, reduce operational cost, and lower the financial risk. For these reasons, an original equipment manufacturer (OEM) usually outsources part of the reverse logistic activities, like used product acquisition or remanufacturing. In this section, we study the decentralized supply chain with vertical structure, in which the reverse logistics activities could be conducted by different supply chain members. The section starts from exploring a typical reverse supply chain consisting of a collector and a remanufacturer, engaged in the acquisition and remanufacturing of used products, respectively. The OEM can choose to outsource the collection or remanufacturing activity to a specific collector or remanufacturer, namely collectordriven or remanufacturer-driven channel, respectively. The decentralized decision problems within these channels are studied in Karakayali et al. (2006) and will be introduced in Sect. 3.1.

Supply Chain Models with Active Acquisition and Remanufacturing

117

Next we discuss the problem on closed-loop supply chain structure, which is regarded as the integration of the forward and reverse channels. The involved entities might be OEM, retailer, and 3PL collector, while the product returns could be through OEM-collection channel, retailer-collection channel or 3PLcollection channel. The related models are investigated in Savaskan et al. (2004) and will be presented in Sect. 3.2.

3.1

Reverse Supply Chain Channels

The reverse supply chain consists of a collector and a remanufacturer. The acquisition of used product and the demand of the remanufactured product are both deterministic and price-sensitive, denoted as r(f) ¼ a + bf and d(p) ¼ a bp, respectively. The collector and the remanufacturer incur unit transaction cost cl and cm, respectively. The salvage value per part is s and per hulk is h. The unit remanufacturing cost is cr. The above assumptions and notations are very similar to Sect. 2, with the difference that the collection and remanufacturing processes are operated by the collector and remanufacturer independently. However, the remanufacturing yield is not considered here. Karakayali et al. (2006) studies the problem on this reverse supply chain channel. Although the main focus is the decentralized interaction, the centralized problem serves as a benchmark for the decentralized one, which is formulated as follows: max f ; p Pðf ; pÞ ¼ ða bpÞðp s cr Þ þ ða þ bf Þðh þ s f cl cm Þ s:t: a bp a þ bf

(10)

It is easy to see this problem is a special case of Sect. 2.2 for R ¼ t(f) 1. For the decentralized scenario, two different channel settings are remanufacturerdriven channel and collector-driven channel. In the remanufacturer-driven/collector-driven channel, the OEM outsources the remanufacturing/collection activity, giving the remanufacturer/collector the leadership role in the channel. The analysis is based on the wholesale price contract under Stackelberg game framework. In the remanufacturer-driven channel, the remanufacturer acts as a leader and the other party responds as a follower. As a result, the collector’s problem is to choose the optimal acquisition price f*(w) for a given wholesale price w, maximizing the following objective: maxPc ¼ ðw f cl Þða þ bf Þ: f

(11)

The remanufacturer’s problem is max w ; p Pr ¼ ða bpÞðp s cr Þ þ ða þ bf ðwÞÞðh þ s w cm Þ s:t: a bp a þ bf ðwÞ:

(12)

118

X. Li and Y. Li

The remanufacturer’s problem can be solved by substituting the simple solution of the collector’s problem and then obtaining the optimal decision. In the collection-driven channel, the collector proposes the wholesale price w and the acquisition price f as the leader, and the remanufacturer correspondingly chooses the selling price of remanufactured product p*(w, f) and the quantity of used products that are not remanufactured t*(w, f). The remanufacturer’s problem is formulated as follows: max p ; t Pr ¼ ða bpÞðp cm cr w þ hÞ þ tðh þ s w cm Þ s:t:

t þ a bp a þ bf

(13)

t 0: The collector sets the wholesale price w and the collection price f accordingly, to maximize her profit function: max w ; f Pc ¼ ðw cl f Þða þ bf Þ s:t: tðw; f Þ þ a bpðw; f Þ a þ bf :

(14)

The solving procedure is similar to the remanufacturer-driven problem. The only delicate notion is that the collector should ensure the remanufacturer’s profit margin to be non-negative, or he is not willing to purchase any unit from the collector. As expected, the centralized channel is shown as the most effective strategy from both environmental and consumer’s perspectives. The acquisition price is highest paid and the selling price lowest charged, which yields the largest quantities of used products collected and remanufactured parts sold. It is also shown that a two-part tariff contract, characterized by a wholesale price and a fixed payment, can be used to coordinate the decentralized supply chain to achieve the collection efficiency attained in the centralized one. For the decentralized channels, a natural question is when the OEM would prefer the collection-driven or remanufacturer-driven channel. Numerical experiment shows that the parameters on both the supply and demand sides can influence such channel choice decision. As a decreases, and/or b increases, and a and/or b increases, it becomes more favorable for an OEM to outsource the collection activity. Otherwise the remanufacturer-driven is more preferable. On the other hand, the OEM can also consider altering the outsourcing decision due to a change in the cost and revenue parameters cr and h. For example, an OEM originally preferring a remanufacturer-driven channel for lower cr may choose a collectiondriven channel as cr increases and exceeds a threshold. Similar behavior could be observed for increasing h. Note that an implicit assumption in the above cases is the homogeneousness of the used product quality. If the used products are heterogeneous, the quality condition of the used product influences the acquisition and remanufacturing costs, and the hulk and part salvage values. Any remanufactured part, however, is

Supply Chain Models with Active Acquisition and Remanufacturing

119

of the same quality level and the same selling price. This setting is the same as in Sect. 2.1. For m quality classes, Karakayali et al. (2006) derives O(m2) iterative algorithms for the centralized and remanufacturer-driven channel problems, and an O(m22m) algorithm for the collector-driven channel problem. The supply chain can also be coordinated by two-part tariff contracts with a proper set of wholesale prices and fixed payments.

3.2

Closed-Loop Supply Chain Channels

We now consider a product in which there is no distinction between a remanufactured item and newly manufactured one, thus OEM could use a hybrid remanufacturing/manufacturing strategy to satisfy the market demand. The goal of this subsection is to present the model on a manufacturer’s reverse channel choice and to discuss its impact on the whole supply chain performance. Specifically, the manufacturer has three options to collect used products for remanufacturing (1) collect directly from customers, called Manufacturer Collection Channel; (2) delegate the downstream retailer to collect, called Retailer Collection Channel; and (3) subcontract the collection to a 3PL collector, like in Sect. 3.1, called 3PL Collection Channel. Suppose that remanufacturing is less costly than manufacturing new product for the manufacturer, i.e., cr < cm. The demand is linear decreasing in the selling price, i.e., d(p) ¼ a bp. The acquisition cost structure, on the other hand, is independent of the collection agent, i.e., the cost of collecting a certain amount of used units is the same for the manufacturer, retailer or 3PL. Let 0 t 1 denote the collection rate, which is the ratio of acquisition quantity of used units to the demand quantity. The total cost of collection C(t) is dependent on the collection rate of used products and given by C(t) ¼ CLt2 + Atd(p), where CL is a scaling parameter and A is the variable collection cost of each unit returned item. The problem is studied in a single-period setting. In the following we present three decentralized closed-loop supply chain channels, and the centralized scenario as well, which serves as a benchmark for the decentralized ones. The objective for the centralized system is to choose the selling price p and collection rate t to maximize max PC ¼ ða bpÞðp cm þ tDÞ CL t2 Atða bpÞ; p;t

(15)

where D ¼ cm cr denotes the economic attractiveness of remanufacturing compared to manufacturing. The analytical solution pC ; tC can be obtained by solving the first-order optimality condition of (15). In the decentralized systems, the manufacturer is assumed to have sufficient channel power over the other entities. She acts as a Stackelberg leader and uses her foresight about the retailer’s and 3PL’s reactions to maximize her own profit. In the

120

X. Li and Y. Li

Manufacturer Collection Channel, for a given wholesale price w, the retailer’s problem is maxp PM R ¼ ðp wÞða bpÞ. Substituting the best response pM ¼ ða þ bwÞ=ð2bÞ, we have the manufacturer’s problem as follows, maxPM M ¼ w;t

a bw a bw ðp cm þ tDÞ CL t2 At : 2 2

(16)

The optimal solutions wM and tM can be obtained by solving the first-order optimality condition. In the Retailer Collection Channel, the manufacturer delegates the retailer to collect used products and pays a transfer price b per unit acquired. Thus the retailer’s problem is maxp;t PRR ¼ ðabpÞðpwÞþbtðabpÞCL t2 AtðabpÞ, and the optimal pR and tR can be solved since the profit function is concave. Given the best response of the retailer, the manufacturer’s problem is R R R R max PM M ¼ ða bp Þðw cm þ t DÞ bt ða bp Þ; w;b

(17)

and the corresponding solution wR and bR can be obtained. An interesting observation is that the manufacturer’s profit is increasing in b, thus it is optimal to set bR ¼ D by the manufacturer. In the 3PL Collection Channel, the manufacturer outsources the collection to an independent 3PL collector and pays a transfer price b per unit collected. In this case, the retailer only engages in the product distribution and chooses the optimal selling price p , for a given wholesale price set by the manufacturer. The 3PL’s problem is R 2 R 3P maxt P3P 3P ¼ btða bp Þ CL t Atða bp Þ, and the optimal solution t can be obtained. As a Stackelberg leader, the manufacturer determines the optimal w3P and b3P by solving R 3P Þ: max P3P M ¼ ða bp Þðw cm þ ðD bÞt w;b

(18)

We have b3P ¼ ðD þ AÞ=2, indicating that b is a direct cost for the manufacturer and the closed-loop supply chain. Savaskan et al. (2004) compares the above cases and obtains the following results on the different closed-loop supply chain channels: 1. tC > tR > tM > t3P : 2. pC < pR < pM < p3P , and consequently dC > dR > d M > d3P . M 3P R M 3P C R M 3P 3. pR M > pM > pM , pR > pR > pR , and consequently p > pT > pT > pT . It shows that the collection rate increases with the 3PL, manufacturer, retailer as collecting agent, respectively, and reaches the highest value in the centralized system. Consequently, the acquisition of used product and sales of remanufactured product are highest in quantity in the centralized system. This result, which has been proved in Sect. 3.1, is validated again in the closed-loop supply chain setting.

Supply Chain Models with Active Acquisition and Remanufacturing

121

On the other hand, the selling price decreases in the same sequence. Moreover, both the profits of the retailer and manufacturer reach the highest values in the Retailer Collection Channel and lowest in the 3PL Collection Channel. In summary, the preferred collecting agent is the retailer, followed by the manufacturer, and then the third-party (3P), no matter from the perspective of the manufacturer, retailer, supply chain or the consumer welfare. Finally, Savaskan et al. (2004) indicates that a simple coordination mechanism can be designed for the Retailer-Collection Channel and thus the maximum of the supply chain profit can be attained.

4 Decentralized Supply Chain of Horizontal Competition Horizontal competition among supply chain members is a common phenomenon. Generally, retailers compete in sales performance, and manufacturers compete for market share. In the closed-loop supply chain area, the horizontal competition is more fierce between the OEM who manufactures the product originally and the small agents who only involve in the used product acquisition and remanufacturing. Such instances can be found in the industry of toner cartridge (Majumder and Groenevelt 2001a) or single-used camera (Ferrer and Swaminathan 2006). In this section, we introduce two models seizing the essential feature of horizontal competition between OEM and local remanufacturer. Both models capture the dynamic nature of the problem by a two-period setting, in which the OEM manufactures in the first period and faces competition with the remanufacturer in the second period. The main differences are that the first model assumes a linear remanufacturing cost and distinguishable product brands between the OEM and remanufacturer, while the second model assumes convex collection and remanufacturing costs and distinguishable product quality between the new product and remanufactured one. Nearly all papers in this field fall into the category of brand competition or quality competition, and this section incorporates typical problems of these two types.

4.1

Distinguishable Brand and Linear Remanufacturing Cost

We first consider a two-period model with two players, an OEM and a local remanufacturer. In the first period, the OEM manufactures and sells new product. A fraction of sold items after use is available for the OEM and remanufacturer to be acquired and remanufactured in the second period, and the OEM can also manufacture new units in addition to remanufacturing used ones. Therefore, the players compete in selling their products to consumers in the second period.

122

X. Li and Y. Li

Since the competition for used product acquisition is not the focus of this problem, the model is analyzed under given allocation mechanisms for returned units. Specifically, the OEM has access to a fraction of the products sold in the first period, namely g, and the remanufacturer has 1 g. Four allocation mechanisms are considered (1) OEM can acquire any used product left over by the remanufacturer, however the remanufacturer cannot do likewise; (2) the remanufacturer can acquire any used product left over by OEM, but OEM cannot do likewise; (3) neither can acquire used product left over by the other; (4) both can acquire used product left over by the other. An important feature is that the two players face a brand competition in selling their products. In other words, there is a perfect substitution between the OEM’s new and remanufactured items like in Sect. 3.2, but customers can distinguish between the remanufacturer and OEM’s products. The demand functions in the second period are of Bertrand-type: Do ðpo ; pl Þ ¼ ao bo po þ co pl ; Dl ðpo ; pl Þ ¼ al bl pl þ cl po ;

(19)

where the subscript o means the variables and parameters of the OEM, and the subscript l means those of the local remanufacturer. We suppose that the quantity of total used products available in the second period is equal to a fraction a of the production quantity in the first period. The problem is analyzed in two stages, starting with Nash game analyze in the second period. Given this equilibrium prediction, the OEM chooses the manufacturing quantity and selling price in the first period. Specifically, the remanufacturer’s problem in the second period is max ql ; pl Pl ¼ ql ðpl rl Þ s:t: ql r; ql Dl ql 0; pl 0

(20)

where ql and pl are the remanufacturer’s decision variables, denoting the remanufacturing quantity and selling price, respectively. rl is unit remanufacturing cost and r is the quantity of attainable used products according to specific allocation rule. Similarly, the OEM’s problem in the second period is max z ; qor ; po Pl ¼ zðpo cÞ þ ðc ro Þqor s:t: qor r; qor z Do qor 0; p0 0

(21)

where qor, z and po are the OEM’s decision variables, denoting the remanufacturing quantity, total selling quantity and the selling price, respectively. The unit

Supply Chain Models with Active Acquisition and Remanufacturing

123

remanufacturing cost is ro and manufacturing cost is c. Here, a crucial assumption is that the variable costs for remanufacturing each unit are constants (though they can be different for the OEM and local remanufacturer, i.e., rO 6¼ rl). Different remanufacturing cost structure will be discussed in the next subsection. Groenevelt and Majumder (2001a) proves that a unique pure strategy Nash Equilibrium exists in the second period game. The total produced quantity (of the newly manufactured and remanufactured units) and the newly manufactured quantity are both nondecreasing in the quantity of returned items R. In fact, as R increases, the remanufacturer no longer uses up all available used products to remanufacture, while the OEM first stops manufacturing new units and then no longer uses up all the used units available for him. In contrast, the first period problem for OEM is difficult to analyze theoretically, as the profit function doesn’t have a convenient structural property. Nevertheless, some important insights can be obtained through numerical experiments. For example, we can consider that OEM is a monopoly without or with remanufacturing option. These cases, called B1 (without remanufacturing) and B2 (with remanufacturing) serve as benchmarks to examine the competitive case. It is shown that a monopoly OEM in B2 produces most in the first period and earns the highest total profit among three cases. However, the comparison of B1 with the competitive case is inconclusive. Under some circumstances the OEM does have higher profit in the competitive case than in B1, indicating that the OEM might prefer to facilitate the remanufacturing even if it would incorporate the reverse channel competition. The effects of parameter changes are also examined. An interesting observation is that although competing with the OEM in remanufacturing, the remanufacturer is better off when the OEM’s remanufacturing cost r0 is reduced, which would induce the OEM to produce more in the first period and thus benefit the remanufacturer. The fraction a is another crucial parameter, the increase of which would enlarge the remanufacturing quantity. However, such an increase does not necessarily raise the OEM’s profit due to the competition effect. Note that the above observations are valid for all allocation mechanisms, indicating the allocation rule is not an essential factor as long as the quantity fraction between players is exogenously imposed.

4.2

Distinguishable Quality and Convex Remanufacturing Cost

Ferguson and Toktay (2006) studies the horizontal competition problem following Majumder and Groenevelt (2001a) in somewhat similar settings. The OEM manufactures new product in the first period, and in the second period faces the entry threat of a local remanufacturer that competes with the OEM by selling the remanufactured product. The quantity of available units to be remanufactured is a fraction g of the production quantity of the first period. Some other key assumptions, which are distinct from the last model, are stated as follows.

124

X. Li and Y. Li

First, the qualities of the new product and remanufactured one are differential for customers. More specifically, each consumer’s willingness-to-pay for a remanufactured product is only a fraction d (∈ (0, 1)) of that for the new one. It is further assumed that the consumer’s willing-to-pay is heterogeneous and uniformly distributed in interval [0,1]. This leads to the demand functions entirely different from Sect. 4.1. Second, the collection/remanufacturing cost is increasing and convex with respect to the collection/remanufacturing quantity, respectively. More specifically, the remanufacturing cost is a quadratic function hq2, for a given remanufacturing quantity q. Finally, instead of an exogenous allocation mechanism in Sect. 4.1, this model incorporates a costly collection and remanufacturing option. More specifically, the firm choosing to remanufacture incurs a fixed cost F ¼ Fc + Fr where Fc and Fr are the fixed costs for building up the collection operation and remanufacturing operation, respectively. Under the above assumptions, it is investigated that whether a monopoly OEM should process the collection and remanufacturing. For a monopoly OEM without remanufacturing, it turns out to be a simple optimization problem maxq(p c)q for each period. For the OEM with remanufacturing, a two-step dynamic programming model is formulated as follows: max q2n ;q2r P2 ðq2n ; q2r jq1 Þ ¼ ðp2n cÞq2n þ ðp2r hq2r Þq2r s:t:

q2r gq1

(22)

and max pðq1 Þ ¼ ðp1 cÞq1 þ p2 ðq1 Þ:

(23)

q1 0

Note that in the above formulation we suppose the fixed cost of remanufacturing F ¼ 0. In this case, remanufacturing is always more profitable than not remanufacturing. However, if F > 0, then there exists a threshold level on the remanufacturing cost factor h, above which the OEM would not remanufacture any unit in the second period. The closed-form solutions of the pricing and quantity decisions can be obtained. Another key issue of this problem is to explore the strategic role of the OEM remanufacturing as an entry deterrent to the local remanufacturer. Ferguson and Toktay (2006) studies the motivation for an OEM without remanufacturing to deter the remanufacturer’s entry into the reverse channel. In fact, an OEM might not remanufacture due to a too high remanufacturing cost factor h or fixed investment F. In this case, however, it is shown that remanufacturing could still be profitable for an external remanufacturer thus the OEM may suffer from it. Specifically, Nash Equilibrium can be obtained by simultaneously solving the following problems faced by the OEM and remanufacturer: q2r Þ ¼ ðp2n ðq2n ; qc max q2n P2 ðq2n jc 2r Þ cÞq2n ;

(24)

Supply Chain Models with Active Acquisition and Remanufacturing

125

and q2r jq2n ; q1 Þ ¼ ðc p2r ðq2n ; qc q2r Þc q2r F max qb P2 ðc 2r Þ hc 2r

s:t:

qc 2r gq1 :

(25)

In the first period, the OEM determines the production quantity q1 expecting the equilibrium result of the second period. By analyzing the above problem, Ferguson and Toktay (2006) characterizes the potential profit loss for the OEM due to the sales competition with the remanufacturer. As a result, the OEM could adopt two entry-deterrent strategies: remanufacturing, and preemptive collection. By the first strategy, the OEM chooses to remanufacture for the sole purpose of discouraging an external remanufacturer to do so, even if the remanufacturing would not be preferable in a monopoly setting. By the second strategy, the OEM would choose to collect returned units with the quantity large enough to deter the remanufacturer from collecting and remanufacturing. The collected used product, however, is never remanufactured by the OEM. They further investigate the impact of collection cost, unit manufacturing cost and consumer willing-to-pay on the choice of which deterrent strategy to use for the OEM. As the collection cost/unit manufacturing cost/willingness-to-pay increases, the profitability of the remanufacturing becomes more prominent. Moreover, if the collection cost increases linearly in the quantity, the collection strategy would also become less attractive, which indicates the significance of an accurate modeling for the collection cost curve.

5 Conclusion The importance of remanufacturing has been widely recognized in terms of environmental sustainability and economical benefit. The topic of this chapter relates to the supply chain models on active acquisition and remanufacturing, with a discussion of the following three classes of problems. The first class is on the centralized system with acquisition price and sale price decisions, faced by remanufacturing firms utilizing a market-driven acquisition channel. The used product supply can be controlled actively by the acquisition price, and the acquisition management is shown as a significant driver of remanufacturing profitability. In the deterministic model, the efficient strategies are developed to balance the supply and demand through the pricing level. In the random yield model, the effect of remanufacturing random yield is explored, and the benefit of delaying pricing decisions to mitigate the yield randomness is analyzed. Both models belong to the class of centralized optimization. It is also common for a dominant party in the supply chain, usually the manufacturer, to dictate terms to other supply chain members to process some activities,

126

X. Li and Y. Li

such as acquisition or remanufacturing. It is the objective of Sect. 3 to investigate this vertical channel design problem for the decentralized supply chain. In the pure reverse logistic setting, it is shown that the channel preference is conditional on some parameters, although an OEM might prefer to outsource the collection activity rather than the remanufacturing activity from a practical perspective. On the other hand, in a closed-loop supply chain setting, the OEM is most likely to assign the collection activity to the retailer and most reluctantly to the 3PL collector. The supply chain coordination issue is also considered in both the cases. Moreover, conflict between entities at the same supply chain level is also prevailing, which is the central issue of Sect. 4. Different settings of product distinction and cost structure lead to different results, while the effects of horizontal competition are explored by comparing with the monopoly OEM case. We believe there is still great potential on the study of supply chain with active acquisition and remanufacturing. Possible further directions include the problem with more random factors, e.g., the problem with random acquisition, random yield, and random demand, etc. In this case, both the centralized and decentralized issues are more complicated, yet more interesting. Another promising research opportunity might be information asymmetry. The information issue in the supply chain field has been a hot topic, but very scarce research has been done related to the product acquisition and remanufacturing. Finally, it is to be studied on negotiation problem between the manufacturer and other supply chain members, within the framework of bargaining theory or other cooperative game theories. Acknowledgment This work is partly supported by National Natural Science Foundation of China (NSFC) Nos. 70971069, 71002077 and 71002106, the Fok Ying-Tong Education Foundation of China (Grant No. 121078), and 2009 Humanities and Social Science Youth Foundation of Nankai University (Grant No. NKQ09027).

References Aras N, Aksen D (2008) Locating collection centers for distance and incentive dependent returns. Int J Prod Econ 111(2):316–333 Aras N, Aksen D, Tanugur AG (2008) Locating collection centers for incentive dependent returns under a pick-up policy with capacitated vehicles. Eur J Oper Res 191(3):1223–1240 Atasu A, Sarvary M, Van Wassenhove LN (2008) Remanufacturing as a marketing strategy. Manage Sci 54(10):1731–1746 Bakal IS, Akcali E (2006) Effects of random yield in reverse supply chains with price-sensitive supply and demand. Prod Oper Manage 15(3):407–420 Debo L, Toktay B, Van Wassenhove LN (2005) Market segmentation and product technology selection for remanufacturable products. Manage Sci 51(8):1193–1205 Dekker R, Fleischmann M, Inderfurth K, Van Wassenhove LN (2004) Reverse logistics: quantitative models for closed-loop supply chains. Springer, New York Ferguson M, Toktay B (2006) The effect of competition on recovery strategies. Prod Oper Manage 15(3):351–368 Ferguson M, Guide VDR Jr, Souza G (2006) Supply chain coordination for false failure returns. Manuf Serv Oper Manage 8(4):376–393

Supply Chain Models with Active Acquisition and Remanufacturing

127

Ferrer G (2003) Yield information and supplier responsiveness in remanufacturing operations. Eur J Oper Res 149:540–556 Ferrer G, Swaminathan J (2006) Managing new and remanufactured products. Manage Sci 52 (1):15–26 Galbreth MR, Blackburn JD (2006) Optimal acquisition and sorting policies for remanufacturing. Prod Oper Manage 15:384–392 Guide VDR Jr, Jayaraman V (2000) Product acquisition management: current industry practice and a proposed framework. Int J Prod Res 38(16):3779–3800 Guide VDR Jr, Van Wassenhove LN (2001) Managing product returns for remanufacturing. Prod Oper Manage 10(2):142–155 Guide VDR Jr, Van Wassenhove LN (2009) The evolution of closed-loop supply chain research. Oper Res 57(1):10–18 Guide VDR Jr, Teunter R, Van Wassenhove LN (2003) Matching supply and demand to maximize profits from remanufacturing. Manuf Serv Oper Manage 5:303–316 Heese HS, Cattani K, Ferrer G, Gilland W, Roth AV (2005) Competitive advantage through takeback of used products. Eur J Oper Res 164:143–157 Ilgin M, Gupta S (2010) Environmentally conscious manufacturing and product recovery (ECMPRO): a review of the state of the art. J Environ Manage 91(3):563–591 Karakayali I, Emir-Farinas H, Akcali E (2006) An analysis of decentralized collection and processing of end-of-life products. J Oper Manage 25(6):1161–1183 Kaya O (2010) Incentive and production decisions for remanufacturing operations. Eur J Oper Res 201:442–453 Liang Y, Pokharel P, Lim GH (2009) Pricing used products for remanufacturing. Eur J Oper Res 193(2):390–395 Majumder P, Groenevelt H (2001a) Competition in remanufacturing. Prod Oper Manage 10 (2):125–141 Majumder P, Groenevelt H (2001b) Procurement competition in remanufacturing. Working paper, Duke University School of Business, Durham, NC Meyer H (1999) Many happy returns. J Bus Strategy 80(7):27–31 Mukhopadhyay S, Setoputro R (2005) Optimal return policy and modular design for build-to-order products. J Oper Manage 23:496–506 Padmanabhan V, Png IPL (1997) Manufacturer’s returns policies and retail competition. Mark Sci 16(4):81–94 Pasternack BA (1985) Optimal pricing and return policies for perishable commodities. Mark Sci 4 (2):166–176 Pokharel S, Mutha A (2009) Perspectives in reverse logistics: a review. Resour Conserv Recycl 53 (4):175–182 Prahinski C, Kocabasoglu C (2006) Empirical research opportunities in reverse supply chains. Omega 34(6):519–532 Qu X, Williams JA (2008) An analytical model for reverse automotive production planning and pricing. Eur J Oper Res 190(3):756–767 Ray S, Boyaci T, Aras N (2005) Optimal prices and trade-in rebates for durable, remanufacturable products. Manuf Serv Oper Manage 7:208–228 Robotis A, Bhattacharya S, van Wassenhove LN (2005) The effect of remanufacturing on procurement decisions for resellers in secondary markets. Eur J Oper Res 16(3):688–705 Savaskan C, Van Wassenhove LN (2006) Reverse channel design: the case of competing retailers. Manage Sci 52(1):1–14 Savaskan C, Bhattacharya S, Van Wassenhove LN (2004) Closed-loop supply chain models with product remanufacturing. Manage Sci 50(2):239–252 Vlachos D, Dekker R (2003) Return handling options and order quantities for single period products. Eur J Oper Res 151:38–52 Webster S, Mitra S (2007) Competitive strategy in remanufacturing and the impact of take-back laws. J Oper Manage 25(6):1123–1140

128

X. Li and Y. Li

Webster S, Mitra S (2008) Competition in remanufacturing and the effects of government subsidies. Int J Prod Econ 111(2):287–298 Wojanowski R, Verter V, Boyaci T (2007) Retail-collection network design under deposit-refund. Comput Oper Res 34:324–345 Yue X, Raghunathan S (2007) The impacts of the full returns policy on a supply chain with information asymmetry. Eur J Oper Res 180:630–647 Zikopoulos C, Tagaras G (2007) Impact of uncertainty in the quality of returns on the profitability of a single-period refurbishing operation. Eur J Oper Res 182:205–225

Part II

Analytical Models for Innovative Coordination under Uncertainty

.

Facilitating Demand Risk-Sharing with the Percent Deviation Contract Matthew J. Drake and Julie L. Swann

Abstract Suppliers do not have much incentive to build capacity for supply chains with stochastic demand in which the buyer bears little or no inventory risk. This hinders the supply chain from satisfying the optimal amount of customer demand from a channel perspective. We describe and analyze the percent deviation contract as an innovative mechanism to improve the overall performance of this type of supply chain. This contract induces a dynamic game of perfect information, and we characterize the subgame-perfect Nash Equilibria under various contract scenarios. We establish ways to set the contract parameters to coordinate the supply chain under uncertainty and show that the percent deviation contract is able to achieve channel coordination in some cases where the quantity flexibility contract fails. In order to aid the implementation of the percent deviation contract in practice, we develop ways to set the parameters to satisfy the buyer’s individual-rationality constraint. Keywords Supply chain coordination • Contracting • Newsvendor • Model • Risksharing

1 Introduction The proliferation of computerized information systems in the 1990s facilitated the establishment of supply chain partnerships in which demand information is shared between firms. The upstream firms can use this information to reduce the traditional M.J. Drake (*) Palumbo-Donahue Schools of Business, Duquesne University, Pittsburgh, PA 15282, USA e-mail: [email protected] J.L. Swann H. Milton Stewart School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, GA 30332-0205, USA e-mail: [email protected] T.-M. Choi and T.C. Edwin Cheng (eds.), Supply Chain Coordination under Uncertainty, International Handbooks on Information Systems, DOI 10.1007/978-3-642-19257-9_6, # Springer-Verlag Berlin Heidelberg 2011

131

132

M.J. Drake and J.L. Swann

demand distortion due to the bullwhip effect. Some firms have also incorporated this information into contracts that induce their supply chain partners to share demand risk, thereby improving supply chain efficiency. Many researchers and practitioners (e.g., Lee 2004; Finley and Srikanth 2005) have advocated demand risk-sharing as a necessary condition for supply chain collaboration efforts to be successful in practice. In this paper we analyze one such contracting mechanism, an innovative structure which we denote as the percent deviation contract. The percent deviation contract is most applicable to supply chains in which the buyer does not traditionally bear any inventory obsolescence risk in satisfying stochastic consumer demand. In these channels the buyer places orders with the supplier only when the consumer demand is known with certainty; that is, the buyer does not carry any excess inventory. This environment occurs in many service operations. One industry that stands to benefit from application of the percent deviation contract is truckload transportation. In fact, a large truckload carrier originally proposed the idea for this particular contract structure but did not know how to set the parameters or whether or not the contract would be beneficial. While these carriers generally have standing weekly orders for loads with their bigger customers, many shippers call dispatch requesting a pickup in a few hours. This limits the carrier’s ability to utilize its equipment effectively by allocating trailers in advance or coordinating backhauls from prior shipments. The percent deviation mechanism is applicable in many traditional manufacturer–retailer channels as well, such as home construction, equipment integrators, window replacement, or door-to-door sales. In all of these industries, the supplier bears most, if not all, of the consumer demand risk in many arrangements. We analyze the strategic properties of the percent deviation contract in which the buyer gives an initial order estimate and the supplier pre-acquires inventory at a low cost. Once the buyer’s consumer demand is realized, the buyer places the actual order, and the supplier fulfills all or a portion of the order, possibly by expediting, at a higher cost. The buyer pays a penalty if the final order is outside of an allowable range established around the initial order estimate. We characterize the subgameperfect Nash Equilibria (SPNE) decisions when the supplier has a fixed expediting capacity and discuss methods of channel coordination to optimize the performance of the entire system. Since the buyer assumes some consumer demand risk under the percent deviation contract, its expected profit may be less than that under a traditional contracting structure; therefore, we develop a method that the supplier can use to satisfy the buyer’s individual-rationality constraint. Our contribution to the existing literature on supply chain collaboration includes analysis of a risk-sharing contract where decisions made by the buyer and supplier explicitly depend on each other and are solvable in the framework of a dynamic, extensive form game. This necessarily results in a more complex contract, but we also show that this contract can be strictly Pareto-improving for both parties. Our contract has a structure similar to the quantity flexibility contract, but ours does not enforce limits on the buyer’s final behavior; thus, this contract can coordinate the supply chain in some cases where quantity flexibility cannot. Many models consider a supply chain with infinite capacity; whereas, the total capacity in our model is a function of the supplier’s decisions as well as an external constraint.

Facilitating Demand Risk-Sharing with the Percent Deviation Contract

133

2 Literature Review The breadth of supply chain contracting literature has grown significantly over the last two decades as researchers and practitioners have examined strategic relationships between supply chain partners. One stream of supply chain contracting literature has proposed and analyzed methods of coordinating decentralized decisions to attain the optimal supply chain profit. See Tsay et al. (1999) and Fugate et al. (2006) for extensive reviews of the supply chain contracting literature. We discuss below the most relevant contracting references, which model a system with multiple, sequential decisions. Tsay (1999) analyzes a quantity flexibility contract in which the retailer commits to purchasing no less than a certain percentage of the initial forecast while the supplier agrees to fulfill up to a certain percentage above the forecast. He also evaluates the sharing of demand risk that produces the coordinated channel. Tsay and Lovejoy (1999) extend these results to a rolling horizon decision environment. More recently, Lian and Deshmukh (2009) develop a finite-horizon dynamic programming model for a quantity flexibility contract where the buyer can adjust previously-determined order quantities in response to updated demand information for a higher price. Bassok and Anupindi (2008) discuss a similar model where the buyer must establish quantity commitments for multiple periods into the future and then can adjust them within a certain tolerance in a rolling horizon fashion. In contrast to quantity flexibility, the percent deviation contract places no limits on the buyer’s final order, although it adds complexity to the decision environment by including additional contract parameters. We show in Sect. 4.3 that this added complexity can be justified because the percent deviation contract succeeds in coordinating the supply chain in several cases where the quantity flexibility contract is known to be unable to coordinate the channel. Donohue (2000) and Cachon (2004) analyze contracts with two-tier pricing structure that induce early commitment from buyers. In both of these contracts the buyer is bound to its order in both periods, whereas in our contract the first order is only an estimate of demand and can be freely adjusted once demand is known. These two papers only consider the full compliance contract regime where the supplier must fulfill the entire order; whereas, we model the supplier’s compliance decision explicitly. Several contracts employ an options framework where the buyer makes a firm order commitment and purchases options for additional goods to be exercised if demand is high. Cachon and Lariviere (2001) consider a single period model with options and forecast sharing. Since the buyer has an incentive to provide a biased forecast, they develop conditions that facilitate the credible sharing of forecasts under both full and voluntary compliance. Barnes-Schuster et al. (2002) extend the options framework using a two-period model with correlated demand between periods. Wang and Liu (2007) develop some structural properties of coordinating options contracts in channels with powerful retailers, and Zhao et al. (2010) show that an options contract can coordinate the supply chain and can be

134

M.J. Drake and J.L. Swann

Pareto-improving compared with a traditional wholesale-price contract. In our model we have no firm commitment and no upper bound on the final order amount, so our contract cannot be reduced to an option-based model. An additional series of studies (see, for example, Erkoc and Wu 2005; Serel 2007) have analyzed reservation fee supply contracts in which the buyer pays a (usually) deductible fee to reserve capacity along with an exercise fee for the final order quantity. The manufacturer builds capacity based on the reservations made, but it can also build excess capacity to offer at a higher spot rate once demand is realized. The aforementioned studies only consider linear reservation fee contracts–where each unit ordered is charged the same prices. The percent deviation contract is a special case of a piecewise-linear reservation fee contract in which the reservation and exercise prices differ for various portions of the order. In addition to contracting, several papers (e.g., Lee et al. 2000; Cachon and Fisher 2000; Balakrishnan et al. 2004) have examined various ways of reducing the bullwhip effect through information sharing in decentralized supply chains. Kulp et al. (2004) study the benefits the manufacturer gains under different degrees of information sharing and collaboration. They find that most of the manufacturers’ benefit from information integration comes from collaborative activities such as vendor-managed inventory and collaborative forecasting instead of simply sharing information. On the contrary, our results suggest that the risk sharing induced by the percent deviation contract enables the supplier and the entire channel to attain higher profit. We develop a model in the next section for the general case where the supplier has an expediting capacity constraint as well as the special case of infinite capacity. In Sect. 4 we identify conditions on the contract parameters that satisfy each party’s participation constraints, detail ways to coordinate the channel in each decentralized scenario, and compare the percent deviation contract with the wellknown quantity flexibility contract. We demonstrate the use of the percent deviation contract on several numerical examples in Sect. 5, and we discuss the study’s conclusions and suggestions for future research in the final section of the chapter.

3 Models and Scenarios The percent deviation contract accommodates the following sequence of decisions. The buyer provides an initial estimate of its final-order demand that will be placed at a later date. The seller can then use this information to acquire goods in advance (e.g., a truckload carrier can preposition trucks or coordinate backhauls to optimize its transportation network) at a low cost in anticipation of this demand. When the buyer’s demand is known with certainty, the buyer places actual order with the supplier. Depending on the contract parameters, the seller can choose to satisfy additional demand by expediting or subcontracting at a high cost or can choose to fulfill only the demand equal to the number of previously-acquired goods. The percent deviation penalty is the mechanism that punishes the buyer for unrealistic

Facilitating Demand Risk-Sharing with the Percent Deviation Contract

135

estimates. If the buyer’s final order is within a certain percentage above or below the initial estimate, no penalty is charged. If the order exceeds the limits, the supplier charges a penalty on all goods ordered outside of the tolerable range.

3.1

Notation and Assumptions

We employ notation adapted from Donohue (2000). The buyer receives r dollars in revenue for each unit and pays a wholesale price, w, to the supplier. We assume that the buyer earns a positive gross margin from these transactions (i.e., r > w). Consumer demand for a period is given by the random variable X, which has a continuous, differentiable probability distribution function, f(x). If the buyer cannot satisfy the customers’ demand (due to lack of product availability), it incurs a customer penalty of b per unit.1 The seller faces a cost of c1 dollars to acquire a unit of inventory in anticipation of demand and must pay c2 dollars to satisfy demand after the firm order has been placed. We assume that c2 > c1, so the c2 can be thought of as an expediting or subcontracting cost. If the supplier has excess inventory at the end of the period, it receives a unit salvage value of v. It is natural to assume that w > v and c1 > v, which ensure that the supplier does not receive too much of a benefit from selling goods for salvage. Since the seller may choose not to satisfy the buyer’s entire order, it must pay the buyer a for each unit ordered but not delivered. We assume that a < b, which signifies that lost customers are more costly for the buyer. The per-unit penalty that the buyer must pay the supplier for orders outside of the allowable deviation range is denoted by p, while d ∈ [0,1] is the percentage that defines the range. For orders above the upper limit of the range, the buyer only pays the percent deviation penalty for the units that the supplier fulfills. The buyer’s initial forecast is given by q1, and the actual order is q2. The number of units the supplier acquires in advance of demand is t1, and the additional goods expedited or subcontracted are denoted by t2. The supplier has a maximum expediting capacity of M units. This particular way of modeling the supplier’s capacity bears further consideration. It is important to note that the capacity for the supplier’s pre-acquisition decision is infinite. By setting the t1 value, the supplier is de facto determining the capacity of the system as a whole, which is equal to t1 + M. This structure is appropriate for buyer–supplier transactions in which the supplier has a lot of capacity in its system but must make the allocation decisions across many customers before production occurs. Therefore, if the supplier knows in advance that demand will be high, it can allocate sufficient capacity to satisfy the large order; closer to the purchase date, however, it can only provide a limited amount of 1 This b could also be viewed as the higher cost from using an alternative supplier not under longterm contract.

136

M.J. Drake and J.L. Swann

excess capacity because the rest of the system is dedicated to fulfilling orders from other customers. We make the following assumptions that improve tractability but are not likely to impede the application of the results. The first assumption is that all costs are constant for each unit of demand for a single product line because we are interested in examining the structure of the incentives. Another assumption is the existence of complete, symmetric cost, capacity, and demand information between the two parties. When the buyer places the final order, it knows the exact demand as is usual in the other relevant service and manufacturer–retailer channels discussed in Sect. 1. If the actual demand exceeds the upper limit of the deviation range, the cost and penalty parameters determine whether or not the buyer’s order equals the full demand. In order for the buyer to order above the deviation threshold, the net cash flow must exceed the shortage penalty owed to the end consumer. r w p> b

(1)

If the inequality holds, then q2 ¼ X, or the actual customer demand. If this inequality is not satisfied (for instance, if the penalty for ordering outside of the deviation range is too high), q2 ¼ min{X,(1 þ d)q1}. The additional assumption w > p assures that if the actual demand is below the lower limit of the deviation range, (1 d)q1, the buyer orders the actual demand.

3.2

General Model with Finite Expediting Capacity

We begin our analysis with the decentralized structure in which each party makes decisions to optimize its individual expected profit. Even though the supplier has an expediting capacity of M units, the cost of expediting these units, c2, might be too high for the supplier to choose to do so. In order for the supplier to use any of this expediting capacity, the cash flow from expediting must be higher than the cost of failing to expedite. These flows are dependent on whether or not the supplier will receive the deviation penalty on some or all of these units. While the two buyer scenarios discussed above, which are dependent on whether or not the buyer is willing to place orders above the upper limit of the deviation range, generate different supplier responses, the derivation and form of these optimal decisions are the same in both cases. Thus, in this chapter we only examine the case in which the buyer is willing to order the entire demand even if it must pay the deviation penalty. (See Drake (2006) for the case in which the buyer will not order above the upper limit of the deviation range.) When the buyer is willing to order the actual demand, the supplier’s expediting decision can be determined a priori, without knowledge of how many units for which the buyer will pay the deviation penalty. If w c2 > a, then the supplier finds it beneficial to expedite whether or not the buyer will pay the deviation penalty

Facilitating Demand Risk-Sharing with the Percent Deviation Contract

137

on any units; consequently, t2 ¼ ðminfq2 t1 ; MgÞþ . We will denote this as Case A. Similarly, if w c2 þ p < a, the supplier would not choose to expedite any units even if the buyer paid the deviation penalty on all of the units; and thus, t2 ¼ 0. This will be Case B. We will use backward induction to solve for the subgame-perfect Nash Equilibria in each scenario. We now formulate the expected profit functions for the buyer and supplier in Case A, where q2 ¼ X and t2 ¼ ðminfq2 t1 ; 0gÞþ . The supplier chooses t1 to maximize the following expected profit function: Z t1 þM Z 1 S PA ¼ w xf ðxÞdx þ ðt1 þ MÞð1 Fðt1 þ MÞÞ a ðx t1 MÞf ðxÞdx Z

t1 þM

0

minft1 þM;ð1dÞq1 g

þp "Z0 þp

t1 þM

ð1þdÞq1

Z

t1

þv

ðminft1 þ M; ð1 dÞq1 g xÞf ðxÞdx # þ

ðx ð1 þ dÞq1 Þf ðxÞdx þ ðt1 þ M ð1 þ dÞq1 Þ ð1 Fðt1 þ MÞÞ Z

ðt1 xÞf ðxÞdx c1 t1 c2

0

t1 þM

ðx t1 Þf ðxÞdx þ Mð1 Fðt1 þ MÞÞ :

t1

(2) There are three separate functions that represent the realizations of the expected profit function in (2) based on the relationship between the total system capacity, t1 þ M, and the boundaries of the deviation range, (1 d)q1 and (1 þ d)q1. The expected profit regions are depicted in Fig. 1. In region S1 the supplier sets capacity so that it cannot even satisfy the lower limit of the deviation range. Region S2 prescribes that the total system capacity lies somewhere in the deviation range. In these first two regions, the buyer will never pay the deviation penalty for orders above the upper limit of the range because these units will never be fulfilled. Region S3 specifies that the system capacity exceeds the upper limit of the deviation range. In each region only one of the three separate expected profit realizations is feasible, regardless of which expected profit is higher in the region. The following observation establishes that the overall expected profit function is continuous. Observation 1 The individual expected profit function realizations that are active in two adjacent feasible regions are equal at the boundary (i.e., and PSA:II: ðð1 þ dÞq1 MÞ PSA:I: ðð1 dÞq1 MÞ ¼ PSA:II: ðð1 dÞq1 MÞ 2 S ¼ PA:III: ðð1 þ dÞq1 MÞ). Fig. 1 Regions of capacity defining the form of the supplier’s expected profit function

S1

S2 (1-d)q1

S3 (1+d)q1

t1+M

The three separate functions (PSA:I: , PSA:II: , and PSA:III: ) are defined in (24)–(26) in the Appendix.

2

138

M.J. Drake and J.L. Swann

Lemma 1. If w þ a > p þ c2, the supplier’s expected profit function in (2) is piecewise-concave, a continuous, piecewise function whose separate segments are individually concave. Each of the individual realizations of (2) has a corresponding maximizing value that is derived from the solution to the following equations, w þ a c1 ðc2 vÞFðtÞ (3) t1A:I: 2 t : Fðt þ MÞ ¼ w þ a c2 p w þ a c1 ðc2 vÞFðtÞ A:II: (4) t1 2 t : Fðt þ MÞ ¼ w þ a c2 t1A:III: 2

t : Fðt þ MÞ ¼

w þ a c1 þ p ðc2 vÞFðtÞ ; w þ a c2 þ p

(5)

which are all dependent on the supplier’s available expediting capacity. It is interesting to note that all of these values are independent of the buyer’s order quantity; they can easily be computed from the exogenous parameters. We can always solve these equations by applying the Intermediate Value Theorem since the left-hand sides are all bounded between 0 and 1. To understand the supplier’s best response to q1, we can consider the individual functional maximizers in (3)–(5) and their relationship to each other and the boundaries of the feasible regions. The following theorem characterizes the supplier’s best response, which is dependent on the specific value of the buyer’s decision, q1, via the feasible region boundary conditions. Theorem 1. The supplier’s best w þ a > p þ c2 is 8 A:I: t1 ; > > > > A:II: > ; t > > >1 > > > > > > > > tA:III: ; > >1 > > < ð1 dÞq1 M; t1 ðq1 Þ ¼ > > > > S > arg maxtA:I > ;t1A:III: PA ; > 1 > > > > arg maxð1dÞq1 M;t1A:III: PSA ; > > > > > > > > > : arg max A:II: A:III: PS ; t1 ;t1 A

response to a given value of q1 when if t1A:I: ð1 dÞq1 M & tA:III: ð1 þ dÞq1 M; 1 if ð1 dÞq1 M tA:II: ð1 þ dÞq1 M & 1 A:III: t1 ð1 þ dÞq1 M; if tA:II: ð1 þ dÞq1 M & tA:III: ð1 þ dÞq1 M; 1 1 if tA:II: ð1 dÞq1 M t1A:I: & 1 ð1 þ dÞq1 M; tA:III: 1 if t1A:I: ð1 dÞq1 M & tA:III: ð1 þ dÞq1 M; 1 if tA:II: ð1 dÞq1 M t1A:I: & 1 ð1 þ dÞq1 M; tA:III: 1 if ð1 dÞq1 M tA:II: ð1 þ dÞq1 M tA:III: : 1 1 (6)

The supplier’s best response function in (6) is admittedly complicated and difficult to interpret. To aid the reader’s understanding of this function, we provide

Facilitating Demand Risk-Sharing with the Percent Deviation Contract

Supplier I

t1 t1III

{S3}

{S1} t1I

{S1,S2} t1III

t1

argmax t1I, t1III

t1III

{S1,S2}

(1-d)q1-M

t1

{S2}

{S3}

t1II I

t1I

t1III t1II

{S1}

II

{S1}

{S1,S2}

argmax (1-d)q1-M, t1III

t1

{S2}

{S1,S2}

{S3}

{S1,S2} t1II

t1III

t1III

{S3}

argmax (1-d)q1-M, t1III t1III

III

II

{S3} t1III

{S 3}

{S3} t1

t1III

{S2} (1-d)q1-M

t1III

t1II

139

t1II

argmax t1II, t1III

argmax t1II, t1III

Fig. 2 Supplier’s best response depiction for Scenario A

a decision tree-type depiction of the supplier’s optimal decision for various problem parameters in Fig. 2. The buyer must choose the q1 that maximizes its expected profit while anticipating the supplier’s response to the chosen value. The buyer’s expected profit function is given by "Z # t1 ðq1 ÞþM B xf ðxÞdx þ t1 ðq1 Þ þ M 1 F t1 ðq1 Þ þ M PA ¼ ðr wÞ Z p "Z0 p

0

minfð1dÞq1 ;t1 ðq1 ÞþMg

t1 ðq1 ÞþM ð1þdÞq1

minfð1 dÞq1 ; t1 ðq1 Þ þ Mg x f ðxÞdx

ðx ð1 þ dÞq1 Þf ðxÞdx

(7)

i þ þ t1 ðq1 Þ þ M ð1 þ dÞq1 1 F t1 ðq1 Þ þ M Z 1 þ ða bÞ x t1 ðq1 Þ M f ðxÞdx: t1 ðq1 ÞþM

The supplier’s best response function in (6) is comprised of four explicit values as well as three situations where the supplier chooses the profit-maximizing quantity from a set of two of the explicit values. We first determine how the buyer should set q1 when the supplier will respond with each of the four possible t1 values. Each of these cases results in a different realization of the buyer’s expected profit function in (7). We apply the Karush–Kuhn–Tucker (KKT) conditions (c.f. Bazaraa et al. 1993: 151–155) over each realization’s feasible range of decisions to determine the constrained optimal values of q1.

140

M.J. Drake and J.L. Swann

Table 1 Possible SPNE pairs and feasibility conditions for Case A’s explicit t1(q1) decisions ðq1 ; t1 ðq1 ÞÞ Feasibility conditions n n A:I: oo A:III: þM t þM t ; t1A:I: Always feasible q1A:I: q : q max 11d ; 1 1þd

A:III: þM 1 maxftA:II: g 1 ;t1 ; t1A:II: Always feasible qA:II: max F1dð0Þ ; 1 1þd ðqA:III: fq : ð1 dÞFðð1 dÞqÞ ¼ 1

qA:III: 1

Þ dÞqÞÞg; tA:III: 1

ð1 þ dÞð1 Fðð1 þ q : Fðð1 dÞqÞ ¼

ðq1:A:IV:

rwaþb ; rwaþbþp

ð1 dÞq1A:IV: MÞ

A:III: þM minftA:II: g 1 ;t1 1þd

n A:II: o t þM tA:III: þM q1A:IV: max 1 1d ; 1 1þd qA:IV: 1

tA:I: 1 þM 1d

Theorem 2. The possible subgame-perfect Nash Equilibrium (SPNE) decision pairs for explicit t1 ðq1 Þ decisions are given in Table 1. We must also determine the buyer’s optimal decision over the regions where it knows that the supplier will be selecting the maximizing argument of a set of two values. From Theorem 1, we establish the following ranges of q1 under which each situation is possible. arg maxtA:I PSA : ;tA:III: 1 1 arg maxð1dÞq1 M;tA:III: PSA: : 1

t1A:I: þ M tA:III: þ M q1 1 (8) 1d 1þd A:I: tA:II: þM t1 þ M t1A:III: þ M 1 (9) q1 min ; 1d 1d 1þd

PSA : q1 arg maxtA:II: ;tA:III: 1 1

tA:II: þ M tA:II: þM tA:III: þ M 1 & 1 q1 1 1d 1þd 1þd

(10)

Since all three situations involve the possible decision t1A:III: , we define the difference function, DðtÞ PSA ðt1A:III: Þ PSA ðtÞ, where t is any other possible supplier pre-acquisition amount. While it is difficult to determine the exact feasible region for q1 that induces each of the possible t1(q1) values, we can use these difference functions to explain how a buyer would determine its optimal decision for a given set of problem parameters. Proposition 1. The structure of the three difference functions for the decisions in (8)–(10) enables us to determine the specific ranges of q1 that induce each of the two possible supplier values for t1. For each instance there are at most seven decision pairs from Table 1 and obtained from the procedure in Proposition 1, but some of these decisions may not be mutually feasible given a set of problem parameters. Since this set contains a maximum of seven elements, the buyer can evaluate its expected profit in (7) with respect to each of the feasible pairs and select the value of q1 that yields the highest

Facilitating Demand Risk-Sharing with the Percent Deviation Contract

141

expected profit to obtain the overall subgame-perfect Nash Equilibrium for this sequential supply chain game. Each of the formulas used in computing the potential decision pairs has an economic interpretation. The buyer always sets the initial order estimate with a goal of minimizing the expected deviation penalty that must be paid under each scenario. In the case where the supplier has system capacity larger than the upper limit of the deviation range, the optimal quantity balances the expected lower and upper deviation penalties. The supplier also seeks to balance the expected revenue from pre-acquiring inventory with the cost of doing so as well as the expected expediting and shortage costs. Even though the resulting formulas are more complicated, the supplier follows the same rationale as in a traditional newsvendor contract. We now consider Case B, where the supplier chooses not to expedite any units after the buyer places the final order because of a high expediting cost. This case is analogous to Case A when M ¼ 0 since the supplier can be viewed as having an effective expediting capacity of zero units if it chooses not to expedite. The maximizing values below correspond with the equations in (3)–(5) with M ¼ 0. t1B:I: 2

w þ a c1 t : FðtÞ ¼ wþavp

tB:II: 1

2

w þ a c1 t : FðtÞ ¼ wþav

t1B:III: 2

t : FðtÞ ¼

(11)

w þ a c1 þ p wþavþp

(12) (13)

Using the relationships in Lemma 4 (stated in the Appendix) to simplify the feasibility conditions, the supplier’s best response in this scenario is characterized by the following theorem. While it may not seem like it at first glance, the feasibility conditions for each of the decisions in (14) correspond to those in (6). Theorem 3. The supplier’s best response to a given value of q1 when w + a > p + v is 8 B:I: t1 ; > > > > tB:II: > 1 ; > > B:III: > t > 1 ; > > < ð1 dÞq1 ; t1 ðq1 Þ ¼ arg max B:I: B:III: PS ; t1 ;t1 B > > > S > B:III: arg max > ð1dÞq1 ;t1 PB ; > > > > > > : S B:III: P ; arg maxtB:II: B 1 ;t1

if t1B:I: ð1 dÞq1 & t1B:III: ð1 þ dÞq1 ; if ð1 dÞq1 tB:II: & t1B:III: ð1 þ dÞq1 ; 1 if tB:II: ð1 þ dÞq ; 1 1 if tB:II: ð1 dÞq1 t1B:I: & t1B:III: ð1 þ dÞq1 ; 1 if t1B:I: ð1 dÞq1 & t1B:III: ð1 þ dÞq1 ; if tB:II: ð1 dÞq1 t1B:I: & 1 t1B:III: ð1 þ dÞq1 ; if ð1 dÞq1 tB:II: ð1 þ dÞq1 t1B:III: : 1 (14)

142

M.J. Drake and J.L. Swann

The possible subgame-perfect Nash Equilibrium decision pairs in Case I.B. are the same as those for Case A (given in Table 1) except with the B supplier decision values replacing the A decisions. The buyer’s decisions are exactly the same as those in Case A since the actual q1 values are independent of M. The buyer can again apply the methods described in the proof of Proposition 1 to determine the optimal value of q1 in the cases in which the supplier’s best response is the value among a set of maximizing arguments for two of the expected profit realizations. Once the feasible set of possible decision pairs is determined, the buyer can again substitute each of them into its expected profit function to find the maximizing decision pair, which is the SPNE. If the contract’s parameters are such that neither of the above scenarios (A or B) apply, then the supplier’s expediting decision is dependent on the magnitude of the final order. The most interesting case is when the supplier will receive the deviation penalty for some of the expedited units and not for others, so we consider the case where q2 > (1 + d)q1 and t1 + M > (1 + d)q1. For it to be Pareto optimal for the supplier to fulfill the entire order, the cash flow from satisfying must be greater than the cash flow from not satisfying. When q2 t1 + M, which means that the supplier has enough capacity to satisfy the full order if it wants to, we must have (w c2) (q2 t1) + p(q2 (1 + d)q1) a(q2 t1). Solving for q2, the supplier satisfies the extra demand if q2

ðw c2 þ aÞt1 þ pð1 þ dÞq1 L1 : w c2 þ a þ p

Formally, the supplier’s expediting decision is ðq2 t1 Þþ ; if q2 L1 t2 ¼ 0; if q2 < L1 : When the actual order exceeds the supplier’s total capacity (i.e., q2 > t1 + M), the supplier can only satisfy an additional M units beyond t1. If the supplier chooses to supply the additional M units, it will have to pay the buyer the a penalty on each of the q2 t1 M that were ordered and not fulfilled. Thus, we must have (w c2)M + p(t1 + M (1 + d)q1) a(q2 t1 M) a(q2 t1) for the supplier to want to supply the extra M units. This condition simplifies to M

pðð1 þ dÞq1 t1 Þ L2 : w c2 þ a þ p

This means that the supplier’s decision to expedite the M additional units is M; if M L2 t2 ¼ 0; if 0

Facilitating Demand Risk-Sharing with the Percent Deviation Contract

143

added supply uncertainty. Knowing that the expediting decision rests with the magnitude of the order might induce the buyer to inflate the final order so that the supplier will fulfill the entire amount. The buyer’s strategic behavior in this case is detrimental for the supplier because it could be induced to expedite when it would not otherwise. To alleviate these difficulties, we recommend that the parties set the negotiated parameters–p and a–such that the contract assumes another case. This could be accomplished by letting a > c2 w, shifting the contract to the A case. Of course, shifts to other scenarios are possible through negotiations, depending on the relative market power of the parties. Since both parties have an incentive to set the contract parameters to move the contract to other cases, we omit this intermediate situation from further analysis and subsequently only consider Cases A and B.

3.3

Infinite Expediting Capacity

We conclude our presentation of the general models by considering a special case in which the supplier’s expediting capacity is infinite (or especially large for practical purposes). These uncapacitated models have an especially simple structure that enables us to develop (quasi) closed-form optimal decisions. Since this extension is based on the expediting capacity, we must only develop models for case analogous to A above, in which the supplier chooses to expedite. It does not matter how much extra capacity the supplier has if it chooses not to use it. Again we only consider the case where the buyer orders the actual demand in all cases (i.e., q2 ¼ X and t2 ¼ ðminfq2 t1 ; MgÞþ ) since (1) holds and w c2 > a. (It is straightforward to extend these models to the case in which the buyer does not order above the deviation range; see Drake (2006) for details.) We denote this scenario as A.1. The supplier’s expected profit function is again the same as in (2) with M ¼ 1, but this substitution results in the simpler function Z PSA:1

1

¼w

Z

0

þp þv

Z

ð1dÞq1

xf ðxÞdx þ p 1

ð1þdÞq1 Z t1

ðð1 dÞq1 xÞf ðxÞdx

0

ðx ð1 þ dÞq1 Þf ðxÞdx

(15) Z

1

ðt1 xÞf ðxÞdx c1 t1 c2

0

ðx t1 Þf ðxÞdx:

t1

The expected profit function in (15) is concave by Lemma 1, so we can solve for the optimal pre-acquisition amount using first order conditions. This yields c 2 c1 tA:1 ; (16) 2 t : FðtÞ ¼ 1 c2 v which is independent of the buyer’s initial order estimate because of the symmetric information assumption and the infinite total system capacity.

144

M.J. Drake and J.L. Swann

Similarly, the buyer’s expected profit function is the same as in (7), but infinite supplier capacity yields the following simplified form. Z PBA:1

¼ ðr wÞ Z 1 p

1

0

ð1þdÞq1

Z

ð1dÞq1

xf ðxÞdx p

ðð1 dÞq1 xÞf ðxÞdx

0

(17)

ðx ð1 þ dÞq1 Þf ðxÞdx

Because the supplier is willing to expedite to satisfy the buyer’s order regardless of its size, the buyer’s expected profit function is no longer dependent on the supplier’s t1 decision. In this case, the t1 decision only affects the supplier’s profitability and not its ability to fulfill the buyer’s order. The buyer’s optimal initial order estimate is given by qA:1 fq : ð1 dÞ 1 Fðð1 dÞqÞ ¼ ð1 þ dÞð1 Fðð1 þ dÞqÞÞg, which corresponds with the optimal decision in Case A.III in which the supplier has the capacity to satisfy orders is the value of q1 above the upper limit of the deviation range. The decision qA:1 1 that equates the marginal expected deviation penalty for demands below the lower limit of the range, p(1 d)F((1 d)q1), to the marginal expected penalty for orders above the upper limit, p(1 + d)(1 F((1 + d)q1)). Since the nominal deviation penalty, p, is the same regardless of whether the deviation was a lower deviation or an upper deviation, it is irrelevant to the buyer’s decision. Of course, if there were two deviation penalties, pl and pu, they would affect the buyer’s decision. A:1 Thus, the SPNE for the A.1 case is ðq1 ; t1 ðq1 ÞÞ ¼ ðqA:1 1 ; t1 Þ for all parameter sets such that w c2 > a. It applies in situations where the supplier always has enough extra capacity in its network to satisfy the buyer’s order. It would be most reasonable when the buyer’s requirements are small compared with the supplier’s capabilities. Consequently, the supplier would only need to utilize the more complicated capacitated contracts for customers who require a large portion of its capacity. Since these buyers are larger, they are presumably more important to the supplier, so it would have more incentive to utilize a more complicated contract for these customers.

4 Economic Analysis and Model Extensions 4.1

Individual Rationality Constraints

The practical implementation of the percent deviation contract is necessarily impacted by the competitive power of the parties. If the buyer has a powerful market presence, it will likely be able to negotiate favorable contract terms by threatening to use another supplier who offers a more traditional agreement. (We assume that the contract is used in a competitive industry, so the buyer can find

Facilitating Demand Risk-Sharing with the Percent Deviation Contract

145

another supplier with comparable service performance and quality.) The terms of the contract, therefore, must satisfy the buyer’s individual-rationality constraint, which says that under the percent deviation contract the buyer must be able to attain an expected profit at least as great as it could under a traditional mechanism. See Tirole (1988) for a detailed discussion of individual-rationality constraints. If this constraint is not satisfied, the buyer will switch to another supplier. In this section we compare our percent deviation contract to the status quo of a traditional wholesale-price contract. In the cases where the supplier’s expediting capacity is limited, the percent deviation mechanism can induce the supplier to preacquire significantly more inventory than it would under the traditional wholesaleprice contract. This additional ability to meet demand is beneficial for both parties, resulting in higher expected profits without further contract modifications. This is the situation demonstrated in the numerical example in Sect. 5.1. In situations where the supplier does not increase its pre-acquisition quantity significantly (i.e., the deviation penalty is not high enough to induce the supplier to pre-acquire much more inventory than under the traditional wholesale-price contract), it is clear that the buyer will earn less expected profit under the percent deviation contract because it now shares additional demand risk by paying the deviation penalty for orders outside of the allowable range. There are several ways in which the parties can adjust the terms of the percent deviation contract to satisfy the buyer’s individual-rationality constraint. The supplier can offer the buyer a fixed transfer payment to share some of its gain. In some cases the supplier can offer a discounted wholesale price, w0 , that gives the buyer the same expected profit as it would attain under the traditional wholesaleprice contract. The remainder of this section illustrates the methodology required to find the requisite discounted wholesale price. The A.1 infinite capacity model is comparable to the traditional newsvendor, wholesale-price (NV) contract, since the supplier chooses and has the capacity to satisfy the entire order. The Rbuyer’s expected profit function under the NV contract 1 is given by PBNV ¼ ðr wÞ 0 xf ðxÞdx. Notice that this function is not dependent on any decision by the supplier, because under a traditional contract in this setting the buyer places orders for exactly the number of units needed with no demand risk. Comparing this expected profit to that under the percent deviation contract in (15) with the qA:1 decision, the contracting parties wish to find w0 such that PBNV ðwÞ 1 PBA:1 ðw0 Þ, to ensure that the buyer earns at least as much expected profit under the percent deviation contract as it does under the original wholesale-price contract. We find that the discounted wholesale price given by 0

2R ð1dÞqA:1

w0 @wp4

1

0

31þ R1 ðð1dÞqA:1 xÞf ðxÞdxþ ð1þdÞqA:1 ðxð1þdÞqA:1 Þf ðxÞdx 1 1 1 5A R1 0 xf ðxÞdx (18)

satisfies the buyer’s participation constraint. The term in brackets represents the percentage of order periods in which the deviation penalty will be paid.

146

M.J. Drake and J.L. Swann

Consequently, the supplier must provide an allowance for this expected penalty if the buyer is to realize the same expected profit as in the newsvendor contract. If the right side of (18) assumes the value of zero, there is no discounted wholesale price mechanism that can satisfy the buyer’s rationality constraint with the given contract parameters. The supplier also has an individual-rationality constraint that should be considered. To induce the buyer to participate in the percent deviation contract in this case, the supplier must offer the discounted wholesale price discussed above. This reduces its expected profit from the high theoretical profit that could be earned under the percent deviation contract with the original newsvendor wholesale price. This is not a problem when we compare the supplier’s expected profit to that of the wholesale-price contract. The percent deviation contract induces the supplier to establish a higher system capacity (t1 + M) than the wholesale-price contract does. This increases the total expected profit of the entire supply chain because the supply chain is able to satisfy more of the consumer demand. If the supplier offers w0 equal to the right-hand side of (18), the buyer’s expected profit under percent deviation will be equal to its expected profit under the wholesale-price contract, and the supplier captures all of the additional supply chain profit. This makes the supplier better off than it was under the wholesale-price contract. Even if the supplier decides to make the buyer strictly better off by offering a slightly smaller value of w0 than the right-hand side of (18) requires, there is a range of values where both parties can improve their position by splitting the increased supply chain expected profit. This is the situation demonstrated in the numerical example in Sect. 5.2.

4.2

Channel Coordination

Supply chain research has shown that the total supply chain profit is maximized by a centralized firm making decisions that are best for the system as a whole. One main objective of supply chain contracts is to align each entity’s own incentives to induce decentralized decisions that attain the maximal centralized supply chain profit. This achievement is commonly referred to as “channel coordination.” We first examine the performance of the centralized channel and then develop mechanisms to coordinate the channel.

4.2.1

Centralized Channel Benchmark

In terms of a centralized channel, the buyer and the seller are viewed as a single entity trying to maximize its own expected profit. Hence, there is no wholesale price (w) paid from the sales department (the buyer) to the manufacturing department (the seller), and the penalties levied under the percent deviation contract (p and a) are not valid. The buyer’s decisions are not relevant either since the single company does not order from itself; the combined firm must only determine the number of units to acquire early and the number to expedite.

Facilitating Demand Risk-Sharing with the Percent Deviation Contract

147

If the cost structure for the centralized channel is such that r c2 > b, the firm will satisfy additional demand beyond the number of pre-acquired goods up to capacity M. In this case the number of units to be expedited is given by t2 ¼ (min {X t1, M})+. The channel’s expected profit function is Z

Z t1 xf ðxÞdx þ ðt1 þ MÞð1 Fðt1 þ MÞÞ þ v ðt1 xÞf ðxÞdx c1 t1 0 0 Z t1 þM c2 ðx t1 Þf ðxÞdx þ Mð1 Fðt1 þ MÞÞ t Z 11 ðx t1 MÞf ðxÞdx: (19) b

PC:I: ¼ r

t1 þM

t1 þM

This newsvendor-type profit function is concave, so first order optimality n C:I: conditions show that the optimal solution for t is t 2 t : Fðt þ MÞ ¼ 1 1 o rþbc1 ðc2 vÞFðtÞ . rþbc2 If r c2 < b the loss from expediting or subcontracting to meet the marginal demand is larger than the cash outlay from the penalty paid to the customer for not satisfying its demand. Accordingly, the centralized channel will not expedite at the higher cost c2 ; formally, we have t2 ¼ 0. The channel expected profit function now becomes Z

t1

PC:II: ¼ r 0

Z

Z xf ðxÞdx þ t1 ð1 Fðt1 ÞÞ þ v 1

b

t1

ðt1 xÞf ðxÞdx c1 t1

0

(20)

ðx t1 Þf ðxÞdx:

t1

The profit function in (20) is very similar to (19), except the expected revenue has been adjusted to reflect the fact that the centralized channel will not satisfy any demandnmore than t1. The o optimal number of units to acquire early is given by

1 tC:II: 2 t : Fðt1 Þ ¼ rþbc 1 rþbv .

4.2.2

Finite Expediting Capacity Channel Coordination

The subgame-perfect Nash Equilibria for the scenarios in which the supplier has finite expediting capacity have a complicated form. As a result, different mechanisms are required for each possible decision pair. Consequently, we consider one possible decision pair to show how the system can be coordinated given that particular decision. The procedure described below is applicable to all other possible decision pairs and case scenarios. We consider Case B, in which the buyer orders the entire demand but the supplier chooses not to expedite, where the corresponding decision pair is q1B:III: ; t1B:III: . The

148

M.J. Drake and J.L. Swann

following theorem contains the channel coordinating condition for this case, which also applies in Case A when the same decision pair is optimal. Theorem 4. The decentralized channel in scenario B (and A) in which the SPNE decision pair is q1B:III: ; t1B:III: will be coordinated if the contract parameters are set such that a þ p þ w ¼ r þ b:

(21)

The left-hand side of (21) comprises parameters that represent payments between the buyer and the supplier. These are set during contract negotiations as opposed to the right-hand side, which only contains parameters that we assumed were exogenous to the contract because they involve an outside party to the contract (the buyer’s customer). The parties can coordinate the channel by setting a, p, and w according to (21).

4.3

Comparison to Quantity Flexibility Contracts

Since the percent deviation contract provides the buyer with order flexibility around an initial order estimate, it is constructive to compare its channel performance with the quantity flexibility contract, which affords the buyer similar flexibility. Tsay (1999) establishes that the quantity flexibility contract cannot coordinate the supply chain when the buyer is not bound by a minimum purchase commitment. The percent deviation contract, on the other hand, does coordinate the channel without establishing a floor on the buyer’s order. Let us consider analysis for a particular case, e.g., the B scenario in which the SPNE is q1B:III: ; t1B:III: . Recall that this scenario can be coordinated by setting a + p + w ¼ r + b. To compare the quantity flexibility and percent deviation contracts, we need to analyze them in a similar framework. We apply the basic quantity flexibility contract structure but modify as follows to correspond to the percent deviation decision environment. We assume that the buyer’s actual order in the quantity flexibility contract is made after the customer demand has been realized, as in the percent deviation scenario. Consequently, the supplier commits to fulfilling a maximum of t1 units. The buyer establishes a minimum purchase commitment of (1 d)q1 units when it provides the initial order estimate, q1. If the buyer ends up being forced to order more units than are ultimately required to satisfy the realized demand (as a result of the minimum purchase quantity), these units generate u dollars per unit as a salvage value. We assume that leftover units of inventory are no more valuable to the buyer than they are to the supplier (i.e., u v). This is practical for several reasons. While it is true that goods generally appreciate in value as they move downstream in a supply chain, the buyer is not physically performing additional functions to add value to the product; consequently, the actual sale price of the salvaged product

Facilitating Demand Risk-Sharing with the Percent Deviation Contract

149

should be no higher than that which the supplier could receive if the good were sold it in the secondary market. Leftover product should be more valuable to the supplier in terms of expected revenue since the supplier could likely use the product to fulfill demand from another buyer while the buyer may have limited outlets to offload the extra product. This is especially true in the market for truckload transportation, which was an inspiration for the percent deviation contract. Carriers would obviously place more value on an unassigned truck than any one particular shipper might. Following the same backward-induction methodology we used in identifying the other equilibria, Theorem 5 provides the equilibrium decision for the quantity flexibility contract. Theorem 5. The SPNE decisions for the quantity flexibility contract are 80

1 rwþba > 1 >

F > > B rwþba C > rvþba > B C; if ðc1 vÞðrþbaÞ > ;F1 > 1d @ > rvþba A < > QF ¼ qQF 1 ;t1 þwvc1 u > > > > ðwþaÞðwuÞ > >

> > wþac > 1 > ; otherwise: : fqjFðð1dÞqÞ¼0g;F1 wþav (22) Suppose the parameter values are such that the quantity flexibility equilibrium decisions are the first pair in (22). We can write the expected total supply chain profit as the sum of the agents’ individual expected profit functions, which reduces to "Z QF # Z tQF t1 1 QF QF QF SC PQF ¼ r xf ðxÞdx þ t1 1 F t1 tQF þu 1 x f ðxÞdx c1 t1 0

Z

b

1 tQF 1

f ðxÞdx: x tQF 1

0

(23) SC Note that if u ¼ v, for any value of t1 we have PSC QF ðt1 Þ ¼ PC:II: ðt1 Þ, where is the centralized supply chain profit in (20).

PSC C:II: ðt1 Þ

Theorem 6. The percent deviation contract coordinates the supply chain in the following cases where the quantity flexibility contract fails to coordinate: 1. When the salvage value is higher at the supplier (u < v), there are cases in which the centralized supply chain profit under the percent deviation contract always exceeds that attainable from the quantity flexibility contract. 2. When the salvage values are equal for both parties (u ¼ v), channel coordination efforts for quantity flexibility require either setting a < 0 or w < c1 , both of which violate the underlying assumptions of the model.

150

M.J. Drake and J.L. Swann

In other supply chain contracting structures such as revenue-sharing agreements, it is possible for suppliers to benefit by selling goods for a wholesale price below their marginal cost of production as the second part of Theorem 6 requires. This strategy is successful because the supplier is receiving part of the buyer’s revenue in addition to the wholesale price. Looking at the supplier’s expected profit function under the quantity flexibility contract in (28), the supplier can either obtain w or v for each of the t1 pre-acquired units. If each of these values are less than c1, the supplier cannot earn positive expected profit by selling below the marginal cost. We have thus shown that there are cases in which the quantity flexibility contract cannot coordinate the supply chain, while the percent deviation contract is able to achieve coordinated performance. The main difficulty the quantity flexibility contract has in this decision environment is that it establishes a minimum purchase commitment for the buyer. The percent deviation contract provides buyers more flexibility by allowing them to choose to pay the penalties associated with ordering outside of the deviation range. Of course, in order to gain this flexibility, the contract must be more complex; therefore, the percent deviation contract would likely be more costly to manage in practice.

5 Numerical Analysis In this section we provide several numerical examples that illustrate the behavior of the percent deviation contract in various decision environments discussed above as well as how parameters can be set to satisfy individual-rationality constraints and to coordinate the channel. We estimated the demand distributions used below from weekly shipping data provided by a major US manufacturer. The demand random variable represents the number of shipments per week required from the supplier to a retailer on a particular origin-destination lane; we consider two such lanes. For one of the lanes, the exponential distribution gave the best fit, and for the other the uniform distribution was appropriate. We failed to reject chi-squared goodness of fit statistics at the 10% significance level for each of the two distributions. For the cost and contract parameters, we constructed values that make relative sense in this manufacturer’s business setting.

5.1

Exponential Demand Example (Case A.)

Consider weekly demand that follows an exponential distribution with l ¼ 0.17297 and the cost parameters listed in Table 2. These parameters define a contract in Case A., since r w + b > p and w + a > c2; all of the supplier’s expected profit function realizations are concave because w + a > c2 + p. Thus, the buyer orders the exact demand, and the supplier chooses to expedite units (up to the capacity of 5). Under a traditional wholesale-price contract with inventory pre-acquisition and

Facilitating Demand Risk-Sharing with the Percent Deviation Contract Table 2 Parameter declarations for numerical examples

Parameter r w c1 c2 a

Exp (.17297) 60 17 7 13 3

a

Unif (0,18) 30 18 6 22 1

151

Parameter b v d p M

Exp (.17297) 12 1 0.2 5 5

Unif (0,18) 4 1 0.2 13 5

b

225

50 25

200

0

175

−25

150

−50

125

−75 30 20 t

10

00

10

20 q

30

30

30

20

20 t

10 00

10 q

Fig. 3 Expected profit functions of (a) the supplier and (b) the buyer for the Exp(0.17297) example

n o 2 vÞFðtÞ , or expediting, the supplier pre-acquires tNV 2 t : Fðt þ MÞ ¼ wc1 ðc wc2 tNV ¼ 4.7667 units. This results in an expected profit of 189.89 for the buyer and 29.21 for the supplier; thus, the total supply chain profit for the wholesale-price contract is 219.10. The centralized-channel pre-acquisition quantity is 10.4930, yielding a maximal channel expected profit of 243.45. The main problem with the wholesale-price contract is that the supplier does not have an incentive to pre-acquire enough inventory because the buyer is not sharing any of the demand risk. This low pre-acquisition amount restricts the total system’s ability to satisfy realized customer demand, which dampens the system’s profit potential. For the same parameter set, the percent deviation contract with (p,d) ¼ (5,0.2) is Pareto-improving for both parties as compared with the wholesale-price contract. Figure 3a, b depict the expected profit functions for the supplier and the buyer, respectively, as a function of the two main decision variables, q1 and t1. Note the piecewise form of these expected profit functions, which reflects the different profit function realizations with their distinct optimal solutions. Applying solu the ; t ðq Þ ¼ tion procedure detailed in Sect. 3.2, the SPNE decision pair is q 1 1 1 A:III: A:III: ¼ ð5:1810; 6:0391Þ. These decisions yield an expected profit of q1 ; t1 192.55 for the buyer and 37.37 for the supplier and a total expected supply chain

152

M.J. Drake and J.L. Swann

profit of 229.92. While both parties are better off in relation to the wholesale-price contract, there are still more gains to be realized because there is a 6% efficiency loss with this solution compared with the centralized solution. We can design the percent deviation contract parameters to ensure that the channel coordination condition in (21) is met. Namely, we need a + w + p ¼ r + b, so we 3 can by setting satisfy this inequality p ¼ 52. This induces an equilibrium pair of A:III: A:III: C:I: ¼ t1 ¼ ð5:1810; 10:4930Þ, which gives channel-optiq1 ; t1 ðq1 Þ ¼ q1 ; t1 mal expected profits of 84.35 and 159.10 for the buyer and supplier, respectively, and a total expected supply chain profit of 243.45, as designed. To induce the supplier to pre-acquire the channel-optimal inventory amount, the buyer has had to relinquish a substantial amount of profit to the supplier. The buyer’s coordinated expected profit does not satisfy its individual-rationality constraint, which requires expected profit of at least 189.89, the buyer’s expected profit from the wholesale-price contract. If the buyer received a fixed transfer payment, then it would be willing to accept the percent deviation contract. In this case, the fixed transfer payment must be larger than 105.54.4 This payment, denoted F, should not be too high, though; or else the supplier would be better off under the original wholesale-price contract as well. Thus, for any fixed supplier-to-buyer transfer payment in the range F ∈ (105.54,129.89), the percent deviation contract is coordinated and strictly Pareto-improving for both parties as compared to the wholesale-price contract.

5.2

Uniform Demand Example (Case B.)

We now consider an example with uniform demand and parameters as defined in Table 2. Since r w + b > p and w + a < c2, a percent deviation contract in this case would fall in scenario B., where the buyer orders the full demand and the supplier chooses not to expedite because it is too expensive.5 Under a traditional wholesale-price contract, the supplier pre-acquires 12.7059 units of inventory. The buyer and supplier expected profits are 95.54 and 76.24, respectively, resulting in a total supply chain expected profit of 171.78. If the firms acted as a centralized

3 Since the deviation penalty is so large, the condition for concavity on the supplier’s first profit function realization is no longer satisfied. This does not matter, though, because the buyer would never choose an equilibrium in this realization, which requires that it pay the deviation penalty for every unit of demand satisfied. 4 Note that, in this case, the channel coordinating condition is a function of the wholesale price, so we do not attempt to satisfy the buyer’s individual-rationality constraint using a wholesale price discount as discussed in Sect. 4.1. 5 We see in Table 2 that the supplier has M ¼ 5 units of available expediting capacity. This number is irrelevant here because regardless of the amount of extra capacity available, the supplier will not use any of it because expediting is too costly.

Facilitating Demand Risk-Sharing with the Percent Deviation Contract

a

153

b

150

50

100

25

50

0

0

15 15

10 t

5 5

0

0

10 q

−25 15

10 t

5

00

5

10 q

15

Fig. 4 Expected profit functions of (a) the supplier and (b) the buyer for the Unif(0,18) example

channel, the pre-acquisition amount would be 15.2727 with a total expected profit of 177.82. Figures 4a, b depict the expected profit functions for the supplier and the buyer under a percent deviation contract in this example. We can apply the solution procedure B:III: B:III:for Case B. to determine the SPNE decision pair of q1 ; t1 ðq1 Þ ¼ q 1 ; t1 ¼ ð10:3846; 15:0968Þ, which results in expected profits of 71.53 and 106.26 for the buyer and the supplier, respectively, and a total supply chain expected profit of 177.79. Note that this decentralized percent deviation contract produces a supply chain profit very close to that of the centralized channel; this is due to the fact that the supplier’s t1 decision value is approximately equal to that of the centralized channel. While the above percent deviation contract is close to coordinated as currently constructed, it does not satisfy the buyer’s individual-rationality constraint when compared with the wholesale-price contract. Consequently, the percent deviation contract must be modified to give the buyer an incentive to accept it over the status quo. If the supplier offers a discounted wholesale price (as discussed in Sect. 4.1) of 15.2346, which represents an approximate discount of 15% off the original price of 18, the equilibrium decision pair becomes q1 ; t1 ðq1 Þ ¼ q1B:III: ; t1B:III: ¼ ð10:3846; 14:1812Þ. This contract results in expected profits of 95.54 and 82.08 for the buyer and the supplier, respectively, and a total supply chain expected profit of 177.62, which is still close to the centralized optimum of 177.82. This percent deviation contract with a discounted wholesale price satisfies the buyer’s participation constraint and provides a higher profit for the supplier in relation to the traditional wholesale-price contract. Thus, for this example the individual-rationality constraint and the Pareto-improving condition are more important than channel coordination since the decentralized percent deviation contracts are close to being coordinated without any additional consideration.

154

M.J. Drake and J.L. Swann

6 Conclusions and Further Research In this chapter we have characterized the subgame-perfect Nash Equilibria of a dynamic supply chain game induced by the percent deviation contract, an innovative mechanism that was motivated by our discussions with a major firm in the transportation industry. Due to the sequential extensive form of this supply chain game, many of the decisions are functions of those decisions made in earlier stages of the game. The main result we have shown is that the percent deviation contract is a viable, albeit somewhat complicated, mechanism whereby the supplier can transfer some of its demand risk to the buyer. The prospect of receiving a deviation penalty for large or small buyer orders induces the supplier to pre-acquire more inventory than it ordinarily would, which increases the total capacity of the system. This extra ability to satisfy end-user demand benefits the entire system, enabling Pareto improvements. Several trajectories exist for future research in this area. The first direction includes relaxing some of the assumptions that we made in these models. A natural extension would be adding some information asymmetry by including one party’s proprietary information on costs or capacity. One could also include nonlinear costs or another pricing policy such as quantity discounts. For completion, it would also be interesting to examine supply chain coordination mechanisms for the other possible decision pairs. More generally, future work incorporating dynamic decision environments could be useful, especially in multi-echelon supply chains. Comparison studies of various contracting mechanisms applied to the same scenario could lead to Pareto-improvements similar to the ones we found. Further analysis is also needed to incorporate the advanced demand information into operational production and transportation network models. Only then will the true value of the percent deviation contract be estimated for the entire system. Acknowledgements This research was funded, in part, by The Logistics Institute Leaders in Logistics Grant from Lucent Technologies and NSF Grants DMI-0223364 and DMI-0348532.

Appendix Proof of Lemma 1. We will define the three realizations of (2) as follows: Z

PSA:I:

Z t1 þM ¼w xf ðxÞdx þ ðt1 þ MÞð1 Fðt1 þ MÞÞ þ p ðt1 þ M xÞf ðxÞdx 0 0 Z t1 þM Z t1 ðt1 xÞf ðxÞdx c1 t1 c2 ðx t1 Þf ðxÞdx þ Mð1 Fðt1 þ MÞÞ þv 0 t1 Z 1 a ðx t1 MÞf ðxÞdx (24Þ t1 þM

t1 þM

Facilitating Demand Risk-Sharing with the Percent Deviation Contract

Z PSA:II: ¼ w

t1 þM

155

xf ðxÞdx þ ðt1 þ MÞð1 Fðt1 þ MÞÞ

0

Z

ð1dÞq1

þp

ðð1 dÞq1 xÞf ðxÞdx Z t1 þM Z t1 ðt1 xÞf ðxÞdx c1 t1 c2 ðx t1 Þf ðxÞdx þ Mð1 Fðt1 þ MÞÞ þv t1 0 Z 1 ðx t1 MÞf ðxÞdx (25Þ a 0

t1 þM

Z PSA:III: ¼ w Z

t1 þM

0 ð1dÞq1

þp "Z0 þp

t1 þM ð1þdÞq1

Z

t1

þv 0

Z a

xf ðxÞdx þ ðt1 þ MÞð1 Fðt1 þ MÞÞ ðð1 dÞq1 xÞf ðxÞdx # ðx ð1 þ dÞq1 Þf ðxÞdx þ ðt1 þ M ð1 þ dÞq1 Þð1 Fðt1 þ MÞÞ Z

t1 þM

ðt1 xÞf ðxÞdx c1 t1 c2

1

t1 þM

ðx t1 Þf ðxÞdx þ Mð1 Fðt1 þ MÞÞ

t1

ðx t1 MÞf ðxÞdx:

(26Þ

The second derivative of (24) taken with respect to t1 is (p + c2 w a)f(t1 + M) (c2 v)f(t1), which is negative for all values of t1 if w + a > p + c2 based on the parameter conditions of scenario A. The second derivative of (25) is (c2 w a)f(t1 + M) (c2 v)f(t1), and the second derivative of (26) is (c2 w a p)f(t1 + M) (c2 v)f(t1). Both of these expressions are negative for all values of t1 without the extra condition. □ Proof of Theorem 1. There are five possible values for the supplier’s best response. Each of the three realizations of the supplier’s expected profit function has an individual maximizer, shown in (3)–(5). In addition, the two points where the pieces of the profit function converge (t1 ¼ (1 d)q1 M and t1 ¼ (1 + d) q1 M) are possible solutions. These solutions would occur when the maximizing t1 values do not lie in their corresponding feasible regions. In order to establish the result in Theorem 1, we first make some observations about the expected profit function that will help us with the main proof. Observation 2 For all values of t1 less than the lower boundary ðð1 dÞq1 MÞ, PSA:II: ðt1 Þ>PSA:I: ðt1 Þ because the term representing the expected value of the lower deviation penalty paid is larger in PSA:II: . For values of t1 greater than the lower boundary, PSA:II: ðt1 Þ

156

M.J. Drake and J.L. Swann

Observation 3 For all values of t1 less than the upper boundary ðð1 þ dÞq1 MÞ, PSA:II: ðt1 Þ > PSA:III: ðt1 Þ because the term representing the expected value of the upper deviation penalty paid in PSA:III: is negative. For values of t1 greater than the upper boundary, PSA:II: ðt1 Þ < PSA:III: ðt1 Þ. The supplier’s best response function depends on the values of the three maximizers relative to the feasible boundaries. There are 27 possible cases because each of the three decisions can potentially lie in three regions; however, the following results show that several of these cases are not possible. Lemma 2. The following results hold in Case A. It is not possible to have t1A:I: < A:III: ð1 dÞq1 M < tA:II: <ð1 þ dÞq1 M < tA:II: 1 , and it is also not possible to have t1 1 . After using Lemma 2 to reduce the number of possible cases, we can determine the overall best response for each given the values of the individual maximizers by applying Observations 1–3. Summarizing all of the scenarios, we obtain the solutiongiven in (6). □ Proof of Proposition 1. We use the difference function to determine when the supplier chooses each t1 for given values of q1 in a range where its best response is known to be the maximizer of its expected profit from a set of two values. We then use this information to characterize explicitly the ranges of q1 that induce each value of t1. If the difference function is positive for a value of q1, then the supplier will choose t1A:III: ; it will select the other possible decision if the function is negative. The number of ranges for q1 is determined by the number of changes of sign in the difference function. The difference function related to the decision in (8) is given by D t1A:I: "Z A:III: # t1 þM A:III: A:III: A:I: A:I: xf ðxÞdxþ t1 þM 1F t1 þM t1 þM 1F t1 þM ¼w þM tA:I: 1

"Z

ð1dÞq1

þp "Z þp

t1A:III: 0

"Z a

t1A:III: þM

c2 Z

t1A:I: þM t1A:I:

Z f ðxÞdx

tA:I: 1

0

xt1A:III: M f ðxÞdx

tA:III: þM 1

tA:III: 1

# f ðxÞdx

ðxð1þdÞq1 Þf ðxÞdxþ t1A:III: þMð1þdÞq1 1F t1A:III: þM

t1A:III: x

1

"Z

t1A:I: þMx

0 t1A:III: þM

þv

t1A:I: þM

ðð1dÞq1 xÞf ðxÞdx

0

ð1þdÞq1

"Z

Z

t1A:I: x

Z

1

t1A:I: þM

#

f ðxÞdx c1 t1A:III: t1A:I:

xt1A:I: M

# f ðxÞdx

xt1A:III: f ðxÞdx

xt1A:I:

# A:I: A:III: f ðxÞdxM F t1 þM þF t1 þM :

#

Facilitating Demand Risk-Sharing with the Percent Deviation Contract

157

This difference function is convex in q1 since @@qD2 ¼ pð1 dÞ2 f ðð1 dÞq1 Þþ 1 difference function at the pð1 þ dÞ2 f ðð1 þ dÞq1 Þ>0. We can begin by evaluating the tA:I: þM tA:III: þM two endpoints of the region defined in (8); that is, q1 ¼ 11d and q1 ¼ 1 1þd . If the difference function is positive for one value and negative for the other, then convexity implies that there exists a single threshold value of q1 in the interval where the difference function changes sign. The buyer can use these supplier decision values to evaluate its best selection of q1 in this region with respect to its expected profit function. If the difference function is positive for both endpoint values of q1, then it is possible that there are zero, one, or two points where the function switches sign. If there are zero or one switching points, then the supplier will choose t1 ¼ t1A:III: for all values of q1 in the region. If there are two switching points, then for values of q1 between these two values, the supplier will choose t1 ¼ t1A:I: . It will choose t1 ¼ t1A:III: for all other values of q1. If the difference function is negative for both endpoint values, then convexity implies that it will be negative for all values of q1; thus, the supplier will always choose t1 ¼ t1A:I: . The difference function related to the decision in (10) is given by A:II: D t1 "Z A:III: t1 þM ¼w xf ðxÞdxþ t1A:III: þ M 1 F t1A:III: þ M 2

t1A:II: þM

þ M 1 F tA:II: þM tA:II: 1 1 "Z A:III: # t1 þM A:III: A:III: þp ðx ð1þ dÞq1 Þf ðxÞdxþ t1 þ M ð1þ dÞq1 1 F t1 þ M ð1þdÞq1

"Z

t1A:III:

þv 0

"Z a c2 Z

1

tA:III: þM 1

"Z

tA:III: 1

Z f ðxÞdx

tA:II: 1

0

x t1A:III: M f ðxÞdx

tA:III: þM 1

t1A:II: þM t1A:II:

t1A:III: x

tA:II: x 1

Z

1 tA:II: 1 þM

#

f ðxÞdx c1 t1A:III: tA:II: 1

x tA:II: M f ðxÞdx 1

#

x t1A:III: f ðxÞdx

# x tA:II: þ M þ F t1A:III: þ M : f ðxÞdx M F tA:II: 1 1

This difference function is convex in q1 since @@qD2 ¼ pð1 þ dÞ2 f ðð1 þ dÞq1 Þ > 0, 1 @D ¼ pð1 þ dÞð1 Fðð1 þ dÞq1 ÞÞ < 0. and it is also decreasing in q1 because @q 1 tA:II: þM This means that if the difference function is negative when q1 ¼ 1 1þd , which is the lower limit of the range defined in (10), then the supplier will always choose t1 ¼ tA:II: if the difference function is positive at the upper endpoint of 1 . Likewise, n A:II: o t þM tA:III: þM , then the supplier will always select the range q1 ¼ min 1 1d ; 1 1þd 2

158

M.J. Drake and J.L. Swann

t1 ¼ t1A:III: . If the difference function is positive for the lower endpoint and negative for the upper endpoint, then there exists exactly one point where the difference function changes sign, and we have two distinct ranges of q1 values where the two t1 decisions are chosen. The difference function related to the decision in (9) is given by Dðð1dÞq1 MÞ "Z A:III: t1 þM xf ðxÞdxþ t1A:III: þM 1F t1A:III: þM ¼w ð1dÞq1

ð1dÞq1 ð1Fðð1dÞq1 ÞÞ "Z A:III: # t1 þM A:III: A:III: þp ðxð1þdÞq1 Þf ðxÞdxþ t1 þM ð1þdÞq1 1F t1 þM "Z

ð1þdÞq1 tA:III: 1

þv

t1A:III: x

0

f ðxÞdx

Z

ð1dÞq1 M

tA:III: þM 1

c2 Z

t1A:III: þM t1A:III:

ðð1dÞq1 M xÞf ðxÞdx

0

c1 t1A:III: ð1dÞq1 þM "Z Z 1 A:III: a xt1 M f ðxÞdx "Z

#

#

1 ð1dÞq1

ðxð1dÞq1 Þf ðxÞdx

xt1A:III: f ðxÞdx

# A:III: ðxð1dÞq1 þMÞf ðxÞdxM Fðð1dÞq1 ÞþF t1 þM :

ð1dÞq1 ð1dÞq1 M

Here one of the potential supplier decisions is an explicit function of the buyer’s q1 decision, so the difference function is more complex. Specifically, the function is not necessarily convex or concave. For a given set of parameters, then the exact switching points can be determined by simple numerical search methods. In many realizations the difference function will be well-behaved; thus, a similar analysis to that performed for the previous two cases above would suffice for these situations. □ Proof of Theorem 4. This case can easily be compared with the centralized Case C.II. in which the centralized firm also does not expedite. The total expected supply chain profit for the voluntary compliance case is Z PSC B: ¼ r

t1

Z xf ðxÞdx þ t1 ð1 Fðt1 ÞÞ þ v

0

t1

ðt1 xÞf ðxÞdx c1 t1

0

Z

1

b t1

ðx t1 Þf ðxÞdx:

(27)

Facilitating Demand Risk-Sharing with the Percent Deviation Contract

159

Comparing (27) with the centralized supply chain profit in (20), it is easily seen that the two profits will be equal if the t1 decisions are equal, which is accomplished rþbc1 1 þp if wþac wþavþp ¼ rþbv . Simplifying this equality yields the channel coordinating condition. □ Proof of Theorem 5. We will solve for the subgame-perfect Nash equilibrium decisions under a quantity flexibility contract via backward induction. The parameters in the B. scenario are such that the buyer orders q2 ¼ maxfX; ð1 dÞq1 g, where X denotes the realized customer demand. The supplier’s expected profit can thus be written as "Z PSQF ¼ w

ð1dÞq1

Z ð1 dÞq1 f ðxÞdx þ

0

"Z

ð1dÞq1

þv Z

#

t1 ð1dÞq1

xf ðxÞdx þ t1 ð1 Fðt1 ÞÞ Z

ðt1 ð1 dÞq1 Þf ðxÞdx þ

0 1

a

#

t1 ð1dÞq1

ðt1 xÞf ðxÞdx c1 t1

ðx t1 Þf ðxÞdx:

(28)

t1

Since the supplier’s expected profit function is concave, first-order optimality wþac conditions imply that the supplier’s optimal decision is t1 ¼ F1 wþav1 . There is one additional consideration, though, since the buyer is guaranteed to order at least (1 d)q1. The supplier should pre-acquire at least the minimum purchase quantity because these n sales are guaranteed. o Thus, the supplier’s optimal decision is 1 tQF 1 ¼ max ð1 dÞq1 ; F

wþac1 wþav

.

The buyer’s expected profit function is given by "Z PBQF

tQF 1

¼r

# xf ðxÞdx þ

0

"Z

ð1dÞq1

w 0

Z

ð1dÞq1

þu 0

tQF 1 ð1

FðtQF 1 ÞÞ Z

ð1 dÞq1 f ðxÞdx þ

tQF 1 ð1dÞq1

# xf ðxÞdx þ

ðð1 dÞq1 xÞf ðxÞdx þ ða bÞ

Z

1 tQF 1

tQF 1 ð1

FðtQF 1 ÞÞ

ðx tQF 1 Þf ðxÞdx: (29)

We can solve for the buyer’s optimal decision, as before, by assuming that the supplier’s decision takes on each of the two possible values and then optimizing the buyer’s profit subject to the constraint the supplier’s decision valid. that makes F1 rwþba rvþba QF 1 rwþba ; t ; F is optimal if ¼ The decision pair qQF 1 1 1d rvþba

160

F1

M.J. Drake and J.L. Swann

F1

, which reduces to (c1 v)(r + b a) + wv c1u QF ¼ (w+ a)(w u). If this inequality is reversed, qQF 1 ; t1

F1 wþac1 wþav 1 . In this case the supplier’s decision is fixed regard; F1 wþac 1d wþav rwþba rvþba

wþac1 wþav

QF less of the value of qQF 1 , so the buyer can reduce its demand risk by offering q1 2 fqjFðð1 dÞqÞ ¼ 0g such that there is no probability of customer demand below the minimum quantity. □ SC Proof of Theorem 6. If u < v, then clearly PSC QF PC:II: for every value of t1, and there exist some values of t1 where the inequality is strict. Consequently, coordination is not possible in these cases because leftover goods are less valuable in the buyer’s possession, which is where they reside under quantity flexibility. SC SC C:II Now let u ¼ v. If tQF 1 ¼ t1 , then PQF ¼ PC:II: , and we would have a coordirþbc1 nated supply chain. Thus, we want to have rwþba rvþba ¼ rþbv . Since a(0), the penalty the supplier pays the buyer for not satisfying units ordered, is the one parameter over which the parties are assumed to have control under quantity flexibility, we solve for the coordinating condition

a¼

ðr þ b vÞðc1 wÞ : c1 v

(30)

Examining the components of (30) individually, we see that the first term in the numerator is greater than zero because r > v and b 0, as is the denominator. So if w > c1 by our initial assumption, then this would require a negative a. We could have a positive coordinating a if we allowed the supplier to sell the goods below cost. □ Proof of Lemma 2. The first result follows directly from Lemma 3. To establish the second result by contradiction, assume that this relationship is true. Since the piecewise functions are concave from Lemma 1, t1A:III: is the single maximum of PSA:III: , and PSA:III: ðt1 Þ is decreasing for values of t1 >t1A:III: . Consequently, S PSA:III: ðt1A:III: Þ>PSA:III: ðð1 þ dÞq1 MÞ>PSA:III: ðtA:II: 1 Þ. Since PA:II: ðð1 þ dÞq1 MÞ ¼ S PA:III: ðð1 þ dÞq1 MÞ from Observation 1, we have PSA:III: ðt1A:III: Þ> S A:II: PSA:II: ðð1 þ dÞq1 MÞ>PSA:III: ðtA:II: 1 Þ. Observation 3 states that PA:III: ðt1 Þ> S S S S A:II: A:III: PA:II: ðt1 Þ, which implies PA:III: ðt1 Þ>PA:II: ðð1 þ dÞq1 MÞ> PA:III: ðtA:II: 1 Þ> S S A:II: PSA:II: ðtA:II: Þ. The statement P ðð1 þ dÞq MÞ>P ðt Þ contradicts the 1 A:II: A:II: 1 1 S maximizes P . □ result that tA:II: A:II: 1 Lemma 3. If w + a > p + c2, then t1A:I: tA:II: 1 . A:I: Proof. Suppose, on the contrary, t1A:I: < tA:II: 1 , which implies that Fðt1 þ MÞb FðtA:II: þ MÞ. Substituting the values given in (3) and (4) and simplifying, we have 1

A:I: A:II: ðc2 vÞðw þ a c2 Þ FðtA:II: 1 Þ Fðt1 Þ p w þ a c1 ðc2 vÞFðt1 Þ : (31)

Facilitating Demand Risk-Sharing with the Percent Deviation Contract

161

The left side of (31) is positive, and the right side is negative since the numerator in (4) must be positive at tA:II: 1 . (The denominator is positive due to the parameter relationship defining Case A.) This leads to a contradiction. □ B:I: B:III: B:II: Lemma 4. If w + a > p + v, then t1 min t1 ; t1 . Proof. The proof follows the same contradiction procedure as in that of Lemma 3. □ Proof of Lemma 5. We will define the four functions resulting from (7) as follows: "Z PBA:I: ¼ ðr wÞ Z

# xf ðxÞdx þ t1 ðq1 Þ þ M 1 F t1 ðq1 Þ

t1 ðq1 ÞþM 0

t1 ðq1 Þ

p

t1 ðq1 Þ þ M x

0

Z f ðxÞdx þ ða bÞ

1

t1 ðq1 Þ

x t1 ðq1 Þ M f ðxÞdx (32)

"Z PBA:II: ¼ ðr wÞ Z

t1 ðq1 ÞþM 0

Z

ð1dÞq1

p

# xf ðxÞdx þ t1 ðq1 Þ þ M 1 F t1 ðq1 Þ

ðð1 dÞq1 xÞf ðxÞdx þ ða bÞ

0

1

t1 ðq1 Þ

x t1 ðq1 Þ M f ðxÞdx (33)

"Z PBA:III:

t1 ðq1 ÞþM

¼ ðr wÞ 0

Z

ð1dÞq1

p

# xf ðxÞdx þ t1 ðq1 Þ þ M 1 F t1 ðq1 Þ Z

ðð1 dÞq1 xÞf ðxÞdx þ ða bÞ

0

"Z p

t1 ðq1 ÞþM ð1þdÞq1

1

t1 ðq1 Þ

x t1 ðq1 Þ M f ðxÞdx

ðx ð1 þ dÞq1 Þf ðxÞdx

þ t1 ðq1 Þ þ M ð1 þ dÞq1 1 F t1 ðq1 Þ þ M : (34) "Z PBA:IV:

ð1dÞq1

¼ ðr wÞ Z p 0

# xf ðxÞdx þ ð1 dÞq1 ð1 Fðð1 dÞq1 ÞÞ

0 ð1dÞq1

Z ðð1 dÞq1 xÞf ðxÞdx þ ða bÞ

1 ð1dÞq1

ðx ð1 dÞq1 Þf ðxÞdx (35)

The concavity result follows the same logic as that used in Lemma 1.

□

162

M.J. Drake and J.L. Swann

Proof of Theorem 2. The following lemma establishes the piecewise-concavity of (7). □ Lemma 5. The buyer’s expected profit function realizations resulting from (7) are concave. Since the buyer’s four individual profit function realizations are concave from Lemma 5, we can use the KKT conditions to solve for the optimal q1 for each function over the region of q1 values where the function is valid, as defined in Theorem 1. t1A:I: þM We first maximize (32) n over theA:I:region o q1 1d . Since (32) is not dependent t þM on q1, any value q1A:I: q : q 11d is optimal. tA:II: þM T tA:II: þM T tA:III: þM We consider (33) over the region q1 1 1d q1 1 1þd q1 1 1þd . Taking the partial derivative and setting it equal to zero yields p(1 d)F((1 d) q1) ¼ 0. Since only the lower deviation penalty exists in this profit function realization, the buyer wants to make the initial order estimate as small as possible to avoid A:II: A:III: F1 ð0Þ maxft1 ;t1 gþM A:II: paying the penalty. Consequently, q1 ¼ q1 max 1d ; . 1þd A:II: A:III: T t þM t þM q1 1 1þd . First order We want to maximize (34) over the region q1 1 1þd A:III: fq : ð1 þ dÞ ¼ ð1 dÞFðð1 dÞqÞþ optimality conditions yield q1 ¼ q1 A:III: þM minftA:II: g 1 ;t1 ð1 þ dÞFðð1 þ dÞqÞg, which is feasible if it is smaller than 1þd A:II: . T t þM Finally, we maximize (35) over the region q1 1 1d q1 A:I: A:III: T t1 þM t1 þM A:IV: q1 1þd . The first order conditions give us q1 ¼ q1 n1d o n A:II: o t þM tA:III: þM rwaþb q : Fðð1 dÞq1 Þ ¼ rwaþbþp , which is feasible if max 1 1d ; 1 1þd qb1

t1A:I: þM 1d .

□

Proof of Theorem 3. The proof of this result follows the same logic as that of Theorem 1, utilizing the results from Lemma 4. □

References Balakrishnan A, Geunes J, Pangburn MF (2004) Coordinating supply chains by controlling upstream variability propagation. Manuf Serv Oper Manage 6(2):163–183 Barnes-Schuster D, Bassok Y, Anupindi R (2002) Coordination and flexibility in supply contracts with options. Manuf Serv Oper Manage 4(3):171–207 Bassok Y, Anupindi R (2008) Analysis of supply contracts with commitments and flexibility. Nav Res Logist 55:459–477 Bazaraa MS, Sherali HD, Shetty CM (1993) Nonlinear programming: theory and algorithms, 2nd edn. Wiley, New York Cachon GP (2004) The allocation of inventory risk in a supply chain: push, pull, and advancepurchase discount contracts. Manage Sci 50(2):222–238 Cachon GP, Fisher M (2000) Supply chain inventory management and the value of shared information. Manage Sci 46(8):1032–1048

Facilitating Demand Risk-Sharing with the Percent Deviation Contract

163

Cachon GP, Lariviere MA (2001) Contracting to assure supply: how to share demand forecasts in a supply chain. Manage Sci 47(5):629–646 Donohue KL (2000) Efficient supply contracts for fashion goods with forecast updating and two production modes. Manage Sci 46(11):1397–1411 Drake MJ (2006) The design of incentives for the management of supply and demand. PhD thesis, Georgia Institute of Technology, Atlanta, GA Erkoc M, Wu SD (2005) Managing high-tech capacity expansion via capacity reservation. Prod Oper Manage 14(2):232–251 Finley F, Srikanth S (2005) Seven imperatives for successful collaboration. Supply Chain Manage Rev 9(1):30–37 Fugate BS, Sahin F, Mentzer JT (2006) Supply chain management coordination mechanisms. J Bus Logistics 27(2):129–161 Kulp SC, Lee HL, Ofek E (2004) Manufacturer benefits from information integration with retail customers. Manage Sci 50(4):431–444 Lee HL (2004) The triple-A supply chain. Harv Bus Rev 82(10):102–112 Lee HL, So KC, Tang CS (2000) The value of information sharing in a two-level supply chain. Manage Sci 46(5):626–643 Lian Z, Deshmukh A (2009) Analysis of supply contracts with quantity flexibility. Eur J Oper Res 196:526–533 Serel DA (2007) Capacity reservation under supply uncertainty. Comput Oper Res 34:1192–1220 Tirole J (1988) Theory of industrial organization. MIT, Cambridge, MA Tsay AA (1999) The quantity flexibility contract and supplier–customer incentives. Manage Sci 45 (10):1339–1358 Tsay AA, Lovejoy WS (1999) Quantity flexibility contracts and supply chain performance. Manuf Serv Oper Manage 1(2):89–111 Tsay AA, Nahmias S, Agrawal N (1999) Modeling supply chain contracts: a review. In: Tayur S, Ganeshan R, Magazine M (eds) Quantitative methods for supply chain management. Kluwer, Norwell, MA Wang X, Liu L (2007) Coordination in a retailer-led supply chain through option contract. Int J Prod Econ 110:115–127 Zhao Y, Wang S, Cheng TCE, Yang X, Huang Z (2010) Coordination of supply chains by option contracts: a cooperative game theory approach. Eur J Oper Res 207:668–675

.

Value-Added Retailer in a Mixed Channel: Asymmetric Information and Contract Design Samar K. Mukhopadhyay, Xiaowei Zhu, and Xiaohang Yue

Abstract With increasing regularity, manufacturers are opening a direct selling channel using internet, keeping their traditional retail channel in place. This mixed channel is attractive to the manufacturers because they retain the advantage of the retailer’s traditional services while increasing their sales base to customers purchasing online. One disadvantage of this model is the potential for channel conflict because they are in direct competition with their own retailers. In this chapter, we propose an innovative way to mitigate this channel conflict, where the manufacturer allows the retailer to add value to the base product so that it is differentiated from their own offering through the direct channel. We model this supply chain where the retailer is also given the authority to price the value added product. Design of an optimal contract from the manufacturer’s point of view is complicated due to the fact that the manufacturer does not know the retailer’s cost of adding value. This chapter develops the closed form solution of the optimal contracts under this information asymmetry. Comparison with channel coordinating contracts is provided. This chapter develops a number of new managerial guidelines and identifies future research topics.

S.K. Mukhopadhyay Graduate School of Business, Sungkyunkwan University, Jongno-Gu, Seoul 110-745, South Korea e-mail: [email protected] X. Zhu (*) College of Business and Public Affairs, West Chester University of Pennsylvania, West Chester, PA 19383, USA e-mail: [email protected] X. Yue Sheldon B. Lubar School of Business, University of Wisconsin-Milwaukee, P.O. Box 742, Milwaukee, WI 53201, USA e-mail: [email protected] T.-M. Choi and T.C. Edwin Cheng (eds.), Supply Chain Coordination under Uncertainty, International Handbooks on Information Systems, DOI 10.1007/978-3-642-19257-9_7, # Springer-Verlag Berlin Heidelberg 2011

165

166

S.K. Mukhopadhyay et al.

Keywords Channel conflict • Information asymmetry • Mixed channel • Valueadding retailer

1 Introduction In addition to their traditional retailer channel, firms are opening direct channels in increasing numbers. This is a new business model facilitated by emerging internet technology. The motivation is the increased control over product distribution and pricing, order capture and customer information. The traditional retail channel is also kept in place because it has important roles to play. These include creating and satisfying demand for the product, engaging in activities to build brand awareness, gathering market information, and providing customer support. The example of this type of mixed channel strategy includes Compaq starting it in 1998. Other firms that have adopted similar strategies include IBM (Nasiretti 1998), HP (Janah 1999), Mattel (Bannon 2000), Nike (Collinger 1998). Balasubramanian (1998) and Levary and Mathieu (2000) suggest that such a strategy could work well. The disadvantage of this model is that the manufacturer is now in direct competition with its channel partners. As Frazier (1999) showed, mixed channel would increase revenue, but would lead to decreased support from the channel partners. In fact, this led to some retailers actually taking action against the manufacturers who opened a direct channel in competition with them. Channel conflict is the biggest deterrent for the manufacturer to go ahead with the mixed channel business model. Because channel conflict is detrimental for the supply chain relationship, there needs to be ways to mitigate this conflict. Some of the ways are separating the brands sold directly and sold through retailers, taking orders over the Internet and then fulfilling the order through the retailers, and sharing a part of the profit from each direct sale with their retailers. They can also maintain the price at par with the retailer so as not to undercut them. Hann (1999) gives an example of Zurich, an insurance company. Another way would be to sell a basic version of the product direct, and let the retailer add further value to the product before selling to the final customer (Fay 1999). In this chapter, we address the mixed channel strategy where the channel conflict is eliminated by the use of a value-adding retailer. We study a business model where the retailer-manufacturer conflict is alleviated by a contract. We explore a number of cases in this scenario (1) a base case, for benchmarking purpose, where the channel is integrated and a joint profit function is maximized; (2) a case where the channel partners are separate but they share full information with each other; and (3) a more general case where there is information asymmetry in the channel. Under the information asymmetry, one partner offers a lump sum side payment to the other to alleviate channel conflict. In all cases we find the optimum price in each channel, the optimum value added by the retailer, and the optimum side payment. This book chapter is based on authors’ original work of Mukhopadhyay et al. (2008a).

Value-Added Retailer in a Mixed Channel

167

2 Literature Survey Supply chain coordination can be accomplished through appropriate contract design. Cachon and Lariviere (2005) study revenue-sharing contracts in a general supply chain model with revenues determined by each retailer’s purchase quantity and price. They compare revenue sharing to a number of other supply chain contracts, like buy-back contracts, price-discount contracts, quantity-flexibility contracts, sales-rebate contracts, franchise contracts, and quantity discounts. Plambeck and Taylor (2008) study how the potential for renegotiation influences the optimal structure of supply contracts. They show that renegotiation can greatly increase the firms’ investments and profits, provided that the contracts are designed correctly. Tsay et al. (1999) and Frazier (1999) survey channel structure and incentive design for performance enhancement. Cohen et al. (1995) study an intermediary who perform specific value-adding functions, and get compensated for this service by the manufacturer or distributors by a side payment. A mixed channel strategy in products that do not provide large value are studied by Chiang et al. (2003) who show that adding a direct channel can mitigate the profit loss. Yao and Liu (2003) study diffusion of customer between two channels and find that, under certain conditions, both channels would enjoy stable demand. Viswanathan (2000) study the mixed channel issue from the product differentiation point of view and conclude that the more different the product is in the two channels, the more the benefit for the channels. Khouja et al. (2010) indicates that the most critical factor in channel selection is the variable cost per unit of product sold using the direct versus the retail channels. There is an increased competition between the manufacturer and retailer (Agatz et al. 2008) as the manufacturer expands his channels to the customers. Though channel structures have been extensively researched in literature (i.e., Hua et al. 2010; Su et al. 2010; Chiang 2010) relatively few have studied mixed channel with value-adding retailer to decreasing the competition between channels. Mukhopadhyay et al. (2008b) find that the retailer would be willing to share information with the manufacturer if her cost of adding value is lower than a threshold value. One of contribution for our research is that we study the mixed channel under asymmetric information, not under full information like most of existing literature. Asymmetric information and supply chain coordination have been the subject of a number of recent studies. Desiraju and Moorthy (1997) study the case of information asymmetry about a price and service-sensitive demand curve. They show that coordination can be achieved by requirement of service performance. Cakanyildirim et al. (2010) find that information asymmetry about manufacturer’s production cost does not necessarily cause inefficiency in the supply chain. Value of information in a capacitated supply chain is derived by Gavirneni et al. (1999). Lee et al. (2000) show that, with a demand process correlated over time, it could be worthwhile to share information about the demand. Corbett and de Groote (2000) derive optimal discount policy for both full and incomplete information cases.

168

S.K. Mukhopadhyay et al.

Corbett et al. (2004) study different types of contracts to coordinate the supply chain for both complete information and asymmetric information. Ha (2001) finds that in case of private information, optimal order quantity is smaller and optimal selling price is higher than for the case with complete information. Mukhopadhyay et al. (2009) designed a contract for the manufacturer to motivate the retail’s marketing effort under asymmetric information of retailer’s sale effort. Section 3 of this chapter presents our mixed channel model. The optimal contracts for the complete information case and asymmetric information are shown in Sect. 4. We will also compare the two cases, and derive a number of managerial insights. In Sect. 5, we report the results of an extensive numerical experimentation to see how the changes in the parameters affect the contracts. Section 6 will conclude the chapter with some further research ideas.

3 The Model Our supply chain consists of a traditional manufacturer and a retailer. There is also a direct channel selling to the same customer pool (see Fig. 1). The retailer augments the basic product by adding value for the customer. Let p1 be the price of the basic product charged by the manufacturer in the direct channel. Let v be the value added to the basic product by the retailer who prices it as p2. The cost to the retailer for adding a value v is cv per unit. It can be assumed that p2 > p1. The effective price to the customer of the augmented product is p2 v because of the additional value compared to the basic product sold in the direct channel. Let the wholesale price charged by the manufacturer to the retailer be w. Customers evaluate both the products and compare their value with the respective prices. Let the equilibrium demands be d1 for the direct channel and d2 for the retailer channel. The decision variables in our model are p1 and w for the manufacturer and p2 and v for the retailer, each maximizing their own profit functions.

MANUFACTURER

(pM) w RETAILER

p1, d1

(pR) p2, cv, d2

Fig. 1 Mixed retail and direct channel distribution system

POTENTIAL CUSTOMERS

Value-Added Retailer in a Mixed Channel

3.1

169

Characterizing the Demands

The retail channel demand, in case of no direct channel is written as: d2 ¼ ða2 bp2 Þ þ bv where a2 is the base demand b is price sensitiveness, and b is the sensitivity of demand with respect to the value added, i.e., it is the increase in demand per unit value added. Similarly, in the absence of the retail channel, the demand from the direct channel is d1 ¼ ða1 bp1 Þ where a1 is the base demand for this channel. Literature in this area uses similar linear demand function (Cotterill and Putsis 2001) and we follow their lead here. We now consider the modified demand when both the channels are operating at the same time. Now, customers would make a purchase decision by considering the two prices p1 and p2, and also the value added v by the retailer. d1 and d2, therefore, would be functions of p1, p2 and v. As a result, there would be migration of customers from one channel to another. We assume that this migration is proportional to the price difference and the additional value. Then the demand of the two channels would be: The direct channel: d1 ¼ ða1 bp1 Þ rðp1 ðp2 vÞÞ ¼ a1 ðb þ rÞp1 þ rðp2 vÞ The retailer channel: d2 ¼ ða2 bp2 Þ þ bv rðp2 v p1 Þ ¼ a2 ðb þ rÞp2 þ ðb þ rÞv þ rp1 where r is the migration effectiveness. To maintain analytical tractability, we assume that a1 ¼ a2 ¼ a, b ¼ b and normalize (b + r) and (b + r) to 1. The demand function is thus simplified as follows. Direct channel: d1 ¼ a p1 þ rðp2 vÞ

(1)

d2 ¼ a ðp2 vÞ þ rp1

(2)

The retailer channel:

We assume that r < 1, so that own channel effects are greater than cross channel effects. a and r are assumed to be common knowledge.

3.2

Value-Adding Cost for the Retailer

When the retailer is allowed to add value to the product, there is a cost denoted by cv per unit. We assume a quadratic cost function for the retailer value-adding process. Specifically, we use the functional form: cv ¼

v2 2

(3)

170

S.K. Mukhopadhyay et al.

where , an efficiency parameter for the retailer’s value added cost, is retailer’s private information. Note that we have defined cv as per unit quadratic cost to capture the phenomenon that adding a large quantum of value is proportionately more costly than adding minimal amount of value. In some cases, there would be a fixed cost (due to infrastructure creation, e.g.) which could be applied to the total sales volume. We are not including this cost here. In this chapter we consider contracts under two information structures. In full information scenario, the retailer shares its private information to the manufacturer. In asymmetric information scenario, the manufacturer does not know . We assume that the manufacturer holds a prior cumulative distribution F() with density function f(), defined on ½ 0 ; 3 , where 0 0 3 1.

3.3

Retailer’s and Manufacturer’s Profit

The practice of a side payment, L, from the manufacturer to the retailer to alleviate the channel conflict is used in some cases. Incorporating this side payment would give the following profit functions: The retailer’s profit function: pR ¼ ðp2 w cv Þd2 þ L

(4)

And the manufacturer’s profit function: pM ¼ p1 d1 þ wd2 L

(5)

Where d1 and d2 are given by (1) and (2), w is the wholesale price charged by the manufacturer to the retailer. We include L as the manufacturer’s decision variable to make the contract more flexible and to achieve the supply chain coordination (Corbett et al. 2004). To maintain analytical tractability, we further assume there are no marginal costs incurred by the manufacturer for selling through direct channel and through the retailer. In reality, both the retailer and the manufacturer have a reservation profit level which they intend to achieve in order for a trade to take place. Let reservation profit levels for the retailer and manufacturer be pR and pM respectively.

4 Two Types of Contracts One type of contract we consider is full information (F) contract. The other type is the asymmetric information (A) contract. An integrated channel (I) provides the base case. Under (I), the contract is designed by maximizing the total profit of

Value-Added Retailer in a Mixed Channel

171

manufacturer and retailer and taking as common knowledge. Under (F), the manufacturer knows the retailer’s and designs the contract taking as common knowledge to maximize its own profit. Under (A), the manufacturer does not know the retailer’s and designs the contract using prior density function f() and cumulative distribution F() defined on ½0 ; 3 .

4.1

Integration Channel Contract (I)

In this case, the channel is integrated and thus would provide the first best solution. It is expected that the profits for the channel would be highest under this scenario and thus, can be used for comparison with other cases. In this case, the two channels are integrated and they together will behave as a single firm and therefore will optimize the joint channel profit. pI ¼ p1 d1 þ ½p2 cv d2

(6)

The optimal prices p1 and p2 and value added v can be obtained by taking first order condition and solve them simultaneously. We get, p1 ¼

a a 3 1 ; p ¼ þ ; v ¼ 2ð1 rÞ 2 2ð1 rÞ 4

The optimal joint profit for an integrated channel is given by a a2 1 þ 2ð1rÞ þ 16 p ¼ 4 2 Even though the two channel partners are integrated, they still need to decide how this total profit, derived above, be divided between the two. Suppose that the retailer has her own reservation profit as pR and the retailer, therefore, would participate in the contract only if the profit pR pR . One possible way of dividing the total profit is that the retailer is given pR to ensure her participation. The manufacturer, therefore, receives the remainder pI pR . A contract like this is proposed by Corbett et al. (2004). It can be shown that pM is a decreasing function of . Therefore, it is possible that when is high enough (approaching 3), pIM could be so low that it would be lower than his reservation profit pM . In that situation, the contract would be unattractive to the manufacturer and there would be no trade. Thus, the contract, to be viable, should be such that the manufacturer is guaranteed at least pM . Let N be that value of above which pIM is lower than pM . N, therefore, is the cut-off point above which there would be no trade, and the manufacturer is said to be following a cutoff policy (Ha 2001). Under this scenario, we also need to find the value of N. This is done in Proposition 1 which gives the complete channel contract under the base case, and the optimum channel profit and its division to the two partners. Proofs of all results are shown in the Appendix, unless stated otherwise. I

172

S.K. Mukhopadhyay et al.

Proposition 1. (a) The optimal contract under channel integration is given by: pI1 ¼

a 2ð1 rÞ

NI ¼

1 qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2a þ 2 4pM þ4pR a2 1þr 1r

pI2 ¼

a 3 þ 2ð1 rÞ 4

1 2 a ð1 þ rÞ where pM þpR 4ð1 rÞ vI ¼

(b) The optimal profits for the retailer and the manufacturer under channel integration are given by: pIR ¼ pR ( pIM ¼ pI ¼

a 4

2

1 a þ 16 2 þ 2ð1rÞ pR pM

N

)

N

2

a a 1 þ þ 4 2ð1 rÞ 162

It is interesting to note that the direct channel price does not depend on the retailer’s cost. Also, when r increases, the manufacturer will increase its direct channel price. Recall that r is the migration factor, and an increasing r will enable the manufacturer to attract more customers away from the retailer. This will enable manufacturer to increase his price, and thus his revenue. This result gives a managerial insight that the manufacturer should try operational and marketing means to increase r. This can be done, for example, by advertising, or by offering incentives like easy return policy for the internet purchase.

4.2

Full Information Contract (F)

The private information held by the retailer about her cost structure (about ) is shared with the manufacturer. The moves of manufacturer and retailer follow a Stackelberg type game: the manufacturer acts as the leader, announcing the p1 and w first; the retailer acts as the follower, announcing the p2 and v after that. The solution of this game follows. The manufacturer decides about his decision variables basing on the retailer’s best response function. This best response function is in terms of the manufacturer’s parameters. This function is obtained by maximizing the retailer’s profit pR with respect to her decision variables, namely p2 and v. Equation (7) gives the retailer’s best response function, as functions of p1 and w. pr2 ¼

3 w a p1 r þ þ þ 4 2 2 2

vr ¼

1

(7)

Value-Added Retailer in a Mixed Channel

173

Next, in stage 1 of the game, the manufacturer derives the optimal p1 and w by maximizing its own profit pM , given in (5), and substituting the optimum values of p2 and v thus making it a function of p1 and w alone. Using the first order conditions, we obtain the manufacturer’s optimal policies as: pF1 ¼

a 2ð1 rÞ

wF ¼

a 1 þ 2ð1 rÞ 4

(8)

In Stage 2 of the game, the retailer uses the manufacturer’s policy announcement given in (8), and maximizes her own profit function to obtain her own optimal policies as: pF2 ¼

ð3 rÞa 7 þ 4ð1 rÞ 8

vF ¼

1

(9)

From (8) and (9), we observe that w p1 . So, the manufacturer sees that selling one unit to the retailer at the wholesale price brings in more revenue than selling in the direct channel, he will have no incentive to open a direct channel, under the full information scenario, unless he wants to do it for reasons other than maximizing profits. These reasons could be to make customers aware of the product, provide product information, not to lose ground to competitors who have web presence, and so on. In that case, the cost penalty for the sub-optimal operation can be thought of as the cost of the above mentioned benefits.

4.3

Asymmetric Information Contract (A)

This is the most realistic case where the manufacturer does not know . As noted earlier, he knows the prior density function f() and cumulative distribution F() defined on ½0 ; 3 . The manufacturer offers the retailer a contract, which is a menu of {p1, w, L} meaning that it offers a number of alternative values for this tupple. The retailer has a choice of not accepting the contracts if none of the alternatives are attractive enough to her. Or she may select one alternative from the menu and decides to accept that. We include a side payment L in this case to formulate a two part nonlinear contract which gives the most flexible contract type (Corbett et al. 2004). Thus the profit for the manufacturer is pAM ¼ wd2 þ p1 d1 L and for the retailer is pAR ¼ ðp2 w cv Þd2 þ L pR . L > 0 is defined as a per-period payment from the manufacturer to the retailer. As noted earlier, this side payment is designed to alleviate the channel conflict in case the retailer is aggravated with the prospect of having competition with the manufacturer. This is also necessary if the retailer is more powerful than the manufacturer. For example, companies like Wal-Mart and Home depot can stop the manufacturer from opening a direct channel. For example, in 1999, Home Depot

174

S.K. Mukhopadhyay et al.

sent mail to more than 1,000 suppliers to ask them to stop online sale (Brooker 1999). We also include the possibility that L can be negative. L < 0 can be interpreted as a payment by the retailer to the manufacturer for getting the opportunity to do the business, as in the case of airline ticket. This can also be applied to the case of franchise, where the retailer has to pay the manufacturer. In most of these cases, there is little gain for the retailer to add value to the product, and the retailer has little power to impede the manufacturer’s sales in the direct channel strategy. The manufacturer offers a menu of contracts which is a function of because is unknown to the manufacturer. Thus the manufacturer offers fp1 ðÞ; wðÞ; LðÞg, and the retailer chooses a ^ to announce. Once the retailer has announced ^, direct Þ, wholesale price wð^ Þ and side payment Lð^Þ are fixed, so the channel price p1 ð^ Þ; wð^ ÞÞ using the best response function retailer will set retail channel price p2 ðp1 ð^ given in (7). This follows from the Revelation Principle of Fudenberg and Tirole (1991). The mechanism by which the retailer should chose which ^ to announce is as follows. First, she uses her own profit function pR and applies a first order condition and a local second order condition given by: @pR ð^ ; Þ ¼ 0 @^

and

@ 2 pR ð^; Þ 0 @^ @

Noting that, by substituting from Proposition 1(a) and earlier deductions, pR ð; ^Þ : ¼ ðpr2 ðp1 ð^ Þ; wð^ ÞÞ wð^ Þ crv Þd2 ðp1 ð^ Þ; wð^ Þ; pr2 Þ þ Lðwð^ ÞÞ 2 a 1 wð^ Þ rp1 ð^ Þ þ þ ¼ þ Lð^ Þ 2 4 2 2 Taking first order condition of pR w.r.t. ^ and solving at ^ ¼ we get: _ LðÞ ¼

1 a þ rp1 w _ _ ðwðÞ r pðÞÞ þ 4 2

This is what is called as the IC or Incentive Compatibility constraint. It can be also shown that _ ÞÞ @ 2 pR ðr p_ ð^ Þ wð^ 0 ð^ ; Þ ¼ 1 @^ @ 42 This is true under the common assumption that F() has decreasing reverse hazard rate, i.e., F()/f() is increasing in . Given that the IC constraint is derived as functions of the manufacturer’s variables, the next step for the manufacturer is to devise his optimal menu of contracts. This is done when the manufacturer maximizes his own profit function over the range of subject to the IC constraint and the individual-rationality (IR) constraint that the retailer will at least recover her own reservation profit. This is given in the following formulation.

Value-Added Retailer in a Mixed Channel

ðN Max

p1 ;w;L;N

0

175

ðwd2 þ p1 d1 LÞf ðÞdþ F

(10)

1 a þ rp1 w _ ðw_ r pÞ þ 4 2

(11)

subject to IC : L_ ¼

IR : pR ¼ ðp2 w cv Þd2 þ L pR

(12)

The first term in (10) gives the expected value of the manufacturer’s profit over the range of from the lowest possible value in the range, viz. 0 and N, the cutoff value explained earlier. For the range of between ½N; 3 , the manufacturer will gain his reservation profit giving him a total amount of F. Equation (11) is the IC constraint, forcing the retailer to truly announce ^ , as derived above. Equation (12) is the IR constraint. The structure of the above formulation fits the standard optimal control formulation with a salvage value. Solution. The above problem is complex as it is, but in this case its intractability is increased even further due to the fact the behavior of F would change the way the problem is solved. The reason is that depending upon the value of F, the “transversality condition” (see Kamien and Schwartz 1981, p. 148 for details) would be free. These points will be elaborated later. Now we enumerate below all possible cases that would arise for the transversality condition. There will be three possible cases depending on how the retailer’s and the manufacturer’s profits are behaving with respect to each other. Case 1. In this case, the manufacturer’s profit decreases in and the retailer’s profit increases in . Then at the cutoff point N, the manufacturer’s profit will hit his reservation profit pM and then remain constant at that value. Therefore F ¼ pM ð1 FÞ and transversality condition (used at cut-off point N) is free (as in case v of Kamien and Schwartz 1981, p. 148). Case 2. This is the case where both the manufacturer’s and the retailer’s profits decrease in . Thus again we have F ¼ pM ð1 FÞ and the transversality condition using when the K(IR) 0 will be required at cut-off point N (as in case vii of Kamien and Schwartz 1981, p. 148). Case 3. The manufacturer’s profit increases in Z and the retailer’s profit decreases profit in . Here the manufacturer’s profit is no longer fixed at pM like in the two earlier cases. Now the salvage value is given by F ¼ ½p1 ðNÞd1 þ wðNÞd2 LðNÞð1 FÞ. The transversality condition using when the K(IR) 0 will be required at cut-off point N (as in case vii of Kamien and Schwartz 1981, p. 148). It is not possible to have a case where both the manufacturer’s and retailer’s profits increase with . Because as increases, the total profit from these two channels is decreasing.

176

S.K. Mukhopadhyay et al.

Next, we will take Case 1, and solve the optimal control problem for this case. This is given in Proposition 2(a) which shows the menu of optimal pricing policies of the manufacturer and his side payment given that the retailer’s IR constraint is satisfied. In Part (b) of Proposition 2, we get the optimal profits for both channels. Proposition 2. (a) The manufacturer’s optimal contract under Asymmetric information for Case 1 is given by: a F ra ; wA ¼ ; 2ð1 rÞ 2f 2 2ð1 rÞ @LA a F 1 2 @wA ¼ ; þ @ @ 2 4f 2 4 pA1 ¼

a 3 F þ þ 2ð1 rÞ 4 4f 2 The optimal profits of the retailer and the manufacturer under asymmetric information are given by: (b) The retailer’s optimal price is given by p2 A ¼

pAR ¼ ðpA2 wA cv A Þd2A þ LA ¼ pAM ¼ pA1 d1A þ wA1 d2A LA ¼

a 1 F þ 2 4 4f 2

2 þ LA

F2 aF F a2 ð1 þ rÞ þ þ þ LA 84 f 2 42 f 83 f 4ð1 rÞ

During the course of the proof (found in Appendix), we obtain: @pAR ¼ 0, 0 satisfies ðpA2 wA cv A Þd2 þ LA ¼ pR , 1 ¼ N A ; 1 satisfies 2 @ F aF F a2 ð1 þ rÞ LðNÞA ¼ pM L(N)A satisfies 4 2 þ 2 þ 3 þ 8N f 4N f 8N f 4ð1 rÞ We derived above the analytical solution to the MEP problem giving the optimal policies of both parties when information asymmetry exists in the supply chain. We have done this for one of the cases, namely, Case 1. Given the complexity of a “twopoint boundary value” problem, this analytical exposition is a significant contribution. For the other cases, we will show our results of numerical solution. We will generate a number of insights into those cases and develop significant guidelines for decision making. These will be reported in Sect. 5. Next, we will study how the information asymmetry affects the optimal policies and the nature of profit from using these strategies by comparing them with those of the first best case, i.e., the case of channel integration.

4.4

Comparison of the Types of Contracts

The profits under the channel integration (I) case and the asymmetric information (A) case are compared here. Note that, intuitively, the channel integration is an

Value-Added Retailer in a Mixed Channel

177

ideal case in a supply chain, where both parties work for the benefit of the supply chain as a whole. But given the reality that the channel partners more often than not are separate entity, how does the more real case of asymmetric channel information compare with the ideal case? This question is answered in the following proposition. Proposition 3. If a supply chain moves from the information asymmetry case to complete channel integration, then (i) The manufacturer’s profit will increase, i.e., pIM pAM (ii) The retailer’s profit will decrease, i.e., pIR pAR (iii) The supply chain profit will increase, i.e., pI pAR þ pAM As we saw, under I, the manufacturer’s profit is decreasing in . The insight we gain here is that the manufacturer would prefer lower or higher v*. The less is the information asymmetry or the more the product is different, the more the manufacturer would benefit. Also, by channel integration, the manufacturer would always gain. He can, therefore, offer incentive to the retailer for being willing to share information. It is interesting to see that when supply chain integration is achieved, the retailer tends to lose some of its profit, even though the total supply chain profit increases. Thus the manufacturer, who was the aggrieved party under the information asymmetry, will benefit more than the total channel benefit, at the cost of the retailer. It intuitively follows that, if the manufacturer wishes to motivate channel coordination, he must offer some incentive to the retailer to make up for her lost profit. The difference of profit of the retailer between the cases of A and I can be thought of as the value of this information to the retailer. The profit realized under supply chain integration is always higher than the sum of the retailer’s and the manufacturer’s profits without such integration. This is generally the same result found in most supply chain research into other aspects of supply chain decision making. Basically, information asymmetry is inefficient for the supply chain as a whole. Assuming some known distribution function for , e.g., uniform, we can show that the more the products are different through the retailer’s value added process, the more the supply chain benefits. It is true for both scenarios, I and A. Next, we analyze the retailer’s optimal policies about its value added process and pricing under the two cases where information is shared and where it is not. We do the same for the manufacturer’s pricing policies. These results are given in the next two propositions. The proofs are straightforward using earlier results and are omitted. Proposition 4. (i) The retailer’s optimal value added amount remains same under both cases of A and I. (ii) The retailer can set higher retail price under A, i.e., pA2 pI2 . Proposition 5. The manufacturer set the same direct channel price p1 under I and A, i.e. pI1 ¼ pA1 . Also, this price is independent of .

178

S.K. Mukhopadhyay et al.

Again, it is interesting to see that the value added process by the retailer becomes unaffected by the level of coordination in the supply chain. It is also seen that the optimum value of the value added depends only on one parameter, namely, . Therefore, if the retailer can use operational means to reduce her cost of adding value, the optimal value-added will be higher. This action will start a ripple effect by allowing the retailer to charge a higher price and increase her profitability. Of course, this needs to be weighed against the cost of such operational means to reduce cost. Our study gives a useful managerial insight as to the retailer’s action about her own cost structure. The manufacturer, on the other hand, does not change his price with the supply chain structure, because his price is not dependent on , and thus on the amount of information sharing. But when this policy is coupled with the retailer’s policy of pA2 pI2 (from Proposition 4), we can see that double marginalization occurs due to the asymmetric information. This, in turn, reduces the whole supply chain efficiency. It will be also interesting to study the behavior of demand for both channels under the two scenarios. We do this in the next proposition. Proposition 6. The retail channel will experience increased demand if the supply chain structure moves from being asymmetric to integrated. At the same time the direct channel demand will decrease. This is a rather surprising finding. We can explain this by using the example of uniform distribution, as shown in the proof of this proposition. With increasing, the retailer channel demand d2 is decreasing and direct channel demand d1 is increasing for both the cases of I and A. The explanation of this can be found in the fact that when increases, the amount of value added decreases. This makes buying from the retailer channel less attractive to the customer. Therefore, more and more customers choose to buy the product from the direct channel. We also find that the cut-off point is higher under I than under A, i.e., N I N A . It means that the manufacturer and retailer can trade longer under I, before it becomes unattractive to either of them to trade (by possibly hitting one of the reservation profit levels). It can be explained as follows. In case of A, the manufacturer does not know . The manufacturer, therefore, would feel safer to trade with retailer only within a small range of . The manufacturer and retailer would both lose some trading opportunities to earn higher profit.

5 Results of Numerical Analysis To validate our analytical results and to gain more insights into the optimal policies, we carried out some numerical analysis. The results are briefly reported here. The numerical values used in this experiment are: a 40

r 0.500

pM 1,100

pR 600

0 0.03

3 0.07

Value-Added Retailer in a Mixed Channel

5.1

179

Variation of Profits Under Different Scenarios

We will start by investigating the three cases detailed in Sect. 4.3. We used Case 1 for our analytical exposition, but here we study the other two cases to see when they occur and what kind of guidelines we can achieve from these cases. We assume a Uniform distribution for with f ðÞ ¼

1 3 0

FðÞ ¼

0 3 0

and

over the interval ½0 ; 3 . The manufacturer’s profit as function of is: pAM ¼

0 0 2 a a0 a2 ð1 þ rÞ þ þ þ LA ðÞ 83 84 4 42 4ð1 rÞ

and the retailer’s profit is pAR ¼ pA1 d1A þ wA1 d2A þ LA ðÞ ¼

a2 0 a 2 þ 2 þ 0 4 þ LA ðÞ: 4 4 16

We plot these two profit expressions as functions of as shown in Fig. 2. This shows that both the manufacturer’s and the retailer’s profit are non-monotone function with . There are three very obvious regions of in the Profit

Region 1 (n0, n1)

Region 2 (n1, n2)

Region 3 (n2, n3)

1250.000 1150.000 1050.000 950.000 850.000 750.000 650.000 n 550.000 0.047 0.050 0.053 0.056 0.059 0.062 0.065 0.068

Fig. 2 The manufacturer and retailer’s profit for various

M (A) R (A)

180

S.K. Mukhopadhyay et al.

graph. We break the range ½0 ; 3 into three regions. In any one of these three regions, the profits show monotone property. Region 1. In this region, in the interval ð0 ; 1 Þ, the manufacturer’s profit decreases and the retailer’s profit increases with . This corresponds to Case 1 in Sect. 4.3. We start with Z0 ¼ 0.03. To obtain 1, we find that 1 satisfies 2 @pAR 0 1 0 ð161 4 þ 41 3 Þ0 2 a þ a ¼ @ 41 3 8 1 2 41 5 321 7 3 ð81 2 þ 1 Þ0 2 þ 0 7 ¼0 161 321 7 This gives 1 ¼ 1:910 ¼ 0:0573. Region 2. The interval ð1 ; 2 Þ gives us Region 2 where both the manufacturer’s and the retailer’s profits decrease with . This corresponds to Case 2 in Sect. 4.3. 1 ¼ 0:0573 as obtained above. We obtain Z2 from: @pAM 0 1 0 2 ð162 4 þ 42 3 Þ0 1 2 ¼ þ þ a þ a @ 42 3 82 2 42 5 322 7 42 2 3 ð162 2 þ 2 Þ0 2 30 ¼0 0 7þ 162 322 7 82 4 Giving

@pAR þ @

a 0 2 30 þ 42 45 84

¼ 0:

We get 2 ¼ 2:010 ¼ 0:0603 Substituting 1 ¼ 0:0573. Region 3. This is the remaining region and the range is given by ð2 ; 3 Þis (0.0603, 0.07). In this region, the manufacturer’s profit increases in and the retailer’s profit decreases. This corresponds to Case 3 in Sect. 4.3. In Region 1, is relatively small, giving the value added (¼1/) as relatively high. The manufacturer’s profit decreases and the retailer’s profit increases in . This can be seen in certain industries like electronics or computer industry. Here the manufacturers are generally locked into working with their retailers. In these industries, the retailers are “unlikely to get dis-intermediated”, says AMR Research’s Bob Parker (Gilbert and Bacheldor 2000). In fact, “Manufacturers are looking to strengthen the channel rather than circumvent it” (Gilbert and Bacheldor 2000). The higher the value the retailer adds to the base product, the higher profit the manufacturer earns. So the manufacturer should cooperate with the retailer when opening a new direct online channel and push the retailer to add more value to the base product on the retail channel. We see this kind of practice from IBM and HP. We also find that the retailer prefers to add small amount of value (with a large ) to avoid heavy cost burden. Another extreme case is shown in Region 3. In the region, is relatively high or value added is relatively low, the manufacturer’s profit increases in and the

Value-Added Retailer in a Mixed Channel

181

retailer’s profit decreases in . This result can explain the competition between manufacturer and retailer in industries such as an airline company and its airline ticket agent. It is hard for the agent (or the retailer) to add any value to the base product. Our study suggests that the airline company should open direct sales channel and compete with the agent. We also see that the less the value added, the more the manufacturer benefit from competition. Region 2 is the middle of the other two extreme cases. We see that both the manufacturer’s and the retailer’s profits decrease in and increase in value added. The high degree of value added would differentiate the product on the two channels and reduce channel conflict. Both the manufacturer and retailer would prefer higher value added and earn higher profits. From the manufacturer’s point of view, he would want to limit within Region 1. There are several reasons. First, when comparing Region 1 with the Region 2, we find that the manufacturer’s profit in Region 1 is dominant to that of Region 2. It implies that the manufacturer should always push the retailer add more value and reduce to bring it back within Region 1. Next, consider Region 3. From the analysis given above, the manufacturer should open a direct sales channel to compete with retail channel. For example, the manufacturer could set wholesale price w equal to or very close to direct sale price p1 and squeeze the retailer out of the market if she could not play any role in the value-adding process. In the airline industry, the ticket agents are facing this fate.

5.2

Behavior of Profits with Varying h

We analyze the behavior of the profits when the parameter is varied. This is shown in Region 1 of Fig. 3. From the figure, we have several findings besides those shown in Propositions 4–6. We can see that pAR pIR and pIM pAM are increasing with . Profit

η0 1250.000

η1

1150.000 1050.000

M (I)

950.000

M (A)

850.000

R (A) R (I)

750.000 650.000 05 7 0.

05 5 0.

05 1

04 9

05 3 0.

0.

0.

04 7 0.

04 5

n

0.

0.

04 3

550.000

Fig. 3 Variation of the manufacturer and retailer’s profit for various

182

S.K. Mukhopadhyay et al.

It means that the value of information is increasing in . With increasing, the retailer earns higher profit because she holds private information, but the manufacturer’s profit decreases. We also see that the cut-off point N I N A . We already know that under A, the manufacturer’s profit drops down to his reservation profit at the cut-off point, i.e., ¼ NA ¼ 1 . While at this point the manufacturer’s profit under I is still higher than his reservation profit, so the manufacturer could continue to trade with the retailer until his profit drops down to pM . At cut-off point NA, the manufacturer and the retailer both earn their reservation profits pM and pR , respectively. At this point, they both have exhausted all trading opportunity. We also carried out sensitivity analyses on the effect on profit of other parameters like base demand a and the migration parameter r. Because of length consideration, we do not report the entire findings here. In short, we find that, with the base demand a increasing, the value of information (given by the difference between the profits in A and I cases) increases. The managerial guideline here is that the retailer should try to increase the base demand by means of, say, advertising or offering better return policy etc. We also found that when the migration parameter r increases, the value of information increases. Again, the guideline for the retailer is that she should use marketing means to influence r.

6 Conclusions An important aspect of supply chain management in the Internet era has been studied in this chapter. More and more firms are introducing a direct channel in addition to the traditional retailer channel. In this new business model, the traditional channel is differentiated by an augmented role for the retailer namely modifying the basic product by adding value for the customer. We have presented a game theoretic formulation for this new business model. We studied two cases: the case with complete information and the case with asymmetric information. We obtained closed form contracts for both the channel partners in terms of market parameters and contrasted the optimum policies with those when the channels are completely integrated. Our study found that the manufacturer – the partner suffering from the asymmetry of the information – would always benefit (increased profit) with more information. We also found that, with information asymmetry, the direct channel price does not change, while the retailer enjoys higher price. One interesting finding is that the quantum of value-added does not change under any scenario and is only dependent on the retailer’s cost structure. Information asymmetry imposes inefficiency to the manufacturer and to the supply chain as a whole. The managerial insight gained from all these results would enable the manufacturer to decide about an information sharing contract which would include suitable incentive for the retailer. Our study showed that the actual values of the decisions variables in the optimum policy depend on the various market parameters like the base demand, and the migration parameter. Our results can be used as a guideline to set decisions

Value-Added Retailer in a Mixed Channel

183

about other variables, like product quality and return policy in order to influence these market parameters to move in the direction which would be beneficial to the channel partners. We also showed the benefit of the complete channel integration. Our model can be extended in many different directions. We can allow the manufacturer to provide a value added service, instead of the retailer. Customer’s special order requirement could be easily delivered to the manufacturer through online direct channel. So some companies let the manufacturers handle the customized order through the direct channel. For example, Disney only takes personalized orders (like putting customer’s name on the product) online and sell standardized product at stores. Another interesting extension could allow both the retailer and manufacturer to offer value added service on both channels, the traditional retailer channel and direct online channel. The value added service through the manufacturer on the direct channel could be modeled by a make-toorder process and the value added service on the traditional retailer channel through the retailer could be modeled by a make-to-stock process and/or a mass customization process. These two approaches of offering variety to the customers, namely, make-to-order and mass customization, could be analyzed and compared. We can expect further research in the mixed channel field with more and more companies adding online stores to their traditional brick-and-mortar stores. We hope that the research methodology and topic presented in this chapter are helpful for future research project in this field.

Appendix Proof of Proposition 1(a) v2 ða p2 þ v þ p1 rÞ pI ¼ p1 d1 þ ðp2 cv Þd2 ¼ p1 ða p1 þ rðp2 vÞÞ þ p2 2 Then we take first order condition with respect to p1, p2 and v, and set them equal to zero, respectively. After that, solving these three equations simultaneously, we can get the desired result. Proof of Proposition 1(b) pFR ¼ ðp2 w Cv Þd2 þ LF ¼ pR 2 2a þ 1 F ) L ¼ pR 4 ( pM ¼

a 1 a2 pR þ þ 4 162 2ð1 rÞ pM

N N

ðaÞ ðbÞ

)

184

S.K. Mukhopadhyay et al.

Setting (a) ¼ (b), we get N¼

1 rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1þr 2a þ 2 4pM þ4pR a2 1r

where

pM þpR

a2 ð1 þ rÞ 4ð1 rÞ

Due to N > 0, we only keep the one with positive value. Proof of Proposition 2(a) The (10)–(12) can be written as: ðN max

mðÞd þ FðNÞ

s:t: _ _ LðÞ ¼ g1 ðÞ; wðÞ ¼ g2 ðÞ;

p_ 1 ðÞ ¼ g3 ðÞ

This is obtained by making the following variable substitution: m : ¼ ðp1 d1 ðpr2 Þ þ wd2 ðpr2 Þ LÞf 2 r r r a 1 w 1 p1 aþ þw þ rwp1 L f ; þ ¼ p1 1 þ 2 2 4 2 4 2 1 a þ rp1 w ðu1 ru2 Þ; g2 ¼ u1 ; g3 ¼ u2 ; þ g1 ¼ 4 2 FðNÞ ¼ pM ð1 FÞ: Using the multiplier equations gives following results: l_ 1 ¼ f

and l1 ¼ f

(13)

_l2 ¼ w þ a þ 1 þ rp1 f þ l1 ðu1 ru2 Þ 2 2 4

(14)

2

_l3 ¼ a 1 þ r þ 2p1 r 1 r þ rw f l1 rðu1 ru2 Þ 2 2 4 2

(15)

Using the optimality conditions gives following results: l1

1 a þ rp1 w þ l2 ¼ 0 þ 4 2

rl1

1 a þ rp1 w þ l3 ¼ 0 þ 4 2

(16)

(17)

Value-Added Retailer in a Mixed Channel

185

Taking derivative on both sides of (16) and using (13), we get 1 a þ rp1 w 1 r u1 f F 2 þ u2 þ l_ 2 ¼ 2 4 2 4 2

(18)

Solving (18) with (14), we get F ¼ fw frp1 22

(19)

Taking derivative on both sides of (17) and using (13), we get l_ 3 ¼ rf

1 a þ rp1 w 1 r u1 þ rF 2 þ u2 þ 2 4 2 4 2

(20)

Solving (20) with (15), we get 2 rF 3r rw ¼ f a þ ar þ þ 2 p 1 2 42 2

(21)

Solving (19) and (21) together, we get desired result pA1 ¼

a F ra ; wA ¼ 2ð1 rÞ 2f 2 2ð1 rÞ

and _ LðÞ ¼ g1 ðÞ ¼

1 a þ rp1 w 1 a þ rp1 w _ ðu1 ru2 Þ ¼ w: þ þ 4 2 4 2

Using the transversality conditions if N is free mðNÞ þ l1 ðNÞg1 ðNÞ þ l2 ðNÞg2 ðNÞ þ l3 ðNÞg3 ðNÞ þ FN ¼ 0 at N we get the following results: ðp1 d1 þ wd2 L pM Þf ¼ 0. Because f 6¼ 0, p1 d1 þ wd2 L pM must equals to 0. The manufacturer can make p1 d1 þ wd2 L pM 0 binding at 1 , 1 ¼ N. Then substitute p1 and w with pA1 (N) and wA (N), we get that L(N)A satisfies

F2 aF F a2 ð1 þ rÞ LðNÞA ¼ pM : þ 2 þ 3 þ 4 2 8N f 4N f 8N f 4ð1 rÞ

0 can be solved by let ðpA2 wA cv A Þd2 þ LA pR binding at 0 .

186

S.K. Mukhopadhyay et al.

Proof of Proposition 3 (i) Manufacturer: Adding pIR þ pIM pAR þ pAM from Proposition 3(ii) and pAR I pR from Proposition 3(iii), we can get pIM pAM . (ii) Retailer: Under (I), the retailer earns her reservation profit through the whole range of . Under (A), it is always higher or equal to her reservation profit. Therefore, pIR pAR for all . As an example, suppose follows a Uniform distribution where F¼

0 1 ; f ¼ : 3 0 3 0

Then, pAR ¼

a 0 þ 2 42

which is decreasing with to a value of pR at ¼ NA. At the same time, profit for the case I is constant at pR for all , so we have pR I pR A . (iii) The supply chain: The supply chain profit under (I) pI ¼

a 1 a2 þ þ 2 2ð1 rÞ 4 16

and the supply chain profit under (A) is pA ¼ pAR þ pAM ¼

a 1 a2 F2 þ þ 4 162 2ð1 rÞ 164 f 2

Obviously, pI > pA. Proof of Proposition 6 Under uniform distribution: d2I ¼

a 1 þ 2 4

and

d2A ¼

a 1 F : So d2I d2A : þ 2 4 4f 2

d1I ¼

a r 2 4

and

d1A ¼

a r rF : So d1I d1A : þ 2 4 4f 2

References Agatz NAH, Fleischmann M, van Nunen JAEE (2008) E-fulfillment and multi-channel distribution – a review. Eur J Oper Res 187(2):339–356 Balasubramanian S (1998) Mail versus mall: a strategic analysis of competition between direct marketers and conventional retailers. Mark Sci 17:181–195

Value-Added Retailer in a Mixed Channel

187

Bannon L (2000) Selling Barbie online may pit Mattel vs. stores. Wall Street J, 17 Nov Brooker K (1999) E-rivals seem to have home depot awfully nervous. Fortune 140(4):28–29 Cachon GP, Lariviere MA (2005) Supply chain coordination with revenue-sharing contracts: strengths and limitations. Manage Sci 51(1):30–44 Cakanyildirim M, Feng Q, Gan X, Sethi S (2010) Contracting and coordination under asymmetric production cost information. http://ssrn.com/abstract¼1084584 Chiang W, Chhajed D, Hess JD (2003) Direct marketing, indirect profits: a strategic analysis of dual-channel supply chain design. Manage Sci 49(1):1–20 Chiang W (2010) Product availability in competitive and cooperative dual-channel distribution with stock-out based substitution. Eur J Oper Res 200(1):111–126 Cohen M, Agrawal N, Agrawal V, Raman A (1995) Analysis of distribution strategies in the industrial paper and plastic industries. Oper Res 43(1):6–18 Collinger T (1998) Lines separating sales channels blur: manufacturers, direct sellers, retailers invade each other’s turf. Advert Age 34, 30 March Corbett C, de Groote X (2000) A supplier’s optimal quantity discount policy under asymmetric information. Manage Sci 46(3):444–450 Corbett C, Zhou D, Tang CS (2004) Designing supply contracts: contract type and information asymmetry. Manage Sci 50(4):550–559 Cotterill RW, Putsis WP (2001) Do models of vertical strategic interaction for national and store brands meet the market test? J Retailing 77(1):83–109 Desiraju R, Moorthy S (1997) Managing a distribution channel under asymmetric information with performance requirement. Manage Sci 43(12):1628–1644 Fay CJ (1999) The direct approach. Best’s Rev 99(11):91–93 Frazier GL (1999) Organizing and managing channels of distribution. J Acad Mark Sci 27 (2):226–241 Fudenberg D, Tirole J (1991) Game theory. MIT, Cambridge, MA Gavirneni S, Kapuscinski R, Tayur S (1999) Value of information in capacitated supply chains. Manage Sci 45(1):16–24 Gilbert A, Bacheldor B (2000) The big squeeze-in suppliers’ rush to sell directly to consumers over the Web, sales agents, distributors, and other channel partners worry that they’ll be pushed out of the picture. Inform Week, 27 March. http://www.informationweek.com/779/channel. htm Ha AY (2001) Supplier-buyer contracting: asymmetric cost information and cutoff level policy for buyer participation. Nav Res Logist 48:41–64 Hann LW (1999) Progressive, zurich show how it’s done. Best’s Rev 99(9):75 Hua G, Wang S, Cheng TCE (2010) Price and lead time decisions in dual-channel supply chains. Eur J Oper Res 205(1):113–126 Janah M (1999) Tech firms slow to use direct web sales. San Jose Mercury News, 24 Feb, C1. http://www.wsj.com Kamien MI, Schwartz N (1981) Dynamic optimization: the calculus of variations and optimal control in economics and management. North Holland, New York Khouja M, Park S, Cai G (2010) Channel selection and pricing in the presence of retail-captive consumers. Int J Prod Econ 125(1):84–95 Lee H, So KC, Tang C (2000) The value of information sharing in a two-level supply chain. Manage Sci 46:626–643 Levary R, Mathieu RG (2000) Hybrid retail: integrating e-commerce and physical stores. Ind Manage 42(5):6–13 Mukhopadhyay SK, Su X, Ghose S (2009) Motivating retail marketing effort: optimal contract design. Prod Oper Manage 18(2):197–211 Mukhopadhyay SK, Zhu X, Yue X (2008a) Optimal contract design for mixed channels under information asymmetry. Prod Oper Manage 17(6):641–650 Mukhopadhyay SK, Yao D, Yue X (2008b) Information sharing of value-adding retailer in a mixed channel hi-tech supply chain. J Bus Res 61:950–958

188

S.K. Mukhopadhyay et al.

Nasiretti R (1998) IBM plans to sell some gear directly to fight its rivals. Wall Street J, 5 June Plambeck EL, Taylor TA (2008) Implications of renegotiation for optimal contract flexibility and investment. Manage Sci 54(12):1997–2011 Su X, Wu L, Yue X (2010) Impact of introducing a direct channel on supply chain performance. Int J Electron Bus 8(2):101–125 Tsay AA, Nahmias S, Agrawal N (1999) Modeling supply chain contracts: a review. In: Tayur S, Ganeshan R, Magazine M (eds) Quantitative models for supply chain management, International series in operations research and management science. Kluwer, Norwell, MA, pp 299–336 Viswanathan S (2000) Competition across channels: do electronic markets complement or cannibalize traditional retailers. International Conference on Information Systems (ICIS) 2000 Proceedings, Brisbane, pp 513–519. http://aisel.aisnet.org/icis2000/51 Yao D, Liu J (2003) Channel redistribution with direct selling. Eur J Oper Res 144:646–658

Capacity Management and Price Discrimination under Demand Uncertainty Using Option Contracts Fang Fang and Andrew Whinston

Abstract This chapter considers the use of option contracts as a price discrimination tool under demand uncertainty to improve supplier profit and supply chain efficiency. Option contracts have long been used to manage demand or supply uncertainty, and the cost of the option is simply considered as the cost to avoid uncertainties. We give an example in a supply chain setting where a supplier has more than one downstream customer with private information. Under such a scenario, our game theoretical model shows that the option price shall be set taking into account the fact that only the customers who are more concerned about the demand uncertainty will purchase. Therefore, the supplier should be able to charge more for a unit of option contract compared to the traditional pricing method where simple expectations are taken. The supplier’s profit improves in three ways. First, the high type customers pay higher marginal prices on average. Second, the high type customers’ demand is satisfied as a first priority, guaranteeing allocation efficiency. Third, the supplier can observe the number of options being purchased and so determine customer types, improving capacity decision efficiency. We compare our results to those of classical second degree price discrimination literature. We show that the use of option contracts guarantee the same level of supplier profit as the level of second degree price discrimination. The overall supply chain efficiency improves to the full information benchmark. Keywords Capacity management • Demand uncertainty • Monopoly revenue management • Option contracts • Price discrimination F. Fang (*) Department of ISOM, College of Business Administration, California State University at San Marcos, 333 S. Twin Oaks Valley Road, San Marcos, CA 92096, USA e-mail: [email protected] A. Whinston Department of IROM, McCombs School of Business, The University of Texas at Austin, 1 University Station B6000, Austin, TX 78712, USA e-mail: [email protected] T.-M. Choi and T.C. Edwin Cheng (eds.), Supply Chain Coordination under Uncertainty, International Handbooks on Information Systems, DOI 10.1007/978-3-642-19257-9_8, # Springer-Verlag Berlin Heidelberg 2011

189

190

F. Fang and A. Whinston

1 Introduction Nowadays, the trend of Globalization has significantly intensified competition among companies. The requirement of effectively managing uncertainties has also been raised to an unprecedent level. Companies are striving to find novel ways to manage any kind of uncertainties. Demand uncertainty, as one of the outstanding example, has attracted a lot of attention in the recent supply chain management literature. A lot of attention has been focused on how to improve forecasting accuracies by sharing information among supply chain partners to reduce the uncertainty (e.g. Cachon and Fisher 2000; Chen 2003; Guo et al. 2006; Li 2002; etc.). In this chapter, we will discuss an innovative method of using option contracts for demand forecast and profit management. Real options contracts has been studied in supply chain management literature as a tool to protect risk-averse partners from potential uncertainties, such as demand and material cost changes (see e.g., Huchzermeier and Cohen 1996). In an incomplete contract setup, option contracts can also improve contracting efficiency by solving the famous hold-up problem (see e.g. Noldeke and Schmidt 1995). In this chapter, we explore using option contracts as a price discrimination tool under demand uncertainty. In the classic economics literature, price discrimination relies on the supplier’s ability to determine customers’ different levels of willingness to pay and hence to charge different prices. When customer types are not observable, the supplier can offer a variety of products with different prices for the customers to choose from. This practice is known as second degree price discrimination (e.g. Tirole 1988). In a classical second degree price discrimination model, the monopoly knows that the customers’ have different valuations over the quality (in some other applications, the bundled quantity) of the product, hence produces a product with two different qualities (or two bundles with different quantities) and charges different prices. The customers with different types will then have to select the product quality/quantity bundles that meet their needs the best. We extend the classical model in a setting where the customers’ demand quantity is uncertain and the supplier’s capacity is tight. In addition, the customers do not obtain higher valuation from the quantity satisfied that exceeds their own need. Example of such scenario could be when customers may want to purchase a certain number of tickets to a game and will not be able to use the extra tickets they do not need. Other cases can be found in the network/telephone service, when a customer may need to send out a certain size of email or text message. Additional network traffic would be of no use. In both cases, demand might be uncertain ex ante when customers cannot decide whether they need to go to the game or to send the email. Under such scenarios, quantity bundling strategy used in traditional second degree discrimination is not feasible. This is because the customers’ desired quantities vary according to their realized demand and would not benefit if obtain a bundle with superior quantity.

Capacity Management and Price Discrimination under Demand Uncertainty

191

In this chapter, we propose using a new form of option contracts to solve such an issue. The main reason that the option contracts would be a good device to use is because of the capacity constrain. In situations when capacity may be tight, customers have an incentive to hedge the risk when their demand cannot be satisfied. The hedging incentive is higher for those customers with higher willingness to pay. Under such circumstance, we suggest the supplier sells a form of option contracts to the customers. Executing one unit of the option contracts guarantees a customer the availability of one unit of demand and meets customers’ hedging incentive. To distinguish customers with different types, we propose that the supplier should price the option contracts in such a way that only those customers with higher willingness to pay will buy the options. The supplier can take into account the fact that those customers who have purchased the options have a higher willingness to pay to the product/service and can charge a higher option price to those customers. Under such a pricing scheme, the high type customers (who will purchase the options) will self-select to pay more than their low type counterparts (who will not purchase the options) to ensure execution of their demand. The supplier would then be able to identify the customers’ types from the purchase of the option contracts. To demonstrate the effectiveness of the option contracts as a price discrimination tool, we present a game theoretic model where one monopolistic supplier or service provider (he) faces two customers each with uncertain future demand. Each customer (she) has private information about their willingness to pay for one unit of the future demand. The supplier has to build capacity before the customers’ demand is realized. Since capacity may be insufficient for the highest level of demand realization, customers suffer from potential demand losses. When demand exceeds capacity, the supplier can only serve the demand randomly. Option contracts can be adopted in the following manner. The supplier opens the option market to the customers before the capacity investment and uncertain demand realization. At that time, customers can purchase options with unit price po. After observing customers’ option purchase decisions, the supplier invests in capacity to prepare for future uncertain demand. Afterwards, the customers find out their actual demand, observe the supplier’s capacity and decide whether to exercise their options. Alternatively, the customers can submit regular demand. Each customer pays a strike price, pe, for each unit of option executed. The amount of demand protected by the options (referred to as the “option demand” in the context) will be satisfied as the first priority. The remaining demand will be satisfied at a unit price p if there is leftover capacity. Our proposed framework improves the supplier’s revenue in three ways. First, customers with a higher willingness to pay will pay a higher unit price when the capacity is tight, increasing the supplier’s overall revenue. Second, the customers’ option demands are executed as a first priority. The remaining demand will be executed only when there is extra capacity. Third, customers’ options purchases reveal their types. This knowledge allows the supplier to more efficiently adjust the capacity levels, better accommodating the potential demand. The last two effects

192

F. Fang and A. Whinston

also improve the supplier’s decision efficiency, leading to an enhanced overall social welfare. In order to successfully induce high type customers into purchasing options, the supplier needs to convince the customers that the capacity could be insufficient. If a supplier is able to change capacity after the options are purchased, however, it may undermine the customers’ initial incentives to purchase them. This is because customers know that the supplier will want to guarantee enough capacity to meet the demand, so as to maximize the revenue by serving as much customer demand as possible. Knowing their own types and the supplier’s capacity cost, a rational customer can always conjecture the expected capacity level that the supplier will invest in contingent on their counterpart’s type. They can therefore calculate the overall benefit of purchasing the options based on these “rationally expected” capacity levels. The supplier cannot mislead the customers. However, the capacity investment levels could be different if the supplier adopts different option contracts (e.g. different po and pe). Hence, the supplier’s pricing strategy of those option contract is critical in our framework. The supplier can decide not to build sufficient capacity to guarantee the execution of the entire option demand. The decision depends on what commitment the supplier makes in the option contracts when he fails to meet the option demand. If the supplier does not compensate unsatisfied customers, the customers will have less incentive to purchase the options. This disappoints the high type customers and reduces their valuation of the options. If the supplier promises too high a compensation, he has to always ensure to overinvest in capacity. This may be inefficient especially when the capacity cost is high. In this paper, we suggest the supplier offer an option buy-back price as a compensation mechanism which leaves the high type customers indifferent between whether to execute their options or to sell them back to the supplier. This buy-back price scheme reduces the high type customers’ strategic decision of exercising the options and induces the supplier’s efficient capacity investment which maximizes ex ante social welfare. Our option framework has the potential to improve revenue management in many industries wherever demand uncertainty and information asymmetry exist. One potential application is in network traffic management. Since business communication relies heavily upon emails and video conferences, network congestion can cause severe economic losses. Companies are willing to pay a premium more than what the regular users pay to ensure important business emails being delivered promptly. Another example application to use the options framework is in the ticket sale business. Many people want to go to a concert or a game but face the risk of not being able to attend ex post. They do not want to pay the full price for the tickets in advance because they do not want to waste money if they are unable to attend. However, if they wait too long, the tickets could be sold out. The option contracts are a good choice for these people. Similar applications can be implemented in airline ticket management, hospital facility management and the hotel reservations business. In hotel reservation business, customers who care more about getting the hotel room can make reservations

Capacity Management and Price Discrimination under Demand Uncertainty

193

before arrival. It bears resemblance to the option framework. However, in practice, reservation is often free (i.e. the option price po is zero). This chapter provides an analytical framework to determine the option price. Supply chain partners can also consider using the option contracts when coordinating with each other. Although some research in supply chain management has proposed using options contracts for risk hedging and improve downstream partners’ ordering efficiency, there has been no study when suppliers have incomplete information about downstream partners’ types. Our proposed framework filled the blank by suggesting that the supplier offer different pricing schemes and allow the downstream partners (e.g. retailers) to self select the type of contracts they prefer. Retailers who are in the regions where the product is more popular would be willing to pay more than other retailers. The supplier should take such fact into consideration when determining the prices of the option contracts, rather than applying traditional options evaluation equations widely used in finance literature. In the options framework, the supplier employs price discrimination by extending the customers’ decision problem into an intertemporal one. That is, the customers have to decide whether and how many options they should purchase to hedge the future risk of demand loss before their demand realization and the supplier’s capacity investment. To make the decision, they have to figure out the capacity level the supplier may invest in and the possibility when they will use the options. When they evaluate the options, rational customers also take into account the fact that their option purchase reveals their types to the supplier and affects supplier’s the capacity level. To illustrate the implementation of option contracts, we present a two-period game-theoretic model. The model has one monopolistic supplier and two potential customers with private valuation of the service. Figure 1 shows the time line of events.

Fig. 1 The time line of the base model

194

F. Fang and A. Whinston

In the first period – the contracting period – the monopolistic supplier announces the option purchase price, po, and the strike price, pe, to maximize his expected overall profit. Then each customer decides the number of options to purchase, Oi, according to their types. At the beginning of the second period – consumption period – the supplier observes the customers’ options purchases and decides on an optimal capacity investment, K. Afterwards, the customers demands, Di, are realized. Each customer decides how many options they are going to exercise (Doi ) based on the observation of the aggregate demand and the capacity level. The supplier satisfies the option demand as a first priority. If Do > K, some of the options cannot be executed and the supplier will compensate the customers. If Do < K, the extra capacity will be used to satisfy the remaining regular demand. The rest of this chapter is organized as follows. A brief literature review is provided in Sect. 2. Section 3 presents the game model and Sect. 4 analyzes the equilibrium strategies of the supplier and two customers. We compare the model outcome to two benchmark cases. In one of the benchmarks, the supplier cannot distinguish the customer types at all. In the other, the supplier can determine each customer’s type and can charge different prices to each of them. In Sect. 5, we discuss the implications and extensions of our model. Section 6 concludes the paper.

2 Literature Review This chapter discusses the use of the options contract for price discrimination under demand uncertainty. The idea of using options contract in supply chain coordination is not a new idea. Cachon (2002) and Lariviere (1999) have surveyed a variety of supply chain contracts, particularly the options contract that induce the supply chain partners to take the right actions under different circumstances. Sethi et al. (2004) formulate a multi-period model to study procurement strategy under an option-like quantity flexible contract with spot market purchasing opportunity. Kleindorfer and Wu (2003) integrated the use of options with operational decisions, such as capacity, technology choice, and production to improve the profitability in supply chain coordination and risk management. In addition, research has also shown that the quantity flexible contracts, buy-back contracts, and pay-to-delay contracts are special cases of a combination of a priceonly contract and a call-option contract (e.g. Cachon and Lariviere 2001; BarnesSchuster et al. 2002; Martı´nez-de-Albe´niz and Simchi-Levi 2005; etc.). Lin and Kulatilaka (2007) introduce an real options approach to evaluate companies’ high-risk investments, e.g. IT outsourcing decisions, IT procurements, etc. Our framework moves one step further to discuss the possible use of option contracts in price discrimination. There has been a rich economics literature on price discrimination. (e.g. Varian 1985; Armstrong and Vickers 2001; Corsetti and Dedola 2005; Mortimer 2007) An

Capacity Management and Price Discrimination under Demand Uncertainty

195

extensive discussion on price discrimination can be found in Tirole (1988) and Mas-Colell et al. (1995). Maskin and Riley (1984) discussed optimal mechanism design that can induce different agents to report truthfully their types. Their model primarily concerns a principle who is trying to allocate limited resources to a number of agents. The assumption was that the resource is rare and demand uncertainty is not an issue. The outstanding result shows that the lowest type of agent will be left no surplus and the higher type agents enjoy certain level of surplus, which is necessary for them to report their true information. Such surplus is called “information rent” in the literature of second degree price discrimination. Deshpande and Schwartz (2002) extend the mechanism into a constrained capacity setup. In this mechanism, non-linear pricing rules are adopted to guarantee incentive compatibility. Boyaci and Ray (2006) discussed the impact of capacity costs on product differentiation in a product delivery model. In our framework, the demand is uncertain so that the allocation for the two types of customers cannot be predeterimined. In this sense, Maskin and Riley’s mechanism cannot be directly applied to solve our problem. An alternative pricing scheme is the spot pricing scheme, which suggests that the price should be dynamically adjusted according to the congestion levels. Gupta et al. (1996, 1999) suggests using priority classes with different spot prices to efficiently allocate network resource. The idea is that the customers with higher delay costs will choose to pay more and send their demand to the network with lower expected delay time. Afeche (2006) suggests to add strategic delay in the queue to further discriminate customers and maximize the revenue of the supplier. However, under the spot pricing mechanism, customers decide whether to execute the demand when observing the spot price. The supplier cannot discover the customers’ type distribution before adjusting the capacity. In addition, consumers will face additional uncertain future spot price due to the different realizations of future. In practice, the spot price is not preferred by both individual consumers and companies due to the management difficulties.

3 Model Development A monopolistic supplier sells to two risk neutral customers (i ¼ 1,2). Each customer can be one of two unknown types. A “high” type customer enjoys higher marginal utility from the satisfied demand than a “low” type customer. Specifically, customer i receives total utility ui ¼ vti Dei mi if she is of type ti ∈ {l, h}. vti represents the customer’s marginal value of satisfied for type ti and vh > vl. Dei is customer i’s demand being satisfied by the supplier, and mi is the total monetary transfer from customer i to the supplier. Each customer knows her own type ti but does not know for certain the other’s type. The supplier can observe neither of the customers’ types. The common belief is that ti ¼ h with probability l ∈ (0,1) and ti ¼ l with probability 1 l. The realizations of t1 and t2 are independent.

196

F. Fang and A. Whinston

Each customer’s demand, Di, is uncertain and could be either DH (with probability a) or DL (with probability 1 a). The realization of (D1, D2) is independent of customers’ type realization (t1, t2). We denote D ¼ ðD1 ; D2 Þ as the demand vector and D ¼ D1 + D2 as the aggregate realized demand. To avoid the trivial case when the supplier will only concern the high demand situation, we assume that DH < 2DL. After observing the demand realization Di, each customer decides how much of the demand should be submitted to the supplier, which we denote as Dsi . In this model, we restrict Dsi bDi , implying that the customer cannot submit a demand higher than the realization of their actual demand. This restriction makes sense when the demand can be verified ex post. Taking the example of network traffic management, customer sends out files with certain sizes. A customer can increase the demand by extending the size of the file. However, she cannot benefit from doing so. Failure to impose the above restriction introduces customers’ strategic behavior when submitting their demand. Cachon and Lariviere (1999) analyzed this kind of strategic behavior under different allocation rules. They show that customers’ order inflation could be an equilibrium strategy and the supplier is worse-off due to the concern of such strategic behavior. However, this is not the focus of our paper. To serve the customer demand, the supplier has to invest in a certain level of capacity K before the demand, D is realized. The marginal cost of capacity investment is c0. Observing the customers’ submitted demand Ds ¼ ðDs1 ; Ds2 Þ, the supplier decides how much demand to execute for each customer, De ¼ ðDe1 ; De2 Þ, ðDei bDsi for i ¼ 1; 2Þ to maximize his expected revenue. The total amount of executed demand, De ¼ De1 þ De2 is constrained by the supplier’s capacity K. To avoid trivial results, we will impose the following three assumptions: Assumption 1. c0 > minfð2a a2 Þvl ; a2 vh g: This condition guarantees that the supplier’s capacity cost is non-trivial so that the supplier cannot simply decide to invest in the highest capacity level possible to avoid congestion. Violation of this condition will result in the trivial case where the customers do not expect any congestion and hence will not purchase any option. Assumption 2. c0 < ð2a a2 Þvh : This condition guarantees that the capacity cost is not prohibitively high so that the supplier can make a profit. Otherwise, the supplier will always choose to invest qﬃﬃﬃﬃﬃﬃﬃﬃﬃ in zero capacity level. h 0 Assumption 3. l < 1 v vc h . This condition reduces our focus to the case where the chance of having a high type customer is small enough so that the supplier will not ignore the possible existence of low type customers. Assuming the supplier is risk neutral and there is no cost for executing the customers’ demand, the potential supplier’s profit P is calculated as P ¼ m1 þ m2 c0 K: We propose that the supplier can use option contracts to manage congestion (i.e. in the event that D > K). When capacity is insufficient to meet the aggregate

Capacity Management and Price Discrimination under Demand Uncertainty

197

demand D, customers can choose to execute their options, guaranteeing that their demand be satisfied. One unit of the option contract guarantees the customer one unit of satisfied demand regardless of congestion. To implement price discrimination, the options are priced such that only high type customers will buy the options because they will suffer more from demand loss than low type customers. By using the options, high type customers can avoid demand loss but may pay a higher unit price on average. In addition, low type customers’ demand will be executed at a lower priority, increasing their chance of facing a demand loss. The supplier benefits from using the option contracts since he can, in effect, charge the high types a higher fee and is able to adjust capacity after observing customers’ actual types. From a social optimal perspective, the allocation efficiency is always guaranteed since those demands with higher marginal value always receive first priority execution.

4 Analysis In order to demonstrate the supplier’s revenue improvement and the overall supply chain efficiency increase, we first introduce two benchmark models to compare the result of the proposed options framework. The first benchmark outlines the problem when the supplier does not have the ability to distinguish customers’ types. He also have to invest in a certain capacity level before the demand uncertainty is resolved and can only charge a linear price, p, for each unit of executed demand. Our proposed options framework should be able to improve situations outlined in the first benchmark. For one, options contracts can help supplier identify the high type customers, if any. The high type customers, with a higher average unit price paid to the supplier, can be guaranteed demand satisfaction when capacity is tight. The second benchmark model is an ideal situation where the supplier has full information. That is, the supplier can observe the customers’ types before the capacity investment. The supplier can also charge different prices ph and pl to the high types and low types, respectively. In this case, allocation efficiency is guaranteed since the supplier always satisfies demand from high type customers first and uses the remaining capacity (if there is any) to serve the low type customers. In addition, the supplier can set the prices that leave no surplus to both types of customers. The capacity investment is efficient in this case since the supplier maximizes his own profit as well as the social welfare. Our options framework, with revenue and efficiency improvement from benchmark 1, would hope to be able to achieve efficiency level demonstrated in the second benchmark. Section 4.3 analyzes the option framework and provide results. In all three frameworks, we examine both the supplier’s expected profit EP and the overall supply chain efficiency W, defined as the sum of the supplier’s expected profit and the expected utilities of both customers. We then compare the outcomes to demonstrate the effectiveness of the proposed options framework.

198

4.1

F. Fang and A. Whinston

Benchmark I: No Discrimination Case

In this section, we examine the case where the supplier can only charge a single linear price p to the customers regardless of their types. In this case, the customers will only be able to submit their demand as regular demand, rather than the option demand. In addition, there is no reason why they will submit their demand fewer than the realized demand. Therefore, we have Dsi ¼ Di for both i ¼ 1,2. However, since the capacity might be constrained, the total satisfied demand would be the minimum between the supplier’s capacity level and the total submitted demand. That is, De ¼ min{K, Ds}. Each customer is charged mi ¼ pDei and the supplier’s profit is P(p,K) ¼ pDe c0K. In this case, we have Dsi ðpÞ

¼

Di 0

if pbvti otherwise:

To decide an optimal price level, the supplier have several alternatives. By charging a price higher than vl, the supplier only serves the high type customers. The supplier can enjoy a higher marginal profit from each unit of demand, but loses business if a customer is of low type. If the price is lower than or equal to vl, both types will be served. The following Proposition 1 concludes that the supplier should serve both types of customers by charging a price p ¼ vl under the conditions we discussed in Sect. 3. The proposition also derives the optimal capacity ND , the supplier’s expected profit EPND, and the overall efficiency decision K ND . The superscript ND represents the no discrimination W ND ¼ E PND þ uND 1 þ u2 case. Proposition 1. Assume that the supplier can only charge a unit price to both customers. The optimal unit price pND ¼ vl and the capacity KND ¼ 2DL. Both types of customers will submit their demand and the supplier’s profit is EPND ¼ ðvl c0 Þ2DL : The overall efficiency is W ND ¼ lvh þ ð1 lÞvl c0 2DL : Note. The proofs of all the lemmas and propositions are provided in Appendix 2. Proposition 1 provides a benchmark outcome when the supplier is least capable of identifying customer types and charge different pricing schemes. Under such conditions, the supplier is conservative in his capacity investment decision and the capacity level is only enough for the least possible level of aggregate demand, K* ¼ 2DL ¼ inf{D}. The low type customers are left no surplus and the high type customers can make strictly positive surplus which equals the product of the difference in the marginal utility of these two types, vh vl, and the capacity 2DL.

Capacity Management and Price Discrimination under Demand Uncertainty

199

In this case, the supplier’s profit (vl c0)2DL equals the surplus it can extract as if the customers are all low type and the demand is always DL. Therefore, the first benchmark is an inefficient case where the supplier cannot take advantage of the fact that there is possible demand increase and higher willingness to pay. In order to improve this situation, the supplier needs to seek effective ways for revenue management and price discrimination.

4.2

Benchmark II: Full Information Case

In the full information benchmark case, we assume that the supplier can distinguish the different types of the customers before the capacity is invested and charge different prices according to the customer types. We use superscript FI to indicate the “full information” case. It is straightforward to show that the optimal prices will be pFI(t ¼ h) ¼ vh and FI p (t ¼ l) ¼ vl. In this case, since the customers only have one venue to submit their demand, the submitted demand Dsi equals to their realized demand Di, for i ¼ 1,2. Under such prices, the supplier leaves no surplus to both types of customers and extracts the maximum profit he can get under capacity constraint KFI. When the capacity is insufficient to meet the total submitted demand Ds ¼ Ds1 þ Ds2 , the supplier will satisfy the high type customers’ demand first to obtain a higher margin. The other decision the supplier has to make is on the capacity level. Since the supplier already knows the types of the customers, the capacity level KFI will then be different according to different customers’ types (t1, t2). Due to symmetry, we have KFI(h,l) ¼ KFI(l,h). Lemma 1 summarizes this optimal capacity level. Lemma 1. In the full information benchmark case, the capacity investment decision is made contingent on the actual types of the two customers. KFI ðt1 ; t2 Þ ¼

DH þ DL 2DL

if t1 ¼ t2 ¼ h otherwise:

The result of Lemma 1 shows that the supplier is able to set a higher capacity level when both customers are of high type. However, when at least one customer is of low type, the supplier will maintain a relatively low capacity level. This is because the supplier is able to identify and charge different prices to customers and to allocate the capacity to serve high type customers first when capacity is tight. When there is at least one low type customer, only the low types who pays a lower unit price suffer. The supplier is expecting less revenue loss which cannot motivate him to increase capacity investment. Proposition 2. When the supplier can observe customer types before making a capacity investment and charges different prices accordingly, the expected supplier

200

F. Fang and A. Whinston

profit equals the overall efficiency. That is, EPFI ¼ WFI. Comparing with EPND and WND yields

EPFI ¼ EPND þ D1 þ D2 þ D3 W FI ¼ W ND þ D1 þ D3

where: 1. D1 ¼ 2lð1 lÞaðDH DL Þðvh vl Þ; 2. D2 ¼ ðvh vl Þl2DL and 3. D3 ¼ l2 ðð2a a2 Þvh c0 ÞðDH DL Þ: It can be shown that D1, D2, and D3 are all strictly positive. Therefore, the results that EPFI > EPND and WFI > WND always hold. Proposition 2 identifies the three sources where the supplier gains higher profit: 1. The profit gain from the supplier’s ability to prioritize the customers’ demand when capacity is tight, represented by D1. 2. The profit gain from the supplier’s ability to charge different prices according to different customer types, represented by D2. 3. The profit gain from the supplier’s ability to invest in capacity levels according to the actual realization of the customer types, represented by D3. The efficiency gain comes in only two parts and does not include D2. This is because the supplier’s ability to charge the high type customers a higher price only affects the monetary transfer among the three supply chain parties. It does not change overall efficiency when adding the profits of all three parties.

4.3

Option-Capacity Game

In this section, we discuss the framework in which the supplier offers the customers a form of option contracts to hedge their demand risk. The customers choose the number of options to purchase at a unit price po before the supplier builds his capacity. The supplier will then observe the number of options purchased by each customer, conjecture the customers types, and decide how much capacity he should build to meet the future demand. After the capacity investment and demand realization, customers observe the supplier’s capacity and the aggregate demand and decide how many options to execute, if they have purchased any. The number of options a customer chooses to execute is called the option demand and is denoted as Doi . For each unit of option demand executed, the customers pay a unit execution price pe. We assume that the customer cannot execute more options than the actual demand, that is Doi bDi . If Doi < Di , the customer’s remaining demand Di Doi will be satisfied randomly. A unit price p is charged if the remaining demand is satisfied.

Capacity Management and Price Discrimination under Demand Uncertainty

201

To successfully discriminate the customers, the supplier must set the option contract parameters (po, pe) in such a way that only high type customers will buy them. The supplier also sets a unit price p ¼ vl for the regular demand. If the aggregate option demand, Do ¼ Do1 þ Do2 , exceeds the capacity K, the supplier needs to buy back some of the options. The option buy back price has to be high enough so that the high type customers are willing to sell it back. Meanwhile, it cannot be too high to make the customers want to sell all the options back instead of executing them. Thus, the buy-back price, pb, should equal vh pe, the marginal benefit the high type customers get from the demand being executed. According to the time line in Fig. 1, the strategic interactions among the supplier and the two customers can be described in a three-stage game. In the first stage, the supplier announces the option contract parameters, po and pe. The customers simultaneously decide the number of options, Oi, to buy based on their own types. We are interested in the equilibrium cases where Oi ðt ¼ lÞ ¼ 0 and Oi ðt ¼ hÞ > 0. With our assumptions on the customer demand, the high type customers have actually two choices: whether to buy Oi ¼ DL for a minimal hedge or to buy Oi ¼ DH for a maximal hedge. The minimum hedge strategy, buying DL units of options, guarantees the customers to execute at least the minimal demand level, when capacity is tight. In our setup, the customers’ demand equals to DL with probability 1 a. When a is small, the customer may consider minimum hedging because the chance of its demand exceeds DL is small. However, the customers also have to consider the real chance when the option is executed. For example, when a is small, the other customers’ demand is also unlikely to be high, so the possibility that the capacity is tight is also very low. Therefore, the customers may not have a strong incentive to choose minimum hedge strategy. Maximum hedge strategy guarantees the customers’ ability to execute all the future realized demand. However, the customers may not need to use all of the options purchased. The option price must be properly set by the supplier to induce the high type customers purchase options with an appropriate hedge strategy. The supplier also need to calculate which customer hedge strategy is the most profitable to him. If the option price is set properly and only the high type customers will purchase them, then the supplier can observe (t1, t2) through the sale of the options and decide the capacity level, K(t1, t2), to maximize his expected profit in the following period, Ep(K). After the demand is realized, the customers simultaneously decide how much demand to submit to the supplier, denoted as Dsi , and how much of Dsi is submitted as option demand, Doi . When the regular price p ¼ vl, Dsi ¼ Di hold for both types of customers. The supplier gathers the total demand (Do, Ds) and decides how to allocate the constrained capacity with the priority of the option demand. Assume that the supplier executes Dei amount of demand from the customers, then if Dei > Doi , the additional demand Dei Doi will be executed as the regular demand at a unit price p ¼ vl. If Dei < Doi , the supplier did not satisfy all the option demand,

202

F. Fang and A. Whinston

and therefore has to buy back the additional option demand Doi Dei at the option buy back price pb ¼ vh pe. The total charge to a customer will be mi ¼ po Oi þ pe Doi þ vl ðDei Doi Þþ vh ðDoi Dei Þþ ; where the last term is the customer’s expected compensation if the option demand is not executed. For the low type customers, since they won’t buy any option in equilibrium, we can simplify the total charge by substituting condition Oi ¼ Doi ¼ 0 and obtain mi ðt ¼ lÞ ¼ vl Dei . The customer’s overall utility: ui ¼

ðvh pe ÞDoi þ ðvh vl ÞðDei Doi Þþ po Oi 0

if t ¼ h if t ¼ l

In the following Sects. 4.3.1–4.3.3, we use backward induction to solve the three-stage game.

4.3.1

The Consumption Period

In period 3, each customer observes her own demand, which could be either DH or DL . After observing the aggregate demand, D, and the capacity, K, the two customers simultaneously decide how many options to execute. Denote O ¼ ðO1 ; O2 Þ. There are six configurations that need to be discussed: O ¼ ðDH ; DH Þ; ðDH ; 0Þ; ðDL ; 0Þ; ðDL ; DL Þ; (0,0) and (DH ; DL Þ. Due to symmetry, we do not need to analyze the cases of (DL, DH), (0, DH) and (0, DL). On the equilibrium path, only the high type customers will buy options. Therefore, supplier will infer that both customers are of high type when O equals (DH, DH), (DL, DL), or (DH, DL). Configurations (DH, 0) and (DL, 0) indicate that only one customer is high type, and configuration (0, 0) indicates both customers are of low type. For the low type customers, it is straightforward that Doi ¼ 0 since they don’t have any options. For the high types, Doi is decided based on the following optimization problem: max

Doi b minfOi ;Di g

ðvh pe ÞDoi þ ðvh vl ÞðDi Doi Þ f

ðK Doi Doi Þþ s:t: f ¼ min 1; D Doi Doi

where Doi is the option demand submitted by the other customer. Doi ¼ 0 if that other customer is of low type. f indicates the probability that regular demand is satisfied. When D < K, there is no congestion and all the demand will be satisfied, f ¼ 1. When Do > K, the option demand along will exceeds the capacity. The supplier does not have additional capacity to serve the regular demand and hence f ¼ 0.

Capacity Management and Price Discrimination under Demand Uncertainty

203

Lemma 2. Denoting D o as the solution set of the above optimization problem, we have Do f0; minfDi ; Oi gg. Lemma 2 suggests that a high type customer will either execute all the options she has purchased up to her realized demand or not execute the options at all, depending on the option execute price pe. As pe increases, the customer pays more on executing the options and hence is increasingly reluctant to do so. Tables 1–6 show the solutions of Doi as functions of the six different realizations of O, respectively. In each table, Doi also varies according to different values of the realized demand D, the option exercise price pe and the available capacity K. From the results summarized in Tables 1–6, we can observe that the amount of options a high type customer will execute decreases as the option execution price pe increases. When pe rvh ðvh vl Þ2DKH , no option contracts will be executed in any possible configuration of O and the demand realization D. This is because that the options are too expensive to execute. Therefore, the options have no value. We can then conclude that the supplier will never charge such an option execution price. In the following discussion, we focus on the cases where the execution price pe < vh ðvh vl ÞDKH when the option contracts will possibly be executed.

Table 1 Option demand when O ¼ ðDH ; DH Þ O ¼ ðDH ; DH Þ

vh ðvh vl Þ minf2DKH ; 1g < pe bvh vh ðvh vl Þ minfDH KþDL ; 1g < pe bvh ðvh vl Þ minf2DKH ; 1g h K v ðvh vl Þ minf2D L ; 1g < h h l pe bv ðv v Þ minfDH KþDL ; 1g K pe bvh ðvh vl Þ minf2D L ; 1g

Table 2 Option demand when O ¼ ðDH ; DL Þ O ¼ ðDH ; DL Þ vh ðvh vl Þ minf2DKH ; 1g < pe bvh vh ðvh vl Þ minfDH KþDL ; 1g < pe bvh ðvh vl Þ minf2DKH ; 1g K vh ðvh vl Þ minf2D L ; 1g < pe bvh ðvh vl Þ minfDH KþDL ; 1g K vl < pe bvh ðvh vl Þ minf2D L ; 1g

p e b vl

D ¼ ðDH ; DH Þ Do1 ¼ 0 Do2 ¼ 0 Do1 ¼ DH Do2 ¼ DH Do1 ¼ DH Do2 ¼ DH Do1 ¼ DH Do2 ¼ DH

(DH,DL)

D ¼ ðDH ; DH Þ Do1 ¼ 0 Do2 ¼ 0 Do1 ¼ DH Do2 ¼ 0

(DH,DL)

Do1 ¼ DH Do2 ¼ 0 Do1 Do2 Do1 Do2

¼ DH ¼0 ¼ DH ¼ DL

Do1 Do2 Do1 Do2 Do1 Do2 Do1 Do2

¼0 ¼0 ¼0 ¼0 ¼ DH ¼ DL ¼ DH ¼ DL

(DL,DH) Do1 Do2 Do1 Do2 Do1 Do2 Do1 Do2

¼0 ¼0 ¼0 ¼0

¼0 ¼0 ¼0 ¼0

Do1 Do2 Do1 Do2

Do1 Do2 Do1 Do2 Do1 Do2

¼ DH ¼ DL ¼ DH ¼ DL ¼ DH ¼ DL

Do1 Do2 Do1 Do2 Do1 Do2

¼0 ¼0 ¼0 ¼0 ¼0 ¼0

¼ DL ¼ DH ¼ DL Do1 ¼ DL ¼ DH Do2 ¼ DL

(DL,DH)

Do1 Do2 Do1 Do2

(DL,DL)

Do1 ¼ DL Do2 ¼ 0

Do1 Do2 Do1 Do2 Do1 Do2

¼0 ¼0 ¼0 ¼0 ¼0 ¼0

¼ DL ¼0 ¼ DL ¼ DL

Do1 Do2 Do1 Do2

¼ DL ¼0 ¼ DL ¼ DL

Do1 Do2 Do1 Do2

¼0 ¼0 ¼0 ¼0

(DL,DL)

204

F. Fang and A. Whinston

Table 3 Option demand when O ¼ ðDH ; 0Þ O ¼ ðDH ; 0Þ v ðv v h

h

l

Þ minf2DKH ; 1g < pe

bv

D ¼ ðDH ; DH Þ Do1 ¼ 0 Do2 ¼ 0 Do1 ¼ DH Do2 ¼ 0

h

vh ðvh vl Þ minfDH KþDL ; 1g < pe bvh ðvh vl Þ minf2DKH ; 1g l v < pe bvh ðvh vl Þ minfDH KþDL ; 1g

Do1 Do2 Do1 Do2

p e b vl

Table 4 Option demand when O ¼ ðDL ; DL Þ O ¼ ðDL ; DL Þ D Þ h vh ðvh vl Þ minfDKþ2ðD H þ2ðD H DL Þ; 1g < pe bv H

L

vh ðvh vl Þ minfDH KþDL ; 1g <

D Þ pe bvh ðvh vl Þ minfDKþ2ðD H þ2ðDH DL Þ; 1g H

L

vl < pe bvh ðvh vl Þ minfDH KþDL ; 1g

Table 5 Option demand when O ¼ ðDL ; 0Þ O ¼ ðDL ; 0Þ 1

vh ðvh vl Þ minf2

KDL D2L þDH DL DL ð2DH DL Þ ; 1g < pe

bvh

vh ðvh vl Þ minfDH KþDL ; 1g <

1 KDL D2 þDH DL

pe bvh ðvh vl Þ minf2 DL ð2DHL DL Þ ; 1g þD D D vh ðvh vl Þ minfKD ðD H þDL ÞDH L

b b

H

H

L

DL

; 1g <

pe v ðv v Þ minfDH KþDL ; 1g L þDH DH DL DL ; 1g vl < pe vh ðvh vl Þ minfKD ðD H þDL ÞDH h

h

l

p e b vl

Table 6 Optimal demand when O ¼ ð0; 0Þ O ¼ ð0; 0Þ D ¼ ðDH ; DH Þ For all pe bvl

Do1 ¼ 0 Do2 ¼ 0

D ¼ ðDH ; DH Þ Do1 ¼ 0 Do2 ¼ 0 Do1 ¼ DL Do2 ¼ DL Do1 Do2 Do1 Do2

p e b vl

¼ DH ¼0 ¼ DH ¼0

¼ DL ¼ DL ¼ DL ¼ DL

(DH,DL) Do1 Do2 Do1 Do2 Do1 Do2 Do1 Do2

¼0 ¼0 ¼0 ¼0 ¼ DH ¼0 ¼ DH ¼0

(DH,DL) Do1 Do2 Do1 Do2 Do1 Do2 Do1 Do2

¼ DL ¼0 ¼ DL ¼0

(DL,DH)

Do1 Do2 Do1 Do2 Do1 Do2 Do1 Do2

¼0 ¼0 ¼0 ¼0 ¼0 ¼0 ¼ DL ¼0

(DL,DL)

¼0 ¼0 ¼0 ¼0

Do1 Do2 Do1 Do2

¼0 ¼0 ¼0 ¼0

¼ DL ¼ DL ¼ DL ¼ DL

Do1 Do2 Do1 Do2

¼ DL ¼ DL ¼ DL ¼ DL

Do1 Do2 Do1 Do2

¼0 ¼0 ¼ DL ¼ DL

(DH,DL)

Do1 ¼ DL Do2 ¼ 0

Do1 ¼ 0 Do2 ¼ 0 Do1 Do2 Do1 Do2

¼0 ¼0 ¼0 ¼0

(DL,DL)

¼0 ¼0 ¼0 ¼0

Do1 Do2 Do1 Do2

¼ DL ¼0 ¼ DL ¼0

Do1 Do2 Do1 Do2 Do1 Do2 Do1 Do2

Do1 Do2 Do1 Do2

D ¼ ðDH ; DH Þ Do1 ¼ 0 Do2 ¼ 0 Do1 ¼ DL Do2 ¼ 0

Do1 Do2 Do1 Do2

(DL,DH)

¼0 ¼0 ¼0 ¼0

¼ DL ¼0 ¼ DL ¼0

(DL,DH) Do1 Do2 Do1 Do2

¼0 ¼0 ¼0 ¼0

(DL,DL) Do1 Do2 Do1 Do2

¼0 ¼0 ¼0 ¼0

Do1 ¼ DL Do1 ¼ 0 Do2 ¼ 0 Do2 ¼ 0 Do1 Do2 Do1 Do2

¼ DL ¼0 ¼ DL ¼0

Do1 Do2 Do1 Do2

¼ DL ¼0 ¼ DL ¼0

(DH,DL)

(DL,DH)

(DL,DL)

Do1 ¼ 0 Do2 ¼ 0

Do1 ¼ 0 Do2 ¼ 0

Do1 ¼ 0 Do2 ¼ 0

Capacity Management and Price Discrimination under Demand Uncertainty

4.3.2

205

Capacity Investment Game

In the above subsection, we have determined both customers’ equilibrium decisions on Doi . This decision is contingent on the number of options both customers have purchased O, the realized demand D, the option strike price pe, and the capacity K. This subsection will analyze the supplier’s optimal capacity decision K*, which is made before the actual demand is realized. As stated previously, the optimal capacity decision K* depends on the supplier’s observation of option purchase by the customers O and the option execution price pe. Based on such information, the supplier shall expect the possible future demand realization D and the customers’ reaction of Doi , i ¼ 1,2. Once the customers have purchased the options, the supplier will only look at the future revenue excluding the revenue from option purchase, po(O1 + O2), to decide the optimal capacity. When K* increases, the chance of congestion decreases. The supplier’s revenue comes more from serving the regular demand. When pe < vl, the supplier has an incentive to prepare enough capacity to avoid the possible customers option demand. However, if pe > vl, the supplier tends to induce the customers into exercising their options by restricting the capacity K. However, restricting capacity may create problems when K < Do because the supplier fails to execute the option demand as he has promised in the option contract. He must compensate those customers whose option demands cannot be executed. The supplier’s capacity decision is to figure out an optimal level of capacity to trade off among the revenue, capacity investment cost, and the possible compensation. Customers are more willing to execute their options as pe decreases. Foreseeing this, the supplier tends to increase the capacity level to guarantee that all the option demand be satisfied. This will minimize the expected compensation. Hence, we predict capacity K* decreases as pe increases. Proposition 3. In a subgame perfect equilibrium, the supplier’s optimal capacity decision is as follows: 1. When O ¼ ðDH ; DH Þ, the optimal capacity 8 L L > if vh ðvh vl ÞDDH < pe bvh < 2D h H L h h l DL e K ¼ vvhp 2DH if vh ðvh vl ÞD2DþD H < pe bv ðv v ÞDH vl > H L : H D þ DL if pe bvh ðvh vl ÞD2DþD H 2. When O ¼ ðDH ; DL Þ, the optimal capacity

K ¼

8 L > < 2D

v pe H þ DL Þ h l ðD > : v v DH þ DL h

if vh ðvh vl ÞDH2DþDL < pe bvh L

if vl < pe bvh ðvh vl ÞDH2DþDL if pe bvl L

3. When O 2 fðDL ; DL Þ; ðDH ; 0Þ; ðDL ; 0Þ; ð0; 0Þg, the optimal capacity K* ¼ 2DL for all pe < vh.

206

F. Fang and A. Whinston K*

DH+DL

K*(D H,D H) K*(DH,D L)

2DL

K*(DL,DL),K*(DH,0),K*(DL,0), and K*(0,0)

h

h

l

L

H

L

v −(v −v )2D /(D +D )

vl

h

h

l

L

H

v −(v −v )D /D h

h

l

H

L

v −(v −v )(D +D )/2D

H

pe

Fig. 2 Optimal capacity levels

Figure 2 summarizes the optimal capacity level K* as a function of pe in all six configurations of O. The capacity is at least the minimal level of the aggregate demand 2DL. It is greater than 2DL only when both customers have purchased the options and at least one of them bought DH units of options. In all other cases, we have O1 + O2 < 2DL and hence the total number of options demand, which is smaller than the total number of options purchased, must be smaller than 2DL. As a result, the supplier will remain unconcerned about the compensation even when he sets the capacity K ¼ 2DL. However, when both customers have purchased options and at least one bought DH units, the supplier has to worry about the possibility that both customers execute all their options and the capacity may be insufficient for satisfying the aggregate option demand. If he stays with the capacity level K ¼ 2DL, the probability of providing compensation will be as high as 2a a2 when pe is low enough. So our assumption that (2a a2)vh > c0 suggests that the supplier should increase the capacity from 2DL to avoid the situation. The assumption that a2vh < c0 suggests that the supplier cannot be better off by increasing the capacity from DH + DL to avoid the possible compensation when D ¼ ðDH ; DH Þ. The optimal capacity is between 2DL and DH + DL.

Capacity Management and Price Discrimination under Demand Uncertainty

4.3.3

207

Optimal Option Prices

Given the optimal decisions of K* and ðDo1 ; Do2 Þ, we can now analyze the first stage of the option-capacity game to derive the optimal option prices (po, pe) and the supplier’s expected profit by using the option contracts. All the decisions analyzed in Sects. 4.3.1 and 4.3.2 are contingent on both the option execution price pe and the customer’s option purchase O. In addition, the customers decide O based on their own types (t1, t2) and how the supplier prices the options. The fundamental question to be addressed is: what is the optimal po and pe that induces the customer to purchase the right number of options and maximizes the supplier’s profit? In this period, the supplier announces the option prices (po, pe). Then the customers submit their option purchase demand Oi to the supplier simultaneously. In order to detect the customer types, the supplier will price the options so that the low type customers will not purchase but the high type customers will. The customers’ valuation of the option contracts is different when the other customer’s option purchase demand, Oi, varies. If we denote the expected value of a unit of option contract as fo(Oi, Oi, pe), then the value of fo can be calculated as follows: fo ðOi ; Oi ; pe Þ ¼

1 h Eui ðOi ; Oi ; pe Þ Euhi ð0; Oi ; pe Þ Oi

where Euhi ðOi ; Oi ; pe Þ represents the expected utility a high type customer gets after purchasing Oi units of options while the other customer has purchased Oi units of options. Lemma 3. fo ðOi ; DH ; pe Þrfo ðOi ; DL ; pe Þrfo ðOi ; 0; pe Þ: Lemma 3 demonstrates an important result that is useful in proving the following Propositions. The lemma mathematically proves the fact that the options contract is more valuable to a high type customer if the other customer also buys more. This is so because the two customers will compete for the limited capacity resource in the consumption period. Buying more options protects them in the competition. Consequently, if the supplier can induce one customer to buy DH units of options, the other customer would want to pay more for the options if she is also a high type. This implies that the supplier prefers to induce the subgame equilibrium where the high type customers choose the maximal hedge strategy. Based on this conclusion, the following Proposition 4 gives the optimal option price and execution price (po, pe). Proposition 4. The supplier maximizes his expected profit when setting pe 2 vl ; H L H H H vh ðvh vl ÞD2DþD H Þ and po ¼ lfo ðD ; D Þ þ ð1 lÞfo ðD ; 0Þ. In such an equilibrium, a high type customer will choose a maximal hedging strategy by purchasing Oi ¼ DH units of options. The expected supplier profit will be EP ¼ ðvl c0 Þ2DL þ 2alðvh vl ÞðDH DL Þ þ l2 ð2a a2 Þvh c0 ðDH DL Þ

208

F. Fang and A. Whinston

and the equilibrium capacity investment K ¼

D H þ DL 2DL

if O1 ¼ O2 ¼ DH otherwise:

The supplier’s capacity investment K* is the same as the one in the full information benchmark case (see Sect. 4.2) given the belief that Oi ðt ¼ hÞ ¼ DH and Oi ðt ¼ lÞ ¼ 0. By using option contracts, the supplier can make a contingent capacity investment based on the customer’s option purchase decision O, which reveals the customer types (t1, t2) in equilibrium. This flexibility improves the capacity investment decision. The option compensation pb ¼ vh pe is important in achieving the efficient capacity level K* ¼ KFI. Under the options framework, the supplier’s incentive to increase the capacity when both customers are of high type is to satisfy as many options as possible while reducing the compensation. That is, the marginal benefit the supplier gets is pe + pb ¼ vh when K < Do. This incentive is well-aligned with the full information benchmark case where the incentive of increasing capacity is to increase the ability of serving high type demand, which yields a marginal profit pFI(t ¼ h) ¼ vh. We can rewrite the supplier’s expected profit as: EP ¼ EPND þ D1 þ D3 þ D4 where EPND is the supplier’s expected profit in the first benchmark. D1 and D3 are defined in Proposition 2. D1 represents the gain from prioritizing the high type customer’s demand over the low type one’s. D3 refers to the gain achieved when the supplier increases capacity after observing two high type customers. D4 ¼ 2al2 (vh vl)(DH DL) > 0. a(vh vl)(DH DL) is the expected loss a high type customer suffers if she doesn’t buy options but competing for capacity with another high type customer who has options. D4 is 2l2 times this expected loss representing the supplier’s expected gain from the competition between two high type customers. Comparing the supplier’s expected profit using the option contracts, EP*, to the supplier’s expected profit in the full information benchmark, EPFI, we can characterize the supplier’s profit loss when he cannot distinguish the high type customer and extract full surplus from them as follows: EPFI EP ¼ D2 D4 ¼ 2l lEuhi ð0; DH Þ þ ð1 lÞEuhi ð0; 0Þ

lEuhi ð0; DH Þ þ ð1 lÞEuhi ð0; 0Þ is the reserve utility a high type customer has when she does not buy options. This reserve utility is also known as the “Information Rent” in the price discrimination literature, (Mas-Colell et al. 1995) representing the cost the supplier pays to induce a high type customer to reveal her type. The expected information rent the supplier pays is exactly D2 D4, which increases as the probability that a customer being high type, l, increases.

Capacity Management and Price Discrimination under Demand Uncertainty

209

Proposition 5. W* ¼ WFI. The capacity investment is efficient in equilibrium and high type customers will always be served as the first priority. Hence, it is not surprising that the option contracts can achieve the same efficiency level as the full information benchmark case. Proposition 5 justifies the optimality of our proposed options contract framework.

5 Discussion and Extensions 5.1

Using Options for Price Discrimination

In the literature of price discrimination with unobservable types, the supplier’s ability to employ a non-linear pricing scheme (e.g. quantity discount and bundling) is critical to his profit. In this paper, we show that with properly priced options, the supplier can achieve the same profit level with a simple linear pricing function using an option framework. Particularly, our framework works when the customer’s demand is uncertain. In our framework, a customer faces two purchase decisions. She chooses the number of options to purchase before the demand realization and the amount of demand (including the numbers of regular demand and option demand) to request afterwards. In both decisions, she has full flexibility to choose the quantities. A linear pricing scheme is applied in all the purchases. The customers prefer such flexibility when they suffer from demand uncertainty. To discriminate the customers, we pay our attention to the case where only high type customers will buy the options. A high type customer’s total charge in equilibrium can be divided into two parts. A fixed payment poDH is charged ex ante regardless of her actual demand realization D to guarantee the priority of their demand execution. In addition, she pays a contingent payment based on her actual demand realization. Adjusting the option price po and pe, the supplier essentially changes the ratio between the ex ante and ex post payments to affect the high type customers hedging incentive and exploit their willingness to pay for the demand. The option framework helps the supplier to conduct price discrimination when the customer’s demand is uncertain. When there is no demand uncertainty (i.e., DH ¼ DL), the capacity will always be enough for the aggregate demand under our assumption. Hence, the option has no value and the supplier cannot discriminate among the customers. Our assumption of the supplier’s marginal capacity cost is critical to derive the result. The discrimination framework is built based on the high type customer’s concern of potential demand loss. If the capacity cost is small, the customers can infer that the supplier will always build enough capacity for the demand and won’t pay money ex ante to hedge the potential demand loss. The discrimination is not implementable in this case. If the capacity cost is large, the supplier will find it

210

F. Fang and A. Whinston

profitable to charge a high regular price (e.g. vh) to exploit the high type customers’ surplus only.

5.2

Multiple Agents and Multiple Types

In this paper, we use a parsimonious model with one supplier and two customers to illustrate price discrimination. The model can be extended to a case where multiple customers with two possible types. In such a model, we could still design the option contracts so that only those high type customers will buy them. The major difference would be a more complicated aggregate demand pattern. In a symmetric equilibrium where all high type customers adopt the same strategy, the supplier can separate the customers into two groups and treat them as two representative agents. The same analysis can then be applied to figure out the optimal option contract. The efficiency level of the second degree price discrimination still holds. The model gets further complicated when customers are of multiple types. In that case, the supplier could design multiple option contracts with different combinations of strike prices pe and compensation pb for each type. The customers could then self-select the options and decide when to exercise them. The demand should then be prioritized according to the supplier’s marginal punishment of not fulfilling the demand (i.e. pb pe). As a result, different demand associated with different option contracts are categorized into different priority levels. Allocation efficiency can be achieved if the customers with higher willingness to pay will always buy the option contracts with higher priority (characterized by pb pe). The challenge is how to price the options to induce the customers to purchase the right options to reveal their types.

5.3

Spot Exchange Options Market

In our setup, customers purchase options to hedge their demand risk. After their demand is realized, they can decide how many of their options to exercise. In this setup, a customer cannot buy additional options from other customer ex post. This raises a question: what if they can exchange their options after observing their demand? On the one hand, an option exchange leads to more efficient option utilization. If a customer is able to sell her extra options after the demand is realized, she is more willing to buy the options ex ante. Thus, the ex post exchange encourages the option purchase ex ante. On the other hand, the customers’ incentive for a maximal hedge decreases with the possibility of option exchange. This is because she may find someone who will sell options to her if her demand is high. Between this conflicting incentives, it is not clear which incentive is stronger in general. However, if we assume that customer types may change ex post, then the existence of an option exchange market helps in some cases. A detailed analysis

Capacity Management and Price Discrimination under Demand Uncertainty

211

of when a spot exchange market helps increase a monopolist’s profit can be found in Geng et al. (2007). Moreover, ex post exchange reduces the customer’s ex post risk of purchasing options. Hence, the existence of an ex post exchange market always improves the option purchase incentive if the customers are risk-averse.

6 Conclusion In this paper, we propose using a form of options contract framework that allows a monopolistic supplier to conduct price discrimination among customers, thereby maximizing his expected revenue under demand uncertainty and a capacity constraint. Our analysis shows that option contracts can benefit the supplier because the high type customers will pay more for hedging potential demand loss. The supplier gains the additional benefit of being able to adjust capacity according to his observation of customer types. We also analyze the strategic interactions among the supplier and customers. We show that, in equilibrium, the efficient capacity level can be induced by setting a compensation price which leaves the high type customers indifferent about whether to exercise their options or to ask for compensation. Overall efficiency is guaranteed and the supplier and the high type customers share the efficiency gain from the efficient capacity investment. Our proposed structure replicates the classical price discrimination outcome where the low type customers do not gain surplus and the high type customers enjoy an information rent. Our proposed structure can easily be adopted in situations where the supplier is not allowed to sell a bundled product with fixed quantity and situations where the actual demand and capacity is not contractible. Our framework has significant revenue management implications for various industrial applications such as network capacity management, airline ticket reservation, and telephone and electricity providers.

Appendix 1: Notation Table i Di D D Dsi Ds Dei De De

Customer index The realized demand of customer i. Di ∈ {DH, DL} D ¼ D1 + D2 the aggregate demand D ¼ ðD1 ; D2 Þ is the demand vector of both customers The demand customer i submits to the supplier Ds ¼ ðDs1 ; Ds2 Þ is the vector of customers’ submitted demand The demand of customer i, being satisfied by the supplier De ¼ De1 þ De2 is the aggregated demand satisfied by the supplier De ¼ ðDe1 ; De2 Þ is the vector of customers’ satisfied demand

212

F. Fang and A. Whinston

Doi Doi Do fo K mi Oi O p po pe pb ti ui vt a l P p

The amount of options executed by customer i when capacity is tight The amount of options executed by the customer other than i Do ¼ Do1 þ Do2 the aggregated option demand submitted by both customers The high type customer’s valuation of one unit of option contract The supplier’s capacity level The amount of monetary transfer made from customer i to the supplier The amount of option contracts customer i will purchase O ¼ ðO1 ; O2 Þ the vector of customers’ options purchase The unit price for the regular demand The option price The option execution price The option buy back price Type of customer i. ti ∈ {l,h} The utility customer i receives The marginal value of demand satisfaction for each type t The probability that a customer’s realized demand is high (Di ¼ DH) The probability that a customer is a high type one The supplier’s profit The supplier’s profit gained after period 1, excluding the sale from the option contracts The probability that regular demand is satisfied under the option framework

f

Appendix 2: Proof of Lemmas and Propositions Proof of Proposition 1. If p ⩽ vl, both customers submit all their demand to the supplier, Dsi

¼ Di ¼

DH DL

with prob ¼ a with prob ¼ 1 a:

The supplier’s expected profit: h i EP ¼ p a2 minfK;2DH gþ2að1aÞminfK;DH þDL Þþð1aÞ2 minfK;2DL g c0 K: Maximizing the profit under the condition p ⩽ vl, we have p* ¼ vl and K* ¼ 2DL. The supplier’s expected profit is EP(p ¼ vl) ¼ (vl c0)2DL. Customer i’s expected utility is ui ¼ (vi vl)DL. If p ∈ (vl, vh), only high type customers will submit the demand. Therefore, Dsi ¼ Di when ti ¼ h and Dsi ¼ 0 for ti ¼ l. When vl < p ⩽ vh, the supplier’s expected profit

Capacity Management and Price Discrimination under Demand Uncertainty

213

h i EPðpÞ ¼ l2 p a2 minfK;2DH gþ2að1agminfK;DH þDL gþð1aÞ2 minfK;2DL g

þ2lð1lÞp aminfK;DH gþð1aÞminfK;DL g c0 K qﬃﬃﬃﬃﬃﬃﬃﬃﬃ h 0 Maximizing the expected profit and applying the assumption l < 1 v vc h , we * h * * h have p ¼ v and K ¼ 0. Thereby, EP (p ¼ v ) ¼ 0 and it is not worthwhile to build capacity and serve high type customers only, due to the low probability of a high type customer’s existence. Compare the two cases, we conclude that the supplier’s best strategy is to set pND ¼ vl and serve both types of customers. The optimal capacity will be KND ¼ 2DL. The expected profit EPND ¼ (vl c0)2DL and the overall efficiency: ND W ND ¼ EPND þ lEuND i ðti ¼ hÞ þ ð1 lÞEui ðti ¼ lÞ

¼ lvh þ ð1 lÞvl c0 2DL :

□ FI

Proof of Lemma 1. The optimal capacity K is made contingent on the customer types T ¼ ðt1 ; t2 Þ. When T ¼ ðh; hÞ, the supplier’s expected profit h EPFI ðK;TÞ¼vh a2 minfK;2DH gþ2að1aÞminfK;DH þDL g i þð1aÞ2 minfK;2DL g c0 K which is maximized when K(h,h) ¼ DH + DL due to the assumption avh < c0 < (2a a2)vh. When T ¼ ðl; lÞ, similarly, we can have KFI(l,l) ¼ 2DL, since (2a a2) l v < c 0 < v l. When T ¼ ðh; lÞ or (l,h) and K > DH, the supplier’s expected profit EPFI ðK; ðh; lÞÞ ¼ a2 vh DH þ vl minfK DH ; DH g þ að1 aÞ vh DH þ vl minfK DH ; DL gÞ þ að1 aÞ vh DL þ vl minfK DL ; DH g þ ð1 aÞ2 vh DL þ vl minfK DL ; DL g c0 K which is maximized when K* ¼ 2DL. It can also be shown that K ⩽ DH cannot be optimal. Therefore, KFI(h,l) ¼ KFI(l,h) ¼ 2DL. □ Proof of Proposition 2. For the supplier, the probability that both customers are high types is l2. The expected profit EPðh; hÞ ¼ vl c0 2DL þ vh vl 2DL þ ð2a a2 Þvh c0 DH DL :

214

F. Fang and A. Whinston

With probability 2l(1 l), one customer is of high type and the other is of low type. The expected profit EP(h, l) ¼ (vl c0)2DL + (vh vl) (aDH + (1 a)DL). With probability (1 l)2, both customers are of low type. The expected profit EP (l,l) ¼ (vl c0)2DL. Since ui(t ¼ h) ¼ ui(t ¼ l) ¼ 0, W FI ¼ EPFI ¼ l2 EPðh; hÞ þ 2lð1 lÞEPðh; lÞ þ ð1 lÞ2 EPðl; lÞ ¼ ðvl c0 Þ2DL þ vh vl 2alð1 lÞ DH DL þ vh vl l2DL þ l2 ð2a a2 Þvh c0 DH DL compared to EPND ¼ (vl c0)2DL and WND ¼ (lvh + (1 l)vl c0)2DL, we can easily conclude that EPFI ¼ EPND þ D1 þ D2 þ D3 W FI ¼ W ND þ D1 þ D3 □ Proof of Lemma 2. The proof is straightforward since we can show that the second order condition of the above objective function is non-negative. It means that the objective function is a convex function. The optimal solution of the maximization □ problem would exist on the boundary. That is, it is either 0 or min{Oi, Di}. Proof of Proposition 3. In this stage, the supplier determines the capacity level to maximize his future revenue less the capacity investment. That is, pðO; D; K; pe Þ ¼ m1 þ m2 po ðO1 þ O2 Þ c0 K ¼ pe Do ðO; D; K; pe Þ þ vl ðminðK; DÞ Do ðO; D; K; pe ÞÞþ vh ðDo ðO; D; K; pe Þ KÞþ c0 K Applying the outcomes from Tables 1–6, we can derive the profit function with parameters O; D; K, and pe. Taking expectation over the realized demand D, we obtain the expected profit function for each O; pe , and K. Maximizing those the expected profit function by choosing K, we can obtain the optimal capacity as stated □ in Proposition 3 as a function of O and option strike price pe. Proof of Lemma 3. From the result of 1. When pe rvh ðvh vl ÞDDH , optimal capacity K* ¼ 2DL for all the possible configurations of O. No options will be exercised for all possible realization of D. Therefore, the option has no value. In other words, fo ðOi ; DH ; pe Þ ¼ fo ðOi ; DL ; pe Þ ¼ fo ðOi ; 0; pe Þ ¼ 0 H h h l DL 2. When vh ðvh vl Þ2DD H DL bpe < v ðv v ÞDH , if the customer has bought Oi ¼ DL, she will never exercise the options no matter what type the other customer is. Therefore, at the first stage, the value of the options contract would be 0 and the customer should not purchase any options with a positive price L

Capacity Management and Price Discrimination under Demand Uncertainty

215

3. When vh ðvh vl ÞDH2DþDL bpe < vh ðvh vl Þ2DD H DL : if the customer has bought Oi ¼ DL units of options, she will only exercise it when D ¼ ðDH ; DH Þ and Doi 6¼ DH . Hence if the other customer is of high type, she is always better off if the other customer has purchased Oi ¼ DH rather than Oi ¼ DL units of options. However, given Oi ¼ DH, customer i will never exercise her options and the value of the options is 0. If the other customer is of low type, the customer’s expected utility from exercising the options are L

H

(a) Oi ¼ DH : ui ðDH Þ ¼ a2 ðvh pe ÞDH þ ð1 a2 ÞDL po DH L H 2 L L (b) Oi ¼ DL : ui ðDL Þ ¼ a2 ðvh pe ÞDL þ ðvh vl Þ2DDH D DL þ ð1 a ÞD po D h l L (c) Oi ¼ 0 : ui ð0Þ ¼ ðv v ÞD Þui ð0Þþpo D D From the results above, we can show that uuiiðD ðDH Þui ð0Þþpo DL > DL , meaning that the customer is better off by purchasing Oi ¼ DH units of options. Following the same calculation, we can show that DL is not the optimal choice when pe > vl. □ H

H

H

Proof of Proposition 4. We need to discuss the profit according to the different pe segments: 1. When pe r vh ðvh vl Þ DDH , no options will be exercised, po ¼ 0 and EP ¼ (vl c0)2DL H L h h l DL 2. When vh ðvh vl ÞD2DþD we have K ðDH ; DH Þ ¼ H bpe < v ðv v ÞDH , vh pe 2DH from Proposition 3. The supplier’s profit vh vl L

EP ¼ Ep þ 2lpo DH

vh pe H 2 2 l 2 l L ¼ l ð2a a Þv c0 h 2D þ ð1 a Þv 2D v vl

þ 2lð1 lÞ a2 ðpe vl ÞDH þ ðvl c0 Þ2DL þ ð1 lÞ2 ðvl c0 Þ2DL

þ 2l Euhi ðDH Þ Euhi ð0Þ which we can show dEP dpe < 0. H L * H H 3. When vl bpe < vh ðvh vl ÞD2DþD H , we have optimal capacity K (D ,D ) ¼ H L * H * L D + D and K (D ,0) ¼ K (0,0) ¼ 2D from Proposition 3. One can get EP ¼ ðvl c0 Þ2DL þ 2alðvh vl ÞðDH DL Þ þ l2 ð2a a2 Þvh c0 ðDH DL Þ: In this case, dEP dpe ¼ 0. In summary, we can conclude that the optimal option exercise price pe should be H L □ vh ðvh vl ÞD2DþD H Proof of Proposition 5. The proof is straightforward from the proof of Proposition 4. □

216

F. Fang and A. Whinston

References Afeche P (2006) Incentive-compatible revenue management in queueing systems: optimal strategic delay and other delaying Tactics. Working paper, The Kellogg School of Management, Northwestern University, Evanston, IL Armstrong M, Vickers J (2001) Competitive price discrimination. Rand J Econ 32:1–27 Barnes-Schuster D, Bassok Y, Anupindi R (2002) Coordination and flexibility in supply contracts with options. Manuf Serv Options Manage 43:171–207 Birge JR (2000) Option methods for incorporating risk into linear capacity planning models. Manuf Serv Operation Manage 2:19–31 Boyaci T, Ray S (2006) The impact of capacity costs on product differentiation in delivery time, delivery reliability, and prices. Prod Oper Manage 15:179–198 Cachon GP, Fisher M (2000) Supply chain inventory management and the value of shared information. Manage Sci 46:1032–1048 Cachon GP, Lariviere MA (1999) An equilibrium analysis of linear, proportional and uniform allocation of scarce capacity. IIE Trans 31:835–849 Cachon GP, Lariviere MA (2001) Contracting to assure supply: how to share demand forecasts in a supply chain. Manage Sci 47:629–646 Cachon GP (2002) Supply coordination with contracts. In: Graves S, Kok T (eds) Handbooks in operations research and management science. North-Holland, Amsterdam Chen F (2003) Information sharing and supply chain coordination. In: Graves SC, De Kok AG (eds) Handbooks in operations research and management science: supply chain management design, coordination and operation. Elsevier, Amsterdam Corsetti G, Dedola L (2005) A macroeconomic model of international price discrimination. J Int Econ 67:129–155 Deshpande V, Schwartz L (2002) Optimal capacity choice and allocation in decentralized supply chains. Technical report, Krannert School of Management, Purdue University, West Lafayette, IN Geng X, Wu R, Whinston AB (2007) Profiting from partial allowance of ticket resale. J Mark 71:184–195 Guo Z, Fang F, Whinston AB (2006) Supply chain information sharing in a macro prediction market. Decis Support Syst 42:1944–1958 Gupta A, Stahl DO, Whinston AB (1996) An economic approach to network computing with priority classes. J Organ Comput Electron Commer 6:71–95 Gupta A, Stahl DO, Whinston AB (1999) The economics of network management. Commun ACM 42:57–63 Huchzermeier A, Cohen MA (1996) Valuing operational flexibility under exchange rate risk. Oper Res 44:100–113 Iyer AV, Deshpande V, Wu Z (2003) A postponement model for demand management. Manage Sci 49:983–1002 Kleindorfer PR, Wu DJ (2003) Integrating long-and-short term contracting via business-to-business exchanges for capacity intensive industries. Manage Sci 49:1597–1615 Lariviere M (1999) Supply chain contracting and coordination with stochastic demand. In: Tayur S, Ganeshan R, Magazine M (eds) Quantitative models for supply chain management. Kluwer, Boston Li L (2002) Information sharing in a supply chain with horizontal competition. Manage Sci 48:1196–1212 Lin L, Kulatilaka N (2007) Strategic Growth Options in Network Industries. In: Silverman B (eds) Advances in Strategic Management. Emerald Group Publishing Limited., pp. 177–198 Martı´nez-de-Albe´niz V, Simchi-Levi D (2005) A portfolio approach to procurement contracts. Prod Options Manage 14:90–114 Mas-Colell A, Whinston MD, Green JR (1995) Microeconomic theory. Oxford University Press, USA

Capacity Management and Price Discrimination under Demand Uncertainty

217

Maskin E, Riley J (1984) Monopoly under incomplete information. Rand J Econ 15:171–196 Mortimer JH (2007) Price discrimination, copyright law, and technological innovation: evidence from the introduction of DVDs. Q J Econ 122:1307–1350 Noldeke G, Schmidt KM (1995) Option contracts and renegotiation: a solution to the hold-up problem. Rand J Econ 26:163–179 ¨ zer O, Wei W (2004) Inventory control with limited capacity and advance demand information. O Oper Res 52:988–1000 Sethi SP, Yan H, Zhang H (2004) Quantity flexible contracts in supply chains with information updates and service-level constraints. Decis Sci 35:691–712 Sodhi M (2004) Managing demand risk in tactical supply chain planning. Prod Oper Manage 14(1):69–79 Tirole J (1988) The theory of industrial organization. MIT, Cambridge, MA Varian HR (1985) Price discrimination and social welfare. Am Econ Rev 75:870–875

.

Dynamic Procurement, Quantity Discounts, and Supply Chain Efficiency Feryal Erhun, Pinar Keskinocak, and Sridhar Tayur

Abstract We study a model with a single supplier and a single buyer who interact multiple times before the buyer sells her product in the end-consumer market. We show that when the supplier uses a wholesale price contract, even under perfect foresight, the supplier, the buyer, and the end-consumers benefit from multiple trading opportunities versus a one-shot procurement agreement. Keywords Advance capacity procurement • Incremental discounts • Strategic interactions • Supply chain coordination

quantity

1 Introduction This chapter studies the benefits of trading more than once while procuring/selling capacity. Consider a simple model with one supplier and one buyer. The buyer produces a product by using the capacity she buys from an uncapacitated supplier. Before the buyer’s selling season begins, there are N periods in which the buyer can procure capacity. The supplier uses a simple wholesale price contract; he

A significant part of the materials in this invited chapter is from the following original article: Erhun F, Keskinocak P, Tayur S (2008) Dynamic procurement, quantity discounts, and supply chain efficiency. Prod Oper Manage 17(5):1–8. F. Erhun (*) Department of Management Science and Engineering, Stanford University, Stanford, CA, USA e-mail: [email protected]edu P. Keskinocak School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, GA, USA e-mail: [email protected] S. Tayur Tepper School of Business, Carnegie Mellon University, Pittsburgh, PA, USA e-mail: [email protected] T.-M. Choi and T.C. Edwin Cheng (eds.), Supply Chain Coordination under Uncertainty, International Handbooks on Information Systems, DOI 10.1007/978-3-642-19257-9_9, # Springer-Verlag Berlin Heidelberg 2011

219

220

F. Erhun et al.

charges a unit capacity price of wn in period n ¼ 1; . . . ; N, which he determines dynamically. Once the buyer procures the capacity, she can produce the product with no additional cost and sell it in an end-consumer market, where the market price of the product is determined by the market clearance assumption. Dynamic procurement, i.e., simple wholesale price contracts repeated over time (possibly with different prices), is a commonly observed practice in a vertical channel. The typical justification for multiple procurement trades is risk hedging. In order to manage the demand risk, a buyer may prefer to procure capacity dynamically over time after receiving some update on demand. Other commonly observed reasons for dynamic procurement include spreading payments over a period of time, minimizing potential capacity risks (supplier’s or buyer’s), supplier’s decreasing cost over time which may translate to lower prices (e.g., as in the electronics industry), and forward buying. We discuss yet another potential impact of dynamic procurement, i.e., as a tool to influence future prices. In a vertical setting, we show that risk hedging is not the only justification for multiple trades. We derive a Pareto improving rationale for the use of additional trading periods in the case of deterministic demand (i.e., when the commonly known and intuitive benefits due to risk hedging are not present) and wholesale price contracts, where all participants (supplier, buyer, and end-consumers) benefit. The additional trading periods inherently create the equivalent of a non-linear pricing scheme, which makes the performance of the decentralized supply chain approach that of a centralized supply chain when the number of trading periods is sufficiently high.

2 Literature Review The paper that is closest to our work is by Allaz and Vila (1993). The authors study a deterministic model where two Cournot duopolists trade in forward markets for delivery in a single period. The authors conclude that even though producers are worse off by forward trading, in equilibrium they will trade forward. In the limit, as the number of forward markets goes to infinity, the competitive outcome is achieved in a duopoly setting. In our model, we look at vertical interactions, as opposed to the horizontal competition of Allaz and Vila. In a setting similar to ours, Anand et al. (2008) study a dynamic model of a procurement contract between a supplier and a buyer in a two-period, uncapacitated, deterministic demand game. The authors eliminate all the classical reasons for inventories, yet show that the buyer’s optimal strategy in equilibrium is to carry inventories, and the supplier is unable to prevent this. The inventories arise for “strategic” reasons. Keskinocak et al. (2003, 2008) extend Anand et al.’s model to limited capacity and limited capacity with backordering, respectively. Research on quantity discounts also relates to our problem. We refer readers to Benton and Park (1996) and Munson and Rosenblatt (1998) for extensive reviews, and to Dolan (1987) for a detailed survey of different variants of the quantity

Dynamic Procurement, Quantity Discounts, and Supply Chain Efficiency

221

discount problem from a marketing research standpoint. Jeuland and Shugan (1983) show that profit sharing mechanisms with quantity discounts can coordinate the supply chain. Following their work, many researchers study the role of quantity discounts as a channel coordination mechanism under different settings; e.g., Weng (1995) and Chen et al. (2001) combine channel coordination with price-sensitive demand and operating cost, Ingene and Parry (1995) introduce competing retailers, Raju and Zhang (2005) study channel coordination with a dominant retailer, and Chen and Roma (2010) consider a single manufacturer offering quantity discounts to competing retailers. Another stream of literature study quantity discounts to improve operational efficiency (Crowther 1964; Monahan 1984; Dada and Srikanth 1987). The dynamic procurement model that we study in this chapter falls in to this category by effectively creating an incremental quantity discount mechanism. Unlike the papers in the literature, the terms of trade are set by both the supplier and the buyer. Beside these two streams of literature that is closely related to our problem, there are three other streams that are related in spirit (1) timing of purchase commitments (2) two-period procurement and risk allocation, and (3) multi-period price and capacity adjustment. These streams consider multiple procurement opportunities in settings where the buyer can accrue additional demand information by postponing his procurement decision. The timing of purchase commitments has been the subject of many studies in the operations management literature. Iyer and Bergen (1997) study a supply chain with a single supplier and a single buyer to compare a traditional system, where the buyer places her order early on, to Quick Response (QR), where the buyer collects demand information before she places her order. The authors assume that the buyer pays the same wholesale price in either case and show that QR is not always Pareto improving. However, quantity discounts and volume commitments across products make QR Pareto improving. Ferguson (2003) and Ferguson et al. (2005) investigate an end-product buyer’s choice of when to commit to an order quantity when there is a demand information update during the supplier’s leadtime. The former paper assumes either all or none of the demand uncertainty is resolved, while the latter relaxes this assumption. The authors find that the buyer is not always better off delaying her quantity commitment and the supplier may prefer delayed commitment depending upon the amount of demand uncertainty resolved during the information update. Taylor (2006) studies a problem similar to the one in Ferguson (2003), however, he considers the sale timing of a supplier. The supplier may sell either early, i.e., well in advance of the selling season, or late, i.e., close to the selling season. Taylor shows that, in considerable generality, the supplier’s profit is greater when he sells late. In a duopolistic environment, Spencer and Brander (1992) identify conditions on demand variability under which the buyers would prefer to postpone their quantity decisions. Cvsa and Gilbert (2002) introduce a supplier to Spencer and Brander’s model and investigate how the supplier can influence the form of competition in the downstream market by offering a precommitment opportunity. In all of these papers, the buyer is limited to one mode of commitment (i.e., early or delayed). However, the buyer may prefer to use both

222

F. Erhun et al.

modes of commitment, which is the subject of the literature on two-period procurement and risk allocation. The literature on two-period procurement and risk allocation allows the buyer two purchasing opportunities: One is before and the other one is after demand realization. Gurnani and Tang (1999) analyze the trade-off between a more accurate demand information and a potentially higher procurement cost at the second period. Donohue (2000) studies a buy-back contract in a two-period setting where the procurement cost at the second period is higher and shows that this contract can coordinate the system. Cachon (2004) and Dong and Zhu (2007) study push (early commitment), pull (delayed commitment), and advance-purchase discount (purchase at a discounted price before the season and at a regular price during the selling season) to study how the inventory ownership impacts supply chain efficiency. Guo et al. (2009) study a three-tier supply chain with two outsourcing structures (delegation and control) to investigate how an OEM can use a two-wholesaleprice contract to increase the available upstream capacity. Erhun et al. (2008) and Li and Scheller-Wolf (2010) extend this literature to a vertical setting by considering the supplier’s pricing decisions as well. Erhun et al. study a capacitated two-tier supply chain and assume that the wholesale price is set by the supplier and the procurement quantity by the buyer. The authors investigate the impact of timing of the decisions and of additional demand information on the supplier’s pricing and the buyer’s procurement decisions. Li and Scheller-Wolf (2010) consider a supply chain composed of a buyer and multi-suppliers with private cost information. The buyer first offers a push or pull contract, and then selects the supplier through a wholesale price auction. The authors numerically find that a push system is more preferable by the buyer if the suppliers’ number is large and the demand level is high, while a pull system is more preferable if demand has high uncertainty and the suppliers’ cost is large. The literature on multi-period price and capacity adjustment seeks answers for when and how much to adjust the price or capacity or both in a dynamically changing environment. In particular, Burnetas and Gilbert (2001) consider a multi-period newsvendor model to study the trade-off between a more accurate demand information and increasing procurement costs. The authors numerically demonstrate that the broker tends to cluster his procurements just before price increases. Elmaghraby and Keskinocak (2003) and Van Mieghem (2003) provide a literature review on dynamic pricing and capacity investment and adjustment issues, respectively.

3 Main Model We study a model where there are N possible periods for capacity procurement before the buyer’s production/selling season begins. The supplier and the buyer maximize their profits. The supplier’s decisions are the wholesale prices for each period, wn, ðn ¼ 1; . . . ; N Þ. The buyer’s decisions are the procurement quantities

Dynamic Procurement, Quantity Discounts, and Supply Chain Efficiency

223

for each period, qn, ðn ¼ 1; . . . ; N Þ, and the production quantity, QN. (We set Q0 ¼ 0 and q0 ¼ 0.) The market is characterized by a linear inverse demand function P(QN) ¼ a bQN, where a is the market potential, b is the price sensitivity, P(QN) is the per-unit market price of the product for QN. We assume that the buyer’s unit cost of production is zero. However, the analysis of a positive, constant production cost (c < a) case is trivial. Since (a bQ)Q cQ ¼ ((a c) bQ)Q, we can simply modify the demand intercept a such that a^ ¼ a c, and the analysis follows. The sequence of events in each period n of the N-period gameis as follows (1) Given previous capacity procurements, qj ; j ¼ 1; . . . ; n 1 , the supplier price wn. (2) Given previous capacity procurements determines the capacity qj ; j ¼ 1; . . . ; n 1 and the current capacity price (wn), the buyer determines her procurement quantity qn. (3) In the last period N, the buyer chooses her production quantity QN and procures extra capacity, if necessary. The market clears only once at the end of the N-th period; i.e., there is only a single selling opportunity to end-consumers. We use backward induction and obtain the pure-strategy Subgame Perfect Nash Equilibrium (SPNE). Proposition 1 characterizes this equilibrium. Proposition 1. The unique pure-strategy SPNE for the N-period dynamic procurement model is the following. For n ¼ 1; . . . ; N 1, let n ¼ N n. Then, a a 2 nþ1 2 n 2 q1 ¼ qn ; wnþ1 ¼ wn ; ; w1 ¼ KN N ; qnþ1 ¼ 4Nb 2 2 n 2 nþ1 QN1 2kþ1 where K1 ¼ 1 and KN ¼ k¼1 The production 2kþ2. PN a a a . QN ¼ n¼1 qn ¼ 2b 4bKN . As N tends to infinity, QN tends to 2b

quantity

is

From Proposition 1, the production quantity for the N-period model is a a QN ¼ 2b 4b KN . As N increases, KN decreases, and the production quantity increases. Therefore, the double marginalization (DM) effect decreases, the efficiency increases and approaches that of the centralized solution. Similar to the argument of Allaz and Vila, a higher N does not necessarily imply that the capacity procurement is over a longer horizon, but rather that procurement occurs more frequently. Even though our model does not include a discount factor due to our interpretation of these N periods, the results of the main model are not sensitive to a discount factor. When we incorporate a discount factor d 1 to our analysis, ^n ¼ dNn wn , n ¼ 1; . . . ; N: The quantities maintain wholesale prices become w their original values. Figure 1 summarizes the prices and quantities in different periods a for the SPNE. 3a 5a The last period’s capacity price decreases & & & & 0 , while the first 2 8 16 9a 75a % % as N increases. The total period capacity price increases a2 % a 16 5a128 11a a production quantity, QN, increases 4b % 16b % 32b % % 2b ; and the market of the product (following the relationship P(QN) ¼ a bQN) decreases price 3a 11a 21a a . Even though the quantity in the first period decreases & & & & 4 16 32 2

224

F. Erhun et al.

n=1

1

a 2

2

9a 16

←⎯ ⎯⎯

3a 8

3

75a 128

←⎯ ⎯ ⎯ 5

15a 32

←⎯ ⎯ ⎯ 3

5a 16

4

1225a 2048

⎯ ←⎯ ⎯ 7

525a 1024

←⎯ ⎯⎯ ⎛ 5⎞

105a 256

…

…

Prices

N

n=2

n=4 …

n=3

n=1

Quantities n=3

n=2

n=4 …

a 8b

⎯⎯ → ⎯ 3

3a 16b

a 12b

⎯⎯ ⎯ → 5

5a ⎯⎯ ⎯→ ⎛×3 ⎞ ⎜ ⎟ 48b ⎝ 2⎠

5a 32b

a 16b

⎯⎯ ⎯ → 7

7a 96b

⎯⎯ ⎯ → 5

35a 384b

…

…

…

…

⎛× ⎞ ⎜ ⎟ ⎝ 6⎠

⎛× ⎞ ⎜ ⎟ ⎝ 4⎠

11a 32b ⎯⎯ ⎯ → 3 ⎛× ⎞ ⎜ ⎟ ⎝ 2⎠

35a 256

93a 256b …

⎛× ⎞ ⎜ ⎟ ⎝ 4⎠

5a 16b

…

⎛× ⎞ ⎜ ⎟ ⎝ 2⎠

…

35a 128

⎜× ⎟ ⎝ 2⎠

a 4b

…

←⎯ ⎯⎯ ⎛ 3⎞

…

⎜× ⎟ ⎝ 4⎠

…

⎛× ⎞ ⎜ ⎟ ⎝ 6⎠

⎛× ⎞ ⎜ ⎟ ⎝ 2⎠

…

⎛× ⎞ ⎜ ⎟ ⎝ 4⎠

…

⎛×3 ⎞ ⎜ ⎟ ⎝ 2⎠

…

a 4b

QN

Fig. 1 Capacity prices and quantities for different N values under dynamic procurement

100 90 80

% of Profits

70 60 50 40 30 20 10 0

1

3

DM Loss 25.00 9.77

5 6.06

7 4.39

9

11

13

15

17

19

3.44

2.83

2.40

2.09

1.85

1.65

Buyer

25.00 31.64 33.38 34.18 34.64 34.94 35.15 35.30 35.42 35.52

Supplier

50.00 58.59 60.56 61.43 61.92 62.23 62.45 62.61 62.73 62.83

Number of Periods, N

Fig. 2 The distribution of profits between the supplier, the buyer, and the double marginalization effect versus the number of periods (N)

a as N increases q1 ¼ 4Nb , the buyer procures capacity in each period. That is, trading occurs in all N periods. The buyer is willing to procure capacity at a given period (at a higher price) because she knows that by doing so the best response for the supplier is to lower the price in the subsequent periods. For a fixed N, the quantity that the buyer procures in each period increases and the capacity price decreases over time. As N increases, the double marginalization effect decreases and the supplier’s and the buyer’s profits both increase (see Fig. 2). Dynamic procurement not only increases the supply chain efficiency, but also naturally allocates the surplus to the supplier and the buyer such that both parties benefit. Independent of the values of a and b, the supplier’s profit converges to approximately 64% of the total profits, and the buyer’s profit converges to

Dynamic Procurement, Quantity Discounts, and Supply Chain Efficiency

225

approximately 36% of the total profits in the presence of additional capacity procurement periods. Even for small values of N, dynamic procurement decreases the inefficiency considerably. For example, for N ¼ 3, the inefficiency is already less than 10% (compared to 25% for N ¼ 1). In our analysis, we assume that N is exogenously determined, i.e., it is an input to the game. We show that as N increases, so do the profits of both players. Hence, the buyer and the supplier can jointly decide on an appropriate N value a priori, based on the marginal benefit of each additional trading period and the possible cost of each trade.

4 Extensions Our main result has two parts (1) as N increases, all participants (supplier, buyer, and end-consumers) strictly benefit and (2) as N goes to infinity, the performance of the decentralized supply chain approaches that of the centralized supply chain. However, we made several simplifying assumptions in our model. Hence we next discuss the implications of these assumptions on our main result and how they can be relaxed.

4.1

Limited Capacity

In this section, we consider the case where the supplier has a capacity of K units that he can sell throughout N periods. Let QN correspond to the total production quantity of an N-period uncapacitated game. For the N-period limited capacity a case, when the capacity is “tight,” i.e., CS b 4b , or when the capacity is “abundant”, i.e., CS ⩾ QN, the results are straightforward and intuitive. In the first case, as the capacity is tight, the supplier does not change his price through the game, so the N-period game is equivalent to a single-period game. When CS ⩾ QN, the problem is equivalent to an unlimited capacity game (Proposition 1). What happens in between these extremes is more interesting. Our main result is as follows. Proposition 2. The SPNE for the N-period capacitated model for CS < QN is as follows: Let N 2 f1; 2; ; Ng be such that QN 1

N X

qj ;

qN ¼

j¼NN þ2

wNN þ1 ¼

NY 1

i¼1

2i þ 1 wN ; 2i

wN ¼ 2

a 2b

a

CS ;

2 bCS ;

qNi ¼

2i qNiþ1 ; 2i þ 1

wNi ¼

2i þ 1 wNiþ1 : 2i

226

F. Erhun et al.

When CS < QN, depending on the capacity, the supplier and the buyer play an N*-period game (N* ⩽ N), the supplier sells all the capacity, and the supply chain is coordinated. Here, period N* corresponds to the smallest n (n 2 f1; 2; . . . ; Ng) for which the supplier’s capacity CS will not be enough for an n-period uncapacitated game. Contrary to the unlimited capacity case, increasing the number of trading periods beyond N* does not increase the profits of either player. The total profits for the buyer and the supplier can be maximized in a finite number of trading periods, N*, which depends on the capacity of the supplier.

4.2

Alternative Price-Sensitive Demand Functions

For most of the insights, we believe that the linear inverse demand function is not a critical assumption. When demand is price-sensitive, i.e., can be modeled by a downward-sloping demand curve, the intuition is as follows: When demand is price-sensitive, the purchasing power of each additional end-consumer is lower. Hence, if the buyer can reduce her marginal cost for each additional unit, she can profitably sell to a larger set of end-consumers. Dynamic procurement allows the supplier to charge lower prices as the number of units purchased by the buyer increases (as in the case of an incremental quantity discount), and hence, it allows the buyer to profitably sell to those end-consumers with lower valuations. Clearly, under a different demand curve, the split of profits between the buyer and the supplier will be different, but dynamic procurement will continue to occur as long as demand is price-sensitive. In order to verify our intuition, we study a multiplicative demand function where D(P) ¼ aP2 and an exponential demand function where D(P) ¼ a exp(P) for a two-period model. (For the multiplicative model we can no longer assume that the production cost of the buyer is zero, hence, for both cases we assume that the unit production cost is c.) Both multiplicative and exponential demand functions lead to similar dynamics to the linear inverse demand function. Therefore, even though we cannot verify the second part of our main result, we continue to show that all participants (supplier, buyer, and endconsumers) strictly benefit as N increases.

4.3

Newsvendor Setting

To test the robustness of our results to price-sensitive demand assumption, we study a setting where a newsvendor procures capacity N times before the demand is realized (we assume that there is no forecast update between periods). We analyze the case where demand is uniformly distributed between 0 and u, the market price is fixed at p per unit, and the buyer’s production cost is c < p per unit. Proposition 3 shows that dynamic procurement continues to increase the quantity that the buyer procures as the supplier is willing to decrease the wholesale price over time.

Dynamic Procurement, Quantity Discounts, and Supply Chain Efficiency

227

Proposition 3 (Newsvendor setting with uniform demand). The unique purestrategy SPNE for the N-period dynamic procurement model is the following. For n ¼ 1; . . . ; N 1, let n ¼ N n. Then, p c 2 nþ1 2 n u p c 2 ; w1 ¼ KN N qn ; wnþ1 ¼ wn ; ; qnþ1 ¼ q1 ¼ p 2 2 n 2 nþ1 2N QN1 2kþ1 The production quantity is where K1 ¼ 1 and KN ¼ k¼1 2kþ2. pc PN pc KN p . QN ¼ n¼1 qn ¼ u 1 2 p . As N tends to infinity, QN tends to u Furthermore, as N goes to infinity, the performance of the decentralized supply chain approaches that of the centralized supply chain. Therefore, our main result continues to hold in a newsvendor-setting with uniform demand distribution. Finally, the uniform distribution leads to the same profit split as in the linear inverse demand case.

4.4

Effort-Dependent Demand

In order to understand the impact of dynamic procurement in case of effort-dependent demand, we study an N-period model where demand is a function of the sales effort in addition to price, i.e., D(P, e) ¼ a P þ e (cost of the effort is e2). Proposition 4 shows that dynamic procurement continues to benefit both parties. Proposition 4 (Effort-dependent demand). The unique pure-strategy SPNE for the N-period dynamic procurement model is the following. For n ¼ 1; . . . ; N 1, let n ¼ N n. Then, a a 2 nþ1 2 n QN ; qn ; wnþ1 ¼ wn ; e ¼ q1 ¼ ; w1 ¼ KN2 N ; qnþ1 ¼ 2 3N 2 2 n 2 nþ1 QN1 2kþ1 The production quantity is where K1 ¼ 1 and KN ¼ k¼1 2kþ2. P a QN ¼ Nn¼1 qn ¼ 2a K . As N tends to infinity, QN tends to 2a 3 3 N 3. Furthermore, as N tends to infinity, both the effort level and the production quantity tend to those of the centralized supply chain. Due to dynamic procurement, the buyer can afford to invest more in the sales effort (as her average procurement cost decreases) and decrease the price of the product; therefore, demand for the product increases. The division of the profits, as well as the improvement of the system performance, mimic those of the setting with linear inverse demand function.

4.5

Information Asymmetry

In practice, the buyer is closer to the end-consumer market and may have more information about the demand compared to the supplier. Hence, we consider

228

F. Erhun et al.

1.5 Increase in the supplier’s 1 expected 0.5 profit 0 0

1 0.8 0.6 0.4

0.2 0.4

α

al /ah

0.2

0.6 0.8 1

0

Fig. 3 The increase in the supplier’s expected profits due to the additional period (ah ¼ 10)

information asymmetry between the supplier and the buyer regarding the market potential. According to the supplier, the market potential can either be “high”, ah, with a probability of a, or “low”, al, with a probability of 1 a. The buyer, on the other hand, knows the exact value of the market potential. (For simplicity, we assume that bi ¼ 1, i ¼ l, h.) Under asymmetric information, depending on the relative values of al and ah, the buyer and the supplier may engage in dynamic procurement. However, despite improving the system performance for a large range of parameters, the additional trading period is not always beneficial, i.e., the result in part (i) can be reversed under asymmetric information. Figure 3 illustrates these facts by plotting the increase in the supplier’s expected profits due to the additional period for different values of al and a. When al is relatively low compared to ah, the supplier does not benefit from the additional period. For higher values of al, there are values of a for which dynamic procurement improves the system performance. Hence, under asymmetric information the additional trading period may continue to enable dynamic procurement depending on the relative values of al and ah. However, our main result may also be reversed under asymmetric information.

4.6

Competition

In this section, we study the impact of dynamic procurement under competition. In this model, we restrict ourselves to two periods and study two competing buyers. Using the capacity they procure from the supplier, the buyers produce similar products, which they sell to end-consumers at the end of the second period. Before production takes place, the buyers can procure capacity in both periods. Firms simultaneously choose their procurement quantities in the first period; these become common knowledge and then firms simultaneously decide how much

Dynamic Procurement, Quantity Discounts, and Supply Chain Efficiency

229

additional capacity to procure in the second period before the market clears. Erhun (2007) studies this setting in detail (when there are more than two periods, more than two buyers, etc.) We present one of their results below to show how dynamic procurement extends to the situations with competition. The following proposition presents the unique SPNE for the dynamic procurement model under competition. Proposition 5. The unique SPNE for the dynamic procurement model is symmetric and is as follows. 9a (i) The wholesale prices for first and second periods are w1 ¼ 31a 60 and w2 ¼ 20, respectively. a . (ii) The buyers’ first period procurement quantity is: q1;1 ¼ q1;2 ¼ 30b 3a . (iii) The buyers’ second period procurement quantity is: q2;1 ¼ q2;2 ¼ 20b

Dynamic procurement continues to increase supply chain efficiency under competition. When N ¼ 1, the supply chain inefficiency is around 11%. With dynamic procurement, this inefficiency decreases to 7% when N ¼ 2. Furthermore, dynamic procurement naturally allocates the surplus to the supplier and the buyers such that they all benefit.

4.7

Demand Uncertainty with Information Update

Even though our goal in this chapter is to characterize a potential impact of dynamic procurement other than mitigating demand risk, as our final extension, we also discuss the dynamics of dynamic procurement when mitigating demand risk is an option. We consider a two-period extension of our main model where the endconsumer market can be in one of two states (indexed by i): “high” demand state, i ¼ h, or “low” demand state, i ¼ l. The probability that the demand market will be in state i is fi, i ¼ l, h, such that fh + fl ¼ 1. The buyer and the supplier learn the state of the demand sometime during the planning horizon. Therefore, the information update splits the planning horizon into two periods, possibly of unequal lengths. The buyer can procure capacity in both periods before the selling season begins. We assume that bh ¼ bl ¼ 1; i.e., the market is characterized by a linear inverse demand function Pi(Qi) ¼ ai Qi, where Pi(Qi) is the price of the product when Qi is the quantity sold by the buyer in the end-consumer market when the demand is in state i and ai is the market potential in state i. We assume that 0 < al ⩽ ah, that is, the market potential is positive in both states and higher under the high demand state. In this two-period model, the supplier announces the first period wholesale price at the beginning of the first period. He announces the second period wholesale price at the beginning of the second period after the demand state is revealed. The buyer chooses the first period procurement quantity before the demand state is revealed. She chooses the second period procurement quantity after the demand state is revealed. The market clears at the end of the second period.

230

F. Erhun et al.

Erhun et al. (2008) study this setting in detail (when there is limited capacity, more than two demand states, etc.). We present one of their results below to show how dynamic procurement extends to the situations with uncertain demand: Proposition 6. When demand is uncertain, the prices and quantities are set as follows: 8 < 9ðfh ah þfl al Þ; fh ah þfl al 16 8 ðw1 ; q1 Þ ¼ : 9fh ah ; ah 16 8 ðw2;i ; q2;i Þ ¼

if fh <

3al ah al

2

otherwise

þ a q þ ai i 1 q1 ; ; 2 4 2

i ¼ h; l:

Depending on the demand state, dynamic prices may increase (advance purchase discount) or decrease (markdown) over time. Dynamic procurement works for the supplier even under uncertain demand; the supplier’s increase with dynamic procurement. However, for the buyer, the value of dynamic procurement depends on when the supplier sets the price in a single procurement situation. If the supplier sets to price after the uncertainty is revealed (no-commitment model), then the buyer’s expected profits are also higher. Since both players benefit, it is a natural consequence that dynamic procurement eliminates supply chain inefficiencies compared to the no-commitment case. If the supplier sets to price before the uncertainty is revealed (early commitment model), the buyer’s expected profits may either increase or decrease under dynamic procurement. Especially when the difference between the market potentials of the high and low demand states is considerable, the buyer becomes worse off with dynamic procurement compared to the early commitment model. However, when the supplier chooses his capacity as well as the prices, for a wide-range of parameters, dynamic procurement is the best alternative for all parties, including the end customers.

5 Conclusion Dynamic procurement is commonly used to mitigate demand risk. However, our research shows that this may not be the only reason why companies use dynamic procurement. During our discussions with a consulting firm for a major manufacturer of finished goods, we observed a strong indication that the manufacturer was able to impact its raw material costs by using a multiple period sourcing approach. Interestingly, there was very little uncertainty in the finished good demand for this particular manufacturer and the supply of the material was constrained. Based on a data analysis over a 4-year horizon, the consulting firm concluded that higher inventory levels at the manufacturer showed strong correlations with reduced sourcing costs on a per-unit basis. A likely reason that the multiple period sourcing

Dynamic Procurement, Quantity Discounts, and Supply Chain Efficiency

231

helped to reduce prices was expressed by many of the manufacturer’s purchasers as “being able to walk away from the table.” The raw material supplier tended to know the inventory position of the manufacturer and understood the economics of the manufacturer’s business such that she could leverage her supply when the manufacturer had immediate shortage concerns. When the manufacturer felt strong enough to walk away, the negotiating position was reversed. In this paper, we analyzed this phenomenon with a stylized model. Similar to the efficiency result of Allaz and Vila, our main result states that, as the number of trading periods increases, the total output of the supply chain increases and approaches that of the centralized supply chain. Contrary to their result, we show that all the parties benefit from multiple trading periods. In equilibrium, dynamic procurement is similar to an incremental quantity discount where the supplier sets the prices and the buyer sets the breakpoints (Fig. 4). It can be viewed as a sequence of bilateral negotiations between the supplier and the buyer, and it provides incentives to both parties to increase the total supply chain profits. There are several directions for future research. In our model, the number of trading periods is known in advance. However, the situation where this information is not common knowledge, either to the buyer and/or to the supplier, would be interesting to analyze. Another possibility is to extend Erhun et al. (2008) and model a setting where the uncertainty is revealed to either player partially. Studying models with signalling or screening would be particularly beneficial to understand the dynamics when the uncertainty revelation is only partial. Finally, further studying a more general model where the wholesale prices can be negotiated and the products can be sold via N possible periods to competing buyers under partial demand revelation would be interesting.

a

b 1500 60

Unit Cost

Total Cost

55 1000

500

50 45 40

0

35 0

10

20 30 Quantity

40

0

10

20

30

40

Quantity

Fig. 4 The total and unit capacity costs versus quantities under dynamic procurement when N ¼ 20 (P(Q20) ¼ 100 Q20)

232

F. Erhun et al.

Acknowledgements The first author was partially supported by NSF Award DMI-0400345 and the second author was supported by NSF Career Award DMII-0093844. The authors would like to express their deepest gratitude to two anonymous reviewers for their constructive comments and suggestions.

References Allaz B, Vila JL (1993) Cournot competition, forward markets and efficiency. J Econ Theory 59 (1):1–16 Anand K, Anupindi R, Bassok Y (2008) Strategic inventories in vertical contracts. Manage Sci 54 (10):1792–1804 Benton WC, Park S (1996) A classification of literature on determining the lot size under quantity discounts. Eur J Oper Res 92(2):219–238 Burnetas A, Gilbert S (2001) Future capacity procurements under unknown demand and increasing costs. Manage Sci 47(7):979–992 Cachon GP (2004) The allocation of inventory risk in a supply chain: push, pull, and advancepurchase discount contracts. Manage Sci 50(2):222–238 Chen R, Roma P (2010) Group buying of competing retailers. Prod Oper Manage 20(2):181–197 Chen F, Federgruen A, Zheng Y (2001) Coordination mechanism for a distribution system with one supplier and multiple retailers. Manage Sci 47(5):693–708 Crowther J (1964) Rationale for quantity discounts. Harv Bus Rev 42(2):121–127 Cvsa V, Gilbert SM (2002) Strategic commitment versus postponement in a two-tier supply chain. Eur J Oper Res 141(3):526–543 Dada M, Srikanth KN (1987) Pricing policies for quantity discounts. Manage Sci 33 (10):1247–1252 Dolan RJ (1987) Quantity discounts: managerial issues and research opportunities. Mark Sci 6 (1):1–23 Dong L, Zhu K (2007) Two-wholesale-price contracts: push, pull, and advance purchase discount contracts. Manuf Serv Oper Manage 9(3):291–311 Donohue KL (2000) Efficient supply contracts for fashion goods with forecast updating and two production modes. Manage Sci 46(11):1397–1411 Elmaghraby W, Keskinocak P (2003) Dynamic pricing in the presence of inventory considerations: research overview, current practices, and future directions. Manage Sci 49 (10):1287–1309 Erhun F (2007) Dynamic pricing in Cournot duopoly with two production periods. Working paper, Department of Management Science and Engineering, Stanford University, Stanford, CA Erhun F, Keskinocak P, Tayur S (2008) Dynamic procurement in a capacitated supply chain facing uncertain demand. IIE Trans 40(8):733–748 Ferguson ME (2003) When to commit in a serial supply chain with forecast updating. Nav Res Logist 50(8):917–936 Ferguson ME, DeCroix GA, Zipkin PH (2005) Commitment decisions with partial information updating. Nav Res Logist 52(8):780–795 Guo P, Niu B, Wang Y (2009) Two-wholesale-price contract in a three-tier supply chain. Working paper, Hong Kong Polytechnic University, Hung Hom, Kowloon Gurnani H, Tang CS (1999) Note: optimal ordering decisions with uncertain cost and demand forecast updating. Manage Sci 45(10):1456–1462 Ingene CA, Parry ME (1995) Channel coordination when retailers compete. Mark Sci 14 (4):360–377 Iyer AV, Bergen ME (1997) Quick response in manufacturer-retailer channels. Manage Sci 43 (4):559–570

Dynamic Procurement, Quantity Discounts, and Supply Chain Efficiency

233

Jeuland AP, Shugan SM (1983) Managing channel profits. Mark Sci 2(3):239–272 Keskinocak P, Charnsirisakskul K, Griffin P (2003) Supply chain procurement with inventory and backordering options. Working paper, School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, GA Keskinocak P, Charnsirisakskul K, Griffin P (2008) Strategic inventory in capacitated supply chain procurement. Managerial Decis Econ 29(1):23–36 Li C, Scheller-Wolf A (2010) Push or pull? Auctioning supply contracts. Prod Oper Manage 20(2): 198–213 Monahan JP (1984) A quantity discount pricing model to increase vendor profits. Manage Sci 30 (6):720–726 Munson CL, Rosenblatt MJ (1998) Theories and realities of quantity discounts: an exploratory study. Prod Oper Manage 7(4):352–369 Raju J, Zhang ZJ (2005) Channel coordination in presence of a dominant retailer. Mark Sci 24 (2):254–262 Spencer BJ, Brander JA (1992) Pre-commitment and flexibility: applications to oligopoly theory. Eur Econ Rev 36(8):1601–1626 Taylor T (2006) Sale timing in a supply chain: when to sell to the retailer. Manuf Serv Oper Manage 8(1):23–42 Van Mieghem JA (2003) Capacity management, investment, and hedging: review and recent developments. Manuf Serv Oper Manage 5(4):269–302 Weng ZK (1995) Channel coordination and quantity discounts. Manage Sci 41(9):1509–1522

.

Coordination of the Supplier–Retailer Relationship in a Multi-period Setting: The Additional Ordering Cost Contract Nicola Bellantuono, Ilaria Giannoccaro, and Pierpaolo Pontrandolfo

Abstract Coordinating supply chains by adopting a centralized decision making approach, which is theoretically desirable, is often practically infeasible, if not ineffective: the high number of involved companies in a supply chain, the lack of adequate contractual power concentrated in the hand of only a few of them, the difficulty to gather all the relevant information by the unique/few decision maker/s, are some of the many reasons preventing supply chains from implementing such a centralized coordination approach. Supply contracts have been proposed in the literature as an alternate way to face such a problem: they let the chain’s partners to autonomously make decision, but at the same time guide them to behave coherently among each other as well as with the chain’s goal. Designing a contract is quite a challenging task, especially under the hypothesis of multi-period settings, which is the assumption considered in this chapter. In the majority of cases, multi-period supply contracts are inherently complex (e.g. many parameters that need to be frequently updated), therefore difficult to be implemented, as well as often designed under hypotheses barely realistic (e.g. null order costs). We propose a supply contract for a two-stage supply chain (supplier–retailer) in a multi-period setting, which tries to overcome such drawbacks. The proposed contract is based on two key mechanisms: additional ordering cost for retailer and price discount offered by the supplier to the retailer. A numerical analysis is finally conducted to identify the conditions that allow the best performance to be achieved. N. Bellantuono (*) • P. Pontrandolfo Dipartimento di Ingegneria dell’Ambiente e per lo Sviluppo Sostenibile, Politecnico di Bari, via De Gasperi s.n., 74100 Taranto, Italy e-mail: [email protected]; [email protected] I. Giannoccaro Dipartimento di Ingegneria Meccanica e Gestionale, Politecnico di Bari, viale Japigia 182, 70125 Bari, Italy e-mail: [email protected] T.-M. Choi and T.C. Edwin Cheng (eds.), Supply Chain Coordination under Uncertainty, International Handbooks on Information Systems, DOI 10.1007/978-3-642-19257-9_10, # Springer-Verlag Berlin Heidelberg 2011

235

236

N. Bellantuono et al.

Keywords Multi-period setting • Supply chain coordination • Supply contracts

1 Introduction A supply chain is a network of organizations that are involved in the different processes and activities to produce value in the form of products and services in the hands of the ultimate consumer (Christopher 1992). The coordination of this network is a key issue in supply chain management. System efficiency is assured when the supply chain is globally managed by a single decision maker who optimizes the performance of the whole system (channel coordination). This approach is usually referred to as centralized strategy (Federgruen 1993). Coordination problem becomes complex when different decision makers coexist within the supply chain, each taking decisions by pursuing their own goals, which are likely to be conflicting against each other (Schneeweiss 2003). These locally rational behaviours result in global inefficiency of the supply chain (Whang 1995). To improve the overall performance, the decision makers should be pushed to behave in the interest of the global supply chain rather than in their own ones. Such a problem is referred to as alignment of incentives in decentralized supply chains (Narayanan and Raman 2004). To align incentives in supply chains, supply contracts should be adopted. They formally rule the transaction between the actors and force them to pursue channel coordination. These mechanisms are based on transfer payment schemes that rule how to split the savings (or the increase in revenues) and let risks be fairly shared (Tsay et al. 1999; Cachon 2003). Transfer payment schemes are designed to increase the system-wide profit so as to make it closer – possibly equal – to the profit resulting from a centralized control (channel coordination). They modify the decision makers’ goals, bringing them to take the same decisions that the single decision maker would take. A further important issue for the contract design is the so-called win-win condition, which occurs if the contract makes every actor gain a profit higher than the one he or she would gain in default of the contract, i.e. under a decentralized setting. Indeed, if the win–win condition is not satisfied, the actor would not be prompted to adopt the contract (Giannoccaro and Pontrandolfo 2004). A number of supply contracts have been developed in the literature. They include quantity/volume discounts (Monahan 1984; Li and Liu 2006), buy back or return policies (Pasternack 1985; Emmons and Gilbert 1998), backup agreements (Eppen and Iyer 1997), allocation rules (Cachon and Lariviere 1999), quantity flexibility contracts (Tsay 1999; Wang and Tsao 2006), and revenue sharing contracts (Giannoccaro and Pontrandolfo 2004; Cachon and Lariviere 2005). For a complete review of supply contract the reader is referred to Tsay et al. (1999), Cachon (2003), and Tang (2006). Most of these contracts address the problem of coordinating supply chains under a single-period setting, namely assuming that the items that flow along the supply

Coordination of the Supplier–Retailer Relationship in a Multi-period Setting

237

chain are perishable or have a quickly decreasing demand. In these cases, the selling season is assumed short enough to impel both new orders be issued once demand has revealed and stocks in excess be held until the following selling season. This assumption permits the adoption of models deriving from the classical newsvendor model (Wadsworth 1959; Hadley and Whitin 1963). On the contrary, few studies have analyzed the same problem under a multi-period setting. In this case, further parameters (e.g. the ordering and inventory costs) must be taken into account, so making more complex not only the design of the contract but also its implementation in practice. The aim of this work is thus to develop an supply contract to coordinate a twostage supply chain under a multi-period setting, which is effective as well as simple to be implemented. In particular, we consider a supply chain in which a supplier provides in a random lead time a single product to a retailer, who in turn serves the market characterized by uncertain demand. Both incur ordering and inventory holding costs. Moreover, when final demand cannot be immediately satisfied, the retailer incurs backorder cost. The contract is designed so as to achieve both channel coordination and win–win condition and is mainly based on two mechanisms (1) the additional ordering cost that the retailer pays to the supplier and (2) the price discount that the supplier gives to the retailer. In the following section a review of contracts for multi-period settings is presented. Then, the supply chain model is described and the contract is designed. A numerical analysis is finally carried out to illustrate how the supply contract works and to identify the scenarios where the contract is more effective.

2 Supply Contracts in Multi-period Settings: A Review Since the 1980s, several studies have been carried out to design contracts for supply chain management, yet only lately research has addressed also multi-period settings. This section reviews the literature on multi-period contracts and describes the most significant contract models. The review proves useful to derive insights and suggestions supporting the design of an innovative contract. The quantity-flexibility multi-period contract by Tsay and Lovejoy (1999) derives from the homonymous contract for the single-period setting (Tsay 1999), which is modified mainly through the adoption of a rolling horizon: at the beginning of each selling season, the retailer and the supplier agree on (1) the wholesale price of the product, (2) the demand forecast for each season up to the planning horizon, and (3) the minimum and maximum quantity for the delivery. In the following seasons, actors update the demand forecast and the contract parameters, to reduce the allowable demand fluctuation for every future season as it becomes closer. Tsay and Lovejoy (1999) argue that their contract enables actors to share the risk related to demand uncertainty: the retailer is guaranteed for a minimum delivery of goods and so she recedes from the rationing game (Lee et al. 1997), i.e. the practice

238

N. Bellantuono et al.

to communicate overestimated forecasts in order to secure the possible delivery of more units than expected as demand. In turn, the supplier is kept from the risk that actual retailer’s orders will be lower than he predicts. Nevertheless, the need to continually update contract parameters restricts the adoption of the quantityflexibility contract to firms willing to cooperate and having human resources able to deal with so complex schemes. Steady-state contracts are simpler to be implemented; indeed, once they have been agreed, they are valid also in future periods without any need for parameters’ updating. Recently, literature has focused on this class of contracts (for a review, see Cachon 2003) and pointed out which clauses can be useful to coordinate the SC. They may include: – Additional penalties for the backorder at each stage of the SC – Grants to boost holdings – Time lag between physical and financial flows Cachon and Zipkin (1999) recur to the game theory to affirm that a supply chain can be coordinated by a contract that uses a linear money transfer from an actor to another to align the Nash equilibrium (i.e. the equilibrium in a decentralized setting) to the global optimum. By adopting such a contract, the retailer is encouraged to increase her stock; the supplier, in turn, increases the punctuality of his deliveries to avoid retailer’s stockouts. A contract similar to that by Cachon and Zipkin (1999) is the additional backlog penalty contract (Lee and Whang 1999): by a shortage reimbursement contract, the supplier pays the retailer when he cannot meet her orders and receives money from her when she has a stockout. Such a mechanism urges actors to increase their holding stock, which otherwise in a decentralized setting would be lower then what the optimal solution prescribes. Lee and Whang (1999) emphasize three properties of the proposed contract, which indeed are common to many others: cost conservation, incentive compatibility, and information decentrability. Cachon (2003) shows that the shortage reimbursement contract substantially agrees with the methodology by Chen (1999) to coordinate systems with a delayed receipt of orders. Also the contract by Porteus (2000), based on the so called responsibility tokens, is similar to the shortage reimbursement contract. In fact, it results in the same transfer payment between actors, although it assigns to the supplier a penalty per late delivered unit, instead of a penalty per occurrence of a – partial or total – late delivery. Lee and Whang (1999) observe that a method to coordinate supply chains based on time discounting consists in abandoning the hypothesis that physical transfers and related financial flows are concurrent. In particular, authors describe the consignment contract, by which retailers give money to the suppliers only once products are sold to the final customers. The steady-state contracts do not need to continually update the parameters, differently from the rolling horizon contracts. Unfortunately, to our knowledge the steady-state contracts so far developed hold under hypotheses that are barely realistic. Their most critical aspect refers to the ordering cost, which in the literature

Coordination of the Supplier–Retailer Relationship in a Multi-period Setting

239

on supply chain inventory management is generally assumed as null (Axs€ater 1990; Axs€ater 1993; Cachon and Zipkin 1999; Lee and Whang 1999) or proportional to order quantity (Shang and Song 2003). In both cases, the optimal replenishment quantity becomes qi ¼ 1 and the search of optimal policies collapses to the simple search – for each actor – of the order point that minimizes the sum of expected holding and backorder costs. By analyzing each cost factor that adds up to the ordering cost, one can demonstrate the ineffectiveness of these assumptions: in fact, the ordering cost is a step-function of the order quantity and thus it can be assumed as constant for a wide interval of values for the latter.

3 The Model Consider a serial two-stage supply chain consisting of a retailer, who faces the market demand, and a supplier, who provides items to the retailer.1 Both actors are risk-neutral, i.e. their utility functions are proportional to the profit (Schweitzer and Cachon 2000), and financial flows occur simultaneously to the corresponding physical flows. At both stages inventories are managed by a continuous review system ruled by the pair ðqi ; ri Þ, the former being the order quantity ðqi > 0Þ and the latter the reorder point ðri > 0Þ. The unmet demand is fully backordered (Fig. 1). The instantaneous demand is a continuous random variable. All the notations are summarized in Table 1. For the sake of simplicity, we prefer to assume that the supplier’s lead time is null, so that the latter can issue his orders at the same time he receives the retailer’s order. In fact, this assumption is usually adopted when it is wished to neglect the effect of suppliers external to the supply chain. This assumption also implies that the demand during the supplier’s lead time is null ðm2 ¼ s2 ¼ 0Þ, as well as the

external supplier

L2

supplier

L1

retailer

(stage 2)

(stage 1)

A2 h2

A1 h1 b

market

d

Fig. 1 The supply chain model

1 For the sake of clarity, we denote the retailer as actor 1 and the supplier as actor 2. Moreover, we use the pronoun “she” for the retailer and “he” the supplier.

240 Table 1 Model notations

N. Bellantuono et al.

Variable D 1 2 mi si Ai hi b qi ri k n ’(k) F(k) () Ci() C() d * c a d

Description Expected annual demand Retailer’s subscript Supplier’s subscript Mean of the demand during the i-th actor’s lead time Standard deviation of the demand during the i-th actor’s lead time Ordering cost at the i-th stage Annual holding cost per unit at the i-th stage Backorder cost per unit Order quantity at the i-th stage Reorder point at the i-th stage Safety factor Nested factor p.d.f. of the standard normal distribution c.d.f. of the standard normal distribution Expected shortage per replenishment cycle Expected annual cost of the i-th actor Expected annual cost of the supply chain Decentralized setting superscript Centralized setting superscript Contract setting superscript Additional ordering cost (contract parameter) Discount per unit sold (contract parameter)

supplier’s reorder point ðr2 ¼ 0Þ and his expected backorder stock. The retailer’s lead time, in turn, is a random positive variable; demand during the retailer’s lead time is normally distributed and its mean m1 and standard deviation are both known and denoted as m1 and s1 , respectively. The probability density function and cumulative distribution function of the standard normal distribution are respectively denoted as ’ðkÞ and FðkÞ. Thus, the retailer’s order point can be expressed in terms of the safety factor, as follows: k¼

r1 m1 : s1

(1)

Therefore, the supplier’s expected annual cost consists in the sum of expected ordering and holding cost: the former is proportional to the expected number of orders per year, whereas the latter is proportional to the units that he holds on average. We assume also that the holding cost of the pipeline stock (units in transit from the supplier to the retailer) is paid by the former. The retailer, in turn, is affected by ordering, holding, and backorder cost. The latter is assumed proportional to the number of units backlogged, irrespective of the time for which the backorder lasts.

Coordination of the Supplier–Retailer Relationship in a Multi-period Setting

3.1

241

Decentralized Setting

Each actor autonomously makes inventory policy decisions, aimed at minimizing his/her own cost. Hence, two separate optimization problems have to be solved (1) finding the pair ðqd1 ; kd Þ that minimizes the retailer’s cost, and (2) identifying the parameter qd2 that minimizes the supplier’s cost, given the retailer’s policy. Under the assumptions above described, the retailer’s expected annual cost is calculated as follows (Hadley and Whitin 1963; Silver et al. 1998)2: C1 ðq1 ; kÞ ¼ A1

hq i D D 1 þ s1 k þ b ðkÞ; þ h1 2 q1 q1

(2)

where the quantity between square brackets is the expected net inventory and: ðkÞ ¼ s1 ½’ðkÞ þ kFðkÞ k

(3)

is the expected shortage per replenishment cycle, i.e. the unmet demand between two consecutive orders [see Appendix A.2]. The optimal retailer’s policy ðqd1 ; kd Þ minimizing (2) can be obtained by the iterative procedure described in Hadley and Whitin (1963) using the following equations [see Appendix A.3]: qd1

rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2D ¼ ½A1 þ bðkd Þ; h1

(4)

and h1 qd1 kd ¼ F1 1 : bD

(5)

To solve the supplier’s optimization problem, we assume that he adopts a nested policy, which allows the computational complexity of the problem to be reduced at the expense of a slight possible decrease of the solution effectiveness (Schwarz and Schrage 1975; Roundy 1985; Axs€ater and Rosling 1993). When a nested policy is used, a stage can issue a order only when the downstream stage does the same. This implies that when the supplier makes a order, his order quantity will be: q2 ¼ nq1 ;

(6)

being the nested factor n a positive integer. 2

The (2) is an approximation but consistent with a wide stream of the literature on inventory management (Hadley and Whitin 1963; De Bodt and Graves 1985; Silver et al. 1998; Mitra and Chatterjee 2004). See Appendix A.1.

242

N. Bellantuono et al.

Therefore, the supplier’s inventory problem is reduced to finding the positive integer nd that minimizes his expected annual cost: C2 ðn; q1 Þ ¼ A2

D n1 þ h2 q1 þ m1 ; nq1 2

(7)

where the quantity between square brackets is the expected net inventory at the supplier’s stage and includes the expected pipeline stock, i.e. the expected quantity in transit from the supplier to the retailer. To this aim, the following procedure is suggested: 1. Compute: qﬃﬃﬃﬃﬃﬃﬃﬃ n^ ¼

2A2 D h2 : d q1

(8)

2. If n^ is an integer, then nd ¼ n^; otherwise: nd ¼ arg min C2 ðn; qd1 Þ; n2fn1 ;n2 g

(9)

where n1 and n2 are the positive integers that surround n^.

3.2

Centralized Setting

Under a centralized setting, the optimal inventory policy is the one that minimizes the expected annual cost of the whole supply chain, whose formula is given by referring to the model by De Bodt and Graves (1985). This model adopts a nested policy using an echelon perspective. The echelon perspective requires to compute (1) the echelon stock of each stage, i.e. the sum of the stock at the stage and all the downstream stages, including the pipeline stock, (2) the echelon holding cost (i.e. the incremental inventory cost at a given stage with reference to the upstream stage), and (3) the echelon order point (i.e. the sum of the order point at the considered stage and those at all the upstream stages). Under the current hypotheses, the expected total supply chain cost is given by De Bodt and Graves (1985). By substituting the echelon cost expressions with the correspondent installation ones, it follows that: hq i A2 D n1 D 1 þ s1 k þ h2 Cðq1 ; k; nÞ ¼ A1 þ þ h1 q1 þ m1 þ b ðkÞ; (10) n q1 2 2 q1 wherein the expected shortage per replenishment cycle ðkÞ is given by (3) and the nested factor n is a positive integer. The retailer’s and supplier’s expected annual costs are given by (2) and (7), respectively.

Coordination of the Supplier–Retailer Relationship in a Multi-period Setting

243

The centralized inventory problem is to find the optimal policy ðq1 ; k ; n Þ that minimizes (10). To solve it, a heuristics based on the continuous relaxation of the problem is proposed, consisting of the following steps (see Appendix A.4): 1. Assume ðkÞ ¼ 0 and compute:

rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ A2 h1 h2 : n~ ¼ A1 h2

(11)

2. Compute: sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2D A2 q1 ð~ nÞ ¼ A1 þ þ bðkÞ n~ h1 þ ð~ n 1Þh2

(12)

3. Compute: 1

kðq1 Þ ¼ F

h1 q1 ð~ nÞ 1 bD

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2A2 D=h2 n~ðq1 Þ ¼ q1

(13)

(14)

4. Iterate steps 2–3 until a suitable approximation is obtained. 5. If n~ðq1 Þ is an integer, then n ¼ n~ðq1 Þ; otherwise denote the positive integers that surround n~ as n1 and n2 . 6. For both n1 and n2 iteratively use (12) and (13) until convergence to compute q1 ðn1 Þ and k ðn1 Þ and, respectively, q1 ðn2 Þ and k ðn2 Þ. 7. Compute (10). The optimal value for the nested factor is: n ¼ arg min C n; q1 ðnÞ; k ðnÞ ; (15) n2fn1 ;n2 g

where q1 ðnÞ and k ðnÞ are the correspondent conditionally optimal values of the other two variables, as determined at step 6.

4 The Additional Ordering Cost Contract The additional ordering cost contract aims to push retailer to make larger orders than she will do under a decentralized setting, so as to led the supply chain to behave like in a centralized fashion. In fact, by comparing the decentralized and the centralized settings, we notice that the inefficiency of the decentralized setting is due to the fact that the retailer makes smaller and more frequent orders. The additional ordering cost contract is then based on a transfer payment from the supplier to the retailer, so defined:

244

N. Bellantuono et al.

FTðq1 ; a; dÞ ¼ dD aA1

D : q1

(16)

It is ruled by two parameters (1) the penalty that the supplier gives to the retailer for each order she issues ðaÞ, and (2) the discount that the suppliers grants to the retailer for each unit sold ðdÞ. Through a suitable design of the values for a and d both the channel coordination and the win–win condition are assured. The retailer’s and the supplier’s expected annual costs under the contract setting are given by the equations, respectively: Cc1 ðq1 ; k; a; dÞ ¼ C1 ðq1 ; kÞ FTðq1 ; a; dÞ hq i D D 1 þ s1 k þ b ðkÞ dD ¼ ð1 þ aÞA1 þ h1 2 q1 q1

(17)

and: Cc2 ðn; q1 ; a; dÞ ¼ C2 ðn; q1 Þ þ FTðq1 ; a; dÞ D n1 D þ h2 ¼ A2 q1 þ m1 þ dD aA1 : nq1 2 q1

(18)

Thus, as in the decentralized setting even under the contract two separate optimization problems have to be solved (1) the retailer’s problem is to identify the pair ðqc1 ; kc Þ that minimizes (17), and (2) the supplier’s problem is to find the positive integer nc that minimizes (18). In particular, from (17) it follows: rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2D c q1 ðaÞ ¼ ½ð1 þ aÞA1 þ bðkc Þ: (19) h1 and 1

k ¼F c

h1 qc1 1 : bD

(20)

Proposition 1. The channel coordination is achieved if the actors agree on an additional ordering cost contract where: a¼

h1 A 2 n

ðn 1Þh2 ½A1 þ bðk Þ : A1 ½h1 þ ðn 1Þh2

(21)

(Proof: See Appendix A.5). Proposition 2. Once the channel coordination is achieved, there exists a range of values for d which assure the win–win condition. (Proof: See Appendix A.6). Observation 1. Channel coordination does not depend on d. Observation 2. The annual expected costs of both actors linearly depend on the value of d.

Coordination of the Supplier–Retailer Relationship in a Multi-period Setting

245

5 Numerical Analysis For illustrative purposes, in this section a numerical analysis is provided to give a measure of the inefficiency of the decentralized approach and to show how the additional ordering cost contract works. Results prove that the contract coordinates the channel and assures a win–win condition. As a measurement of the inefficiency of the decentralized setting, the competition penalty (Cachon and Zipkin 1999) is defined as follows: C qd1 ; kd ; nd C q1 ; k ; n CP ¼ 100 : C q1 ; k ; n

(22)

The higher CP, the higher the penalty in terms of increase of the supply chain expected cost. Data used in the numerical analysis are shown in Table 2. The latter consists of 27 scenarios designed by varying the value ðs1 =m1 Þ, the ratios between the ordering costs ðA1 =A2 Þ, and the annual holding costs per unit ðh2 =h1 Þ. The sensitivity analysis aims to identify the scenarios where the contract is more effective. Table 3 shows the retailer’s, the supplier’s, and the system-wide expected annual costs both in the decentralized and in the centralized setting, as well as the corresponding competition penalty in all scenarios. As we expected, CP is always positive, which means that the centralized setting provides better system-wide performances than the decentralized one. Furthermore, the retailer’s performance gets worse moving from the decentralized to the centralized setting, so explaining why the retailer has no incentive to agree on system-wide optimal policy and why a supply contract is thus necessary. The data presented in Table 3 are analyzed in Table 4, where for each value of s1 , A1 , and h2 the means of the competition penalty obtained for the three levels of the other two variables are reported. Results show that CP is positively affected by h2 and to a smaller extent by A1 , whereas the effects of s1 on CP are in general negligible, in spite of the differences in the expected costs. Therefore, the contract proves very useful especially when the holding costs per unit at both stages are similar, irrespective of the demand variability or the difference between retailer’s Table 2 Values used in the numerical analysis.

Variable D m1 s1 A1 A2 h1 h2 b

Levels 1 1 3 3 1 1 3 1

Values 1,000 100 10, 20, 30 5, 20, 35 50 1 0.5, 0.7, 0.9 10

246

N. Bellantuono et al.

Table 3 Retailer’s, supplier’s and system-wide expected annual costs in decentralized and centralized settings for different values of s1, A1, and h2, given D ¼ 1,000, m1 ¼ 100, A2 ¼ 50, h1 ¼ 1, and b ¼ 10 A1 h2 Decentralized setting Centralized setting CP (%) s1 10

5

20

35

20

5

20

35

30

5

20

35

0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9

C

C1

375.01 426.05 470.80 447.82 488.20 528.57 497.95 543.99 563.99 400.12 452.24 495.10 470.64 511.39 552.14 520.67 564.39 584.39 425.35 477.41 519.57 493.43 534.57 575.71 543.37 584.77 604.77

126.59 126.59 126.59 224.18 224.18 224.18 287.71 287.71 287.71 153.07 153.07 153.07 248.28 248.28 248.28 310.79 310.79 310.79 179.43 179.43 179.43 272.32 272.32 272.32 333.82 333.82 333.82

Table 4 Mean competition penalty for different values of s1, A1, and h2, given D ¼ 1,000, m1 ¼ 100, A2 ¼ 50, h1 ¼ 1, and b ¼ 10

A1 Mean (CP) h2 Mean (CP) s1 mean (CP)

C2 248.42 299.45 344.21 223.65 264.02 304.39 210.24 256.28 276.28 247.05 299.17 342.03 222.35 263.10 303.86 209.88 253.60 273.60 245.91 297.98 340.14 221.11 262.25 303.39 209.56 250.95 270.95

C 369.68 413.79 443.93 440.86 465.96 485.96 483.71 503.71 523.71 394.93 438.17 466.15 464.26 487.70 507.70 505.07 525.07 545.07 420.14 462.50 488.32 487.61 509.41 529.41 526.39 546.39 566.39

5 3.56% 0.5 1.93% 10 4.91%

C1 132.56 145.72 204.95 227.97 263.75 263.75 312.21 313.63 313.63 159.01 170.83 228.93 251.76 286.91 286.91 333.31 336.17 336.17 184.63 195.87 252.85 275.50 310.01 310.01 354.36 358.67 358.67

C2 236.36 268.07 238.98 212.89 202.21 222.21 170.08 190.08 210.08 235.92 267.34 237.22 212.49 200.80 220.80 168.90 188.90 208.90 235.51 266.63 235.47 212.12 199.40 219.40 167.73 187.73 207.73

20 5.00% 0.7 5.16% 20 4.83%

1.44 2.96 6.05 1.58 4.77 8.77 2.94 8.00 7.69 1.31 3.21 6.21 1.37 4.86 8.75 3.09 7.49 7.21 1.24 3.22 6.40 1.19 4.94 8.75 3.23 7.02 6.78

35 5.94% 0.9 7.40% 30 4.75%

and supplier’s ordering costs. Moreover, as ordering costs at both stages become similar, the competition penalty increases ceteris paribus, thus the possible benefit deriving from the adoption of the contract grows. Finally, keeping equal the ratios A1 =A2 and h2 =h1 , an increase in demand variability results in a CP slightly decreasing. However, it does not mean that the additional ordering contract

Coordination of the Supplier–Retailer Relationship in a Multi-period Setting

247

becomes ineffective or unnecessary: indeed, since the expected annual cost increase in demand variability, even if CP reduces the savings that can be obtained through the contract are significant. From Table 5, which gives the optimal values for q1 ; k; and n in both settings, we can see that in all the scenarios the optimal retailer’s order quantity is higher in the centralized policy than in the decentralized one. This justifies why the additional ordering cost contract has been designed so as to make the retailer increase her order quantity, by a penalty paid whenever she issues an order. Table 6 illustrates the value of a that allows the channel coordination be achieved in all the scenarios and the corresponding range for d where also the win–win condition is satisfied. In particular, below the minimum value of such an interval the contract is not convenient for the retailer (namely, it increases her expected cost compared to the decentralized setting), whereas above the maximum value it is not convenient for the supplier. Table 5 Optimal values for q1 ; k; and n in decentralized and centralized settings for different values of s1, A1, and h2, given D ¼ 1,000, m1 ¼ 100, A2 ¼ 50, h1 ¼ 1, and b. ¼ 10 A1 h2 Decentralized setting Centralized setting s1 10

5

20

35

20

5

20

35

30

5

20

35

0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9

qd1

kd

nd

q1

k

n

103.5 103.5 103.5 203.7 203.7 203.7 268.4 268.4 268.4 107.1 107.1 107.1 207.5 207.5 207.5 272.3 272.3 272.3 110.8 110.8 110.8 211.4 211.4 211.4 276.3 276.3 276.3

2.3136 2.3136 2.3136 2.0461 2.0461 2.0461 1.9294 1.9294 1.9294 2.3006 2.3006 2.3006 2.0385 2.0385 2.0385 1.9231 1.9231 1.9231 2.2876 2.2876 2.2876 2.0308 2.0308 2.0308 1.9168 1.9168 1.9168

4 4 3 2 2 2 2 1 1 4 4 3 2 2 2 2 1 1 4 3 3 2 2 2 2 1 1

149.0 190.0 335.6 247.5 378.2 378.2 416.4 416.4 416.4 150.8 192.2 339.6 250.0 382.3 382.3 420.5 420.5 420.5 152.6 194.4 343.7 252.6 386.4 386.4 424.7 424.7 424.7

2.1728 2.0748 1.8308 1.9643 1.7766 1.7766 1.7320 1.7320 1.7320 2.1680 2.0701 1.8255 1.9599 1.7716 1.7716 1.7273 1.7273 1.7273 2.1632 2.0654 1.8201 1.9555 1.7667 1.7667 1.7227 1.7227 1.7227

3 2 1 2 1 1 1 1 1 3 2 1 2 1 1 1 1 1 3 2 1 2 1 1 1 1 1

248

N. Bellantuono et al.

Table 6 Optimal value for a and minimum and maximum value for d that let the additional ordering cost contract achieve channel coordination and win–win condition for different values of s1, A1, and h2, given D ¼ 1,000, m1 ¼ 100, A2 ¼ 50, h1 ¼ 1, and b ¼ 10 A1 h2 a dmin dmax s1 10 5 0.5 1.1253 0.0437 0.0498 0.7 2.4723 0.0842 0.0964 0.9 10.0000 0.2273 0.2542 20 0.5 0.4844 0.0429 0.0499 0.7 2.5000 0.1718 0.1940 0.9 2.5000 0.1718 0.2144 35 0.5 1.4286 0.1446 0.1602 0.7 1.4286 0.1460 0.1863 0.9 1.4286 0.1460 0.1863 20 5 0.5 1.0597 0.0411 0.0463 0.7 2.4137 0.0805 0.0946 0.9 10.0000 0.2231 0.2520 20 0.5 0.4685 0.0410 0.0473 0.7 2.5000 0.1694 0.1931 0.9 2.5000 0.1694 0.2139 35 0.5 1.4286 0.1414 0.1599 0.7 1.4286 0.1443 0.1836 0.9 1.4286 0.1443 0.1836 30 5 0.5 1.0040 0.0381 0.0433 0.7 2.3537 0.0770 0.0919 0.9 10.0000 0.2189 0.2501 20 0.5 0.4522 0.0390 0.0448 0.7 2.5000 0.1671 0.1922 0.9 2.5000 0.1671 0.2134 35 0.5 1.4286 0.1383 0.1596 0.7 1.4286 0.1426 0.1810 0.9 1.4286 0.1426 0.1810

6 Concluding Remarks This work has proposed an innovative contract to manage supplies in decentralized two-stage supply chains characterized by: random, independent demand and lead time; infinite planning horizon; continuous review of inventory; total backorder of the unmet demand. The contract is innovative because it takes into account ordering costs, which are usually neglected in the literature on multi-period supply contracts. It ensures both the system-wide efficiency and the win–win condition. Furthermore, the proposed contract is straightforward to be implemented, since it requires that the actors agree on two parameters only, which control how costs are split up among the actors. In particular, the contract is ruled by the parameter a, which is the penalty that the retailer imposes to the supplier for each order, and the parameter d, which specifies the discount per unit that the supplier grants to the retailer.

Coordination of the Supplier–Retailer Relationship in a Multi-period Setting

249

Finally, the contract has two interesting properties (1) the parameter a is enough to drive the system to efficiency; (2) the parameter d has a linear effect on the expected cost of each actor. As a result, each actor is able to assess the benefits that the contract provides not only to him/her but also to his/her counterpart. In this way typical cautious behaviour that characterize the negotiation phase on the contract parameters should be mitigated. We believe that there are several directions to which this field of study can be extended. Further research would be addressed to the application of the additional ordering cost contract to more complex supply chains: in particular, it can be interesting to analyze how to extend its application field to distribution supply chain, characterized by arborescent topologies, as well as to supply chains having more than two stages. Another possible extension of this research would consist in analyzing if the actors perceive as fair the agreement of the contract: to this aim the research could encompass both on field studies and laboratory experiments. Finally, the performance of contract can be compared to the ones assured by other coordination schemes. Acknowledgments This work has been supported by Regione Puglia (APQ PS025 - ICT supporting logistics services: a model of organized market).

Appendix: Proofs and Discussions Approximations in Equation (2) Two approximations are made in (2). They refer to the computations of the expected on hand inventory and the expected annual shortage, which respectively affect the expected annual inventory cost and backorder cost. We discuss both approximations in the followings. Let us denote the retailer’s lead time as L1 and the probability density function of the demand during L1 as f ðxjL1 Þ. By definition, the expected on hand inventory: OHðq1 ; r1 Þ ¼

q1 þ 2

ð r1

ðr1 xÞf ðxjL1 Þdx

(23)

0

is equal to the expected net inventory plus the expected backorder stock. However, if the backorder cost per unit is high, the expected backorder stock is negligible compared to the expected net inventory. Thus, the expected on hand inventory can be assumed equal to: NIðq1 ; r1 Þ ¼

q1 þ 2

ð þ1 0

ðr1 xÞf ðxjL1 Þdx ¼

q1 þ r 1 m1 : 2

(24)

250

N. Bellantuono et al.

If the demand during the retailer’s lead time is normally distributed, by recurring to the safety factor defined in (1), the expected net inventory can be also expressed as: NIðq1 ; kÞ ¼

q1 þ ks1 : 2

(25)

The expected annual shortage is equal to the number of replenishments per year ðD=qi Þ times the expected shortage per replenishment cycle ðkÞ. The approximation made in (2) consists in assuming that the latter is equal to the expected backorder stock when an order arrives, which is exact only if all backorders are satisfied within one cycle.

Proof of Equation (3) The expected shortage per replenishment cycle is: ðr1 Þ ¼

ð þ1

ðx r1 ÞfðxjL1 Þdx:

(26)

r1

By assuming the demand during the retailer’s lead time normally distributed, being z ¼ ðx m1 Þ=s1 and k as in (1), we obtain that x ¼ m1 þ s1 z and r1 ¼ m1 þ s1 k, and observe that dx ¼ s1 dz. Therefore, (26) becomes: ðkÞ ¼

ð þ1

s1 ðz kÞ’ðzÞdz ¼ s1 ½’ðkÞ þ kFðkÞ k:

(27)

k

Proof of Equations (4) and (5) To minimize (2), we impose the first order conditions: 8 @ D h1 D > > < 0 ¼ @q C1 ðq1 ; kÞ ¼ A1 q þ b q ðkÞ 2 1 1 1 ; > @ D > : 0 ¼ C1 ðq1 ; kÞ ¼ h1 s1 þ b s1 ½FðkÞ 1 @k q1

(28)

where we have observed that, in case of normally distributed demand, since: @’ðkÞ @ 1 k2 =2 p ﬃﬃﬃﬃﬃ ﬃ e ¼ k’ðkÞ; ¼ @k @k 2p

(29)

Coordination of the Supplier–Retailer Relationship in a Multi-period Setting

251

then: @ ðkÞ ¼ s1 ½FðkÞ 1: @k

(30)

Rearranging (28), both (4) and (5) derive.

Discussion on the Heuristics for the Centralized Setting The continuous relaxation of the problem consist in finding the minimum of the following equation: h i ~ 1 ; k; n~Þ ¼ A1 þ A2 þ bðkÞ D þ h1 q1 þ s1 k þ h2 n~ 1 q1 þ m1 : (31) Cðq q1 n~ 2 2 It is equal to (10) except for the variable n~, which is a positive real number instead of a positive integer as n. The first order condition consists in imposing that the first derivatives of (31) are null: @ ~ A2 D 1 Cðq1 ; k; n~Þ ¼ A1 þ þ bðkÞ 2 þ ½h1 þ ðn 1Þh2 0¼ @q1 n~ q1 2 0¼

(32)

@ ~ bD Cðq1 ; k; n~Þ ¼ h1 þ ½FðkÞ 1 @r1 q1

(33)

@ ~ A2 D h2 q1 Cðq1 ; k; n~Þ ¼ þ 2 @ n~ q1 n~2

(34)

0¼

which respectively result in (12)–(14). Rearranging (14) and combining it with (12), we obtain: sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ A2 h1 h2 n~ ¼ : A1 þ bðkÞ h2

(35)

Assuming ðkÞ ¼ 0, from (35) we derive (11), which can be used as starting value for the heuristics. It can be observed that (11) is equal to the optimal choice for n in the continuous relaxation of the deterministic problem, as described in Silver et al. (1998). By recursively calculating (12)–(14), the optimal solution for the relaxed problem (31) is obtained. To derive the one for the original problem, as in De Bodt and Graves (1985) we conjecture that (10) is unimodal in q1 ðnÞ; k ðnÞ; n .

252

N. Bellantuono et al.

Proof of Proposition 1 The achievement of the channel coordination implies that the actors autonomously define their policies so as to allow the expected annual system-wide cost be minimized. By summing (17) and (18), it can be observed that the expected annual system-wide cost is the same as the one in default of the contract – see (10) – and does not depend on the contract parameters. Therefore, a sufficient condition is to assure that: qc1 ¼ q1 ;

(36)

k1c ¼ k1 ;

(37)

nc ¼ n :

(38)

and

To prove (37), it is enough to observe that (13) and (20) have the same analytical expression, and are identical if (36) holds. Moreover, combining (12) and (20), (36) can be written as: rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2D 2D A2 c ½ð1 þ aÞA1 þ bðk Þ ¼ A1 þ þ bðk Þ : n h1 h1 þ ðn 1Þh2

(39)

When (20) holds, (39) can be rearranged so as to obtain (21). Once both (36) and (37) are satisfied, to obtain the channel coordination it is enough to choose nc so as to satisfy (38), too.

Proof of Proposition 2 The win–win condition is assured if each actor incurs lower cost under the contract setting than under the decentralized setting. Thus, it follows that:

Cc1 qc1 ; kc ; a; d C1 qd1 ; kd : Cc2 qc1 ; nc ; a; d C2 qd1 ; nd

(40)

Let us remember that, when the actors agree on the additional ordering cost contract, the expected annual retailer’s and supplier’s costs can be expressed in terms of the cost they sustain in default of the contract and the transfer payment, as shown in (17) and (18). Therefore, (40) can be written also as:

Coordination of the Supplier–Retailer Relationship in a Multi-period Setting

8 d d D > > < C1 q1 ; k dD þ aA1 q C1 q1 ; k 1 d d ; > D > : C2 q1 ; n þ dD aA1 C2 r1 ; n q1

253

(41)

which can be rearranged so as to obtain: C1 q1 ; k C1 qd1 ; kd C2 qd1 ; nd C2 q1 ; n aA1 aA1 þ d þ : q1 q1 D D

(42)

To prove that such a range for d is defined in a consistent domain, we consider the first and third members of (42), which can be rearranged to obtain: C1 q1 ; k þ C2 q1 ; n C1 qd1 ; kd þ C2 qd1 ; nd :

(43)

C q1 ; k ; n C qd1 ; kd ; nd :

(44)

which means:

The inequality above is always true by definition.

References Axs€ater S (1990) Simple solution procedures for a class of two echelon inventory problems. Oper Res 38(1):64–69 Axs€ater S (1993) Exact and approximate evaluation of batch-ordering policies for two level inventory systems. Oper Res 41(4):777–785 Axs€ater S, Rosling K (1993) Notes: installation vs. echelon stock policies for multilevel inventory control. Manage Sci 39(10):1274–1280 Cachon GP (2003) Supply chain coordination with contracts. In: Graves S, de Kok T (eds) Handbooks in operations research and management science: supply chain management. North-Holland, Amsterdam Cachon GP, Lariviere MA (1999) Capacity choice and allocation: strategic behaviour and supply chain performance. Manage Sci 45(8):1091–1108 Cachon GP, Lariviere MA (2005) Supply chain coordination with revenue sharing contracts: strength and limitations. Manage Sci 51(1):30–44 Cachon GP, Zipkin PH (1999) Competitive and cooperative inventory policies in a two stage supply chain. Manage Sci 45(7):936–953 Chen F (1999) Decentralized supply chains subject to information delays. Manage Sci 45(8): S221–S234 Christopher M (1992) Logistics and supply chain management. Pitman, London De Bodt MA, Graves SC (1985) Continuous review policies for a multi-echelon inventory problem with stochastic demand. Manage Sci 31(10):1286–1299 Emmons H, Gilbert S (1998) Return policies in pricing and inventory decision for catalogue goods. Manage Sci 44(2):276–283

254

N. Bellantuono et al.

Eppen GD, Iyer AV (1997) Backup agreements in fashion buying: the value of upstream flexibility. Manage Sci 43(11):1469–1484 Federgruen A (1993) Centralized planning models for multi-echelon inventory systems under uncertainty. In: Graves S, Rinnooy Kan A, Zipkin P (eds) Logistics of production and inventory. North Holland, Amsterdam, pp 133–173 Giannoccaro I, Pontrandolfo P (2004) Supply chain coordination by revenue sharing contracts. Int J Prod Econ 89:131–139 Hadley G, Whitin TM (1963) Analysis of inventory systems. Prentice-Hall, Englewood Cliffs, NJ Lee H, Padmanabhan V, Whang S (1997) Information distortion in a supply chain. The bullwhip effect. Manage Sci 43(4):546–558 Lee H, Whang S (1999) Decentralized multi-echelon supply chains: incentives and information. Manage Sci 45(5):633–639 Li J, Liu L (2006) Supply chain coordination with quantity discount policy. Int J Prod Econ 101:89–98 Mitra S, Chatterjee AK (2004) Echelon stock based continuous review (R, Q) policy for fast moving items. Omega 32:161–166 Monahan JP (1984) A quantitative discount pricing model to increase vendor profits. Manage Sci 30(6):720–726 Narayanan VG, Raman A (2004) Aligning incentives in supply chains. Harv Bus Rev 82 (11):94–102 Pasternack BA (1985) Optimal pricing and returns policies for perishable commodities. Mark Sci 4(2):166–176 Porteus E (2000) Responsibility tokens in supply chain management. Manuf Serv Oper Manage 2:203–219 Roundy R (1985) 98%-effective integer-ratio lot-sizing for one-warehouse multi-retailer systems. Manage Sci 31(11):1416–1430 Shang KH, Song JS (2003) Newsvendor bounds and heuristic for optimal policies in serial supply chains. Manage Sci 49(5):618–638 Schneeweiss C (2003) Distributed decision making – a unified approach. Eur J Oper Res 150:237–252 Schwarz LB, Schrage L (1975) Optimal and system myopic policies for multi-echelon production/ inventory systems. Manage Sci 21(11):1285–1294 Schweitzer ME, Cachon GP (2000) Decision bias in the newsvendor problem with a known demand distribution: experimental evidence. Manage Sci 46(3):404–420 Silver EA, Pyke DF, Peterson R (1998) Inventory management and production planning and scheduling. Wiley, New York Tang CS (2006) Perspectives in supply chain risk management. Int J Prod Econ 103(2):451–488 Tsay AA (1999) Quantity-flexibility contract and supplier–customer incentives. Manage Sci 45 (10):1339–1358 Tsay AA, Lovejoy WS (1999) Quantity flexibility contracts and supply chain performance. Manuf Serv Oper Manage 1(2):89–111 Tsay AA, Nahmias S, Agrawal N (1999) Modeling supply chain contracts: a review. In: Tayur S, Ganeshan R, Magazine M (eds) Quantitative models for supply chain. Kluwer, Norwell, MA, pp 299–336 Wadsworth GP (1959) Probability. In: Wadsworth GP (ed) Notes on operations research. Technology Press, Cambridge, MA Wang Q, Tsao D (2006) Supply contract with bidirectional options: the buyer’s perspective. Int J Prod Econ 101:30–52 Whang S (1995) Coordination in operations: a taxonomy. J Oper Manage 12:412–422

Use of Supply Chain Contract to Motivate Selling Effort Samar K. Mukhopadhyay and Xuemei Su

Abstract Selling of a product is often delegated by the Original Equipment Manufacturers (OEM) to another firm called sales agent. The OEM needs to devise a mechanism to motivate the agent to exert higher marketing effort in order to boost her sales revenue. She also needs to design a profit allocation scheme, a complex task because of the fact that she has incomplete information about the agent’s marketing cost. In this chapter, two important contract forms are analyzed, compared and the OEM’s strategy are developed. Closed form solutions have been derived for three decision variables: marketing effort, order quantity and retail price for both forms of contracts. The revelation principle has been applied in that derivation which find inefficiency and “distribution distortion” due to information asymmetry. We show that the two contract forms perform differently, and each party’s preference toward a particular contract form is linked with the total reservation profit level and/or the sales agent’s cost type. We find that full trading opportunity, as in the full information case, cannot be achieved by any of the two contracts and the OEM suffers due to information deficiency. The chapter also identifies guidelines for the OEM to exert higher control or be more flexible. Further research avenues are also identified. Keywords Distribution channel • Game theory • Retail contracts • Sales agent • Supply chain

S.K. Mukhopadhyay (*) Graduate School of Business, Sungkyunkwan University, Jongno-Gu, Seoul 110–745, South Korea e-mail: [email protected] X. Su College of Business Administration, California State University Long Beach, 1250 Bellflower Blvd, Long Beach, CA 90840, USA e-mail: [email protected] T.-M. Choi and T.C. Edwin Cheng (eds.), Supply Chain Coordination under Uncertainty, International Handbooks on Information Systems, DOI 10.1007/978-3-642-19257-9_11, # Springer-Verlag Berlin Heidelberg 2011

255

256

S.K. Mukhopadhyay and X. Su

1 Introduction A common practice among Original Equipment Manufacturers (OEM) is to delegate the sales of the product to another firm variously called sales agent, franchisee, or sales representative. The motivation from the OEM’s point of view is to concentrate their effort to product design and manufacturing while leaving the sales and marketing to another firm with suitable expertise. This mode is one of the prominent methods in product distribution (Kaufmann and Dant 2001). This is especially true in global distribution when the OEM wants to introduce their products in foreign markets. In that situation, a local firm with its knowledge of local market would be invaluable and in some cases, inevitable. The sales agent provides services like presale advice, after sales service and advertising. These services are selling efforts that would enhance demand for the product. In the industrial goods market, this will also include customer information sessions, product demonstrations, trade shows and so on. It is, therefore, common for an OEM to use incentives to increase an agent’s effort (Lafontaine and Slade 1997). These incentives are formalized in sales contracts between the OEM and the sales agent. Two common types of contracts used in supply chain franchising are the Franchise Fee (FF) contract and the Retail Price Maintenance (RPM) contract. The FF contract is characterized by a variable wholesale price per unit and a fixed franchise fee. Thus, the FF contract is a two-part-tariff contract. The agent is free to set the retail price. When the RPM contract is employed, it is the OEM that sets the retail price and also the order quantity. Then a cost-plus payment from the agent to the OEM is specified. We study these two types of contracts in this chapter. Other forms of contracts that are used in a supply chain are revenue-sharing contract (Foros et al. 2009; Cachon and Lariviere 2005), quantity discounts contract (Raju and Zhang 2005), channel rebates contract (Taylor 2002), buy-back contract (Zhao et al. 2010), quantity flexibility contract (Krishnan et al. 2004; Tsay 1999) and optimal contracts via mechanism design (Laffont and Martimort 2000; Watson 2007). This chapter investigates how an OEM can use the FF and RPM contracts to motivate the sales agent to put in more efforts which in turn increases the demand for her product and thus her revenue. To design an effective contract, a number of parameters are needed to be specified. Note that the agent’s sales effort cannot be effectively monitored and therefore cannot be put in the contract as a parameter. We also recognize the fact that the agent’s cost of selling effort is only known to himself. So the OEM designs the contract without this information. We will devise the optimal contract design by the OEM under this information asymmetry and will identify the conditions under which one type of contract is preferred over the other. Our model includes “reservation profits” for both the OEM and the sales agent. The reservation profit of each party is the level of the profit they expect from their respective outside opportunities. The sales agent, therefore, would refuse to enter into a contract with the OEM if the expected profit under any contract is less than his reservation profit. The same is true for the OEM. As will be seen later, we

Use of Supply Chain Contract to Motivate Selling Effort

257

uncover an important role for the total of two reservation profits. We find that given the information asymmetry suffered by the OEM, her preference to one contract form or the other depends on the total reservation profits. Also, when the total is within certain ranges, the OEM has a dominant strategy because her preference does not depend on the agent’s cost. This book chapter is based on the authors’ original work of Mukhopadhyay et al. (2009). Some of the research work that studied contract design in the presence of an agent’s service effort are cited below. Desai and Srinivasan (1995) investigate a franchising channel where an informed principal (the contract designer) signals the market demand to an agent whose effort cannot be monitored. Unlike their paper, we assume that the contract designer is less informed and a screening game is played. Desiraju and Moorthy (1997) study how the requirements set by the manufacturer on retail price or service or both may improve the working of a distribution channel. The agent’s service is not contractible in our study. Blair and Lewis (1994) investigate optimal retail contracts that can be used by the manufacturer to encourage dealer promotion, and conclude that the optimal contract exhibits a form of resale price maintenance and quantity fixing. Our study develops new insights for helping an OEM to make a judicious choice between two contract forms under different conditions. Chen et al. (2010) study the coordination mechanism for the supply chain with leadtime consideration and price-dependent demand. Zhu and Mukhopadhyay (2009) study contract design in call-center outsourcing where the agent determines the service level. Dukes and Liu (2010) study the effects of retailer in-store media (ISM) on distribution channel relationships. They show that ISM is important in coordinating a distribution channel on advertising volume and product sales. Information asymmetry is considered in research by Gal-Or (1991). In that study the retailer has private information about demand and retailing cost. Buyer’s marginal cost is private information in the study by Ha (2001). Better information as value to the supplier is characterized by Corbett et al. (2004). Supplier’s cost is private information in Gan et al. (2005) who find that supply chain coordination can be achieved only when the supplier’s reservation profit decreases with production cost. Co-ordination can be achieved, as Krishnan et al. (2004) find, when buy backs can be combined with promotional cost-sharing agreements. In Cakanyildirim et al. (2008), production cost is private information. The retailer designs a menu of contracts specifying the order quality and profit percentage. Yang et al. (2009) study a manufacturer that faces a supplier privileged with private information about supply disruptions. Information asymmetry is also studied by Mukhopadhyay et al. (2008) and Su et al. (2010) in dual-channel distribution. Our study includes the effect of the OEM’s incentive to motivate the agent’s effort to increase sales. Agent compensation literature typically includes moral hazard (selling effort not observable to the firm) and adverse selection (the salesperson has superior information about the market prior to contracting with the firm). Kreps (1990) and Laffont and Martimort (2001) devise a menu of contracts offered to the agent as a typical solution to these types of problems. Laffont and Tirole (1986) and Gibbons (1987) show that in some cases a menu of linear contracts would be optimal. Chen (2005)

258

S.K. Mukhopadhyay and X. Su

studies Gonik’s (1978) scheme and compares it with a menu of linear contracts in a model where the market information possessed by the salesforce is important for the firm’s production and inventory-planning decisions. Ayra et al. (2009) study a Quasi-Robust multiagent model in which the mechanism must be designed before the environment is as well understood as is usually assumed. Wu et al. (2008) argue that people not only care about outcomes, but also about the process that produces these outcomes. They analytically show why fair process is not always used even though fair process enhances both employee motivation and performance. A comprehensive review of salesforce compensation problems can be found in Coughlan (1993). An emerging research stream studies contracting in a complex supply chain with multiple manufacturers and/or multiple retailers. Cui et al. (2008) proposes a trade promotion model that can price discriminate between a dominant retailer and small independents. Krishnan and Winter (2010) study a channel where a manufacturer distributes a product through retailers who compete on both price and fill rate. Cachon and Kok (2010) study a contracting scenario where multiple manufacturers compete for a retailer’s business, and conclude that the same contractual form can exhibit quite different properties from that seen in a one-manufacturer supply chain. Majumder and Srinivasan (2008) show that contract leadership, as well as the position in the supply chain network, affect the performance of the entire supply chain. This chapter is organized as follows. Section 2 introduces our model and derives contracts under full information. Section 3 derives the contracts under asymmetric information, compares the two forms of contracts and discusses the OEM’s strategies. Section 4 concludes the chapter including some avenues for the future research.

2 Contract Under Full Information Our supply chain consists of an OEM who sells her product through a sales agent. The sales agent uses a selling effort denoted by e aimed at increasing demand. We assume that the cost of exerting marketing effort is a convex, increasing function of e, say 12 ke2 . The constant k denotes the agent’s cost type and reflects how efficiently the agent conducts the marketing effort. The OEM’s unit production cost is s. The reservation profits of the OEM and the agent are pM and pR respectively. The reservation profits are lowest level of profits expected by the parties and represent the amount of profits that can be obtained from outside opportunities. Thus, neither party would enter the contract if the expected profit is below their respective reservation profit. The sales agent can choose to either sign the contract or reject it. Negotiation is not allowed. The demand function is: q ¼ a bp þ e

(1)

Where p is the retail price, a is the base demand that depends on factors not included in our model, and b is the sensitiveness of demand with respect to price.

Use of Supply Chain Contract to Motivate Selling Effort

259

Notice that the sales effort e positively impacts the demand. a and b are constants and common knowledge. The linear demand function is widely used in the literature (Desiraju and Moorthy 1997; Lal 1990; Gal-Or 1991).

2.1

The FF and RPM Contracts

In this subsection, we will model the two types of contracts. 2.1.1

FF Contract

In the Franchise Fee (FF) contract, the OEM specifies the unit wholesale price w and a fixed fee L, paid by the agent. Wimmer and Garen (1997) and Federal Trade Commission’s Guide (2005) gives a comprehensive guide of franchising and its fees. L can possibly be negative in which case it is the OEM who makes the payment to the agent, presumably to subsidize the agent’s marketing effort. Positive L, though, is more common. The agent’s total payment to the OEM is wq + L for an order quality q. The OEM’s profit is, pM ¼ ðw sÞq þ L

(2)

1 pR ¼ ðp wÞq ke2 L 2

(3)

The agent’s profit is,

2.1.2

RPM Contract

In Retail Price Maintenance (RPM) Contract, the OEM specifies the retail price. These types of contracts are widely adopted in practice, for example, in the fashion and luxury goods industry, companies such as Gucci set the retail price of their goods for sale through both vertically integrated and independent retailers. Nike requires its retailers to not sell their shoes below a suggested retail price (Gurnani and Xu 2006). RPM contracts are characterized by three parameters: the retail price p, the order quantity q, and a cost plus payment amount R. Thus the total payment to OEM is s q þ R, where s q covers the OEM’s total production cost, and R is her profit. If the agent decides to accept the contract, each party’s profit is pM ¼ R

(4)

1 pR ¼ pq ke2 sq R 2

(5)

260

2.2

S.K. Mukhopadhyay and X. Su

The Full Information Case

In this section, we derive the optimal contract under full information. Here, the agent’s cost type k is common knowledge. The OEM maximizes her profit by maximizing the total channel profit and letting the agent earn his reservation profit pR , thereby extracting the rest of the channel profit for herself. The joint profit maximization function is: 1 Max pT ðp; q; e; kÞ ¼ ðp sÞq ke2 p;q;e 2

(6)

Equation (6) is maximized to obtain the optimum values of the decision variables as follows: e ¼

ða bsÞ ; 2bk 1

q ¼

bkða bsÞ ; 2bk 1

p ¼

kða þ bsÞ s ; 2bk 1

pT ¼

kða bsÞ2 2ð2bk 1Þ

These optimum values are the “first best” solutions, because other solutions, due to information asymmetry, would be inferior to these solutions. With full information, the OEM can maximize the channel profit by specifying that the sales agent adopt the fist-best solutions for marketing effort level, sales level and retail price. Then, the OEM can extract the whole channel profit by specifying L (in case of FF contract) or R (in case of RPM contract). It is intuitive that these first best solutions are all decreasing in k. This means that a cost-inefficient agent cannot provide optimal marketing effort, leading to optimal sales, and costumers would not likely pay a high price for a low service level. An inefficient sales agent, therefore, brings sluggish channel profit. Obviously, the two parties will enter into a contract only if pM þ pR pT , which mandates that the agent’s cost type k bT . bT , a threshold value, is called the cutoff point, and is derived as: bT ¼

2ðpM þ pR Þ 4bðpM þ pR Þ ða bsÞ2

We will use, “1” and “2” as subscripts or superscripts to represent FF contract and RPM contract respectively. Table 1 shows the solutions for both contract forms. Table 1 Equilibrium results under complete information

FF contract w¼s L ¼ pT pR b1 ¼ bT pT pR if k bT p1M ¼ pM if bT k p1R ¼ pR

RPM contract p2 ¼ p ; q2 ¼ q R ¼ pT pR b2 ¼ bT pT pR if k bT p2M ¼ pM if bT k p2R ¼ pR

Use of Supply Chain Contract to Motivate Selling Effort

261

3 The Asymmetric Information Case In this section, we consider the problem of contract design when k is unknown to the OEM. She only has a prior knowledge that k is somewhere within the range [k; k], with a distribution denoted by FðkÞ. f ðkÞ is the probability density function. Without knowing the exact value of k, the OEM has no way of determining optimal values of the parameters w, L, and R. In these cases, it is customary to offer a “menu” of contracts. This menu is a set of options for the agent to choose from based on his cost type k, known only to himself.

3.1

FF Contract Menu

The OEM’s menu of contracts consists of a number of tuples {w(k),L(k)}, each item consisting of parameters w and L for a given value of k. By virtue of the “revelation principle” (Myerson 1979), the OEM hopes that the agent would declare the true value of k because the menu is designed in such a way that a truthful revelation of the information would yield highest profit for the agent. Define pR ðk~jkÞ as the profit ~ LðkÞg ~ from the of an agent who is of a cost type k and chooses a contract fwðkÞ; menu. The agent solves the problem: ~ wðkÞÞ ~ q/ ðwðkÞÞ ~ 1 k e/ ðwðkÞÞ ~ 2 LðkÞ ~ R1 : Max pR ðk~jkÞ :¼ ðp/ ðwðkÞÞ 2 k~ ~ 2 kða bwðkÞÞ ~ ¼ LðkÞ 2ð2kb 1Þ Where “k” is the true cost type, and k~ is the announced cost type by the agent. ~ as obtained by using the First p ; q/ and e/ are the agent’s best responses to wðkÞ Order Condition (FOC) on (3). Revelation principle requires that pR ðk~jkÞ be concave in k~ and achieves the maximum at k~ ¼ k. Only then it will be to the agent’s interest to reveal k. Depending on the range of k and the value of b, some common types of distributions like uniform, beta and truncated normal meet this requirement. We, therefore, can write the OEM’s problem as /

ð b1 M1 Max

wðkÞ;LðkÞ

k

pM ðk; qðkÞÞf ðkÞdk þ

ðk b1

pM f ðkÞdk

(7)

S:t: IC : qðkÞ ¼ arg max pR ðk; qÞ

(8)

1 IR : pR ðk; qðkÞÞ ¼ ðp wðkÞÞqðkÞ ke2 LðkÞ pR 2

(9)

262

S.K. Mukhopadhyay and X. Su

pM ðk; qðkÞÞ ¼ ðwðkÞ sÞqðkÞ þ LðkÞ

(10)

The OEM profit in (7) depends on the quantity q ordered by the agent, which in turn depends on the agent’s cost type k. Constraint (8) is called the agent’s “Incentive-Compatibility” constraint. This constraint ensures that, an agent with cost type k will choose q to maximize its profit. Constraint (9) represents the agent’s “Individual Rationality” constraint. This states that the agent’s profit must be no less than his reservation profit pR for him to agree to trade. The quantity b1 in the objective function (7) is a value of k 2 ½k; k, such that when k ¼ b1 either of the two parties hits their respective reservation profit. Therefore, for the values k > b1 , no contract is signed between the two parties and the OEM would earn her reservation profit (pM ) elsewhere. As pM is a decreasing function of k (see Corollary 1c,) we would have pM ðk; qðkÞÞ pM for k b1 . The formulation given in (7) through (10) fits the optimal control formulation with variable endpoint conditions and salvage value (see Kamien and Schwartz 1981, pp. 143–148). We use the methodology therein to solve the problem M1 . The solution to this problem is given in proposition 1. Proofs of all propositions, unless otherwise stated, are given in the Appendix. We use the notation x?½l; u :¼ max fl; min fx; ugg as the projection of x on the interval [l; u]. Proposition 1. Under asymmetric information the optimal values of the OEM’s parameters in the franchise fee contract is given by: w¼sþ

ða bsÞFðkÞ bFðkÞ þ bkð2bk 1Þf ðkÞ

L(k) is given by the solution of @L bk2 ða bsÞf ðkÞ @w ¼ @k FðkÞ þ kð2bk 1Þf ðkÞ @k Lðb1 Þ

satisfies

Lðb1 Þ ¼ pM

8k k b1

ða bsÞ2 b21 Fðb1 Þf ðb1 Þ 2ðFðb1 Þ þ b1 ð2bb1 1Þf ðb1 ÞÞ2

The resulting cutoff point is given by b1 ¼ b0 ?½k; k where b0 is the solution of: 2ðpM þ pR Þ b2 f ðbÞ ¼ FðbÞ þ bð2bb 1Þf ðbÞ ða bsÞ2 The second column of Table 2 gives a summary of equilibrium results for FF contract. Corollary 1 summarizes major properties of the equilibrium results.

Use of Supply Chain Contract to Motivate Selling Effort

263

Corollary 1. Under the FF contract with asymmetric information, (a) As k increases, w increases, and L decreases (b) At equilibrium, e1 ; q1 and p1 are all decreasing in k, and e1 < e ; q1 < q ; p1 > p , for any k 2 ðk; k. (c) pM ðkÞ and pR ðkÞ are decreasing in k, until the cutoff point b1 ; where b1 bT . (d) For any k 6¼ k, the agent’s profit is higher and the OEM’s profit is lower compared to their counterparts under full information. Total channel profit is lower than that under full information. We observe the following from Corollary 1. A higher fixed fee is associated with a lower unit wholesale price and vice versa. As seen in Fig. 1. As also reported by Wimmer and Garen (1997), factors that increase the franchisee’s effort (e here), would lower the recurring fee (w here), and increase the franchise fee (L here). The insight here is that a cost-efficient sales agent (with low k) will enjoy a discounted wholesale price, would exert higher marketing effort to gain higher demand, and can charge customers a higher price. All of these actions would contribute to higher profits for both the OEM and the agent. This is an important finding of this chapter. We also see that the cut-off point, if it exists, is unique. It is possible for the franchise fee to go negative for a high cost type agent. In that case, the “franchise fee” is from the OEM to the sales agent meaning that the OEM subsidizes an inefficient sales agent, or simply because the effort required is costly, and the OEM offers to cover part of the investment. Proposition 1 requires the revelation principle to work. The design of the menu of contract must ensure that w is increasing in k, and L is decreasing in k. This will make w > s (the production cost), giving rise to the double marginalization problem. This double marginalization phenomenon was first identified by Spengler (1950). In our case, double marginalization is reflected as higher retail price, lower 250

L

200

150

100

50

0 2.00

2.17

2.35

2.52

2.69

2.87 k

Fig. 1 Optimal w and L for varying k

3.04

3.21

3.39

3.56

w

264

S.K. Mukhopadhyay and X. Su

Table 2 Results of FF and RPM contracts under asymmetric information FF contract: fwðkÞ; LðkÞg Unit transfer price Retail price Marketing effort Sales Total channel profit

RPM contract: fpðkÞ; qðkÞ; RðkÞg

ða bsÞFðkÞ bFðkÞ þ bkð2bk 1Þf ðkÞ ða bsÞðFðkÞ þ bk2 f ðkÞÞ p1 ¼ s þ bFðkÞ þ bkð2bk 1Þf ðkÞ ða bsÞkf ðkÞ e1 ¼ FðkÞ þ kð2bk 1Þf ðkÞ ða bsÞbk2 f ðkÞ q1 ¼ FðkÞ þ kð2bk 1Þf ðkÞ

sþ

p1T ¼

s ða bsÞðFðkÞ þ kf ðkÞÞ 2bðFðkÞ þ kf ðkÞÞ f ðkÞ ða bsÞf ðkÞ e2 ¼ 2bðFðkÞ þ kf ðkÞÞ f ðkÞ ða bsÞbðFðkÞ þ kf ðkÞÞ q2 ¼ 2bðFðkÞ þ kf ðkÞÞ f ðkÞ p2 ¼ s þ

ða bsÞ2 k2 f ðkÞð2FðkÞ þ kð2bk 1Þf ðkÞÞ 2ðFðkÞ þ kð2bk 1Þf ðkÞÞ

2

p2T ¼

ða bsÞ2 ðFðkÞ þ kf ðkÞÞ 4bðFðkÞ þ kf ðkÞÞ 2f ðkÞ

ða bsÞ2 ðk þ zðkÞÞ 4bðk þ zðkÞÞ 2 ð 1 k ða bsÞ2 dx pR 2 k ð2bðx þ zðxÞÞ 1Þ2 ðk 1 ða bsÞ2 p2R ¼ pR þ dx 2 k ð2bðx þ zðxÞÞ 1Þ2 p2M ¼

Profit of the OEM

p1M ¼

Profit of the sales agent p1R ¼

ða bsÞ2 k2 FðkÞf ðkÞ ðFðkÞ þ kð2bk 1Þ f ðkÞÞ2 2 3

ða bsÞ k ð2bk 1Þ f ðkÞ

þL

2

2ðFðkÞ þ kð2bk 1Þ f ðkÞÞ2

L

sales, and less marketing effort compared to the first-best. The inefficiency caused by information asymmetry is further reflected on the cutoff point. Since b1 bT , the FF contract cannot fully explore all the trading opportunities as presented under the full information case. Part (d) of the corollary shows that the OEM is worse off and the sales agent is better off under asymmetric information. This phenomenon is called information rent, that is, the benefit earned due to holding private information. To visualize this, we use the results in the Table 2 to draw Fig. 2, to show the variation of p1M pM ; p1R pR and p1T pT with respect to k respectively. Of these values, p1R pR measures the information rent. As we can see from Fig. 2, the lower the cost type, the higher the information rent. Gal-Or (1991) refers to this as a “distributional distortion”, since it relates to the distribution of the surplus between the OEM and the agent. Obviously, the channel as a whole is worse off compared to the full information case. Figure 2 shows that p1T pT is negative and decrease as k increases, but almost close to zero, which means the channel’s profit loss due to information asymmetry is trivial. We will further discuss it in Sect. 3.3.

3.2

The RPM Contract Menu

In the RPM contract, the menu consists of a tuple fpðkÞ; qðkÞ; RðkÞg. Each item on the menu is intended for an agent of a specific cost type. The profit of the agent of cost type k declaring a cost type k~ is given by ~ sÞ qðkÞ ~ 1 keðkÞ ~ ¼ pT ðpðkÞ; ~ qðkÞ; ~ kÞ RðkÞ ~ ~ 2 RðkÞ pR ðk~jkÞ ¼ ðpðkÞ 2

Use of Supply Chain Contract to Motivate Selling Effort

265

250 200 150 100 50 0 2.00 -50

2.17

2.35

2.52

2.69

2.87 k

3.04

3.21

3.39

3.56

-100 -150 -200 Series 2

-250

Series 1

Series 3

Fig. 2 Information rent under FF contract1

Where pT is defined in function (6). The OEM’s problem M2 of designing the optimal contract is: ð b2 M2

max

pðkÞ;qðkÞ;RðkÞ

RðkÞf ðkÞdk þ

k

ðk b2

pM f ðkÞdk

~ pR ðkjkÞ ~ S:t: pR ðkÞ pR ðk~jkÞ and pR ðkÞ

(11)

k k; k~ k;

pR ðkÞ pR qðkÞ 0 Lemma 1 provides a characterization of problem M2, to be used for deriving the optimal contract menu. Lemma 1. A solution fpðkÞ; qðkÞ; RðkÞg is feasible for problem ðM2 Þ if and only if Rk (a) pR ðkÞ ¼ pR þ 12 k eðxÞ2 dx (b) eðkÞ is decreasing in k (c) qðkÞ 0 The design problem M2 can then be reformulated as: ðk max

pðkÞ;qðkÞ0

k

ðpT ðpðkÞ; qðkÞ; kÞ

zðkÞ eðkÞ2 ÞdFðkÞ 2

Data series 1, 2, 3 denote p1M pM ; p1R pR and p1T pT respectively.

1

(12)

266

S.K. Mukhopadhyay and X. Su

We need two constraints: (1) qðkÞ is nonnegative and (2) eðkÞ is decreasing in k (Proof is provided in the Appendix). zðkÞ is defined as zðkÞ ¼ FðkÞ=f ðkÞ. If we ignore the feasibility requirements, we can maximize the integrand in function (12) for each k like in (6) with a cost factor k þ zðkÞ for the agent. So, we substitute pðkÞ ¼ p ðk þ zðkÞÞ and qðkÞ ¼ q ðk þ zðkÞÞ where p and q are the first-best solutions. To have a feasible solution, we need eðk þ zðkÞÞ to be decreasing in k and zðkÞ ¼ FðkÞ=f ðkÞ increasing in k. We now consider the contract menu ^ k k kg with f^ pðkÞ; q^ðkÞ; RðkÞ q^ðkÞ ¼ q ðk þ zðkÞÞ p^ðkÞ ¼ p ðk þ zðkÞÞ 1 ^ ¼ pT ð^ pðkÞ; q^ðkÞ; kÞ RðkÞ 2

ðk k

e^ðxÞ2 dx pR

The optimal solutions are shown in Proposition 2. Proposition 2. If zðxÞ ¼ FðxÞ=f ðxÞis increasing in x, ^ (a) f^ pðkÞ; q^ðkÞ; RðkÞg is the optimal solution to M2 ^ðkÞs (b) p^ðkÞ; q^ðkÞ; and e^ðkÞ, are all decreasing in k, and e^ðkÞ ¼ pkþzðkÞ < e ðkÞ; q^ðkÞ < q ðkÞ; p^ðkÞ < p ðkÞ for any k 2 ðk; k (c) pM ðkÞ and pR ðkÞ are decreasing in k until the cutoff point. (d) The agent’s profit is higher and the OEM’s profit is lower compared to their counterparts under full information. Total supply chain profit is lower than that under full information. (e) The cutoff point b2 ¼ b0 ?½k; k where b0 is the solution of: b þ zðbÞ 2bðb þ zðbÞÞ 1

ðk b

dx ð2bðx þ zðxÞÞ 1Þ

2

¼

2ðpM þ pR Þ ða bsÞ2

The results are shown in the third column of Table 2. Under RPM contract, the OEM makes all the decisions using a cost factor of k þ zðkÞ instead of k. Compared to the first best solution, the OEM is worse off and the agent is better off; the agent orders less and exerts less marketing effort; and the retail price is lower. The channel as a whole is also worse off, for every cost type k. We show the variation of p2M pM , p2T pT and p2R pR with respect to k in Fig. 3. Like earlier, the information rent decreases with the agent’s cost type k. Both parties’ profits and the total channel profit are monotonically decreasing in k till k ¼ b2 . The cutoff point b2 bT . This shows that the RPM contract cannot fully explore all the trading opportunities compared to the full information case. There are two insights from the monotonic property: a low cost type agent benefits both the agent and the OEM, and therefore the channel; and, that the cutoff point is unique.

Use of Supply Chain Contract to Motivate Selling Effort

267

60

40

20

0 2.00

2.17

2.35

2.52

2.69

2.87 k

3.04

3.21

3.39

3.56

-20

-40

-60

Series1

Series2

Series3

Fig. 3 Information rent under RPM contract2

3.3

FF and RPM Contracts Compared

In the above analysis, we found that, under information asymmetry, both FF and RPM contract forms induce less marketing effort provisions, realize less sales, and generate lower channel profit compared to the full information case. Also, neither contract explores all the trading opportunities. Under full information, the trade stops only when both are earning no more than their respective reservation profits. However, under asymmetric information, the trade stops as soon as either party hits their respective reservation profit first. We will now compare these two contract forms, using subscripts 1 and 2 to denote the result of FF and RPM contracts respectively. Also, to simplify notations we use p2 ; q2 , and e2 for the RPM contract, instead of p^ðkÞ; q^ðkÞ; and e^ðkÞ respectively. The comprehensive comparison of the two forms of contracts is given in Corollary 2, while Fig. 4 shows a visual portray. Corollary 2. (a) Marketing level comparison: e e1 e2 . FF Contracts exerts more marketing effort than RPM contracts and both are less than the first best effort. (b) Price comparison: p1 p p2 . Price is highest in FF contract. RPM contract price is lower than the first best price. (c) Sales level comparison: q q2 q1 . FF sales level is lowest of all. First best sales level is highest. Data series 1, 2, 3 denote p2M pM ; p2R pR and p2T pT respectively.

2

268

S.K. Mukhopadhyay and X. Su

50

p1

e1

q1

p2

e2

q2

45 40 35 30 25 20 15 10 5 0 2.00

2.17

2.35

2.52

2.69

2.87 k

3.04

3.21

3.39

3.56

Fig. 4 Optimal p, e and q for two contract forms

The equalities only hold when k ¼ k. An agent with a cost type k 6¼ k, given an FF contract, will price the product higher, provide higher marketing effort, but sell less compared to when given a RPM contract. Some more observations follow. 1. The FF contract provides a better mechanism to motivate marketing effort provisions. The sales agent is free to choose the marketing effort level and set the retail price. This flexibility motivates him to exert more marketing effort compared to that of an RPM contract, and enables him to charge higher retail price. This flexibility becomes even more critical when the agent is of a high cost type. Recall that, in the RPM contract, the OEM specifies that e2 ðkÞ ¼ e ðk þ zðkÞÞ which is greatly distorted down from the first-best level when k is high because zðkÞ increases in k and e decreases in k. In the FF contract, it is true that a high cost type agent will face a high unit wholesale price, but the detrimental effect of the increase in unit wholesale price is softened when the increase in the unit wholesale price is combined with the offer of a more dramatic decrease in franchise fee. As a result, the agent still has the room to exert reasonable level of marketing effort. The decrease in marketing effort is not seen as dramatic as it is in the RPM contract. Relatively speaking, FF contract provides the agent higherpowered incentives. 2. It is notable that higher retail price in an FF contract results from not only the higher marketing effort as seen above but also from “double marginalization”. The double marginalization hurts the channel profit and the OEM’s profit as well. In contrast, double marginalization is avoided in the RPM contract because the OEM acts like a central planner and dictates a retail price and order quantity to the agent. However, with incomplete information, this centralized decision is

Use of Supply Chain Contract to Motivate Selling Effort

269

not trouble free. With RPM contract, both the retail price and the marketing effort level are distorted down from the first-best solutions ðp2 < p and e2 < e Þ for inducing truthful information reporting. These distortions hurt the channel profit too. Further discussions will follow Corollary 3. 3. It seems counter-intuitive that the FF contract realizes lower sales level with more marketing effort compared to the RPM contract. The high retail price in the FF contract will shed light on this. It is to the benefit of the agent to increase profit through pricing higher, instead of selling more units under the FF contract. There are two reasons for this. One is that customers would like to pay higher price given better services. The other is the sales agent finds it harder to induce more demand through price cut, which is limited by the double marginalization problem. Next, we examine the effect on the channel profit for each of the two contract forms. Corollary 3 summarizes the results and Fig. 5 visually depicts the channel profit under each of the two contract forms. Corollary 3. With information asymmetry, the total channel profit for each contract form is equal at k ¼ k, and decreases monotonically with k. For any k > k, the total channel profit of an FF contract is higher than that of the RPM contract. The difference in channel profit increases as k increases. Note that the conclusion drawn in Corollary 3 is based on the assumptions presented earlier in Sects. 3.1 and 3.2. Those assumptions guarantee that the profit functions are concave and revelation principle can be applied. Corollary 3 shows that for any cost type k, the RPM contract generates less channel profit and this worsens when the agent is of a high cost type. This is because the way the RPM contract is designed. In a RPM contract, the OEM acts like a central planner and directly specifies the agent’s order quantity and retail price. The marketing effort

Profit--FF

Profit--RPM

500

450

400

350 2.00

2.17

2.35

2.52

2.69

2.87 k

3.04

Fig. 5 Channel profits under FF contract and RPM contract

3.21

3.39

3.56

270

S.K. Mukhopadhyay and X. Su

level is indirectly specified by the OEM through specifying order quantity and retail price according to (1). However, with asymmetric information, this tight control comes with a cost. It requires distorting price, output and marketing effort level to induce the agent’s truthful information reporting. Specifically, q^ðkÞ ¼ q ðk þ zðkÞÞ; p^ðkÞ ¼ p ðk þ zðkÞÞ; and e^ðkÞ ¼ e ðk þ zðkÞÞ. Note that q ; p and e are decreasing in k, and zðkÞ is increasing in k. As a result, e^ðkÞ, q^ðkÞ and p^ðkÞ will be quite off the first-best solutions of e ðkÞ, q ðkÞ and p ðkÞ if the agent is of a high cost type. As a result, the channel profit will be greatly reduced. Figure 3 displays that the channel profit loss, p2T pT ; is substantial when the agent is of a high cost type. By comparing Fig. 2 with Fig. 3, we can see that the channel profit loss is minimal with the FF contract. We do recognize that the channel profit gets hurt as double marginalization problem is introduced into the design of the FF contract. But because the agent has the freedom to choose its retail price, the agent has the motivation to exert more marketing effort so as to charge customers a higher price. In addition, due to the arrangement that the fixed fee charge decreases as the agent’ cost type k increases, even a high cost type retailer is encouraged to exert a reasonable amount of marketing effort. With this flexibility in design, the FF contract can better align the agent’s interests with that of the channel.

3.4

Profit Allocation Mechanisms and the OEM’s Strategy

We treat the allocation of channel profit as a two-phase process. Note that no matter which contract form is offered, each party has to get at least their reservation profit before they would enter a contract. First, each party takes their reservation profit (pM or pR ) out of the total channel profit; Second, the two parties share the rest of the channel profit, the allocation of which is dependent on the contract form. This profit allocation mechanism in FF contract is analytically complex. We do the analysis for the RPM contract. The term “allocable profit” is defined as the profit in excess of the total of reservation profits i.e. Allocable profit ¼ total channel profit ðpM þ pR Þ The optimal strategy for contract offering is guided by the value of the allocable profit and, therefore, the total reservation profit. This observation is one of the main contributions of this chapter. Recall that the agent’s profit under the RPM contract is, 1 pR ðkÞ ¼ pR þ 2

ðk

eðxÞ2 dx ðLemma 1ðaÞÞ

(13)

k

Ðk Then the agent’s marginal utility from entering the RPM contract is 12 k eðxÞ2 dx. For a particular k, the realized total channel profit is fixed (refer to Table 2), and the

Use of Supply Chain Contract to Motivate Selling Effort

271

Ðk agent’s share of the allocable profit, 12 k eðxÞ2 dx, is also a fixed amount. If pM þ pR is small, the allocable profit is large and the OEM’s share of allocable profit will be large by restricting the agent’s share. Therefore, the RPM contract is favorable to the OEM. When pMÐ þ pR is large, the allocable profit is small. But the OEM still k has to allocate 12 k eðxÞ2 dx to the agent. This makes the RPM contract less attractive to the OEM. We now study the question of what contract the OEM would prefer based on an agent’ cost type for a fixed pM þ pR . For a low k, the realized channel profit, and hence the allocable profit would be large. The agent would benefit because equilibrium eðkÞ is larger for a smaller k, and the agent’s share of allocable profit is higher according to (13). But as the allocable profit generated by this cost efficient agent is large, the OEM may benefit even more. Thus, the RPM contract is favorable to the OEM when k is small, and becomes less attractive when k is large. However, this discussion holds only when pM þ pR is moderately small. If pM þ pR is very large, the allocable profit is very small, leaving the OEM almost nothing even with a cost efficient agent. This is summarized below. Observation 1. For a given combination of ðpM þ pR ; kÞ, where k is a value below cutoff points, and for a large allocable profit, the profit allocation mechanism of a RPM contract is favorable to the OEM. The OEM should offer RPM contract to the agent. For small allocable profits, the OEM should offer FF contract to the agent. When k is high enough so that either b1 or b2 comes into play, the monotonic property of either party’s profit with respect to k will be interrupted. Each party’s preference toward a certain contract type may change accordingly. Figure 6 summarizes the impact of cutoff points, pM þ pR and k on the OEM’s choice of a certain contract form. Figure 6 depicts the OEM’s preferred contract forms as a function of pM þ pR . It plots the tracks of kM , b1 and b2 where kM is a possible cost type of the agent, at which the OEM switches its preference between the two contract forms. Below kM , the combinations of ðpM þ pR ; kÞ result in adequate allocable profit, which makes the RPM contract attractive to the OEM. Above the tracks of b1 and b2 , no trading is possible either because the total reservation profit is too high, or because the agent is too inefficient, or both. For the combinations of ðpM þ pR ; kÞ which are between the tracks kM and b1 , the FF contract is more attractive to the OEM. For the small area above the track of b1 but below the track of b2 , the OEM prefers RPM contract because no trading is possible for FF contract.

3.5

The OEM’s Dominant Strategy

The OEM’s choice of a certain contract form are dependent on the different combinations of ðpM þ pR ; kÞ and the cutoff points. However, the OEM does not know the agent’s cost type k at the time of deciding about the contract form. Recall the sequence of events. First, the OEM chooses a contract form, FF or RPM.

272

S.K. Mukhopadhyay and X. Su

pM + pR

Z

hY,Z

Y

hX,Y

X k

k increase

k

k

Preference is RPM contract

Track of kM

Preference is FF contract

Track of b1

Indifferent between the two contract forms

Track of b2

Fig. 6 The OEM’s preferred contract form

Second, for the selected contract form, the OEM provides a menu of contracts, each of which is intended for an agent of a particular cost type k. Finally, k is revealed when the agent selects one contract from that menu. It would, therefore, be practically meaningful to investigate if the OEM can make a choice between the two contract forms before knowing the agent’s cost type k. Noting that the total reservation profit pM þ pR is common knowledge, the area in Fig. 6 is divided into three regions X; Y and Z with two thresholds of X;Y and Y;Z . The threshold X;Y is the value of pM þ pR where kM ¼ k and Y;Z is the value of pM þ pR where kM ¼ k. The values of X;Y and Y;Z can be uniquely determined and X;Y < Y;Z . Proposition 3 summarizes the OEM’s dominant strategy. Proposition 3. If pM þ pR X;Y , the OEM prefers the RPM contract regardless of the value of k; If pM þ pR Y;Z , the OEM prefers the FF contract or no contract, regardless of the value of k. Proposition 3 provides some clear cut strategies for the OEM even when the agent’s marketing cost information is not known. For moderate level of total

Use of Supply Chain Contract to Motivate Selling Effort

273

reservation profit, i.e., X;Y < pM þ pR Y;Z the OEM’s optimal strategy depends on the unknown k, so any choice has some risks.

4 Conclusion and Further Research When an OEM is dependent on the sales agent for the marketing of her product, she needs to devise a mechanism to motivate the agent to exert higher marketing effort in order to boost her sales revenue. She also needs to design a profit allocation scheme, a complex task because of the fact that she has incomplete information about the agent’s marketing cost. In this chapter, two important contract forms are analyzed, compared and the OEM’s strategy are developed. Closed form solutions have been derived for three decision variables: marketing effort, order quantity and retail price for both forms of contracts. The revelation principle has been applied in that derivation which find inefficiency and “distribution distortion” due to information asymmetry. Full trading opportunity, as in the full information case, cannot be achieved by any of the two contracts and the OEM suffers due to information deficiency. The operational differences of the two contract forms and marginal guidelines are fully identified in this chapter. For example, the FF contract motivates more marketing effort and generates more channel profit. This chapter identifies the role of total reservation profit for selecting a suitable contract form. Figure 6 highlights the guidelines in selecting a contract form for varying k and different values of the total reservation profit. It also identifies ranges where the OEM’s strategy, surprisingly, does not depend on the value of k, thereby making the lack of cost information irrelevant. Double marginalization is a concern in supply chain coordination literature. This chapter finds some further insight into the problem. It is present in the FF contract, but not in the RPM contract. But, it has been proved that, in FF contract, channel profit is always higher, and so is the OEM’s profit under certain conditions. Therefore, double marginalization need not to be viewed as detrimental. As per the RPM contract is concerned, the OEM has higher control of even determining the agent’s order quantity and retail price. But it still does not guarantee higher profit for her – a fact that is counter intuitive. In this form of contract, larger value of k results in reduced channel profit and the allocable profit, and it hurts the OEM more than the agent. The chapter identifies guidelines for the OEM to exert higher control or to be more flexible. We now identify avenues for further research. We have identified regions in Fig. 6 where the OEM’s choice is not clear cut and a risk is involved in selecting a contract form. A coordination plan can be developed for such cases. Two possible directions could be to devise an incentive plan for the agent to divulge private information, and to offer both contract forms with the provision of compensation for the OEM. Contract forms, other than the two studied here, can also be examined and designed. A further research area will be examining multiple agents.

274

S.K. Mukhopadhyay and X. Su

Appendix Proof of Proposition 1 We can rewrite the OEM’s problem as, bð1

M1

mðkÞdk þ Fðb1 Þ

Max

wðkÞ;LðkÞ;b1 k

s:t: L ðkÞ ¼ g1 ðkÞ w ðkÞ ¼ g2 ðkÞ where mðkÞ :¼ ððw sÞq þ LÞf ðkÞ ¼ ððw sÞ bkðabwÞ 2bk1 þ LÞf ðkÞ g1 ðkÞ :¼

bkða bwÞ u1 2bk 1

g2 ðkÞ :¼ u1 ¼ w

and

Fðb1 Þ :¼ pM ð1 FðkÞÞ

Using the multiplier equations

@mðkÞ @g1 @g2 þ l2 Þ ¼ f ðkÞ þ l1 @L @L @L ) l1 ¼ FðkÞ

l1 ¼ ð

@mðkÞ @g1 @g2 þ l2 Þ þ l1 @w @w @w bkða bwÞ b2 k b2 k ¼ f ðkÞ ð Þ þ f ðkÞ ðw sÞ l1 u1 2bk 1 2bk 1 2bk 1

(14)

l2 ¼ ð

(15)

Using the optimality conditions @mðkÞ @g1 @g2 bkða bwÞ þ l2 ¼ 0 þ l1 þ l2 ¼ 0 ) l1 @u1 2bk 1 @u1 @u1

(16)

Because both pM and pR are decreasing in k (we will verify it later), IRM and IRR need only hold at k ¼ b1 . They will then be satisfied at all k b1 . Write: Kðb1 Þ :¼

b1 ða bwÞ2 Lðb1 Þ pR 0 2ð2bb1 1Þ

The transversality conditions then require that there exists p such that: l1 ðb1 Þ ¼

@F @K þp ¼ p @L @L

(17)

Use of Supply Chain Contract to Motivate Selling Effort

l2 ðb1 Þ ¼

275

@F @K a bw þp ¼ pbb1 @w @w 2bb1 1

mðb1 Þ þ l1 ðb1 Þg1 ðb1 Þ þ l2 ðb1 Þg2 ðb1 Þ pM f ðb1 Þ þ p

(18) @K ¼0 @b1

p 0; Kðb1 Þ 0; pKðb1 Þ ¼ 0

(19) (20)

Taking derivative on both sides of (16) and using (14)

) l2 ¼ FðkÞð

bða bwÞ

b2 ku1 bkða bwÞ Þ f ðkÞ þ 2 2bk 1 2bk 1 ð2bk 1Þ

(21)

Solving (21) and (15) ) w¼sþ

ða bsÞFðkÞ bFðkÞ þ bkð2bk 1Þf ðkÞ

From (19) ððwðb1 Þ sÞ

bb1 ða bwðb1 ÞÞ p ða bwðb1 ÞÞ2 þ Lðb1 ÞÞf ðb1 Þ pM f ðb1 Þ ¼0 2bb1 1 2 ð2bb1 1Þ2 2

@K 1 ÞÞ ¼ ðabwðb and p ¼ F) (Note that @b 2ð2bb1 1Þ2 1 Plug in wðb1 Þ

) Lðb1 Þ ¼ pM

ða bsÞ2 b21 Fðb1 Þf ðb1 Þ 2ðFðb1 Þ þ b1 ð2bb1 1Þf ðb1 ÞÞ2

(22)

or f ðb1 Þ ¼ 0 FðkÞ=f ðkÞ is increasing in k (one of the assumptions), so f ðb1 Þ ¼ 0 can only occur at b1 ¼ k or b1 ¼ k. For k < b1 < k, using p ¼ F(b1 Þ > 0 gives Kðb1 Þ ¼ 0, which, combined with (22) )

2ðpM þ pR Þ b21 f ðb1 Þ ¼ Fðb1 Þ þ b1 ð2bb1 1Þf ðb1 Þ ða bsÞ2

Proof of Corollary 1 Part (b) e e1 ¼ ¼

ða bsÞ ða bsÞkf ðkÞ 2bk 1 FðkÞ þ kð2bk 1Þf ðkÞ ða bsÞFðkÞ >0 ð2bk 1ÞðFðkÞ þ kð2bk 1Þf ðkÞÞ

276

S.K. Mukhopadhyay and X. Su

q q1 ¼ ¼

p p 1 ¼

ða bsÞbkFðkÞ >0 ð2bk 1ÞðFðkÞ þ kð2bk 1Þf ðkÞÞ

kða þ bsÞ s ða bsÞðFðkÞ þ bk2 f ðkÞÞ ðs þ Þ 2bk 1 bFðkÞ þ bkð2bk 1Þf ðkÞ

¼

q1

bkða bsÞ ða bsÞbk2 f ðkÞ 2bk 1 FðkÞ þ kð2bk 1Þf ðkÞ

ða bsÞðbk 1ÞFðkÞ <0 bð2bk 1ÞðFðkÞ þ kð2bk 1Þf ðkÞÞ

bkða bwÞ 0 bða bwÞ b2 k w ¼ð < 0ðsince w > 0Þ Þk ¼ 2 2bk 1 2bk 1 ð2bk 1Þ

Proof of Corollary 1 Part (c)

p1M ¼ ðw sÞq þ L ¼ ðw sÞ

bkða bwÞ þL 2bk 1

@p1M bðw sÞ ¼ ða bw þ bkð2bk 1Þ wÞ < 0ðsince w > 0Þ 2 @k ð2bk 1Þ

1 kða bwÞ2 L p1R ¼ ðp wÞq ke2 L ¼ 2ð2bk 1Þ 2 @p1R ða bwÞ2 bkða bwÞ ¼ < 0ð recall that L ¼ wÞ 2 @k 2bk 1 2ð2bk 1Þ

Since both p1M and p1R are decreasing in k, p1T ¼ p1M þ p1R is decreasing in k.

Proof of Corollary 1 Part (d) Under complete information, pM ¼ pT pR Under asymmetric information p1M ¼ p1T p1R As has been approved, the agent’s profit is monotonically decreasing in k, until hitting its reservation profit pR . However, under complete information, the agent

Use of Supply Chain Contract to Motivate Selling Effort

277

always earns its reservation profit pR . Hence, the agent’s profit is always better off under asymmetric information, i.e., p1R > pR For any k > k, p1T pT ¼ Hence p1M < pM

kða bsÞ2 FðkÞ2 2ð2bk 1ÞðFðkÞ þ kð2bk 1Þf ðkÞÞ2

<0

Proof of Lemma 1 ~2 eðkÞ ~ ~ ~ pR ðkjkÞ ¼ pR ðkÞ þ ðk kÞ 2 Let ~ þ ðk~ kÞ pR ðk~jkÞ pR ðkÞ ) pR ðkÞ

~ eðkÞ pR ðkÞ 2 2

(23)

For the same reason, 2

~ eðkÞ pR ðkÞ ~ pR ðkÞ þ ðk kÞ 2

(24)

2 ~ eðkÞ ~ ðk~ kÞ eðkÞ pR ðkÞ pR ðkÞ 2 2

(25)

From (23) and (24), ðk~ kÞ

2

Divided by k~ k, and take limitation k~ ! k ) pR ðkÞ ¼ eðkÞ 2 Part (a)

2

pR ðkÞ ¼

ðk k

pR ðkÞdk ¼

ðk k

pR ðkÞdk

ðk k

pR ðkÞdk ¼ pR þ

Part (b) 2 ~2 from (25): ðk~ kÞ eð2kÞ ðk~ kÞ eðkÞ 2 ~ 2 eðkÞ2 ) eðkÞ ~ eðkÞ when k~> k ) eðkÞ 2 2 ~ eðkÞ ~ ~ when k < k ) eðkÞ eðkÞ ) eðkÞ This finishes the proof that eðkÞ is decreasing in k

1 2

ðk k

eðkÞ2 dk

(26)

278

S.K. Mukhopadhyay and X. Su

Derivation of Equation (12) The objective function (11) can be written as: ðk

RðkÞdFðkÞ ¼

k

ðk

where ðk ðk

ðpT ðpðkÞ;qðkÞ;kÞ pR ðkÞÞdFðkÞ ¼

k

1 2

eðxÞ2 dxdFðkÞ ¼

k k

ðk ðk k k

ðk

ðk

pT ðpðkÞ;qðkÞ;kÞdFðkÞ

k

eðxÞ2 dxdFðkÞ pR

FðkÞeðxÞ2 dk

k

¼

ðk

ðpT ðpðkÞ;qðkÞ;kÞ

1 FðkÞ eðkÞ2 ÞdFðkÞ pR 2 f ðkÞ

ðpT ðpðkÞ;qðkÞ;kÞ

zðkÞ eðkÞ2 ÞdFðkÞ pR 2

k

¼

ðk k

where zðkÞ ¼

FðkÞ f ðkÞ

Proof of Proposition 2, Part (c)

p2R ¼

e^ðkÞ2 ða bsÞ2 ¼ <0 2 2ð2bðk þ zðkÞÞ 1Þ2

2

ðabsÞ ð1þzðkÞÞ p2T ¼ 2ð2bðkþzðkÞÞ1Þ 2 < 0 total supply chain profit is decreasing in k p2M

¼

p2T

p2R

ða bsÞ2 1 þ zðkÞ 1 ð ¼ Þ 2 2 ð2bðk þ zðkÞÞ 1Þ2 ð2bðk þ zðkÞÞ 1Þ

¼

ða bsÞ2 zðkÞ 2ð2bðk þ zðkÞÞ 1Þ2

< 0:

where zðkÞ ¼

FðkÞ and zðkÞ > 0; f ðkÞ

The OEM’s profit is also monotonically decreasing in k.

Use of Supply Chain Contract to Motivate Selling Effort

279

Proof of Proposition 2, Part (e) Since the retailer earns a profit strictly higher than pR for any k 6¼ k the cutoff point should be where the OEM’s profit hits pM if there is one on the support ½k; k. Let p2M

ð ða bsÞ2 ðb þ zðkÞÞ 1 k ða bsÞ2 ¼ dx pR ¼ pM 2ð2bðb þ zðkÞÞ 1Þ 2 b ð2bðx þ zðxÞÞ 1Þ2 ðk 2ðpM þ pR Þ b þ zðkÞ 1 ) dx ¼ 2 2bðb þ zðkÞÞ 1 ða bsÞ2 b ð2bðx þ zðxÞÞ 1Þ

References Ayra A, Demski J, Glover J, Liang P (2009) Quasi-robust multiagent contracts. Manage Sci 55(5):752–762 Blair BF, Lewis TR (1994) Optimal retail contracts with asymmetric information and moral hazard. Rand J Econ 25(2):284–296 Cachon GP, Kok AG (2010) Competing manufacturers in a retail supply chain: on contractual form and coordination. Manage Sci 56(3):571–589 Cachon GP, Lariviere MA (2005) Supply chain coordination with revenue-sharing contracts: strengths and limitations. Manage Sci 51(1):30–45 Cakanyildirim M, Feng Q, Gan X, Sethi SP (2008) Contracting and coordination under asymmetric production cost information. SSRN Working Paper Series. http://ssrn.com/abstract¼1084584 Chen F (2005) Salesforce incentives, market information, and production/inventory planning. Manage Sci 51(1):60–75 Chen H, Chen Y, Chiu CH, Choi TM, Sethi SP (2010) Coordination mechanism for the supply chain with leadtime consideration and price-dependent demand. Eur J Oper Res 203(1):70–80 Corbett CJ, Zhou D, Tang CS (2004) Designing supply contracts: contract type and information asymmetry. Manage Sci 50(4):550–559 Coughlan AT (1993) Salesforce compensation: a review of MS/OR advances. In: Eliashberg J, Lilien GL (eds) Handbooks in OR&MS, vol 5. Elsevier, Amsterdam, pp 611–651 Cui TH, Raju JS, Zhang ZJ (2008) A price discrimination model of trade promotions. Mark Sci 27(5):779–795 Desai PS, Srinivasan K (1995) Demand signaling under unobservable effort in franchising: linear and nonlinear price contracts. Manage Sci 41(10):1608–1623 Desiraju R, Moorthy S (1997) Managing a distribution channel under asymmetric information with performance requirements. Manage Sci 43(12):1628–1644 Dukes A, Liu Y (2010) In-store media and distribution channel coordination. Mark Sci 29(1): 94–107 Federal Trade Commission (2005) Consumer guide to buying a franchise. Purch World 37(4):17 Foros Ø, Hagen KP, Kind HJ (2009) Price-dependent profit sharing as a channel coordination device. Manage Sci 55(8):1280–1291 Gal-Or E (1991) Vertical restraints with incomplete information. J Ind Econ 39(5):503–516 Gan X, Sethi SP, Yan H (2005) Channel coordination with a risk-neutral supplier and a downsiderisk-averse retailer. Prod Oper Manage 14(1):80–89 Gibbons R (1987) Price-rate incentive schemes. J Labor Econ 5(4):413–429 Gonik J (1978) Tie salesmen’s bonuses to their forecasts. Harv Bus Rev 56(3):116–122

280

S.K. Mukhopadhyay and X. Su

Gurnani H, Xu Y (2006) Resale price maintenance contracts with retailer sales effort: effect of flexibility and competition. Nav Res Logistics 53(5):448–463 Ha AY (2001) Supplier–buyer contracting: asymmetric cost information and cutoff level policy for buyer participation. Nav Res Logistics 48:41–64 Kamien MI, Schwartz N (1981) Dynamic optimization: the calculus of variations and optimal control in economics and management. North-Holland, New York Kaufmann PJ, Dant RP (2001) The pricing of franchise rights. J Retail 77:537–545 Kreps D (1990) A course in microeconomic theory. Princeton University Press, Princeton, NJ Krishnan H, Winter A (2010) Inventory dynamics and supply chain coordination. Manage Sci 56(1):141–147 Krishnan H, Kapuscinski R, Butz DA (2004) Coordinating contracts for decentralized supply chains with retailer promotional effort. Manage Sci 50(1):48–62 Laffont JJ, Martimort D (2000) Mechanism design with collusion and correlation. Econometrica 68(2):309–342 Laffont JJ, Martimort D (2001) The theory of incentives: the principal-agent model. Princeton University Press, Princeton, NJ Laffont JJ, Tirole J (1986) Using cost observation to regulate firms. J Polit Econ 94(3):614–641 Lafontaine F, Slade ME (1997) Retail contracting: theory and practice. J Ind Econ 45(1):1–25 Lal R (1990) Manufacturer trade deals and retail price promotions. J Mark Res 27(4):428–444 Majumder P, Srinivasan A (2008) Leadership and competition in network supply chains. Manage Sci 54(6):1189–1204 Mukhopadhyay SK, Zhu X, Yue X (2008) Optimal contract design for mixed channels under information asymmetry. Prod Oper Manage 17(6):641–650 Mukhopadhyay SK, Su X, Ghose S (2009) Motivating retail marketing effort: optimal contract design. Prod Oper Manage 18(2):197–211 Myerson RB (1979) Incentive compatibility and the bargaining problem. Econometrica 47:61–74 Raju J, Zhang ZJ (2005) Channel coordination in the presence of a dominant retailer. Mark Sci 24(2):254–304 Spengler JJ (1950) Vertical integration and antitrust policy. J Polit Econ 58:347–352 Su X, Wu L, Yue X (2010) Impact of introducing a direct channel on supply chain performance. Int J Electron Bus 8(2):101–112 Taylor TA (2002) Supply chain coordination under channel rebates with sales effort effects. Manage Sci 48(8):992–1007 Tsay A (1999) Quantity flexibility contract and supplier-customer incentives. Manage Sci 45(10):1339–1358 Watson J (2007) Contract, mechanism design, and technological detail. Econometrica 75(1):55–81 Wimmer BS, Garen JE (1997) Moral hazard, asset specificity, implicit bonding, and compensation: the case of franchising. Econ Enquiry 35(3):544–555 Wu Y, Loch CH, Heyden LV (2008) A model of fair process and its limits. Manuf Serv Oper Manage 10(4):637–653 Yang Z, Aydin G, Babich V, Beil DR (2009) Supply disruptions, asymmetric information, and a backup production option. Manage Sci 55(2):192–209 Zhao Y, Choi TM, Cheng TCE, Sethi SP, Wang S (2010) Buyback contracts with a stochastic demand curve. SSRN Working Paper Series. http://ssrn.com/abstract¼1559475 Zhu X, Mukhopadhyay SK (2009) Optimal contract design for outsourcing: pricing and quality decisions. Int J Revenue Manage 3(2):197–217

Price and Warranty Competition in a Duopoly Supply Chain Santanu Sinha and S.P. Sarmah

Abstract This chapter analyzes the coordination and competition issues in a twostage distribution channel where two different retailers compete on their retail price and warranty policy to sell two substitutable products in the same market. The demand faced by each retailer not only depends on its own price and warranty duration, but also on the price and warranty duration set by the other. Mathematical models have been developed to analyze the dynamic competition and coordination mechanism for three different cases where retailers compete (1) exclusively on price; (2) exclusively on warranty duration; (3) both price and warranty duration. The mathematical models show that under price/warranty competition, the steady state equilibrium is dynamically stable in nature under certain condition(s). Further, it has been shown that the channel profit for each case is higher under coordination than that of under competition and the maximum channel profit is achieved when retailers coordinate each other to adopt a centralized policy to set both price and warranty duration. However, it has been observed that though coordination enhances overall supply-chain profitability, it may make consumers worse-off due to higher product prices. The model is illustrated with suitable numerical examples. Keywords Competition • Coordination • Game-theory • Pricing • Stability • Supply-chain management • Warranty

S. Sinha Complex Decision Support Systems, Tata Consultancy Services, Akruti Trade Centre, MIDC, Andheri (E), Mumbai 400093, India e-mail: [email protected] S.P. Sarmah (*) Department of Industrial Engineering and Management, Indian Institute of Technology, Kharagpur 721302, India e-mail: [email protected] T.-M. Choi and T.C. Edwin Cheng (eds.), Supply Chain Coordination under Uncertainty, International Handbooks on Information Systems, DOI 10.1007/978-3-642-19257-9_12, # Springer-Verlag Berlin Heidelberg 2011

281

282

S. Sinha and S.P. Sarmah

1 Introduction The landscape of business environment has experienced significant changes in recent years. Among many factors, globalization of business, increased market competition, awareness of customers, and increased demand for the value added products/services have largely contributed to the change in the shift. The changing face of business environment has compelled academic researchers and industry leaders to rethink about how to manage business operations more efficiently and effectively. Since, the scope for improvement within an organization is restricted with limited resources; the researchers and practitioners are looking for newer alternatives. In this sense, the importance of integrating business activities, both inside and outside the organization’s boundary have been realized by all. The concept of integrating business functions beyond the organization’s boundary has led to the development of the theory and practices of Supply Chain Management (SCM) and one of the important issues in SCM is coordination. There is a growing body of academic researchers and practitioners from a variety of disciplines who focus on different issues of supply chain coordination and strive to establish potential coordination mechanisms to eliminate sub-optimization within a supply chain and enhance overall performance. An important finding from the existing body of literature is that in most of the coordination models, the buyers are assigned to the supplier(s) exogenously, i.e. products are considered independent. However, when there are many vendors in the market who can supply similar type of the product to the buyers, there is a pricecompetition among the vendors. In real-life business, there are many substitutable products in different market place where the respective vendors/retailers have to compete the others to sell the products. For example, Pepsi and Coca-Cola in soft drink market, Sotheby’s and Christie’s in diamond auctions; Kodak and Fuji-film in motion picture film stock market; ABC, CBS, and NBC in US television (before FOX); GM, Ford and Chrysler in auto industry (before the 1970s); etc. Under such scenario, development of coordination mechanism and analysis of competition is an important area of study (Sinha and Sarmah 2010). There are numerous papers on monopolistic and duopolistic competition in marketing and operations management literature; for example, Moorthy (1988), Choi (2003), Yao and Liu (2005), etc. The authors have studied several issues on price competition in supply-chain distribution channels under different contexts. In most of these models, demand of a product is assumed to be a function both its own price as well as the price of the other. However, one critical observation is that in addition to price, consumers also look for additional “value” from the various nonprice attributes, such as quality, service, delivery flexibility, etc. In this sense, the suppliers may also consider the non-price attribute(s) as a competitive tool in their marketing strategy and tend to compete on both price and non-price attribute(s). Several researchers have brought many dimensions of such competition, for example, quality (Banker et al. 1998), service (Tsay and Agrawal 2000), delivery frequency (Ha et al. 2003), etc.

Price and Warranty Competition in a Duopoly Supply Chain

283

The end consumers’ value perception and purchase decisions are also significantly influenced by warranty duration. This is perfectly related with the growing consumption pattern of FMCG, engineering instruments, manufacturing/electronics products where customers not only look for comparable prices but also associated product value/risk. Thus, offering free repair/replacement during the warranty/protection period often enhances the purchase decision of buyers. Many manufacturers/retailers offer warranties to the end consumers in different forms to boost up the overall sales demand. For example, automobile manufacturer Hyundai is well known for their extensive warranty coverage. GM and Ford have extended the powertrain warranty for their 2007 vehicles from 3 years/36,000 miles to 5 years/100,000 miles and 5 years/60,000 miles respectively (Scherer 2006). This has resulted in higher sales, bringing greater profit (Connelly 2006). Following GM, several imported brands have offered broader powertrain coverage on their 2007 vehicles (Hu 2008). Mitsubishi has started offering warranty coverage over 10 years or 100,000 miles. Similarly Suzuki vehicles have powertrain warranty coverage for 7 years or 100,000 miles (Scherer 2006). Similar to automobile industry, FMCG industry have also applied various forms of warranty to escalate the overall market demand. Similarly, several other firms have also used warranty as a marketing weapon to boost up the product demand – along with pricing strategy. However, the issue of price and warranty competition among competing vendors is still an unaddressed research question which requires further analysis. In this chapter, we consider the issues of price and warranty competition between two different retailers. The two competing retailers obtain a product from a common manufacturer, add some value on it, and finally sell it to the market. Three different cases are considered here where the retailers compete (1) exclusively on price; (2) exclusively on warranty duration; (3) both price and warranty duration. For each case, we have developed the steady state equilibrium for dynamic competition and system-wide solution under integrated/coordination mechanism. The main features that make a distinction of this work from the existing related literature is that the formulations and equilibrium strategies of our models explicitly depend on the pricing and warranty policy of competing retailers. We have considered here the inherent dynamics associated with the process of price adjustment while modeling competition. Static modeling of retail price competition can derive the equilibrium but the adjustment of retail price to equilibrium does not occur instantaneously. Like most of the dynamic economic systems, the mechanism of dynamic adjustment is an iterative process converging to equilibrium over a period of time. This chapter analyzes the stability of such equilibrium. Here, the term “stability” means that whether the process of dynamic adjustment of price/warranty duration will eventually converge to equilibrium over a period of time and there is no further divergence from that “fixed/stable” point. We have derived the conditions for the equilibrium to be dynamically stable. This chapter has been organized as follows. A brief review of literature is included in Sect. 2. Section 3 includes the notation and modeling assumptions. The mathematical models are developed in Sect. 4. Section 5 illustrates the dynamics of price, warranty, and simultaneous price and warranty competition.

284

S. Sinha and S.P. Sarmah

The impact of coordination has been analyzed in Sect. 6. Further, numerical illustration has been included in Sect. 7. Finally, the conclusion of the chapter is given in Sect. 8 along with a few possible future research directions.

2 Review of Literature In this section we have provided a brief review of literature related to the work. From the perspective of economic theory, a large number of research papers are available on market competition. Most of the papers deal with either quantitycompetition (Cournot 1938) or price-competition (Bertrand 1883) and their primary focus is on applying game theory to derive equilibria under varied assumptions. On the other hand, in marketing and operations management literature, there are many papers on monopolistic and duopolistic competition. The aspects of coordination and competition have received a considerable amount of attention from the researchers. Moorthy (1988) has considered two identical firms competing on quality and price and analyzed the role of consumer preferences, firms’ costs and price competition in determining a firm’s equilibrium product strategy. Rao (1991) has developed a modelling framework to derive the equilibrium in a duopoly market where the members compete on price and promotions. Competition between direct and indirect channels has been analyzed by Choi (2003). Further, Yao and Liu (2005) have developed competitive equilibrium pricing policies under the Bertrand and the Stackelberg competition model between a mixed e-tail and retail distribution channel. They have shown that introduction of e-tail into a manufacturing distribution system not only generates competitive pricing and payoffs, but also encourages cost-effective retail services. They have also proposed a strategic approach for the manufacturer to add an e-tail channel. However, most of these models have focused only on price competition. Apart from price competition, there are several other models where the authors have studied the aspect of non-price competition. For example, Ha et al. (2003) have considered a supply chain in which two suppliers compete on price and delivery frequency to supply to a customer. They have shown that the customer is better off under delivery competition, while the suppliers are better off under price competition. However, the model did not consider any coordination aspects within the channel and the demand was assumed to be price-independent. Banker et al. (1998) have studied both price and quality competition and addressed the question of how quality is influenced by competitive intensity in an oligopoly market. So (2000) has studied the aspect of price and delivery time as the competition attributes and illustrated how different firms and market characteristics might affect the price and delivery time competition in the market. Tsay and Agrawal (2000) have studied a distribution system in which a manufacturer supplies a common product to two independent retailers, who in turn use service as well as retail price to directly compete for end customers. They have examined the drivers of each firm’s strategy, and the consequences for total sales, market share, and profitability. Finally, it has

Price and Warranty Competition in a Duopoly Supply Chain

285

been shown that the relative intensity of competition with respect to each competitive dimension plays a key role, as does the degree of cooperation between the retailers. Another important non-price attributes is warranty policy – which is also a very popular marketing tool. Most common form of warranty policy may include free replacement of failed product, coverage of parts/labor work, repair of failed products within a specified interval – known as warranty duration. As a common practice in industry, warranties have received the attention of researchers from many diverse disciplines. Authors like, Menezes and Currim (1992) and Padmanabhan (1993) have justified how application of warranty policy could be a marketing tool to differentiate from competitors. A comprehensive review of related research in different application domains can be found in Blischke and Murthy (1996). Several researchers have explored various aspects of warranties, such as warranty type, product failures during warranty period, warranty claims, warranty costs, and warranty logistics. A review has been provided by Hu (2008). However, though there is a stream of literature that has focused exclusively on design of warranty policy (Murthy 2006; Wu et al. 2009), there are not much research on warranty competition. Further, most of the competition models have dealt with deriving static equilibrium, and not studied the aspects of dynamic process adjustment. Thus, in order to address the issues, in this chapter, we extend the aspects of price and warranty competition in a duopoly market. We also extend the analysis to include the dynamics of such competition under various scenarios.

3 Notation and Assumptions The following notations are used to develop the mathematical model (where i ¼ 1, 2) Di Demand of product i pi Unit retail price of manufacturer i ti Warranty length of product i Unit repair cost of product i ci w Per unit wholesale price of the manufacturer pi Profit of retailer i

3.1

Demand Function

In this study, a two-stage distribution channel is considered where a manufacturer distributes a single product to two different retailers. The retailers add some value to the product and sell it to the end customers. The end products are substitute to each other. Further, in addition to product prices of the product itself and those of

286

S. Sinha and S.P. Sarmah

related substitute products, demand also depends on a variety of other non-price factors such as quality, delivery, service, etc. In this study we consider warranty length as another factor that influences demand. In this sense, the demand function is assumed to take the following linear form (Banker et al. 1998): Di ¼ a bpi þ gpj þ y ti mtj

ði; j ¼ 1; 2; i 6¼ jÞ

(1)

The price and cross-price elasticity parameters b and g move independently. It is assumed that a; b>0 so that demand for each product declines with the product’s own price. Further, b>jgj so that “own-price effect” always dominates the “crossprice effect”. The assumed demand function is downward slopping, a variation of a class of more general linear demand functions used in many previous studies (McGuire and Staelin 1983; Choi 1991).

3.2

Warranty Cost

Under the warranty policy, retailer i is responsible to repair each failed product during the warranty length ti with no extra charge to the customer. Following Wu et al. (2009), let us consider that N(t) is the number of failures for the product over a warranty duration of t. Thus, assuming the failure time of the products is independently and identically distributed with the cumulative distribution function F(ti), the expected value of N(ti) is given by E½N ðti Þ ¼ Mðti Þ, where M(ti) is the expected number of renewals of the product with warranty length ti. Further, considering Fðti Þ as a Weibull distribution with failure rate li and a n shape parametern, Fðti Þ ¼ 1 eðli ti Þ . Accordingly, the expected number of failures of product with warranty length ti can be obtained as: Mðti Þ ¼ ðli ti Þn (Wu et al. 2009). Thus, the warranty-related cost of product i in the warranty duration can be given as, Ci ¼ ci ðli ti Þn Di .

4 Mathematical Models Let us consider a typical market model with two competing retailers where each retailer procures the same material from the sole manufacturer at a certain wholesale price w, adds some value on it, and finally sells it in the market – as shown in Fig. 1. Depending on the nature of competition between the two retailers, three different cases have been considered here: (i) Price competition where both the retailers compete each other on price only. (ii) Warranty competition where both retailers compete each other on warranty length. (iii) Price and Warranty competition where both retailers compete each other on price and warranty duration simultaneously.

Price and Warranty Competition in a Duopoly Supply Chain

287

Manufacturer

w

Retailer 1

{p1,t1}

Retailer 2

{p2,t2}

D1

D2

MARKET

Fig. 1 The distribution channel

4.1

Equilibrium in Price Competition

The expected profit function of retailer i can be given as: pi ðpi Þ ¼ Di ½pi w ci ðli ti Þn ¼ a bpi þ gpj þ y ti mtj ½pi w ci ðli ti Þn ; ði; j ¼ 1; 2; i 6¼ jÞ The objective of retailer i, Max pi ðpi Þ

(2)

Since, @@ppi 2i ¼ 2b<0, pi ðpi Þ is concave in nature and the optimal solution can be i achieved from first order conditions: @p @pi ¼ a 2bpi þ gpj þ y ti mtj þ wbþ n ci ðli ti Þ b ¼ 0 Or, 2

pi ¼

a þ wb þ gpj þ y ti þ ci ðli ti Þn b mtj 2b

(3)

This is known as the best response function1 of retailer i. The objective of each retailer is to find out the best response function against each other’s retail price and accordingly find out the Pareto-optimal solution. It is obvious that a change in the retail price of a product can influence demand for other product. Since both retailers are profit maximizer, each retailer will adjust her retail price in response to the

1 A retail price p1 ¼ p1 ðp2 Þ is defined as the best response function against retail price p2 if and only if p1 ðp1 ðp2 Þ; p2 Þ p1 ðp1 ; p2 Þ, for any p1 . Similarly, a retail price p2 ¼ p2 ðp1 Þ is defined as the best response function against retail price p1 if and only if p1 ðp1 ðp2 Þ; p2 Þ p1 ðp1 ; p2 Þ, for any p2 (Shy 2003).

288

S. Sinha and S.P. Sarmah

p1(p2)

p2

p2(p1)

{p1*, p2*}

p1

Fig. 2 Best response functions in price competition

other. The process of price-adjustment then leads to further price rise/fall till it reaches the equilibrium (Fig. 2). This equilibrium is a Pareto-optimal solution derived by the intersection(s) of the best response functions beyond which no retailer has an incentive to unilaterally change its strategy given the other’s equilibrium strategy and hence it is a Paretooptimal solution. In this particular case, the Pareto-optimal solution is Nash–Bertrand equilibrium.2 Proposition 1. (a) Under the general retail price competition, both retailers adjust their retail prices in the similar direction of the change (i.e. simultaneously either increase or decrease). (b) Under the general retail price competition, given retailer j sets her retail pricepj , retailer i will increase her retail price pi , if a þ gpj þ y ti mtj þ wb þ ci ðli ti Þn b >pi . 2b Proof. The proof is given in Appendix A. The Nash–Bertrand equilibrium ðp1 ; p2 Þ can be derived by solving the best response functions as given by (3) and accordingly, pi ¼

n 2abþaggmti þ2byti þ2b2 ci ðli ti Þn 2bmtj þgyt2 þbcj g lj tj þ2b2 wþbgw 4b2 l2 (4)

2 A Nash–Bertrand equilibrium point is a pair of retail prices ðp1 ; p2 Þ offered by the retailers, each of which is a best response of the other: p1 ¼ p1 ðp2 Þ and p2 ¼ p2 ðp1 Þ (Shy 2003).

Price and Warranty Competition in a Duopoly Supply Chain

289

Finally, ½pi p ¼ Di ðp1 ; p2 Þðpi w ci ðli ti Þn Þ

(5)

Proposition 2. Under the general retail price competition, the Nash–Bertrand equilibrium increases with failure rate li and warranty length ti . However, the Nash–Bertrand equilibrium increases with shape factor n if 2b2 ci ðli ti Þn lnðli ti Þþ n bcj g lj tj ln lj tj >0. Proof. The proof is given in Appendix B.

4.2

Equilibrium in Warranty Competition

In this case, both retailers are assumed to compete each other on warranty length. Thus, the objective of retailer i, Max pi ðti Þ

(6)

It is straightforward to derive that, @ 2 pi ¼ ci nli n ðti Þn2 ½2y ti þ Di ðn 1Þ @ti 2 Since, ci nli n ðti Þn2 >0, for Di >0 and n 1, @@tip2i <0. Thus, pi ðti Þ is concave in nature and the optimal solution can be achieved from first order conditions: 2

@pi ¼ ci li n ðti Þn1 y ti þ a bpi þ gpj þ y ti mtj n pi y þ wy ¼ 0 @ti Or, ðti Þn1 ¼

ðpi wÞy ; ci li n ðy ti þ Di nÞ

where

Di ¼ a bpi þ gpj þ y ti mtj :

(7)

The intermittent values of ti can be derived by iterative computations and finally, the Nash equilibrium ðt1 ; t2 Þ can be derived by solving, ðti Þn1 ¼

ðpi wÞy ; where Di ¼ a bpi þ gpj þ yti mtj ci li ðy ti þ Di nÞ n

(8)

Accordingly, ½pi w ¼ Di ðt1 ; t2 Þðpi w ci li ti Þ

(9)

290

S. Sinha and S.P. Sarmah

t1(t2)

t2

t2(t1)

{t1*,t2*}

t1

Fig. 3 Best response functions in warranty competition

Following (7), it has been observed that the best response functions ti tj , ði; j ¼ 1; 2; i 6¼ jÞ are nonlinear in nature due to the shape parameter n. Thus, the dynamic adjustment of warranty duration, starting from an initial condition, could further go uphill or downhill to reach the Nash equilibrium ðt1 ; t2 Þ – as shown in Fig. 3 as an example. However, it is to be noted that the slopes and the trajectory of any such best response functions and the number of intersection points (equilibrium) depend on the functional form and type of the profit functions.

4.3

Equilibrium in Price and Warranty Competition

In this case, both the retailers compete each other on warranty length. Thus, the objective of retailer i, Max pi ðpi ; ti Þ

(10)

The Hessian matrix of pi ðpi ; ti Þ is given as, 2

@ 2 pi 6 @pi 2 Hi ¼ 6 4 @ 2 pi @ti @pi

3 @ 2 pi 2b @pi @ti 7 7 ¼ @ 2 pi 5 y þ bci nli n ti n1 @ti 2

y þ bci nli n ti n1 n ci nli ðti Þn2 ½2y ti þ Di ðn 1Þ

Price and Warranty Competition in a Duopoly Supply Chain

291

Here, the members of the principal diagonal are negative; thus, the objective function pi ðpi ; ti Þ is a concave function if jHi j<0, 2 Or, ðy þ bci nli n ti n1 Þ >2bci nli n ðti Þn2 ½2y ti þ Di ðn 1Þ: The Nash equilibrium under simultaneous price and warranty competition ðp~i ; ~ti Þ can be derived by solving the following set of best response functions, 9 a þ wb þ gpj þ y ti þ ci ðli ti Þn b mtj > > = 2b > ðpi wÞy > ; ti n1 ¼ n ci li ðy ti þ Di nÞ

pi ¼

(11)

The Nash equilibrium ðp~1 ; ~t1 Þ and ðp~2 ; ~t2 Þ can be found by simultaneously solving the sets of equations in (11). Finally, ½pi p;w ¼ D~i ðp~i ; ~ti Þðp~i w ci li~ti Þ

(12)

5 Dynamics of Competition In the earlier section, we have derived the Nash–Bertrand equilibrium under three different cases where the retailers compete each other on (1) price, (2) warranty duration, (3) both price and warranty duration. However, the adjustment of initial price or warranty duration gradually leads toward the Nash–Bertrand equilibrium following an iterative process, where at each step; each retailer chooses a policy which maximizes the individual profit based on the expected policy set by her opponent. Hence, at each time period every retailer depends on an expectation of the other retailer’s policy in the next time period to determine the corresponding profit-maximizing policy for that period. This leads to a dynamic adjustment of price and warranty duration which finally reaches to the Nash–Bertrand equilibrium. Here, we analyze the behavior of such dynamic adjustment process of price and warranty competition under two different scenarios (1) naı¨ve expectation and (2) adaptive expectation. In the former case, each player assumes the last values taken by the competitors without estimation of their future reactions in each step. However, in case of adaptive expectation, each retailer revises her beliefs according to the adaptive expectations rules which compute the outputs with weights between last period’s outputs and her reaction function. A related discussion is included in Agiza and Elsadany (2003) and Shone (2001). In this section, we develop mathematical models to capture the scenario where retailers compete each other in (1) price, (2) warranty duration, (3) both price and warranty duration under both naı¨ve and adaptive .expectation. We have investigated the stability condition(s) corresponding to each case. The main objective of the

292

S. Sinha and S.P. Sarmah

models is to investigate the dynamic behavior of a duopoly game with price and warranty competition under different expectation rules. Through a numerical we further explain the movement of the system variables over a period of time.

5.1

Dynamics of Price Competition

If we denote the retail price of product i at time period t by pi ðtÞ, then the retail price pi ðt þ 1Þ for the period ðt þ 1Þ is decided by solving the two optimization problems (Agiza and Elsadany 2003): p1 ðt þ 1Þ ¼ arg max p1 ðp1 ðtÞ; p2 e ðt þ 1ÞÞ p1

p2 ðt þ 1Þ ¼ arg max p2 ðp1 e ðt þ 1Þ; p2 ðtÞÞ;

(13)

p2

where the function pi ð:Þ denotes the profit of the retailer i and pj e ðt þ 1Þ represents the expectation of retailer i about the pricing decision of retailer j, ði; j ¼ 1; 2; j 6¼ iÞ. We consider that the retailers could be naı¨ve or adaptive players – depending on their adjustment process.

5.2

Dynamics of Price Competition with Naı¨ve Expectation

We assume that both the retailers are naı¨ve. With the assumption, we can express the process of duopoly game which can be defined as, 9 a þ wb þ gp2 ðtÞ þ y t1 þ c1 ðl1 t1 Þn b mt2 > > p1 ðt þ 1Þ ¼ = 2b > a þ wb þ gp1 ðtÞ þ y t2 þ c2 ðl2 t2 Þn b mt1 > ; p2 ðt þ 1Þ ¼ 2b

(14)

In equilibrium, pi ðt þ 1Þ ¼ pi ðtÞ and thus we are interested in the solution of the system with non-negative equilibrium points defined as, 9 a þ wb þ gp2 ðtÞ þ y t1 þ c1 ðl1 t1 Þn b mt2 > > = 2b a þ wb þ gp1 ðtÞ þ y t2 þ c2 ðl2 t2 Þn b mt1 > > ; p2 ðtÞ ¼ 2b

p1 ðtÞ ¼

The equilibrium point ðp1 ; p2 Þ has already been derived as,

(15)

Price and Warranty Competition in a Duopoly Supply Chain

pi ¼

293

n 2abþaggmti þ2byti þ2b2 ci ðli ti Þn 2bmtj þgyt2 þbcj g lj tj þ2b2 wþbgw 4b2 g2

It is clear that the Nash equilibrium ðp1 ; p2 Þ is located at the intersection of the two reaction functions. In this study, we are interested in studying the local stability of equilibrium point at ðp1 ; p2 Þ. This can be analyzed by the eigen values of the Jacobian matrix of the system (14) on the complex plane. The Jacobian matrix of (14) at the point ðp1 ; p2 Þ has the following form, 2

@f1 6 @p1 JpN ðp1 ; p2 Þ ¼ 6 4 @f2 @p1

3 @f1 @p2 7 7; @f2 5

(16)

@p2

a þ wb þ gpj ðtÞ þ y ti þ ci ðli ti Þn b mtj ¼ fi 2b 2 g 3 0 6 2b 7 Thus, JpN ðp1 ; p2 Þ ¼ 4 g 5 0 2b We now investigate the local stability The characteristic of Nash equilibrium. @f1 @f2 2 N N N equation is given as, f ðeÞ ¼ e Tr Jp e þ Det Jp ,where Tr Jp ¼ @p þ @p 1 2 @f1 @f2 @f1 @f2 N is the trace and Det Jp ¼ @p1 @p2 @p2 @p1 is the determinant of the Jacobian where, pi ðt þ 1Þ ¼

matrix defined in (16). 2 2 N J Now, Tr JpN ¼ 0 and Det JpN ¼ g 4Det JpN ¼ 2 <0. Thus, Tr p 4b 2 g >0. b2 Since, Tr2 JpN 4Det JpN >0, the eigen values of Nash equilibrium are real. Following a standard stability analysis, the necessary and sufficient condition for the stability of Nash equilibrium at ðp1 ; p2 Þ is that the eigen values of the Jacobian matrix JpN ðp1 ; p2 Þ are inside the unit circle of the complex plane. This is true if and only if the following conditions are hold (Puu 2002; Agiza and Elsadany 2004): 1. 1 Tr JpN þ Det JpN >0 2. 1 þ Tr JpN þ Det JpN >0 3. Det JpN 1<0 2 g Since, Det JpN 1 ¼ 4b 2 þ 1 <0, the third condition is already satisfied. g2 The first and second conditions imply, 1 4b 2 >0. Thus, the dynamic adjustment of retail price eventually reaches to a stable Nash equilibrium for b2 > g4 . 2

294

5.3

S. Sinha and S.P. Sarmah

Dynamics of Price Competition with Adaptive Expectation

Assuming both the retailers to be adaptive, the dynamic equation of the adaptive expectation can be defined as below, pi ðt þ 1Þ ¼ ð1 vi Þpi ðtÞ þ

vi a þ wb þ gpj ðtÞ þ y ti þ ci ðli ti Þn b mtj ; 2b

(17)

ði; j ¼ 1; 2; i 6¼ jÞ Here, vi 2 ½0; 1 is the speed of adjustment of the adaptive players. It can easily be noted that, if vi ¼ 1, it reduces to the form of naı¨ve expectation. This implies that naive expectation is a special case of adaptive expectations behavior. Now, we look for the equilibrium of the system (17) and discuss their stability properties. The fixed points of the map (17) are obtained as nonnegative solutions of the algebraic system by setting pi ðt þ 1Þ ¼ pi ðtÞ. This implies, vi a þ wb þ gpj ðtÞ þ y ti þ ci ðli ti Þn b mtj 2pi b ¼ 0; 2b

(18)

ði; j ¼ 1; 2; i 6¼ jÞ Since, v1 ; v2 > 0; the equilibrium point ðp1 ; p2 Þ can be derived as,

pi ¼

n 2abþaggmti þ2byti þ2b2 ci ðli ti Þn 2bmtj þgyt2 þbcj g lj tj þ2b2 wþbgw 4b2 g2

This shows that, the Nash equilibrium does not change with adaptive expectation; however the speed to reach Nash equilibrium depends on the speed of adjustment. Here, we are interested in studying the local stability of equilibrium point at ðp1 ; p2 Þ which can be analyzed by the eigen values of the Jacobian matrix of the system (17) on the complex plane – as shown below, 2

@g1 6 @p1 JpAD ðp1 ; p2 Þ ¼ 6 4 @g2 @p1

3 @g1 @p2 7 7; @g2 5

(19)

@p2

vi where, pi ðt þ 1Þ ¼ ð1 vi Þpi ðtÞ þ 2b a þ wb þ gpj þ y ti þ ci ðli ti Þn b mtj ¼ gi 2 v1 g 3 ð1 v1 Þ 6 2b 7 Thus, JpAD ðp0 1 ; p0 2 Þ ¼ 4 v2 g 5 ð 1 v2 Þ 2b v2 g2 AD Now, Tr Jp ¼ 2 v1 v2 and Det JpAD ¼ ð1 v1 Þð1 v2 Þ v14b 2 . For stable equilibrium, the following conditions have to be fulfilled,

Price and Warranty Competition in a Duopoly Supply Chain

295

C1. 1 Tr JpAD þ Det JpAD >0 C2. 1 þ Tr JpAD þ Det JpAD >0 C3. Det JpAD 1<0 Proposition 3. The Nash equilibrium is dynamically stable in price competition 2 under adaptive expectation if b2 > g4 . Proof. The proof is given in Appendix C.

5.4

Dynamics of Warranty Competition

Similar to the earlier model of dynamic price competition, the simultaneous game of warranty competition through warranty length adjustment can be represented in the following form, 9 t1 ðT þ 1Þ ¼ arg max p1 ðt1 ðTÞ; t2 e ðT þ 1ÞÞ = t1

t2 ðT þ 1Þ ¼ arg max p2 ðt1 e ðT þ 1Þ; t2 ðTÞÞ ;

(20)

t2

where warranty length offered for product i at time period T and T+1 is represented by ti ðTÞ and ti ðT þ 1Þ respectively. Here, we analyze the adjustment process and predict whether the Nash equilibrium is globally stable. The term “stability”, means whether during the process of dynamic price adjustment, an initial combination of warranty length will eventually converge to equilibrium in the long run without further deviation. The present dynamic system can be represented by (Ferguson and Lim 1998): t_1 ¼ c1 ðt1 t1 Þ

(21)

t_2 ¼ c2 ðt2 t2 Þ;

(22)

where, t_1 ¼ dt1 =dT, t_2 ¼ dt2 =dT and c1 ; c2 > 0 are adjustment coefficients representing the speed of the adjustment. The terms ðt1 ; t2 Þ and ðt1 ; t2 Þ denote the Nash equilibrium and the actual level of warranty length at any time-period (T) respectively. Since the present system is non-linear, we apply the following theorem developed by Olech (1963) to check the dynamic stability of the system given by (21). Theorem 1. Consider an autonomous system x_ ¼ f ðx; yÞ y_ ¼ gðx; yÞ

) (23)

296

S. Sinha and S.P. Sarmah

where x_ ¼ dx=dt, y_ ¼ dx=dt, and ðx; yÞ 2 R2 . The functions f and g are assumed to be of class C1 onR2 . Suppose that there is a unique equilibrium point ðx; yÞ on R2 , i.e., a point such that f ðx; yÞ ¼ 0 and gðx; yÞ ¼ 0. If the following conditions are satisfied, the equilibrium is asymptotically stable in the large (i) Trace J ðx; yÞ fx þ gy <0, for all ðx; yÞ in R2 : for all ðx; yÞ in R2 : (ii) Det J ðx; yÞ fx gy fy gx >0, (iii) Either fx gy 6¼ 0, for all ðx; yÞ in R2 : for all ðx; yÞ in R2 . or fy gx 6¼ 0, where, fx ð@ [email protected]Þf ðx; yÞ and fy , gx , and gy are similarly defined. A proof of the theorem is given in Olech (1963). We have considered this theorem as given and proceed to look for conditions which guarantee the global stability of the mentioned system. It is straight-forward to derive the following differential equations: 2

1 !n1

ðp1 wÞy t_i ¼ ci 4 n ci li y ti þ a bpi þ gpj þ y ti mtj n

3 ti 5;

ði; j ¼ 1; 2; i 6¼ jÞ (24)

Conditions under which the above nonlinear system will be dynamically stable have been derived and shown in the form of the following proposition. Proposition 4. The Nash equilibrium under non-coordinated warranty length competition, is globally asymptotically stable for the following: (a) n 1,

i h ih i h n n n Di >0 and K1 ðy t1 þ D1 nÞn1 þ 1 K2 ðy t2 þ D2 nÞn1 þ 1 > L1 ðyt1 þ D1 nÞn1 h i n L2 ðy t2 þ D2 nÞn1

(b) For n<1

h i n n Di >0, Ki ðy ti þ Di nÞn1 < 1, and K1 ðy t1 þ D1 nÞn1 þ 1 h i h ih i n n n K2 ðy t2 þ D2 nÞn1 þ 1 > L1 ðy t1 þ D1 nÞn1 L2 ðy t2 þ D2 nÞn1

1

yðn þ 1Þ pi y wy n1 nm pi y wy n1 1 and L ¼ where, Ki ¼ : i ðn 1Þ ci li n n1 ci l i n

Proof. The Jacobian matrix of t_i (i ¼ 1, 2) is given as, 2

@ t_1 6 @t 1 J¼6 4 @ t_2 @t1 It is straight-forward to derive,

3 @ t_1 @t2 7 7 ¼ J1 @ t_2 5 J3 @t2

J2 J4

Price and Warranty Competition in a Duopoly Supply Chain

h i9 n @ t_i > ¼ ci Ki ðy ti þ Di nÞn1 þ 1 > = @ti h i n > @ t_i > ; ¼ ci Li ðy ti þ Di nÞn1 @tj

297

ði; j ¼ 1; 2; i 6¼ jÞ

(25)

(I) From stability condition (i), • for n 1 and Di >0, Trace J ðx; yÞ fx þ gy ¼ J1 þ J4 <0 • for n<1 and Di >0, Trace J ðx; yÞ fx þ gy ¼ J1 þ J4 <0 n Ki ðy ti þ Di nÞn1 < 1

if

(II) From stability condition (ii), h ih i n n Det ¼ c1 c2 K1 ðy t1 þ D1 nÞn1 þ 1 K2 ðy t2 þ D2 nÞn1 þ 1 h ih i n n c1 c2 L1 ðy t1 þ D1 nÞn1 L2 ðy t2 þ D2 nÞn1 If Det J ðx; yÞ>0, then, h

ih i h i n n n K1 ðy t1 þ D1 nÞn1 þ 1 K2 ðy t2 þ D2 nÞn1 þ 1 > L1 ðy t1 þ D1 nÞn1 h i n L2 ðy t2 þ D2 nÞn1

(III) Since no element in the matrix J is zero the other conditions of the theorem are also satisfied. □

Hence proved.

5.5

Dynamics of Warranty Competition with Adaptive Expectation

Assuming both the retailers to be adaptive, the dynamics of warranty length adjustment can be defined as below,

ti ðT þ 1Þ ¼ ð1 wi Þti ðTÞ þ wi

ðpi wÞy ci li n ½y ti ðT þ 1Þ þ Di n

1 n1

ði ¼ 1; 2Þ

298

S. Sinha and S.P. Sarmah

Here, wi 2 ½0; 1 is the speed of adjustment of the adaptive players which considers both previous warranty policy and the best response function. Now, we look for the equilibrium of the system (25). The fixed points of the map (25) are obtained as nonnegative solutions of the algebraic system by setting ti ðT þ 1Þ ¼ ti ðTÞ. This implies,

ti ðTÞ ¼

ðpi wÞy ci li n ½y ti ðT þ 1Þ þ Di n

1 n1

ði ¼ 1; 2Þ

(26)

This clearly shows that Nash equilibrium remains unchanged irrespective of the speed of adjustment. Under adaptive expectation, the warranty competition leads to the same Nash equilibrium and the Nash equilibrium is dynamically stable under the conditions shown in Proposition 4.

5.6

Dynamics of Price and Warranty Competition

In this case, we consider that both retailers dynamically adjust both retail price and warranty length in each time period. The retail price pi ðT þ 1Þ and warranty length ti ðT þ 1Þ for the period ðT þ 1Þ is decided by solving the two optimization problems (Agiza and Elsadany 2003): 9 p1 ðT þ 1Þ ¼ arg max p1 ðp1 ðTÞ; p2 e ðT þ 1ÞÞ > > p1 > > > e > p2 ðT þ 1Þ ¼ arg max p2 ðp1 ðT þ 1Þ; p2 ðTÞÞ > = p 2

t1 ðT þ 1Þ ¼ arg max p1 ðt1 ðTÞ; t2 e ðT þ 1ÞÞ > > > t1 > > > > e t2 ðT þ 1Þ ¼ arg max p2 ðt1 ðT þ 1Þ; t2 ðTÞÞ ;

(27)

t2

Here, the function pi ð:Þ denotes the profit of the retailer i and ð:Þj e ðT þ 1Þ represents the expectation of retailer i about the strategy of retailer j, ði; j ¼ 1; 2; j 6¼ iÞ. The dynamic adjustment of retail price and warranty length adjustment can be represented as follows (Ferguson and Lim 1998): 9 p_ 1 ¼ t1 ðp1 p1 Þ > > > p_ 2 ¼ t2 ðp2 p2 Þ = t_1 ¼ t3 ðt1 t1 Þ > > > ; t_2 ¼ t4 ðt2 t2 Þ

(28)

where, p_ i ¼ dpi =dT, t_i ¼ dti =dT and t1 ; t2 ; t3 ; t4 >0 are adjustment coefficients representing the speed of the adjustment. The terms ðp1 ; p2 Þ, ðt1 ; t2 Þ and

Price and Warranty Competition in a Duopoly Supply Chain

299

ðt1 ; t2 Þ, ðp1 ; p2 Þ denote the Nash equilibrium and the actual level of price and warranty length respectively at any time-period (T). We now check the dynamic stability of the system given by (28). The dynamic process can be represented by the following differential equations: p_ i ¼ ti

n 2ab þ ag gmti þ 2byti þ 2b2 ci ðli ti Þn 2bmtj þ gytj þ bcj g lj tj þ 2b2 w þ bgw 2 t_i ¼ t2þi 4

4b2 l2

! pi

3 !1 n1 ðp1 wÞy ti 5 ci li n y ti þ a bpi þ gpj þ y ti mtj n (29)

The Jacobian matrix of the dynamic system (28) is given as, 2

@ p_ 1 6 @p1 6 6 @ p_ 2 6 6 @p1 J ðp; tÞ ¼ 6 6 @ t_1 6 6 @p1 6 4 @ t_2 @p1

@ p_ 1 @p2 @ p_ 2 @p2 @ t_1 @p2 @ t_2 @p2

@ p_ 1 @t1 @ p_ 2 @t1 @ t_1 @t1 @ t_2 @t1

3 @ p_ 1 @t2 7 7 @ p_ 2 7 7 @t2 7 7 @ t_1 7 7 @t2 7 7 @ t_2 5 @t2

(30)

Further derivation of the Jacobian matrix J ðp; tÞ is given in Appendix D. Proposition 5. The Nash equilibrium under price and warranty length competition is globally asymptotically stable for the following: • For n 1: Di >0 and Det ¼ jJ ðp; tÞj>0 n • For n<1: Di >0, Ki ðy ti þ Di nÞn1 < 1, and Det ¼ jJ ðp; tÞj>0

Proof. The proof is straightforward from Theorem 1.

6 Channel Coordination In this section, we consider the different aspects of channel coordination between the retailers. A typical case may occur where both the retailers, understanding their inter-dependence, coordinate each other to set the optimal values of price/warranty duration that maximize the overall system/channel profit and thereby the individual pay-offs. The centralized policy thus includes deciding globally optimal retail price and warranty duration. The retailers can choose to decide system-wide optimal (1) retail price, (2) warranty duration, (3) both retail price and warranty duration.

300

S. Sinha and S.P. Sarmah

However, the current discussion excludes the aspect of sharing of coordination benefit.

6.1

Coordinated Policy to Set Retail Price

The system/channel profit is the total profit of the two retailers as follows, Pch ¼ p1 þ p2 Or, Pch ¼ ða bp1 þ gp2 þ yt1 mt2 Þ½p1 w c1 ðl1 t1 Þn þ ða bp2 þ gp1 þ yt2 mt1 Þ½p2 w c2 ðl2 t2 Þn . From the first order conditions, n @Pch ¼a 2bpi þ 2gpj þ yti mtj þ wb þ ci bðli ti Þn wg cj g lj tj ¼ 0; @pi ði;j ¼ 1; 2;j 6¼ iÞ

(31)

Solving, Ab þ Bg pi cp ¼ 2 2 b g2

(32)

n where, A ¼ a þ yti mtj þ wb þ ci bðli ti Þn wg cj g lj tj and, B ¼ a þ ytj n mti þ wb þ cj b lj tj wg ci gðli ti Þn ; where i; j ¼ 1; 2; i 6¼ j. The super-script “cp” indicates coordinated retail price. Finally, the system profit can be represented as, Pch cp ¼ p1 ðp1 cp ; p2 cp Þ þ p2 ðp1 cp ; p2 cp Þ.

6.2

Coordinated Policy to Set Warranty Duration

The objective of this model is to find out the system-wide optimal warranty duration for both the retailers. Since, the system profit, Pch ¼ p1 þ p2 is the total profit of the two retailers as follows, from the first order conditions, @Pch ¼0 @ti Or, "

1 n #n1 y½pi w ci ðli ti Þn m pj w cj lj tj ; ti ¼ ci li n Di n

ði; j ¼ 1; 2; j 6¼ iÞ (33)

Price and Warranty Competition in a Duopoly Supply Chain

301

The explicit solution of (33) is intractable; however iterative computation can be applied to derive the optimal solutionðt1 cw ; t2 cw Þ. The super-script “cw” indicates coordinated warranty length. The system profit is given as, Pch cw ¼ p1 ðt1 cw ; t2 cw Þ þ p2 ðt1 cw ; t2 cw Þ

6.3

(34)

Global Coordination

We use the term “global coordination” to mention a typical case where the retailers take a centralized decision to set both retail price and warranty duration to maximize system/channel profit. The system profit is the total profit of the two retailers as follows, Pch ðp1 ; p2 ; t1 ; t2 Þ ¼ p1 ðp1 ; p2 ; t1 ; t2 Þ þ p2 ðp1 ; p2 ; t1 ; t2 Þ From the first order conditions, @Pch @Pch @pi ¼ 0 and @ti ¼ 0 9 a þ 2gpj þ y ti mtj þ wb þ c1 bðli ti Þn wg cj g lj tj > > pi ¼ > > = 2b 1 " # n n1 > y½pi w ci ðli ti Þn m pj w cj lj tj > > > ti ¼ ; n ci nli Di

(35)

ði; j ¼ 1; 2; j 6¼ iÞ Solving the above simultaneous equations, the optimal ðpi g ; ti g Þ can be derived. Here, the super-script “g” indicates global coordination policy. Accordingly, Pch g ¼ p1 ðt1 g ; t2 g Þ þ p2 ðt1 g ; t2 g Þ.

7 Numerical Illustration A numerical illustration has been included to validate the mathematical models. The following data are considered for the numerical example. The data are very similar to Banker et al. (1998). a ¼ 1; 000; b ¼ 10; g ¼ 8:8; y ¼ 6; m ¼ 5:4; l1 ; l2 ¼ ½0:5; 6:0; c1 ¼ 2:5; c2 ¼ 2; w ¼ 5; n ¼ ½1:5; 5:0.

302

7.1

S. Sinha and S.P. Sarmah

Price Competition

Let us consider a typical case with, n ¼ 2:5, l1 ¼ l2 ¼ 0:5, t1 ¼ 0:5, and t2 ¼ 1:5. Let us further assume that, at t ¼ 0, retailer 1 and retailer 2 have the following offering in the market: fp1 ð0Þ ¼ 35; t1 ð0Þ ¼ 0:5g and fp2 ð0Þ ¼ 40; t2 ð0Þ ¼ 1:5g. If both the retailers compete only on the retail price, the price equilibrium is achieved at: fp1 ; p2 g ¼ f93:92; 94:63g.

7.2

Dynamics of Price Competition

Let us consider at typical case with, n ¼ 2:5, l1 ¼ l2 ¼ 0:5, t1 ¼ 0:5, and t2 ¼ 1:5. The dynamics of the price competition under naı¨ve expectation has been shown below (Table 1; Fig. 4): Dynamics of price competition with adaptive expectation has been shown below under varying speed of adjustment. It has been found that irrespective of the speed, the dynamical system converges to the same equilibrium. However, with faster speed, the equilibrium is reached faster and if v1 ¼ v2 ¼ 1, the system behaves like a dynamical system under naı¨ve expectation (Fig. 5).

7.3

Price Competition: Sensitivity Analysis

The following table shows the sensitivity of the price equilibrium on different parameters. In this experiment we have assumed the basic initial data: n ¼ 2:5, l1 ¼ l2 ¼ 0:5, t1 ¼ 0:5, and t2 ¼ 1:5 (Table 2). Table 1 Dynamics of price competition with naı¨ve expectation Sr. p1 p2 D1 p1 D2 1 35 40 997 29,829.12 914 2 69.88 68.70 901 58,366.70 934 3 82.51 84.05 909 70,420.82 892 4 89.27 89.61 891 74,993.68 896 5 91.71 92.58 892 77,318.90 888 6 93.02 93.66 889 78,168.52 888 7 93.49 94.23 889 78,618.36 887 8 93.75 94.44 889 78,781.51 887 9 93.84 94.55 889 78,868.59 887 10 93.89 94.59 888 78,900.13 887 11 93.90 94.61 888 78,916.99 887 12 93.91 94.62 888 78,923.09 887 13 93.92 94.62 888 78,926.35 887 14 93.92 94.63 888 78,927.53 887 15 93.92 94.63 888 78,928.17 887

p2 31,109.72 58,604.03 69,636.99 74,916.21 76,867.88 77,888.30 78,259.24 78,456.73 78,528.28 78,566.52 78,580.36 78,587.76 78,590.44 78,591.87 78,592.39

Pch 60,938.83 116,970.73 140,057.81 149,909.89 154,186.78 156,056.82 156,877.60 157,238.24 157,396.88 157,466.64 157,497.35 157,510.85 157,516.79 157,519.41 157,520.56

Price and Warranty Competition in a Duopoly Supply Chain

303

100

pi

75

50

p1 p2

25

0 0

5

10

15

20

25

Time period

Fig. 4 Dynamics of price competition with naı¨ve expectation 100 90 80 70

p1 p2 p1 p2 p1 p2 p1 p2

Pi

60 50 40 30 20

(v (v (v (v (v (v (v (v

= = = = = = = =

0.25) 0.25) 0.5) 0.5) 0.75) 0.75) 1.0) 1.0)

10 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43

Time period

Fig. 5 Dynamics of price competition with adaptive expectation

It shows that, the equilibrium price fp1 ; p2 g increases with increase in li and ti . However, the equilibrium price decreases with increase in n. Further, it has been observed that the channel profit increases with increase in n and decreases with increase in ti .

7.4

Price Coordination

Assuming, n ¼ 2:5, l1 ¼ l2 ¼ 0:5, t1 ¼ 0:5, and t2 ¼ 1:5, under integrated pricing policy, the optimal retail prices can be derived as: p1 cp ¼ 419:30 and p2 cp ¼ 420:05. Accordingly, the demand and profits are as below (Table 3),

304

S. Sinha and S.P. Sarmah

Table 2 Sensitivity analysis: price competition p2 Parameter Value p1 n 1.5 94.15 94.89 2.0 94.01 94.74 2.5 93.92 94.63 3.0 93.86 94.54 3.5 93.82 94.46 4.0 93.78 94.40 4.5 93.76 94.34 5.0 93.74 94.30 li 0.5 93.92 94.63 1.0 95.38 97.54 1.5 98.50 103.75 2.0 103.66 114.01 2.5 111.17 128.96 3.0 121.32 149.15 3.5 134.34 175.08 4.0 150.49 207.21 5.0 192.98 291.77 6.0 250.36 405.97 0.5 93.84 93.84 ti 1.0 94.17 94.14 1.5 94.85 94.77 2.0 95.95 95.78 2.5 97.55 97.24 3.0 99.69 99.21 3.5 102.43 101.72 4.0 105.82 104.84 4.5 109.91 108.59 5.0 114.73 113.01 Table 3 Price coordination p2 cp D1 cp p1 cp 419.30 420.05 498

p1 cp 206,427.95

p1 78,927.79 78,948.91 78,928.59 78,891.70 78,850.10 78,809.26 78,771.50 78,737.59 78,928.59 80,894.41 85,167.67 92,470.06 103,642.42 119,742.08 142,124.85 172,520.80 266,577.51 425,957.74 78,790.74 78,733.86 78,558.83 78,239.11 77,753.07 77,082.58 76,212.36 75,129.73 73,824.48 72,288.85

D2 cp 495

p2 78,487.47 78,529.12 78,592.85 78,661.91 78,728.66 78,789.95 78,844.78 78,893.15 78,592.85 75,738.65 69,827.89 60,586.82 48,300.93 33,915.78 19,160.67 6,681.53 4,546.26 72,296.33 78,808.85 78,836.28 78,840.93 78,817.57 78,761.86 78,669.99 78,538.57 78,364.53 78,145.05 77,877.56

p2 cp 205,227.87

Pch 157,415.26 157,478.03 157,521.44 157,553.60 157,578.76 157,599.21 157,616.28 157,630.74 157,521.44 156,633.07 154,995.56 153,056.88 151,943.35 153,657.86 161,285.52 179,202.33 271,123.78 498,254.06 157,599.59 157,570.14 157,399.76 157,056.68 156,514.93 155,752.57 154,750.93 153,494.26 151,969.52 150,166.40

Pch cp 411,655.82

A comparison between price competition and global coordination has been illustrated through the Fig. 6. This shows that price coordination can generate significantly higher profit as compared to that of under price competition.

7.5

Warranty Competition

We consider a case with, n ¼ 2:5, l1 ¼ l2 ¼ 0:5, p1 ¼ 35, and p2 ¼ 40. Let us further assume that, at t ¼ 0, retailer 1 and retailer 2 have the following offering in

Price and Warranty Competition in a Duopoly Supply Chain

305

450000 400000

Profit

350000

Profit of R1 (Price Competition)

300000 250000

Profit of R2 (Price Competition)

200000

Channel Profit (Price Competition)

150000

Profit of R1 (Price Coord.)

100000

Profit of R2 (Price Coord.) Channel Profit (Price Coord.)

50000 0 1

3

5

7

9

11

13

15

17

19

Time Period

Fig. 6 Price competition vs. price coordination: a comparison Table 4 Dynamics of price competition with naı¨ve expectation Sr. t1 t2 D1 p1 D2 1 0.500 t2 997 29,829.12 914 2 0.299 64.30 1,002 30,025.94 909 3 0.298 81.15 1,002 30,026.00 909 4 0.298 88.76 1,002 30,026.00 909 5 0.298 92.02 1,002 30,026.00 909

p2 31,109.72 31,775.00 31,775.22 31,775.22 31,775.22

Pch 6,0938.83 61,800.94 61,801.22 61,801.22 61,801.22

the market: t1 ð0Þ ¼ 0:5 and t2 ð0Þ ¼ 1:5. If both the retailers compete only on the warranty length, the equilibrium is achieved at:ft1 ; t2 g ¼ f0:30; 0:41g.

7.6

Dynamics of Warranty Competition

Assuming, n ¼ 2:5, l1 ¼ l2 ¼ 0:5, p1 ¼ 35, p2 ¼ 40, t1 ð0Þ ¼ 0:5 and t2 ð0Þ ¼ 1:5, the dynamics of the warranty length competition under naı¨ve expectation has been shown below (Table 4; Fig. 7): Dynamics of warranty competition with adaptive expectation has been shown below under varying speed of adjustment. It has been found that irrespective of the speed, the dynamical system converges to the same equilibrium – similar to price competition. However, with faster speed, the equilibrium is reached faster and if w1 ¼ w2 ¼ 1, the system behaves like a dynamical system under naı¨ve expectation – as observed earlier in the case of price competition (Fig. 8).

7.7

Warranty Competition: Sensitivity Analysis

The following table shows the sensitivity of the warranty equilibrium with respect to different parameters. In this experiment the basic initial data have been assumed as: n ¼ 2:5, l1 ¼ l2 ¼ 0:5, p1 ¼ 35, and p2 ¼ 40. The following table shows the sensitivity of the price equilibrium on different parameters (Table 5).

306

S. Sinha and S.P. Sarmah 1.600 1.400 1.200 t1

t(i)

1.000

t2

0.800 0.600 0.400 0.200 0.000 1

2

3 Time period

4

5

Fig. 7 Dynamics of warranty length competition with naı¨ve expectation

1.6

t1 (v = 0.25) t2 (v = 0.25)

1.4

t1 (v = 0.5) 1.2

t2 (v = 0.5) t1 (v = 0.75)

1.0 ti

t2 (v = 0.75) 0.8

t1 (v = 1.0) t2 (v = 1.0)

0.6 0.4 0.2 0.0 1

2

3

4

5

6

7

8 9 10 Time period

11

12

13

14

15

16

Fig. 8 Dynamics of warranty competition with adaptive expectation

It shows that, the equilibrium price ft1 ; t2 g increases with increase in nand pi and decreases with increase in li . The channel profit is found to increase with increase in li and pi .

7.8

Coordinated Warranty Policy

Assuming, n ¼ 2:5, l1 ¼ l2 ¼ 0:5, p1 ¼ 35, p2 ¼ 40, under integrated warranty policy, the optimal warranty length are: t1 cw ¼ 0 and t2 cw ¼ 0:15. Accordingly, the demand and profits are derived as below (Table 6),

Price and Warranty Competition in a Duopoly Supply Chain

307

Table 5 Sensitivity analysis: warranty competition t2 p1 Parameter Value t1 n 1.5 0.02 0.05 30,053.41 2.0 0.14 0.23 30,035.53 2.5 0.30 0.41 30,026.00 3.0 0.44 0.55 30,022.68 3.5 0.56 0.67 30,022.53 4.0 0.66 0.77 30,023.92 4.5 0.75 0.86 30,026.02 5.0 0.82 0.93 30,028.42 li 0.5 0.30 0.41 30,026.00 1.0 0.09 0.13 30,049.27 1.5 0.05 0.07 30,054.54 2.0 0.03 0.04 30,056.62 2.5 0.02 0.03 30,057.67 25.0 0.23 0.27 19,387.63 pi 35.0 0.31 0.36 28,715.50 45.0 0.37 0.43 37,800.12 50.0 0.41 0.47 42,251.28 60.0 0.47 0.54 50,971.38 70.0 0.53 0.61 59,448.60 80.0 0.59 0.68 67,683.02 90.0 0.64 0.75 75,674.67 100.0 0.70 0.81 83,423.57 125.0 0.84 0.97 101,733.84 150.0 0.97 1.13 118,526.83

p2 31,779.86 31,777.11 31,775.22 31,774.99 31,775.77 31,777.06 31,778.58 31,780.17 31,775.22 31,778.50 31,779.24 31,779.53 31,779.67 19,394.32 28,728.75 37,821.68 42,277.62 51,008.46 59,497.98 67,746.21 75,753.15 83,518.82 101,877.44 118,728.02

Pch 61,833.26 61,812.63 61,801.22 61,797.67 61,798.30 61,800.98 61,804.60 61,808.59 61,801.22 61,827.77 61,833.78 61,836.15 61,837.34 38,781.96 57,444.26 75,621.80 84,528.89 101,979.84 118,946.59 135,429.22 151,427.82 166,942.39 203,611.28 237,254.84

Table 6 Integrated warranty policy t2 cw D1 cw p1 cw t1 cw 0 0.15 1001 30,035.25

p2 cw 31,809.15

Pch cw 61,844.40

D2 cw 909

A comparison between warranty competition and warranty coordination has been illustrated through Fig. 9. This shows that integrated warranty policy can generate higher profit as compared to that of warranty competition.

7.9

Price and Warranty Competition

In this case, we illustrate a case where both retailers simultaneously compete on retail price and warranty length. For this example, we consider n ¼ 1:5 and 2:5, l1 ¼ l2 ¼ 0:5. Let us further assume that, at t ¼ 0, retailer 1 and retailer 2 have the following offering in the market: p1 ð0Þ ¼ 35, p2 ð0Þ ¼ 40 and t1 ð0Þ ¼ 0:5, t2 ð0Þ ¼ 1:5. Accordingly, the equilibrium price and warranty length have been derived as below (Table 7):

308

S. Sinha and S.P. Sarmah

Ch Profit (Warranty Competition)

Profit R2 (Warranty Competition)

Profit R1 (Warranty Competition)

Ch Profit (Warranty Coord.)

70000 60000

Profit

50000 40000 30000 20000 10000 0 1

2

3 Time period

4

5

Fig. 9 Warranty competition vs. warranty coordination: a comparison Table 7 Price and warranty competition ~t1 ~t2 D1 p~2 n p~1 1.5 93.83 0.20 93.89 0.32 887 2.5 93.92 0.67 93.97 0.77 888

p1 78,756.68 78,778.10

D2 888 888

p2 78,784.02 78,823.38

Pch 157,540.70 157,601.48

This shows that channel profit increases with increase in shape parameter n. Below, Fig. 10 represents the dynamics of the equilibrium for n ¼ 1:5; 2:5, l1 ¼ l2 ¼ 0:5 and the initial condition p1 ð0Þ ¼ 35, p2 ð0Þ ¼ 40 and t1 ð0Þ ¼ 0:5, t2 ð0Þ ¼ 1:5.

7.10

Global coordination

Assuming, n ¼ 2:5, l1 ¼ l2 ¼ 0:5, under integrated price and warranty policy, the optimal solutions have been derived as below (Table 8), A comparison between warranty competition and warranty coordination has been illustrated through Fig. 11.

7.11

Global Coordination: Sensitivity Analysis

Here, further experiments have been conducted to study the impact of n, Li, and ci on global coordination. The basic parameters assumed as, n ¼ 2:5, l1 ¼ l2 ¼ 0:5, c1 ¼ 2:5 and c2 ¼ 2. The results have been tabulated in Table 9. It shows that increase in shape parameter n increases channel profit, retail price, and warranty duration. However, increase in failure rate li and repair cost ci decreases channel profit, retail price, and warranty duration.

p1 (n=1.5)

p1 (n=2.5) p2 (n=2.5)

p2 (n=1.5)

309

t1 (n=1.5)

t1 (n=2.5)

t2 (n=1.5)

t2 (n=2.5)

100.0

1.6

90.0

1.4

80.0

1.2

Profit

70.0 60.0

1.0

50.0

0.8

40.0

0.6

30.0

0.4

20.0

Warranty Length

Price and Warranty Competition in a Duopoly Supply Chain

0.2

10.0 0.0

0.0 1

2

3

4

5

6

7 8 9 Time period

10

11

12

13

14

15

Fig. 10 Dynamics of price and warranty competition Table 8 Optimal price and warranty length under global coordination n p1 g t1 g p2 g t2 g D1 g p1 g D2 g p2 g 2.5

419.4

0.59

419.41

0.68

497

205,856.3

497

206,015.22

Pch g 411,871.56

450000 400000 350000 Profit R1 (Price & Warr. Comp.)

Profit

300000

Profit R2 (Price & Warr. Comp.)

250000

Channel Profit (Price & Warr. Comp.)

200000

Profit R1 (Price & Warr. Coord.) Profit R2 (Price & Warr. Coord.)

150000

Channel Profit (Price & Warr. Coord.)

100000 50000 0 1

2

3

4

5

6

7

8

9

10 11 12 13 14 15

Time period

Fig. 11 Price and warranty policy: competition vs. global coordination

8 Conclusion This chapter analyzes the coordination and competition issues in a two-stage distribution channel where two different retailers compete each other on their retail price and warranty policy to sell two substitute products in the same market.

310 Table 9 Sensitivity analysis: global coordination t1 g p2 g t2 g Parameter Value p1 g n 1.5 419.2 0.14 419.26 0.22 2.0 419.3 0.40 419.36 0.50 2.5 419.4 0.59 419.41 0.68 3.0 419.4 0.73 419.44 0.82 3.5 419.4 0.84 419.46 0.92 5 419.5 1.06 419.48 1.12 Li 0.05 428.6 27.29 430.44 31.82 0.1 422.1 8.61 422.71 10.01 0.25 419.8 1.87 419.93 2.17 0.5 419.4 0.59 419.41 0.68 1 419.2 0.19 419.24 0.22 2.5 419.2 0.04 419.18 0.05 2.5 419.4 0.59 419.37 0.59 ci 5 419.3 0.37 419.30 0.37 7.5 419.3 0.28 419.26 0.28 10 419.2 0.23 419.25 0.23 15 419.2 0.18 419.23 0.18

S. Sinha and S.P. Sarmah

D1 g 497 497 497 497 497 497 494 496 497 497 497 497 497 497 497 497 497

D2 g 497 497 497 497 497 498 511 501 498 497 497 497 497 497 497 497 497

p1 g 205,810.7 205,822.7 205,856.3 205,890.3 205,921.6 205,836.9 206,510.6 206,061.0 205,889.5 205,856.3 205,845.6 205,841.9 205,928.7 205,896.2 205,883.1 205,875.7 205,867.5

p2 g 205,901.2 205,970.8 206,015.2 206,047.7 206,072.2 206,279.9 214,038.8 208,402.0 206,395.3 206,015.2 205,895.8 205,852.8 205,928.7 205,896.2 205,883.1 205,875.7 205,867.5

Pch g 411,711.9 411,793.5 411,871.6 411,938.0 411,993.8 412,116.8 420,549.3 414,463.0 412,284.7 411,871.6 411,741.5 411,694.7 411,857.5 411,792.4 411,766.2 411,751.4 411,734.9

The demand faced by each retailer not only depends on its own price and warranty duration, but also on the price and warranty duration set by the other. Mathematical models have been developed to analyze the dynamic competition and coordination mechanism for three different cases where retailers compete (1) exclusively on price; (2) exclusively on warranty duration; (3) both price and warranty duration. The adjustment of initial price or warranty duration during dynamic competition gradually leads toward the Nash–Bertrand equilibrium following an iterative process, where at each step each retailer chooses a policy which maximizes the individual profit based on the expected policy set by her opponent. Further, we analyze the behavior of such dynamic adjustment process of price and warranty competition under two different scenarios (1) naı¨ve expectation and (2) adaptive expectation – depending on the adjustment of expectation function of each retailer. Finally, it has been shown that under non-cooperative price/warranty competition, the steady state equilibrium is dynamically stable in nature under certain conditions. It has been shown here that the channel profit for each case is higher under coordination that of under competition. The channel profit is found to be the maximum under global coordination where retailers adopt centralized policy to set price and warranty duration. However, it has been observed that though coordination enhances overall supply-chain profitability, it may make consumers worseoff due to higher product prices. The model is illustrated with suitable numerical examples. The model can significantly help industry practitioners to visualize and understand the dynamic nature of price and non-price (warranty) competition. It also can predict the overall pay-off in case of centralized or coordinated strategy.

Price and Warranty Competition in a Duopoly Supply Chain

311

Accordingly, a delicate balance between coordination and competition can be achieved in case the existing business model fails to meet profitability expectation. Thus, the industry practitioners can take a pro-active role in choosing the attribute to compete and decide when to coordinate with their competitor. The model could be extended in several directions. Various other forms of demand function could be used to replicate more realistic scenarios. Also, in most of the industrial cases, price/warranty competition takes place under asymmetric information. Further, there could be more number of players in the market; and one interesting dimension towards further research is the “entry” and “exit” decisions of a firm in the market. Finally, all retailers may not set price simultaneously. There are cases where one firm takes the role of a price-leader while the others are followers. Such type of competition model under Stackelberg game framework is also worth mentioning for future research.

Appendix A: Proof of Proposition 1 g i Proof. 1(a). It is straightforward to derive, @p @pj ¼ 2b >0, which shows both retailers change their retail price in the similar direction. (b). It has already been derived,

@pi ¼ a 2bpi þ gpj þ y ti mtj þ wb þ ci ðli ti Þn b ¼ 0 @pi @pi @pi

Given retailer j sets her retail price pj , retailer i will increase her retail price pi , if >0, a þ gpj þ y ti mtj þ wb þ ci ðli ti Þn b >pi . Or, 2b Hence proved. □

Appendix B: Proof of Proposition 2 It is straight forward to derive,

@pi 2b2 ci nðti Þn ðli Þn1 ¼ >0 @li 4b2 g2

@pi gm þ 2by þ 2b2 ci nðli Þn ðti Þn1 ¼ >0 @ti 4b2 g2 Since, g < b, m < y, then gm þ 2by > 0. Hence,

@pi @ti

>0

312

S. Sinha and S.P. Sarmah

n n @pi 2b2 ci ðli ti Þ lnðli ti Þ þ bcj g lj tj ln lj tj ¼ @n 4b2 g2

Thus,

@pi @n

n >0 if 2b2 ci ðli ti Þn lnðli ti Þ þ bcj g lj tj ln lj tj >0

Appendix C: Proof of Proposition 3 Substituting Tr JpAD and Det JpAD in condition C1, C2 and C3, (i)

v1 v2 g2 1 Tr JpAD þ Det JpAD ¼ 1 ð2 v1 v2 Þ þ ð1 v1 Þð1 v2 Þ 4b2 g2 Or, 1 Tr JpAD þ Det JpAD ¼ v1 v2 1 4b 2 . Since, v1 v2 >0, the first condi-

g 2 g tion yields 1 4b 2 >0, or, b > 4 . 2

(ii)

2

v1 v2 g2 1 þ Tr JpAD þ Det JpAD ¼ 1 þ ð2 v1 v2 Þ þ ð1 v1 Þð1 v2 Þ 4b2

g2 Or, 1 Tr JpAD þ Det JpAD ¼ 4 2ðv1 þ v2 Þ þ v1 v2 1 4b 2 . Since,0

v1 v2 g2 Det JpAD 1 ¼ ð1 v1 Þð1 v2 Þ 1 4b2

v2 g Let, f ðvÞ ¼ ð1 v1 Þð1 v2 Þ v14b 2 g2 C ¼ 1 4b . 2 0 The Hessian matrix for f ðvÞ, Hv ¼ C following the first order conditions, v1 ¼ v2

2

if C>0 or, b2 > g4 . □ Hence Proved. 2

1 ¼ ðv1 v2 þ v1 v2 CÞ,

where

C shows that f ðvÞ is concave. Thus, 0 ¼ C1 and f ðvÞ ¼ 1 C . Thus, Max: f ðvÞ<0

gmþ2byþ2b2 c1 nl1 n t1 n1

!

2bmþgyþbc2 gnl2 n t2 n1

!3

where, X1 ¼

p1 y wy p2 y wy ; Y ; Y2 ¼ ½y t2 þ D2 n. ¼ ½ y t þ D n , X ¼ 1 1 1 2 c 1 l1 n c2 l2 n

t1 t1 0 t1 6 7 6 7 4b2 l2 4b2 l2 6 ! !7 6 7 6 2bmþgyþbc1 gnl1 n t1 n1 gmþ2byþ2b2 c2 nl2 n t2 n1 7 6 7 0 t t t2 1 2 6 7 6 7 4b2 l2 4b2 l2 6 7 2 3 6 7 1 J ðp;tÞ ¼ 6

h i h i 7 n1 2n 1 1 n 1 n n t p ywy 6 3 7 1 4 5 X1 n1 Y1 n1 þX1 n1 bnY1 n1 t3 K1 ðyt1 þD1 nÞn1 þ1 t3 ½gnn1 t3 L1 ðyt1 þD1 nÞn1 6 7 n 6 n1 7 c1 l1 6 7 6 7 2 3 6 7 1

h i h i 6 7 n1 1 2n 1 1 n n n p ywy t 2 4 4 5 4 5 n1 n1 n1 n1 n1 n1 n1 X2 Y2 t4 t4 K2 ðyt2 þD2 nÞ t4 L2 ðyt2 þD2 nÞ ðgnÞ þX2 bnY2 þ1 n1 c2 l2 n

2

Appendix D: The Jacobian Matrix J ðp; tÞ

314

S. Sinha and S.P. Sarmah

References Agiza HN, Elsadany AA (2003) Nonlinear dynamics in the Cournot duopoly game with heterogeneous players. Phys A 320:512–524 Agiza HN, and Elsadany AA (2004) Chaotic dynamics in nonlinear duopoly game with heterogeneous players, Applied Mathematics and Computation 149(3):843–860 Agliari A, Gardini L, Puu T (2000) The dynamics of a triopoly Cournot game, Chaos, Solitons and Fractals 11(15):2531–2560 Bertrand J (1883) ‘Theorie mathematique de la Richesse Sociale par Leon Walras and Recherches sur les principes mathematique de la theorie des dicheses par Augustin Coumot’, Journal des Savants, pp. 499–508, translated by Friedman, J.W. (1988). In: Daughety AR (ed) Coumot oligopoly. Cambridge University Press, Cambridge, pp 73–81 Blischke W, Murthy D (1996) Product warranty handbook. Marcel Dekker, New York Choi SC (1991) Price competition in a channel structure with a common retailer, Marketing Science 10(4):271–296 Choi SC (2003) Expanding to direct channel: market converges as entry barrier. J Interact Mark 17(1):25–40 Connelly M (2006) Dealers: extended factory warranties boost profits. Automot News Detroit 81:46 Cournot A (1938) Researches into the mathematical principles of the theory of wealth, translated by Bacon NT (1960). Kelley, New York Ferguson BS, Lim GC (1998) Introduction to Dynamic Economic models. Manchester University Press, Manchester, M13 9NR, UK Ha A, Li L, Ng SM (2003) Price and delivery logistics competition in a supply chain. Manage Sci 49(9):1139–1153 Hu W-T (2008) Supply chain coordination contracts with free replacement warranty. Doctoral dissertation, Drexel University, Philadelphia, PA McGuire TW, Staelin R (1983) An industry equilibrium analysis of downstream vertical integration, Marketing Science, 2 (Spring), 161–191 Menezes MAJ, Currim IS (1992) An approach for determination of warranty length. Int J Res Mark 9:177–196 Moorthy KS (1988) Product and price competition in a duopoly. Mark Sci 7(2):141–168 Murthy NP (2006) Product warranty and reliability. Ann Oper Res 143:133–146 Olech C (1963) On the global stability of an autonomous system on the plane, in Contributions to Differential Equations. In: Lasalle JP, Diaz JB. (Eds.), Vol.1, Wiley, New York Padmanabhan V (1993) Warranty policy and extended service contracts: theory and an application to automobiles. Mark Sci 12:230–248 Rao CR (1991) Pricing and promotions in asymmetric duopoly. Mark Sci 10(2):131–144 Scherer K (2006) GM warranties take big leap. Automot News Detroit 81:30 Shone R (2001) An introduction to economic dynamics. Cambridge University Press, New York, pp 1–5 Shy O (2003) How to price. Cambridge University Press, Cambridge Sinha S, Sarmah SP (2010) Coordination and price competition in a duopoly common retailer supply chain. Comput Ind Eng 59(2):280–295 So KC (2000) Price and time competition for service delivery. Manuf Serv Oper Manage 2(4):392–409 Tsay A, Agrawal N (2000) Channel dynamics under price and service competition. Manuf Serv Oper Manage 2(4):372–391 Wu C-C, Lin P-C, Chou C-Y (2009) Optimal price, warranty length and production rate for free replacement policy in the static demand market. Omega 37(1):29–39 Yao DQ, Liu J (2005) Competitive pricing of mixed retail and e-tail distribution channels. Omega 33:235–247

Supply Chain Coordination for Newsvendor-Type Products with Two Ordering Opportunities Yong-Wu Zhou and Sheng-Dong Wang

Abstract This chapter discusses supply chain coordination issues for a newsvendor-type product with two ordering opportunities. We consider a twoechelon supply chain consisting of one manufacturer and one buyer, where the manufacturer sells his product through the buyer who faces a random demand. The manufacturer does not hold inventory but activates production with a fixed production setup cost to respond to the buyer’s order. At the start of the selling period, the buyer’s first order is delivered. By the end of the sales period, an urgent second order is allowed to meet the willingly-backordered demand if the buyer shares the manufacturer’s setup cost incurred by the second order. We discuss two parties’ optimal order policies in the decentralized setting, and examine the impact of pool schemes of the second setup cost on the decentralized system performance. We show that the decentralized system would perform best if the manufacturer covers utterly the second production setup cost. Also, we prove that under the twice-order framework in the chapter the expected profit of the centralized system is not equal to but greater than the sum of two members’ expected profits in the decentralized system, which is not consistent with our expectation. In order to maximize the expected profit of the channel, two coordinated policies are proposed to achieve perfect coordination: a two-part-tariff policy for the special case that the buyer pays all the manufacturing setup cost, and a revised revenue-sharing contract for the case that two parties share the manufacturing setup cost. Keywords Supply chain coordination • Newsvendor model • Two ordering opportunities • Partial backlogging • Revised revenue-sharing contract Y.-W. Zhou (*) School of Business Administration, South China University of Technology, Guangzhou, Guangdong, P.R. China e-mail: [email protected] S.-D. Wang Department of Mathematics, Hefei Electronic Engineering Institute, Hefei, Anhui, P.R. China e-mail: [email protected] T.-M. Choi and T.C. Edwin Cheng (eds.), Supply Chain Coordination under Uncertainty, International Handbooks on Information Systems, DOI 10.1007/978-3-642-19257-9_13, # Springer-Verlag Berlin Heidelberg 2011

315

316

Y.-W. Zhou and S.-D. Wang

1 Introduction Like our human beings, products have also their own life cycle. From introduction to decline products pass through various stages: introduction, growth, maturity, saturation and decline. Nowadays, with the rapid improvement of technology, the life cycle of products becomes shorter and shorter. It makes more and more products, like USB flash drivers, personal computers, mobile phones, fashion apparel, etc., to have the attributes of fashion or seasonal goods. We call all of short life-cycle products as fashion products or newsvendor-type products. For nonfashion products, like most of fast moving consumer products, we all know that there exists substantial progress in theory and application of supply chain management. But the same success has not been achieved for fashion products. Fisher and Raman (1999) pointed out three possible reasons. First, the demand patterns for these products make the estimation of demand and demand variability extremely difficult. Second, traditional demand forecasting methods usually assume that at least 1 year of demand history is available, which is not possible for short life-cycle products since their life cycles are generally less than 1 year. Finally, the cost of carrying inventory is much higher for short life-cycle products because of the risk of obsolescence. The challenge in managing supply chains for newsvendor-type products is to ensure product availability while keeping leftover products as low as possible. In the past five decades, both researchers and practitioners paid much attention on supply chain management for newsvendor-type products. A lot of models on this topic can be found in literature (see Sect. 2). Most of the existing models, which studied supply chain coordination problems for single-period items under the newsvendor framework, assumed that products could be ordered (or produced) only once during the whole selling period. Consequently, the decision maker is unable to take advantage of the subsequent information that becomes available as the season draws closer or after the season begins. In reality, however, newsvendortype products can be replenished many times in the selling season. For example, in the 1980s Benetton, the fashion retail giant, significantly reduced markdowns and leftover inventory by employing a second order opportunity around the start of the season. This phenomenon leads to a lot of researches on the newsvendor models with two order opportunities. Most of these researches, such as Lau and Lau (1998) and Li et al. (2009b), considered the following twice-order framework: the first order is placed at the start of the preseason and delivered at the start of the selling season; the second order is placed at or after the start of the selling season for subsequent delivery. Besides, there is yet another class of models, in which twice orders are placed before the start of the season. However, the models just mentioned only considered how retailers make their twice-order decisions but neglected the coordination issues of supply chains. The main aim of this chapter is to discuss coordination issues of a manufacturerbuyer supply chain for newsvendor-type products with two ordering opportunities. Unlike the two twice-order frameworks mentioned above, we consider such

Supply Chain Coordination for Newsvendor-Type Products

317

a twice-order framework, in which the first order is placed at the start of the season and the second one (if needed) is placed at the end of the season. This framework is first employed by Weng (2004). Its benefit is mainly in that placing the second order at the end of the season can obtain accurate demand information. Under the framework, we will consider such a situation that a manufacturer supplies a newsvendor-type product to a buyer who faces a stochastic demand. The manufacturer does not hold inventory but activates production with a fixed production setup cost to respond to the buyer’s order. At the beginning of the selling period, the buyer’s first order is delivered from the manufacturer to the buyer in order to meet random demand. Due to the uncertainty of demand, the actual demand during the selling period may deviate from the ordered lot size. By the end of the sales period, if the demand exceeds the ordered quantity, the buyer would have an opportunity to place an urgent second order from the manufacturer to meet the willinglybackordered demand as long as the buyer shares the manufacturer’s second setup cost incurred by the second order. The main questions we concern are the following (1) What is the first order lot size of the buyer and whether should he/she place the second order? (2) Whether should the manufacturer activate an urgent second production in response to the buyer’s second order? (3) How do we design an effective mechanism to coordinate the whole supply chain? The coordination problem considered in this chapter generalizes that in Weng (2004), where the manufacturing setup cost of the second order (if happened) is paid utterly by the buyer and the demands exceeded the first order quantity are all backordered. One of the aims of this chapter is to observe the impact of the fraction that the buyer shares the second setup cost on the decentralized system performance. Our research shows that the performance of the decentralized system decreases as the proportion of the second production setup cost shared by the buyer increases. It implies that the decentralized system performs best if the second setup cost is utterly paid by the manufacturer itself but worst if the second setup cost is paid only by the buyer, like in Weng (2004). Another difference between Weng (2004) and this chapter is that Weng (2004) considered the sum of the two members’ expected profits as the total expected profit of the centralized system, whereas this chapter has shown that the expected profit of the centralized system is always greater than the sum of the two members’ expected profits. Consequently, the quantity discount mechanism shown in Weng (2004) could not coordinate the decentralized channel really to the “centralized” system. We present in the chapter two perfect coordination scenarios: a two-part-tariff policy for the special case that the buyer pays all the manufacturing setup cost, and a revised revenue-sharing contract for the general case that the two parties share the manufacturing setup cost. The two coordination mechanisms optimize the expected profit of the whole supply chain rather than the sum of two parties’ expected profits. Hence, they achieve the perfect coordination of the decentralized system. While our work directly generalizes the work of Weng (2004), it also nicely complements the work of Lau and Lau (1998) and Milner and Kouvelis (2005). The key differences are that (1) they do not allow for partial backorders, (2) they focus on determining the buyer’s twice order policies, whereas we focus on how the buyer

318

Y.-W. Zhou and S.-D. Wang

sets the twice-order policies but also on how to design the perfect coordination mechanism of the whole channel, and (3) two different twice-order frameworks are considered. The remainder of the chapter is organized as the following. Section 2 reviews briefly the related literature. Section 3 describes the problems discussed in the chapter and introduces notations and assumptions needed by developing the model. While Sect. 4 shows the mathematical model for the decentralized system, Sect. 5 develops the mathematical model for the centralized system. Also, Sect. 5 compares the optimal policies between the decentralized and centralized system and discusses the property of the decentralized system performance. Section 6 presents a two-part-tariff model that can achieve perfect coordination of the system under the special case that the buyer pays all the manufacturing setup cost, and a revised revenue-sharing contract that can realize perfect coordination of the channel under the case that two parties share the manufacturing setup cost. In Sect. 7, two numerical examples are shown to illustrate the model. Conclusions are given in Sect. 8.

2 Literature Review Over the past five decades, both researchers and practitioners have had great interest in operations management issues for newsvendor-type products. A lot of models on this topic can be found in publications, of which the earlier ones mainly focused on how to find the buyer’s appropriate ordering policies to maximize (minimize) his or her expected profit (cost) (see, e.g., Whitin 1955; Goodman and Moody 1970; Kabak and Schiff 1978; Ismail and Louderback 1979; Atkinson 1979; Lau 1980; Nahmias and Schmidt 1984). An excellent review about these earlier researches can be seen in Khouja (1999). Due to the globalization of market and competition, the idea of supply chain management becomes very popular. Many researchers have shifted their attention to coordination issues of the supply chain for newsvendor-type products. For example, through the returns policy, Pasternack (1985) presented a supply chain coordination model in which a single manufacturer sold a single newsvendor-type product to a single retailer. He pointed out that it was an effective coordinating policy to take back residual products from the retailer at the end of selling period. Emmons and Gilbert (1998) further studied the role of returns policies in pricing and inventory decisions for catalogue goods. Taylor (2002) employed a one-period model with one supplier and one retailer to explore the rebate policy. Cachon (2003) developed a supply chain coordination model with price dependent demand. Webster and Weng (2008) presented an ordering and pricing model in a manufacturing and distribution supply chain for newsvendor-type products. Arcelus et al. (2008) developed a two-echelon supply chain model to depict the profitability of a secondary market to a profit-maximizing manufacturer, who was offering to the retailer a buyback policy for the unsold merchandise left at the end of

Supply Chain Coordination for Newsvendor-Type Products

319

the selling season. Due to the difficulty of obtaining the analytical results, they resorted to the numerical analysis. Li et al. (2009a) considered the supply chain coordination and decision making issue under consignment contract with revenue sharing. Chen et al. (2010) studied a coordination contract for a supplier–retailer channel where a fashionable product with a stochastic price-dependent demand is produced and sold. They formulated a two-stage optimization problem in which the supplier first decides the amount of capacity reservation, and the retailer then determines the order quantity and the retail price after observing the demand information. Other related studies included Chen et al. (2006), Wong et al. (2009), etc. However, all the researches mentioned above ignored demand information updating. That is, they assumed that the retailer orders only once during the whole selling period (including the lead time). The existing literature concerning the newsvendor model with two ordering opportunities can be divided into three categories. The first category assumed that the retailer’s two ordering opportunities take place before the season. Following the initial order, additional market information is obtained. Then, based on an improved demand forecast, the second-order amount is decided before the selling period. For example, Donohue (2000) formulated an efficient supply contract for fashion goods with forecast updating before the season and two production modes. Choi et al. (2003) developed a two-stage optimal ordering model with Bayesian information updating. Serel (2009) investigated a quick response system in which retailers place separate orders for a product at two different times before the selling season. The second one assumed that the retailer’s first order happens at the beginning of the season and the other within the season. Lau and Lau (1997, 1998) were possibly the first to consider the situation in which a second order is placed within the selling period. Through dynamic programming, Milner and Kouvelis (2005) also proposed a single-period inventory model with two ordering opportunities. Yet they focused on examining the interplay between the value of information and flexibility in their order decisions. Li et al. (2009b) further conducted a comprehensive analysis of the stocking problem with two order opportunities, with focus on elucidating the optimal ordering policy structure, whereas Pan et al. (2009) presented a two-period pricing and ordering model for the dominant retailer in a two-echelon supply chain with demand information updating. However, these researches considered retailers’ ordering and pricing issues only. In the third category, it is assumed that retailers have two ordering opportunities happening at the beginning and the end of the season, respectively. For instance, Weng (2004) developed a newsvendor-type coordination model for a single-manufacturer single-buyer supply chain with two ordering opportunities. Zhou and Li (2007) discussed an issue similar to that in Weng (2004) by considering the buyer’s inventory cost but neglecting the buyer’s backorder cost. Wang et al. (2010) developed a supply chain coordination model for newsvendor-type products with two ordering opportunities. In their model, they allowed the manufacturer to use two different production modes to respond to the retailer’s twice orders, and employed the general functions of different forms to model the expected market demand that is dependent on price and advertising expenditure.

320

Y.-W. Zhou and S.-D. Wang

3 Notations and Assumptions Consider a two-echelon supply chain where a manufacturer sells a newsvendor-type product through a buyer, who faces a stochastic demand. The manufacturer does not hold inventory of finished products but activates production with a fixed production setup cost and the negligible manufacturing or assembling time to respond to the buyer’s order. At the beginning of the selling period, an initial order is delivered from the manufacturer to the buyer. Then, the buyer uses the initial order delivered to meet random demand. Due to the uncertainty of demand, the actual demand during the selling period may deviate from the ordered lot size. By the end of the sales period, if the demand exceeds the ordered quantity, the buyer would have an opportunity to place an urgent second order from the manufacturer to meet the willingly-backordered demand as long as the buyer shares the manufacturer’s second setup cost incurred by the second order. Our key questions are the following (1) What is the optimal lot size of the first order? (2) Whether should a second order be placed and an urgent second production activated? (3) What is the impact of pool schemes of the urgent setup cost on the decentralized system performance? (4) How do we design coordination mechanism of the supply chain? A number of industries fall into our model. For example, in the mobile telecom industry, each mobile phone manufacturer (such as Motorola, Nokia or Sumsung) may sell through its downstream telecom service companies who use its equipment. For a new-model mobile phone, telecom service providers would tend to purchase an initial lot size of the new equipment from the mobile phone manufacturer, and then bind some special service with the new-style mobile phone. While the newstyle mobile phone is out of stock during the selling period, part of customers may be willing to wait till the next order is delivered due to their preference to the service and brand. The model we will present in the chapter can be applied to this case. In order to develop our model, the following notations and assumptions are used. Some other notations are given where they are needed. Assumptions 1. The buyer faces a random demand. 2. Each of the two parties has complete information about the other’s cost structure. 3. The manufacturer does not hold inventory and the production time for any lot size is neglected. 4. The buyer has complete demand information before she places her second order and all excess demand (after the first order) is observed but partially backlogged. 5. Both the manufacturer and the buyer share the second production setup cost once the second production happened at the end of the selling season. 6. Among many alternative objectives, we use the common expected profit function as the objectives of two parties. Notations x The random demand for the buyer with increasing concave CDF F(x), PDF f(x) and mean m

Supply Chain Coordination for Newsvendor-Type Products

p c r b

b k Q ti si l

v

321

The buyer’s unit sales price The buyer’s unit purchase price, or the manufacturer’s wholesale price The buyer’s salvage value (e.g., unit discount sales price) The backlogging rate (0 b 1), which means a fraction b of the excess demand during the stock out period is backordered, and the remaining fraction (1 b) is lost The buyer’s unit backorder cost The buyer’s unit shortage cost for the lost sales The buyer’s (first) order quantity The buyer’s ordering and transportation cost for the i-th (i ¼ 1,2) order The manufacturer’s production setup cost for the i-th (i ¼ 1,2) setup The proportion of the manufacturer’s second production setup cost shared by the buyer once the second production happened. This can be explained as a compensation for the manufacturer’s loss incurred by changing the production plan or an incentive for the manufacturer to activate the second production when the buyer place a second order, where 0l1 The manufacturer’s unit production cost

To avoid unrealistic and trivial cases, assume that the following relationship is kept: p > c > v > r and p + k c b > 0.

4 Decentralized Order Policies Consider first the decentralized decision-making system. In this setting, the profitmaximizing buyer would determine independently the initial order quantity Q for the whole sales period, and whether to place a second order to satisfy the part of the unfilled demand at the end of the selling period if he is required to share the setup cost incurred by his second order. The manufacturer would then determine whether to activate a second production in response to the buyer’s second order. Since an order of Q units is placed by the buyer at the start of the sales period, then by the end of the selling period, if demand x exceeds the order quantity Q, there would be a shortage of (x Q) units, of which b(x Q) units could be backordered. Therefore, at the end of the sales period, the buyer will have two alternatives: either place a second order of b(x Q) units, or do not. If the buyer chooses the former, her or his profit will be (p c b)b(x Q) k(1 b) (x Q) t2 ls2. In contrast, if the buyer selects the latter, the corresponding profit is k(x Q). Generally, the buyer is willing to place a second order if the former can have the buyer get more profits than does the latter, i.e., ðp c bÞbðx QÞ kð1 bÞðx QÞ t2 ls2 kðx QÞ or bðx QÞ qb ¼ ðt2 þ ls2 Þ=ðp þ k c bÞ: It means that qb is the buyer’s threshold quantity beyond which the buyer places a second order. On the other hand, as pointed out by Weng (2004), the manufacturer

322

Y.-W. Zhou and S.-D. Wang

would be usually willing to satisfy the second order if it yields positive profit. Hence, the manufacturer is willing to activate a second production in response to the buyer’s second order as long as (c v)b(x Q) (1 l)s2 0 or b(x Q) qm ¼ (1 l)s2/(c v). It shows that qm is a threshold quantity beyond which the manufacturer makes a second production. Generally speaking, qb and qm are probably unequal. That is to say, the manufacturer may not be willing to activate a second production while the buyer is willing to place a second order. Likewise, the buyer may not be willing to place a second order yet while the manufacturer is willing to activate a second run production. In what follows, for convenience, we first derive the expected profits of both the manufacturer and the buyer respectively in terms of two cases: qb qm and qb qm. We then develop the buyer’s optimal ordering policy in the decentralized system.

4.1

Case with qb qm

In this case, by the end of the selling period, there might be four situations pertaining to the practical demand x. Figure 1 shows a sketch of the four situations. Situation 1: The practical demand x does not exceed the order quantity Q. In such a situation, the buyer has (Q x) units left at the end of the selling period. Hence, the buyer’s and manufacturer’s profits will be respectively given by BP1 ðx; QÞ ¼ px þ rðQ xÞ cQ t1 ;

MP1 ðx; QÞ ¼ ðc vÞQ s1 :

Situation 2: Q < x < Q + qb/b. Under this situation, the backordered demand b(x Q) is less than the buyer’s threshold quantity qb. So the buyer does not place a second order and the demand of (x Q) units is lost. Thus, the profits of both parties can be expressed as BP1 ðx; QÞ ¼ ðp cÞQ kðx QÞ t1 ;

MP1 ðx; QÞ ¼ ðc vÞQ s1 :

Situation 3: Q + qb/b x < Q + qm/b. In this situation, the backordered demand b(x Q) exceeds the buyer’s threshold quantity but does not reach the manufacturer’s. It means that the buyer is willing to place a second order but the manufacturer is not willing to activate a second production if the buyer orders only b(x Q) units. If to want the second order satisfied, the buyer has to order at least qm units. However, the order quantity qm is larger than the backlogged demand, which means that the buyer would bear the loss in purchasing costs of unsold items of (qm b(x Q)) units. If the buyer places a second order of qm units, the buyer’s profit is ðp c bÞbðx QÞ kð1 bÞðx QÞ ðc rÞ½qm bðx QÞ t2 ls2 : 1

2 Q

3 Q+ qb /b

Fig. 1 Four possible situations of the practical demand

4 Q+ qm /b

x

Supply Chain Coordination for Newsvendor-Type Products

323

In contrast, if the buyer does not place a second order, the corresponding profit is k(x Q). It is obvious that the buyer is willing to place a second order of qm units if the difference of these profits is nonnegative, i.e., (p c b)b(x Q) which k(1 b)(x Q) (c r)[qm b(x Q)] t2 ls2 k(x Q), is equivalent to b(x Q) q0 ¼ [t2 þ ls2 þ (c r)qm]/(p þ k r b) or x Q þ q0/b. Therefore, when Q þ qb/b x < Q þ q0/b, there is no second transaction happened and both parties’ profits still are BP1(x, Q)¼(p c)Q k(x Q) t1 and MP1(x, Q) ¼ (c v)Q s1; whereas when Q þ q0/b x < Q þ qm/b, the buyer will place a second order of qm units and the manufacturer is also willing to reproduce qm units. Then, when Q þ q0/b x < Q þ qm /b, the profits of the buyer and manufacturer will respectively be BP1 ðx; QÞ ¼ ðp cÞQ þ ðp c bÞbðx QÞ kð1 bÞðx QÞ ðc rÞ½qm bðx QÞ t1 t2 ls; MP1 ðx; QÞ ¼ ðc vÞ½Q þ qm s1 ð1 lÞs2 : Situation 4: x Q+qm/b. That is, the backlogged demand b(x Q) is larger than both partners’ threshold quantities. Thus, the buyer will place a second order with quantity b(x Q), and the manufacturer will also quickly activate a second production in order to ensure the ordered items satisfied in time. Hence, under such a situation, both parties’ profits are given by BP1 ðx; QÞ ¼ ðp cÞQ þ ðp c bÞbðx QÞ kð1 bÞðx QÞ t1 t2 ls2 : MP1 ðx; QÞ ¼ ðc vÞ½Q þ bðx QÞ s1 ð1 lÞs2 : Based on the above analysis, one easily derived that the buyer’s expected profit is: BP1 ðQÞ ¼

ðQ 0

þ

BP1 ðx; QÞf ðxÞdx þ

ð Qþqm =b Qþq0 =b

ð Qþq0 =b Q

BP1 ðx; QÞf ðxÞdx þ

¼ ðp þ k c bÞ½Q þ b þ ðp þ k rÞ

ðQ

BP1 ðx; QÞf ðxÞdx

ð þ1

ð þ1 Qþq0 =b

Qþqm =b

BP1 ðx; QÞf ðxÞdx

ðx QÞf ðxÞdx

ðx QÞf ðxÞdx þ bQ km

0

t1 ðt2 þ ls2 Þ½1 FðQ þ q0 =bÞ ð Qþqm =b ½qm bðx QÞf ðxÞdx: ðc rÞ Qþq0 =b

(1)

324

Y.-W. Zhou and S.-D. Wang

The manufacturer’s expected profit is given by MP1 ðQÞ ¼

ð Qþq0 =b 0

þ

MP1 ðx; QÞf ðxÞdx þ

ð þ1 Qþqm =b

ð Qþqm =b Qþq0 =b

MP1 ðx; QÞf ðxÞdx

MP1 ðx; QÞf ðxÞdx:

¼ ðc vÞ½Q þ b

ð þ1 Qþq0 =b

(2)

ðx QÞf ðxÞdx

s1 ð1 lÞs2 ½1 FðQ þ q0 =bÞ ð Qþqm =b þ ðc vÞ ½qm bðx QÞf ðxÞdx Qþq0 =b

4.2

Case with qb qm

Similarly, when qb qm, if by the end of the selling period the practical demand x is less than or equal to Q, there are (Q x) units left at the end of the sales period. Hence, the buyer does not need to place a second order. When Q < x < Q þ qb/b, the backlogged demand b(x Q) is less than the buyer’s threshold quantity. So the buyer has no incentive to place a second order. If x Q þ qb/b, the backordered demand b(x Q) is larger than the threshold quantities of both parties. Thus, the buyer will place a second order with quantity b(x Q) whereas the manufacturer will also activate a second production for the buyer’s second order. Figure 2 describes graphically the buyer’s second order decision. Under the above twice-ordering strategy, one can obtain that the buyer’s expected profit is: BP2 ðQÞ ¼ ðp þ k c bÞ½Q þ b þ ðp þ k r Þ

ðQ

ð þ1 Qþqb =b

ðx QÞf ðxÞdx (3)

ðx QÞf ðxÞdx

0

þ bQ km t1 ðt2 þ ls2 Þ½1 FðQ þ qb =bÞ:

No shortage

Q

No second order

Q+ qm /b

Fig. 2 The buyer’s second decision

Place a second order of b(x-Q) units Q+ qb /b

x

Supply Chain Coordination for Newsvendor-Type Products

And, the manufacturer’s expected profit is given by ð þ1 ðx QÞf ðxÞdx s1 ð1 lÞs2 MP2 ðQÞ ¼ðc vÞ½Q þ b Qþqb =b

325

(4)

½1 FðQ þ qb =bÞ: Summarizing the above two cases will give the expected profits of the buyer and the manufacturer in the decentralized system respectively as MP1 ðQÞ qb qm BP1 ðQÞ qb qm and MPðQÞ ¼ : (5) BPðQÞ ¼ BP2 ðQÞ qb qm MP2 ðQÞ qb qm From (5), one can derive the property of the manufacturer’s expected profit function, MP(Q). Property 1. The manufacturer’s expected profit under the decentralized system, MP(Q), is a monotone increasing function with respect to Q. Proof. (i) If qb qm, the first-order derivative of MP1(Q) with respect to Q will be MP01 ðQÞ ¼ ðc vÞ½1 b þ bFðQ þ qm =bÞþ½ð1 lÞs2 ðc vÞqm f ðQ þ q0 =bÞ ¼ ðc vÞ½1 b þ bFðQ þ qm =bÞ > 0; which means that MP1(Q) is a monotone increasing function of Q. (ii) If qb qm, the first-order derivative of MP2(Q) with respect to Q is MP02 ðQÞ ¼ ðc vÞ½1 b þ bFðQ þ qb =bÞþ½ð1 lÞs2 ðc vÞqb f ðQ þ qb =bÞ:

(6)

Since f 0 (x) < 0, then FðQ þ qb =bÞ > ðQ þ qb =bÞf ðQ þ qb =bÞ:

(7)

Substituting (7) into (6) gives MP0 2(Q) > (c v)(1 b) þ [(c v)bQ þ (1 l)s2]f(Q þ qb/b) > 0. Hence, MP2(Q) is also a monotone increasing function of Q. □ Based on Property 1, we present the buyer’s optimal ordering policies in the decentralized system in Theorem 1. Theorem 1. For any increasing concave CDF F(.), the buyer’s unique optimal ordering policy, Qb, that maximizes the buyer’s expected profit is given by Qb1 ; qb qm Qb ¼ ; Qb2 ; qb qm

326

Y.-W. Zhou and S.-D. Wang

where Qb1 and Qb2 are respectively given by ðp þ k rÞFðQb 1 Þ þ ðp þ k r bÞbFðQb 1 þ q0 =bÞ ðc rÞbFðQb 1 þ qm =bÞ þ ðp þ k cÞð1 bÞ þ bb ¼ 0 ðp þ k rÞFðQb 2 Þ þ ðp þ k c bÞbFðQb 2 þ qb =bÞ þ ðp þ k cÞð1 bÞ þ bb ¼ 0:

(8)

(9)

Proof. (i) For the case with qb qm, taking the first- and second-order derivatives of BP1(Q) shown in (1) with respect to Q will respectively give

BP01 ðQÞ ¼ ðp þ k rÞFðQÞ þ ðp þ k r bÞbFðQ þ q0 =bÞ ðc rÞbFðQ þ qm =bÞ þ ðp þ k cÞð1 bÞ þ b

(10)

BP001 ðQÞ¼ðpþkrÞ½f ðQÞbf ðQþq0 =bÞbbf ðQþq0 =bÞðcrÞbf ðQþqm =bÞ: (11) Since F00 (x) < 0, i.e., f 0 (x) < 0, f(Q) > f(Q + q0/b). Hence, from (11) one has BP00 1(Q) < 0. Additionally, from (10) one easily derives that limQ !þ1 BP01 ðQÞ ¼ ðc rÞ < 0 and limQ !0þ BP01 ðQÞ ¼ ðp þ k r bÞbFðq0 =bÞ ðc rÞbFðqm =bÞ þ ðp þ k cÞð1 bÞ þ bb: It is obvious that if (p þ k r b)bF(q0/b) (c r)bF(qm/b) þ (p þ k c) (1 b) þ bb 0, then BP0 1(Q) 0, i.e., BP1(Q) is a monotone decreasing function of Q. Thus, the buyer’s optimal order quantity will be Qb1 ¼ 0, which implies that no business happens between the manufacturer and the buyer. In order to avoid such unrealistic and trivial cases, we assume in the subsequent analysis that if qb qm, ðp þ k r bÞbFðq0 =bÞ ðc rÞbFðqm =bÞ þ ðp þ k cÞð1 bÞ þ bb > 0: (12) Hence, there exists a unique positive root Qb1 to equation BP0 1(Q) ¼ 0. Since BP00 1(Q) < 0, BP1(Q) reaches its maximum at Qb1. (ii) If qb qm, the first- and second-order derivatives of BP2(Q) given in (3) with respect to Q will be respectively BP02 ðQÞ ¼ ðpþk rÞFðQÞþðpþk cbÞbFðQþqb =bÞþðpþk cÞð1bÞþbb

Supply Chain Coordination for Newsvendor-Type Products

327

and BP002 ðQÞ ¼ ðp þ k rÞf ðQÞ þ ðp þ k c bÞbf ðQ þ qb =bÞ: Since F00 (x) < 0 and 0 b 1, one has BP00 2(Q) < 0. It is easy to check that limQ !þ1 BP02 ðQÞ ¼ ðc rÞ < 0 and limQ !0þ BP02 ðQÞ ¼ ðp þ k c bÞbFðqb =bÞ þ ðp þ k cÞð1 bÞ þ bb > 0: Thus, there exists a unique positive root Qb2 to BP0 2(Qb2) ¼ 0. Hence, Qb2 is the maximum point of BP2(Q). □ Substituting Qb1 into (1) and (2) if qb qm and Qb2 into (3) and (4) if qb qm will give the optimal expected profits of the buyer and the manufacturer, BP(Qb) and MP(Qb).

5 Centralized Order Policy Consider now a situation where both the manufacturer and the buyer are willing to cooperate to pursue the centralized optimal ordering policy. Hence, unlike in the decentralized channel, the objective in this setting is to maximize the expected total profit of the system. In the subsequent analysis, we first formulate the expected total profit of the system. As described in Sect. 4, the second transaction between the two parties in the decentralized system will occur only if both parties have profits higher than those in the case without the second order (or production). In the centralized system, however, even if the second transaction results in the decrease of one party’s profit, it will still occur if it can lead to the increase of the channel’s profit. That is to say, the occurrence of the second transaction will be subject only to the following condition ðp c bÞbðx QÞ kð1 bÞðx QÞ t2 þ ðc vÞbðx QÞ s2 kðx QÞ or bðx QÞ qc ¼ ðt2 þ s2 Þ=ðp þ k v bÞ: It indicates that qc is the threshold quantity of the centralized system, beyond which the second transaction will occur.

328

Y.-W. Zhou and S.-D. Wang

Similarly, for the centralized system, if the practical demand during the selling period is x, the profit of the system is given by 8 px þ rðQ xÞ vQ t1 s1 ; if x Q > > > > > < ðp vÞQ kðx QÞ t1 s1 ; if Q < x < Q þ qc =b JPðx; QÞ ¼ > ðp vÞQ þ ðp v bÞbðx QÞ kð1 bÞðx QÞ > > > > : t1 t2 s1 s2 ; if x Q þ qc =b Hence, the expected total profit of the system, JP(Q), will be given by ð þ1 JPðQÞ ¼ ðp þ k v bÞ½Q þ b ðx QÞf ðxÞdx þ ðp þ k rÞ

ðQ

Qþqc =b

ðx QÞf ðxÞdx

(13)

0

þ bQ km t1 s1 ðt2 þ s2 Þ½1 FðQ þ qc =bÞ: Maximizing JP(Q) will give Theorem 2. Theorem 2. For any increasing concave CDF F(.), the unique optimal ordering policy, QJ, for the centralized system is given by ðp þ k rÞFðQJ Þ þ ðp þ k v bÞbFðQJ þ qc =bÞ þ ðp þ k vÞð1 bÞ þ bb ¼ 0: (14) Proof. Taking the first- and second-order derivatives of JP(Q) with respect to Q, we obtain JP0 ðQÞ ¼ ðp þ k rÞFðQÞ þ ðp þ k v bÞbFðQ þ qc =bÞ þ ðp þ k vÞð1 bÞ þ bb JP00 ðQÞ ¼ ðp þ k rÞf ðQÞ þ ðp þ k v bÞbf ðQ þ qc =bÞ: Since f(Q) > f(Q + qc/b) and p + k r > p + k v b, we have JP00 (Q) < 0. It is easy to check that limQ!0+JP0 (Q) ¼ (p + k v b)bF(qc/b) + (p + k v)(1 b) + bb > 0 and limQ!+1JP0 (Q) ¼ (v r) < 0. Hence, there exists a unique positive root QJ to JP0 (QJ) ¼ 0, and JP(Q) reaches its maximum at QJ. □ As to a two-echelon supply chain for newsvendor-type products with a single order opportunity, a common fact is that the expected profit of the centralized system is exactly equal to the sum of two members’ expected profits in the decentralized system. Many researchers like Taylor (2002), Cachon (2003), etc., have presented a lot of effective coordination mechanisms by employing successfully this fact. Weng (2004) applied directly this common fact to a supply chain for newsvendor-type products under the twice-order framework defined in this chapter. Then, he presented a quantity discount scheme that could maximize the expected

Supply Chain Coordination for Newsvendor-Type Products

329

profit of his so-called centralized channel. However, Theorem 3 shows that the above common fact does not hold in the supply chain under the twice-order framework considered in the chapter. We find out that the expected profit function [shown in (8)] of the centralized channel is always greater than the sum of the expected profits of two members in the decentralized system. Before giving Theorem 3, we need to show the following lemma. Lemma 1. (i) If qb qm, then qc q0; (ii) if qb qm, then qc qb. Proof. (i) If qb qm, one has qc q0 ¼ ðt2 þ s2 Þ=ðp þ k v bÞ ½t2 þ ls2 þ ð1 lÞs2 ðc rÞ=ðc vÞ=ðp þ k r bÞ ¼ ½t2 þ ls2 ð1 lÞs2 ðp þ k c bÞ=ðc vÞðv rÞ=½ðp þ k v bÞ ðp þ k r bÞ: Since qb qm, t2 þ ls2 (1 l)s2(p þ k c b)/(c v). From assumptions presented in Sect. 2 that p > c > v > r and p þ k c b > 0, one can easily derive p þ k r b > p þ k v b > p þ k c b > 0. Hence, we have qc q0. (ii) If qb qm, one can get (t2 + ls2)/(p + k c b) (1 l)s2/(c v). Therefore, one has qc ¼ ½t2 þ ls2 þ ð1 lÞs2 =ðp þ k v bÞ ½t2 þ ls2 þ ðt2 þ ls2 Þðc vÞ=ðp þ k c bÞ=ðp þ k v bÞ ¼ qb : □ Lemma 1 means that, the threshold value beyond which the centralized system implements the second transaction is always less than or equal to its counterpart in the decentralized system. From (1)–(4) and Lemma 1, one can derive Theorem 3. Theorem 3. The expected profit of the centralized channel is always greater than the sum of the expected profits of two members in the decentralized system. Proof. For the case of qb qm, due to (1) and (2), one can easily derive the expected total profit of the decentralized system as BP1 ðQÞ þ MP1 ðQÞ ¼ ðp þ k v bÞ½Q þ b þ ðp þ k rÞ

ðQ

ð þ1 Qþq0 =b

ðx QÞf ðxÞdx

ðx QÞf ðxÞdx

0

þ bQ km ðt1 þ s1 Þ ðt2 þ s2 Þ½1 FðQ þ q0 =bÞ ð Qþqm =b ðv rÞ ½qm bðx QÞf ðxÞdx Qþq0 =b

(15)

330

Y.-W. Zhou and S.-D. Wang

Due to (13), the expected profit for the centralized system can be expressed as JPðQÞ ¼ ðp þ k v bÞ½Q þ b þ ðp þ k rÞ

ðQ

ð þ1 Qþq0 =b

ðx QÞf ðxÞdx þ b

ð Qþq0 =b Qþqc =b

ðx QÞf ðxÞdx

ðx QÞf ðxÞdx þ bQ km ðt1 þ s1 Þ

0

ðt2 þ s2 Þ½1 FðQ þ q0 =bÞ ðt2 þ s2 Þ½FðQ þ q0 =bÞ FðQ þ qc =bÞ (16) Combining (15) and (16) gives JPðQÞ ¼ BP1 ðQÞ þ MP1 ðQÞ þ f ðxÞdx þ ðv rÞ

ð Qþq0 =b Qþqc =b

ð Qþqm =b Qþq0 =b

½ðp þ k v bÞbðx QÞ t2 s2 (17)

½qm bðx QÞf ðxÞdx

Similarly, for the case of qb qm, combining (3) and (4) with (13), the expected profit for the centralized system can be rewritten as JPðQÞ ¼ BP2 ðQÞ þ MP2 ðQÞ ð Qþqb =b ½ðp þ k v bÞbðx QÞ t2 s2 f ðxÞdx þ

(18)

Qþqc =b

From (17) and (18), one can observe that the expected profit function of the centralized system is not simply equal to, but larger than, the sum of the expected profit functions of the two partners in the system. □ In the following, we give the explanation of this phenomenon. In fact, under the case of qb qm, as analyzed earlier, if the practical demand x is less than Q þ qc/b (of course, also less than Q þ q0/b due to Lemma 1), the buyer, no matter whether in the centralized or decentralized system, does not place a second order. If the practical demand x is greater than Q þ qm/b(of course, also greater than Q + qc/b due to Lemma 1), a second order of b(x Q) units is implemented in both the centralized and the decentralized system. It implies that, under such two situations, the profit for the centralized system is just equal to the sum of two parties’ profits in the decentralized system. However, if the practical demand x satisfies Q þ qc/b x < Q þ q0/b, the centralized system is willing to activate a production to supply a second order of b(x Q) units. This second transaction brings the system the profit of (p v b)b(x Q) k(1 b) (x Q) t2 s2. In contrast, the second transaction will not occur in the decentralized system. Consequently, the sum of two parties’ profits is equal to k (x Q). The difference of these two profits is equal to (p þ k v b)b(x Q) t2 s2, which represents the increment in the channel profit yielded by two

Supply Chain Coordination for Newsvendor-Type Products

331

partners’ cooperation when the practical demand x falls into [Q þ qc/b, Q þ q0/b]. If the practical demand x satisfies Q þ q0/b x < Q þ qm/b, the second order of b(x Q) units will occur in the centralized system, which brings the system the profit of (p v b)b(x Q) k(1 b)(x Q) t2 s2. In the decentralized system, however, the second order of qm units will happen. The sum of two members’ profits resulted from the second transaction is given by ðp c bÞbðx QÞ kð1 bÞðx QÞ ðc rÞ½qm bðx QÞ t2 þ ðc vÞqm s2: The difference of the two profits is equal to (v r)[qm b(x Q)], which denotes the increment in the system profit incurred by the cooperation between the manufacturer and the buyer when the practical demand x belongs to [Q + q0/b, Q + qm/b]. Thus, the third and fourth terms in (17) exactly represents the expected increment of the system profit when two members in the channel are willing to make a decision jointly. Similarly, one can also obtain the intuitive explanation of (18). The above analysis has revealed that if we simply consider the sum of the expected profit of two parties in the considered decentralized system as the jointly decision-making objective or the expected profit of the centralized system, then this cooperative-looking system does not actually reach perfect coordination or complete cooperation state. The main reason is that in the cooperative-looking centralized system, the second order decision made by the buyer is still based on the buyer’s benefit rather than on the channel’s benefit. Hence, we refer to this type of cooperation as incomplete cooperation (ic for brevity) and this cooperative-looking system as an ic system. Let JPic(Q) be the expected profit of this ic system. Then, one has JPic1 ðQÞ; qb qm JPic ðQÞ ¼ JPic2 ðQÞ; qb qm where JPic1(Q) ¼ BP1(Q) þ MP1(Q) and JPic2(Q) ¼ BP2(Q) þ MP2(Q). It is obvious from (17) and (18) that JP(Q) JPic(Q) for any given Q. Theorem 4 shows the optimal ordering policies under the ic system just mentioned. Theorem 4. For any increasing concave CDF F(.), the unique optimal ordering policy, QJic, that maximizes the sum of the expected profit of two members, i.e., JPic(Q), will be given by ic1 QJ ; qb qm QJ ic ¼ Qic2 J ; q b qm where QJic1 and QJic2 satisfy respectively ðp þ k rÞF QJ ic1 þ ðp þ k r bÞbF QJ ic1 þ q0 =b ðv rÞbF QJ ic1 þ qm =b þ ðp þ k vÞð1 bÞ þ bb ¼ 0

(19)

332

Y.-W. Zhou and S.-D. Wang

ðp þ k rÞF QJ ic2 þ ðp þ k v bÞbF QJ ic2 þ qb =b þ ðp þ k vÞð1 bÞ þ bb ½ðp þ k v bÞqb ðt2 þ s2 Þ f QJ ic2 þ qb =b ¼ 0: (20) Proof. (i) If qb qm, taking the first- and second-order derivatives of JPic1(Q) with respect to Q, we have JP0ic1 ðQÞ ¼ ðp þ k rÞFðQÞ þ ðp þ k r bÞbFðQ þ q0 =bÞ ðv rÞbFðQ þ qm =bÞ þ ðp þ k vÞð1 bÞ þ bb; JP00ic1 ðQÞ ¼ ðp þ k rÞ½f ðQÞ bf ðQ þ q0 =bÞ bbf ðQ þ q0 =bÞ ðv rÞ bf ðQ þ qm =bÞ: Since f(Q) > f(Q + q0/b), one has JP00 ic1(Q) < 0. It is easy to check that limQ !þ1 JP0ic1 ðQÞ ¼ ðv rÞ < 0 and limQ !0þ JP0ic1 ðQÞ ¼ ðp þ k r bÞbFðq0 =bÞ ðv rÞbFðqm =bÞ þ ðp þ k vÞ ð1 bÞ þ bb: From (12), one can derive ðp þ k r bÞbFðq0 =bÞ ðv rÞbFðqm =bÞ þ ðp þ k vÞð1 bÞ þ bb ¼ ðp þ k r bÞbFðq0 =bÞ ðc rÞbFðqm =bÞ þ ðp þ k cÞð1 bÞ þ bb þ ðc vÞbFðqm =bÞ > ðc vÞbFðqm =bÞ > 0; which implies limQ!0+ JP0 ic1(Q) > 0. Hence, there exists a unique positive root QJic1 to equation JP0 ic1(QJic1) ¼ 0, at which JPic1(Q) reaches its maximum. (ii) If qb qm, the first- and second-order derivatives of JPic2(Q)with respect to Q will be JP0ic2 ðQÞ ¼ ðp þ k rÞFðQÞ þ ðp þ k v bÞbFðQ þ qb =bÞ þ ðp þ k vÞð1 bÞ þ bb ½ðp þ k v bÞqb ðt2 þ s2 Þf ðQ þ qb =bÞ; JP00ic2 ðQÞ ¼ ðp þ k rÞf ðQÞ þ ðp þ k v bÞbf ðQ þ qb =bÞ ½ðp þ k v bÞqb ðt2 þ s2 Þf 0 ðQ þ qb =bÞ:

Supply Chain Coordination for Newsvendor-Type Products

333

Since f(Q) > f(Q + qb/b), from Lemma 1 one knows that qc qb. Thus, one can derive JP00ic2 ðQÞ ðv þ b rÞf ðQ þ qb =bÞ ½ðp þ k v bÞqc ðt2 þ s2 Þf 0 ðQ þ qb =bÞ ¼ ðv þ b rÞf ðQ þ qb =bÞ 0: Due to limQ !þ1 JP0ic2 ðQÞ ¼ limQ !þ1 fðv rÞ ½ðp þ k v bÞqb ðt2 þ s2 Þ f ðQ þ qb =bÞg < ðv rÞ < 0 and limQ !0þ JP0ic2 ðQÞ ¼ ðp þ k v bÞbFðqb =bÞ þ ðp þ k vÞð1 bÞ þ bb ½ðp þ k v bÞqb ðt2 þ s2 Þf ðqb =bÞ > ðp þ k v bÞqb f ðqb =bÞ þ ðp þ k vÞð1 bÞ þ bb ½ðp þ k v bÞqb ðt2 þ s2 Þf ðqb =bÞ > 0; there exists a unique positive root QJic2 to equation JP0 ic2(QJic2) ¼ 0, at which JPic2(Q) reaches its maximum. □ Before comparing optimal ordering policies under both decentralized and centralized system, we introduce Lemma 2. Lemma 2. If qb qm, then ½FðQ þ q0 =bÞ FðQ þ qc =bÞ=½FðQ þ qm =bÞ FðQ þ qc =bÞ ðv rÞ= ðp þ k r bÞ: Proof. From the integral mean value theorem, one has ð Qþq0 =b

FðQ þ q0 =bÞ FðQ þ qc =bÞ ¼

Qþqc =b

FðQ þ qm =bÞ FðQ þ qc =bÞ ¼

f ðxÞdx ¼ f ðx1 Þðq0 qc Þ=b;

ð Qþqm =b Qþqc =b

f ðxÞdx ¼ f ðx3 Þðqm qc Þ=b

and ð Qþqm =b Qþqc =b

f ðxÞdx ¼

ð Qþq0 =b Qþqc =b

f ðxÞdx þ

ð Qþqm =b Qþq0 =b

f ðxÞdx

¼ f ðx1 Þðq0 qc Þ=b þ f ðx2 Þðqm q0 Þ=b; where x1 2 ðQ þ qc =b; Q þ q0 =bÞ, x2 2 ðQ þ q0 =b; Q þ qm =bÞand x3 2 ðQ þ qc = b; Q þ qm =bÞ.

334

Y.-W. Zhou and S.-D. Wang

Thus, one has f(x3)(qm qc) ¼ f(x1)(q0 qc) + f(x2)(qm q0). Since f(x) is a decreasing function, f(x3)(qm qc) < f(x1)(q0 qc + qm q0) ¼ f(x1)(qm qc) due to x1 < x2. It leads to f(x3) < f(x1). Noting that q0 may be equal to qc, one can obtain ½FðQ þ q0 =bÞ FðQ þ qc =bÞ=½FðQ þ qm =bÞ FðQ þ qc =bÞ ¼ f ðx1 Þðq0 qc Þ=½f ðx3 Þðqm qc Þ ðq0 qc Þ=ðqm qc Þ:

(21)

ðv rÞðp þ k c bÞ ðqm qb Þ and ðp þ k v bÞðp þ k r bÞ ðpþk cbÞ ðpþk cbÞ ðqm qb Þþðq0 qc Þ ¼ qm qc ¼ qm q0 þq0 qc ¼ pþk r b pþk vb ðqm qb Þ, one easily derives Due to q0 qc ¼

ðq0 qc Þ=ðqm qc Þ ¼ ðv rÞ=ðp þ k r bÞ: From (21) and (22), it is obvious to have Lemma 2. Based on Lemma 2, one can obtain the following results.

(22) □

Theorem 5. (i) QJic QJ, QJic > Qb; (ii) JP(QJ) JPic(QJic) > BP(Qb) + MP (Qb). Proof. (i) We first prove QJic QJ. If qb qm, one can get from Lemma 2 that ðp þ k r bÞFðQ þ q0 =bÞ ðv rÞFðQ þ qm =bÞ ðp þ k v bÞ FðQ þ qc =bÞ: Noting (17) and (23), one has 0 ¼ ðp þ k rÞF QJ ic1 þ ðp þ k r bÞbF QJ ic1 þ q0 =b ðv rÞbF QJ ic1 þ qm =b þ ðp þ k vÞð1 bÞ þ bb ðp þ k rÞF QJ ic1 þ ðp þ k v bÞbF QJ ic1 þ qc =b

(23)

(24)

þ ðp þ k vÞð1 bÞ þ bb; which is equivalent to JP0 (QJic1) 0. In addition, from the proof of Theorem 2, one can see that JP0 (QJ) ¼ 0 and JP00 (Q) < 0. Hence, it is clear to have QJic1 QJ . If qb qm, then from Lemma 1 one has qc qb. Letting G(q) ¼ the left side of (13), one can derive dGðqÞ=dq ¼ ½ðp þ k v bÞq t2 s2 f 0 QJ ic2 þ q=b =b: When qc q qb, it is obvious to have (p + k v b)q t2 s2 (p + k v b)qc t2 s2¼0. Since f 0 (x) < 0, dG(q)/dq 0 if qc q qb, which

Supply Chain Coordination for Newsvendor-Type Products

335

means that G(q) is a monotone increasing function of q in [qc, qb]. Therefore, one has G(qc) G(qb) ¼ 0, i.e., ðp þ k rÞF QJ ic2 þ ðp þ k v bÞbF QJ ic2 þ qc =b þ ðp þ k vÞð1 bÞ þ bb 0;

(25)

which is just equivalent to JP0 (QJic2) 0. This together with JP0 (QJ) ¼ 0 and JP00 (Q) < 0 will give QJic2 QJ. The proof of QJic QJ is completed. Next, we prove QJic > Qb. If qb qm, one can know from (13) that the buyer’s optimal order quantity Qb1 satisfies: ðp þ k rÞFðQb 1 Þ þ ðp þ k r bÞbFðQb 1 þ q0 =bÞ ðc rÞbFðQb 1 þ qm =bÞ þ ðp þ k cÞð1 bÞ þ bb ¼ 0; which gives ðp þ k rÞFðQb 1 Þ þ ðp þ k r bÞbFðQb 1 þ q0 =bÞ ðv rÞb FðQb 1 þ qm =bÞ þ ðp þ k vÞð1 bÞ þ bb > 0:

(26)

(26) can be rewritten as JP0 ic1(Qb1) > 0. Additionally, from the proof of Theorem 4 one has JP00 ic1(Q) < 0. It implies that JP0 ic1(Q) is monotone decreasing. Hence, after noting JP0 ic1(QJic1) ¼ 0, one can derive QJic1 > Qb1. Similarly, one can prove QJic2 > Qb2. The proof is omitted. (ii) From Theorem 4, we know that JPic(Q) reaches its maximum at QJic. Since QJic > Qb, JPic(QJic) > BP(Qb) + MP(Qb). In addition, (17) and (18) have implied that JP(Q) > BP(Q) + MP(Q) ¼ JPic(Q) for any Q (>0). Hence, it is obvious to have JPic(QJic) < JP(QJic) < JP(QJ). Theorem 5 indicates that, for the considered two-echelon supply chain, the cooperation between two parties in decision making, even the aforesaid incomplete cooperation, will lead to an increase in the system’s expected profit, and that the buyer’s optimal order quantity in the ic setting is greater than the counterparts in both centralized and decentralized setting.

5.1

Property of the ic and Decentralized System Performance

From the definitions of qb, qm, and q0, one can easily obtain that all these threshold quantities have to do with l. It implies that JPic(Q) depends on l as well. Property 2 shows the monotonity of JPic(Q, l) with respect to l.

336

Y.-W. Zhou and S.-D. Wang

Property 2. For any given Q, JPic(Q, l) is a decreasing function with respect to l. Proof. (i) Prove that JPic1(Q, l) is a monotone decreasing function of l. For a given Q, the first-order partial derivative of JPic1(Q, l) with respect to l is @JPic1 ðQ; lÞ[email protected] ¼ ½ðp þ k v bÞq0 þ t2 þ s2 þ ðv rÞðqm q0 Þ ½f ðQ þ q0 =bÞ=b dq0 =dl

(27)

þ ½FðQ þ qm =bÞ FðQ þ q0 =bÞdqm =dl From definitions of qm and q0, one has ðp þ k v bÞq0 þ t2 þ s2 þ ðv rÞðqm q0 Þ ¼ ½t2 þ ls2 þ ðc rÞqm þ ðv rÞqm þ t2 þ s2 ¼ ðc vÞqm þ ð1 lÞs2 ¼ 0: (28) Substituting (28) into (27) will give @JPic1 ðQ; lÞ[email protected] ¼ ½FðQ þ qm =bÞ FðQ þ q0 =bÞdqm =dl:

(29)

Since q0 qm (see definitions of q0 and qm) and dqm/dl ¼ s2/(c v), (29) means ∂JPic1(Q, l)/∂l 0, i.e., JPic1(Q, l) is a decreasing function of l. (ii) Prove that JPic2(Q, l) is a decreasing function of l. For a given Q, the first-order partial derivative of JPic2(Q, l) about l is @JPic2 ðQ; lÞ[email protected] ¼ ½ðp þ k v bÞqb þ t2 þ s2 ½f ðQ þ qb =bÞ=b dqb =dl: (30) Noting that qb qm and definitions of qb and qm, one can get ðp þ k v bÞqb þ t2 þ s2 ¼ ðp þ k c bÞqb ðc vÞqb þ t2 þ ls2 þ ð1 lÞs2 ¼ ðc vÞqb þ ð1 lÞs2

(31)

ðc vÞqm þ ð1 lÞs2 ¼ 0: Since dqb/dl ¼ s2/(p þ k c b) > 0, substituting (31) into (30) gives ∂JPic2(Q, l)/∂l 0. Namely, JPic2(Q, l) is a monotone decreasing function of l. □ Property 2 indicates that the bigger the value of l, the smaller the expected profit of the ic system. That is to say, the expected profit of the ic system depends on how two parties share the manufacturer’s second production setup cost. Especially, if the manufacturer independently pays all of the second production setup cost, i.e., l ¼ 0, the expected profit of the ic system will be always maximal for any given order quantity Q. In contrast, the buyer’s payment for all of the second setup cost will lead to the minimal expected profit of the ic system. Therefore, the best option

Supply Chain Coordination for Newsvendor-Type Products

337

for the ic system is to let the manufacturer pay all of the second setup cost. This is exactly opposite to the conclusion announced by Weng (2004) that “the general results obtained on the effect of coordination do not depend on how the manufacture’s production setup cost in the second order is allocated between two parties (whether it is paid by the buyer, paid by the manufacturer, or shared by both parties )” (Weng 2004, p. 151). A direct corollary of Property 2 is the following. Corollary 1. The sum of the optimal expected profits of two parties in the decentralized system decreases as l increases. Proof. As known in Sect. 4, the sum of the optimal expected profits of both entities is BP(Qb) þ MP(Qb), which is exactly equal to JPic(Qb, l). Hence, Property 2 also means Corollary 1. □ This corollary explains that sharing the manufacturer’s second production setup cost other than utterly paid by the retailer can increase the decentralized system performance. Moreover, the decentralized system would perform best if the manufacturer covers the second production setup cost completely.

5.2

A Special Case

If l ¼ 1 and b ¼ 1, all excess demand is completely backordered, and the second production setup cost is utterly paid by the buyer. It means that the threshold value that the manufacturer is willing to activate a second production in the decentralized system will be equal to zero, i.e., qm ¼ 0. Under such a case, the expected profit of the ic system, denoted as JPic(Q, l, b), becomes JPic ðQ; l ¼ 1; b ¼ 1Þ ¼ BP2 ðQ; l ¼ 1; b ¼ 1Þ þ MP2 ðQ; l ¼ 1; b ¼ 1Þ; which is just equal to the system’s expected profit (2.7) defined in Weng (2004). For notational convenience, let JPw(Q) denote the system’s expected profit and QJw be the corresponding optimal coordinated ordering quantity in Weng (2004). Then, JPw(Q) ¼ JPic(Q, l ¼ 1, b ¼ 1). Hence, Corollary 2 can also be derived directly from Property 2. Corollary 2. JPw ðQJw Þ JPic ðQJ ic ; l; b ¼ 1Þ: Proof. From the analysis presented in the second paragraph in Sect. 5.2, one has JPw(QJw) ¼ JPic(QJw, l ¼ 1, b ¼ 1). And Theorem 4 means JPic(QJw, l ¼ 1, b ¼ 1) JPic(QJic, l ¼ 1, b ¼ 1). Since JPic(Q, l, b) is a monotone decreasing function with respect to l, one can get JPic(QJic, l ¼ 1, b ¼ 1) JPic(QJic, l, □ b ¼ 1), where 0 l 1. Hence, one has JPw(QJw) JPic(QJic, l, b ¼ 1).

338

Y.-W. Zhou and S.-D. Wang

Corollary 2 further verifies that for any l (0 l < 1), the optimal expected profit of the system in Weng (2004) is always less than that of the ic system in the present model, of course, also less than that of the system under complete cooperation in this chapter.

6 Possible Perfect Coordination Scenarios By designing a simple quantity discount policy, Weng (2004) realized coordination of the ic system under the special case with l ¼ 1 and b ¼ 1. Following the way in Weng (2004), one can also achieve coordination of the ic system under the case with 0 l 1 and 0 b 1 but cannot realize perfect coordination of the system, even if under the special case with l ¼ 1 and b ¼ 1. Maybe, a more complicated quantity discount policy could achieve perfect coordination of the whole channel, but designing such a policy is out of our ability. Then, we pay our attention to a widely-used effective coordination mechanism: two-part tariff (hereafter, TPT for brevity), characterized by a two-tuple parameter (ct, K) in which the manufacturer sells the product to the buyer at the unit wholesale price ct ¼ v and charges the buyer a fixed franchise fee K. For the special case with l ¼ 1, it can be easily shown that for any given K, the buyer’s optimal order quantity would be Qt ¼ QJ, the counterpart of the centralized system, if the buyer accepts the TPT. Hence, the TPT achieves perfect coordination of the channel. Thus, as long as the manufacturer sets a suitable K-value that makes both parties’ benefits greater than before, both parties would accept the TPT that realizes perfect coordination of the channel. However, for the general case with l 6¼ 1, a common TPT mechanism is not able to achieve perfect coordination of the chain. Next, we move to another widely-used effective coordination mechanism: Revenue-Sharing Contract (hereafter, RSC for brevity), proposed by Cachon and Lariviere (2000). It is described by two parameters (cr, F), i.e., the manufacturer charges the buyer a unit wholesale price cr, lower than the unit marginal cost v, in exchange for a percentage (1 F) of the buyer’s revenue. Unfortunately, we find out that RSC also fails to coordinate the supply chain presented in our model. However, a revised revenue-sharing contract (hereafter, RRSC for brevity) would be able to complete perfect coordination of the supply chain. Before describing the RRSC, we need define the buyer’s generalized revenue as follows: Definition 1. Buyer’s generalized revenue ¼ Buyer’s revenue (Buyer’s shortage cost + Buyer’s backorder cost) The considered RRSC is characterized by three-tuple parameters (cr, F, T). Parameters cr and F are used to achieve the supply chain coordination, whereas parameter T is adopted to split the expected profit of the coordinated system between two parties. In such a RRSC, the manufacturer charges the buyer a unit wholesale price cr so that the threshold quantities of both the manufacturer and the

Supply Chain Coordination for Newsvendor-Type Products

339

buyer are equal to the counterpart of the centralized system, then selectively requires from the buyer a percentage (1 F) of his generalized revenue to keep the buyer’s optimal order quantity consistent with the centralized system’s, and finally gives the buyer a return profit T to compensate the buyer’s possible loss for accepting the RRSC. As explained above, under the RRSC, the optimal wholesale price should be chosen to make the threshold quantities of both parties equal to one of the centralized system. That is, cr satisfies qb ¼ qm ¼ qc, which leads to the manufacturer’s optimal wholesale price as: cr ¼ v þ ð1 lÞs2 ðp þ k v bÞ=ðt2 þ s2 Þ:

(32)

It is clear from (32) that this optimal wholesale price is larger than the unit marginal cost v, which is opposite to the counterpart in common RSC. If the buyer places an order of Q units at the wholesale price cr* given by (32), the generalized revenue of the buyer will be GRðQÞ ¼ ðp þ k cr bÞ½Q þ b

ðQ

ð þ1 Qþqc =b

ðx QÞf ðxÞdx þ ðp þ k rÞ

ðx QÞf ðxÞdx þ bQ km:

(33)

0

Thus, for any return profit T set by the manufacturer, the expected profits of the buyer and the manufacturer under the RRSC are respectively BPr ðQ; F; T Þ ¼ FGRðQÞ t1 ðt2 þ ls2 Þ½1 FðQ þ qc =bÞ þ T;

(34)

MPr ðQ; F; T Þ ¼ ð1 FÞGRðQÞ þ MCr ðQÞ s1 ð1 lÞs2 ½1 FðQ þ qc =bÞ T; (35) where MCr ðQÞ ¼ ðcr vÞ½Q þ b

ð þ1 Qþqc =b

ðx QÞf ðxÞdx:

Due to f 0 (x) < 0 and cr* > v > r, one can easily get that BPr(Q,F,T) is a concave function about Q. Hence, to achieve supply chain coordination, the manufacturer should select a F so that under the RRSC the buyer’s optimal order quantity Qr is just equal to the optimal order quantity QJ of the centralized system. That is, ðQ;F;TÞ * this F should satisfy @[email protected] Q¼QJ ¼ 0, which gives the optimal fraction, F , of the generalized revenue kept by the buyer as F ¼ ðt2 þ ls2 Þf ðQJ þ qc =bÞ=½ðt2 þ ls2 Þf ðQJ þ qc =bÞ þ BCr ðQJ Þ;

(36)

340

Y.-W. Zhou and S.-D. Wang

where BCr ðQÞ ¼ ðp þ k rÞFðQÞ ðp þ k cr bÞbFðQ þ qc =bÞ ðp þ k cr Þð1 bÞ bb: From (14) and cr* > v > r, it is not difficult to show that 0 < F* < 1, which means that the F* is indeed a feasible fraction of the generalized revenue. Thus, for any given T, the RRSC, (cr*, F*, T), had actually achieved perfect coordination of the supply chain if it were accepted to implement. However, whether this RRSC can be implemented would depend on whether both parties gain more expected profits under the RRSC or what values the parameter T takes. Suppose that the manufacturer is willing to offer the RRSC only if her expected profit under the RRSC increases by e 100% (e 0) as compared to her original expected profit (MP(Qb)), and that the buyer is willing to accept the RRSC only when it can let the buyer’s expected profit increased by d 100% (d 0). Then, it is easy to show that the values of T available to both parties should satisfy Tmin T Tmax, where Tmax is the manufacturer’s largest endurable return profit and Tmax ¼ ð1 F ÞGRðQJ Þ þ MCr ðQJ Þ s1 ð1 lÞs2 ½1 FðQJ þ qc =bÞ (37) ð1 þ eÞMPðQb Þ Tmin is the buyer’s smallest acceptable return profit and Tmin ¼ ð1 þ dÞBPðQb Þ F GRðQJ Þ þ t1 þ ðt2 þ ls2 Þ½1 FðQJ þ qc =bÞ (38) It can be easily derived from (13), (33), (37) and (38) that Tmin T Tmax is equivalent to ð1 þ dÞBPðQb Þ þ ð1 þ eÞMPðQb Þ JPðQJ Þ

(39)

Thus, if (39) holds, the manufacturer certainly offers the buyer a return profit so that the buyer keeps his reservation profit only, because she, as the designer of the contract, will always want to capture the lion’s share of the channel profit. So, the optimal return profit set by the manufacturer is T* ¼ Tmin. Furthermore, whether (39) holds will depend on what values of d and e (required by the retailer and the manufacturer, respectively) take. For example, for some e specified by the manufacturer, if dmax ¼ {JP(QJ) (1 + e) MP(Qb)}/BP(Qb) 1 0, then (39) will hold as long as the value of d required by the retailer does not exceed dmax. If dmax < 0, (39) does not hold for any d 0. This implies that the manufacturer has asked for a too big e. To sum up, we have the following. Theorem 6. (i)The necessary condition that there exists any feasible RRSC is given by (39). (ii) If the necessary condition is satisfied, then the optimal RRSC that can achieve perfect coordination of the channel will be (cr*, F*, T*), where cr*, F* and T* are given by (32), (36) and (38), respectively.

Supply Chain Coordination for Newsvendor-Type Products

341

7 Numerical Examples In order to illustrate the model, we show a numerical example for each of two cases: qb < qm and qb > qm. Example 1. Case with qb < qm The parameters of the model are listed below: p ¼ 10, c ¼ 6.5, v ¼ 3, b ¼ 1, k ¼ 4, r ¼ 0.5, t1 ¼ 50, t2 ¼ 150, s1 ¼ 200, s2 ¼ 300, l ¼ 0, b ¼ 1 and d ¼ 0. The random demand x is assumed to follow the exponential distribution with m ¼ 150. Following the model presented in this chapter, one can obtain that (1) the optimal first order quantity in the decentralized setting is Qb1 ¼ 45.3, the expected profits of the manufacturer and the buyer are respectively MP(Qb1) ¼ 177.8 and BP (Qb1) ¼ 203.0, and the sum of two parties’ expected profits is 380.8; (2) the optimal first order quantity of the centralized system is QJ ¼ 133.6, the optimal RRSC is (cr*, F*, T*) ¼ (9.7, 0.062, 325.2), which achieves perfect coordination of the supply chain and enhances the system’s expected profit to JP(QJ) ¼ 466.1. However, for the same values of parameters given in Example 1, the coordinated order policy in Weng (2004) enhanced the system’s expected profit only to JPw(167.7) ¼ 460.6. Example 2. Case with qb > qm The parameters of the model are listed below: p ¼ 11, c ¼ 7, v ¼ 1.5, k ¼ 1, b ¼ 0.5, t2 ¼ 260. Other parameters are kept the same as in Example 1. Based on the presented solution procedure in this chapter, the following can be obtained (1) In the decentralized setting, the optimal first order quantity is Qb2 ¼ 39.2, the expected profits of the manufacturer and the buyer are respectively MP(Qb2) ¼ 457.0 and BP(Qb2) ¼ 295.5, and the sum of two parties’ expected profits is 752.5; (2) the optimal first order quantity of the centralized system is QJ ¼ 229.4, the optimal RRSC is (cr*, F*, T*) ¼ (6.9, 0.054, 390.4), which can perfectly coordinate the whole channel and enhance the system’s expected profit to JP(QJ) ¼ 945.6. However, for the same values of parameters given in Example 2, the coordinated order policy in Weng (2004) enhanced the system’s expected profit only to JPw(195.7) ¼ 929.2.

8 Conclusions Most of the literature on coordination issues of the supply chain with single period products assumed that only one order happened during the whole period. However, in practice, buyers probably choose to place more than once order in the selling period because they know more exact information about demand as time moves ahead. In this chapter, we further generalize the newsboy-type order coordination issue considered by Weng (2004) for a two-echelon supply chain with two ordering opportunities, and extend it to cover the case with two-party-shared second setup

342

Y.-W. Zhou and S.-D. Wang

cost and partial backlogging. We prove that the ic system and decentralized system would perform best if the manufacturer covers utterly the second production setup cost. We find out that the expected profit of the centralized system is not always equal to the sum of two members’ expected profits in the decentralized system, which is not consistent with our intuitive expectation and those in the existing related literature, like Cachon (2003), Weng (2004), Zhou and Li (2007), etc. In order to achieve perfect coordination of the considered channel, we try three widely-used effective mechanisms: simple quantity discount, two-part tariff and revenue-sharing contract. Consequently, both simple quantity discount and revenue-sharing contract is not able to achieve the channel’s perfect coordination. Neither can the two-part tariff except the special case that the buyer pays all the manufacturing setup cost. The chapter then presents a RRSC policy that completes the perfect coordination of the supply chain. Worthwhile to mention is that for simplicity the chapter only considers a constant backordered fraction of the unfilled demand in the sales period. In reality, however, this backordered fraction may probably influence the buyer’s expected profit directly. In that case, it would be beneficial for the buyer to choose a suitable second order quantity. This problem will be considered in our future research. Other possible extensions of the model include: considering multiple manufacturers, multiple buyers with price or quantity competition, random demand with unknown probability distribution, etc.

References Arcelus FJ, Kumar S, Srinivasan G (2008) Evaluating manufacturer’s buyback policies in a singleperiod two-echelon framework under price-dependent stochastic demand. Omega 36(5): 808–824 Atkinson AA (1979) Incentives, uncertainty and risk in newsboy problem. Decis Sci 10:341–357 Cachon G, Lariviere M (2000) Supply chain coordination with revenue sharing: strengths and limitations. Working paper, University of Pennsylvania, Philadelphia, PA Cachon G (2003) Supply chain coordination with contracts. In: de Kok AG, Graves SC (eds) Handbooks in operations research and management science, Chap 6, vol 11. Elsevier, Amsterdam Chen H, Chen J, Chen Y (2006) A coordination mechanism for a supply chain with demand information updating. Int J Prod Econ 103:347–361 Chen H, Chen Y, Chiu CH, Choi TM, Sethi S (2010) Coordination mechanism for the supply chain with leadtime consideration and price-dependent demand. Eur J Oper Res 203(1):70–80 Choi TM, Li D, Yan H (2003) Optimal two-stage ordering policy with Bayesian information updating. J Oper Res Soc 54:846–856 Donohue KL (2000) Efficient supply contracts for fashion goods with forecast updating and two production modes. Manage Sci 46(11):1397–1411 Emmons H, Gilbert SM (1998) The role of returns policies in pricing and inventory decisions for catalogue goods. Manage Sci 44(2):276–283 Fisher M, Raman A (1999) Managing short life-cycle products. Ascet 1 Goodman DA, Moody KW (1970) Determining optimal price promotion quantities. J Mark 34: 31–39 Ismail B, Louderback J (1979) Optimizing and satisfying in stochastic cost-volume-profit analysis. Decis Sci 10:205–217

Supply Chain Coordination for Newsvendor-Type Products

343

Kabak I, Schiff A (1978) Inventory models and management objectives. Sloan Manage Rev 10: 53–59 Khouja M (1999) The single-period (news-vendor) problem: Literature review and suggestions for future research. Omega 27(5):537–553 Lau HS (1980) The newsboy problem under alternative optimization objectives. J Oper Res Soc 31:525–535 Lau HS, Lau AH (1997) Reordering strategies for a newsboy-type product. Eur J Oper Res 103: 557–572 Lau AH, Lau HS (1998) Decision models for single-period products with two ordering opportunities. Int J Prod Econ 55:57–70 Li S, Zhu Z, Huang L (2009a) Supply chain coordination and decision making under consignment contract with revenue sharing. Int J Prod Econ 120(1):88–99 Li J, Chand S, Dada M, Mehta S (2009b) Managing inventory over a short season: models with two procurement opportunities. Manuf Serv Oper Manage 11(1):174–184 Milner JM, Kouvelis P (2005) Order quantity and timing flexibility in supply chains: the role of demand characteristics. Manage Sci 51(6):970–985 Nahmias S, Schmidt C (1984) An efficient heuristic for the multi-item newsboy problem with a single constraint. Nav Res Logistics Q 31:463–474 Pan K, Lai KK, Liang L, Leung SCH (2009) Two-period pricing and ordering policy for the dominant retailer in a two-echelon supply chain with demand uncertainty. Omega 37:919–929 Pasternack BA (1985) Optimal pricing and return policies for perishable commodities. Mark Sci 4(2):166–176 Serel D (2009) Optimal ordering and pricing in a quick response system. Int J Prod Econ 121(2):700–714 Taylor TA (2002) Supply chain coordination under channel rebates with sales effort effects. Manage Sci 48(8):992–1007 Wang SD, Zhou YW, Wang JP (2010) Supply chain coordination with two production modes and random demand depending on advertising expenditure and selling price. International Journal of Systems Science, 2010, 41(10):1257–1272 Webster S, Weng ZK (2008) Ordering and pricing policies in a manufacturing and distribution supply chain for fashion products. Int J Prod Econ 114(2):476–486 Weng ZK (2004) Coordinating order quantities between the manufacturer and the buyer: a generalized newsvendor model. Eur J Oper Res 156:148–161 Whitin TM (1955) Inventory control and price theory. Manage Sci 2:61–68 Wong WK, Qi J, Leung SYS (2009) Coordinating supply chains with sales rebate contracts and vendor-managed inventory. Int J Prod Econ 120(1):151–161 Zhou Y, Li DH (2007) Coordinating order quantity decisions in the supply chain contract under random demand. Appl Math Model 31(6):1029–1038

.

Part III

Channel Power, Bargaining and Coordination

.

Bargaining in a Two-Stage Supply Chain Through Revenue-Sharing Contract Jing Hou and Amy Z. Zeng

Abstract We focus on a bargaining problem between one supplier and one retailer that are coordinated by a revenue-sharing contract. The suppler is assumed to have the ability to influence the retailer’s profit by setting his/her target inventory level, which in turn determines the lead time. We examine the cases under which either the supplier or the retailer is dominant in the bargaining process. The key contract parameter, the acceptable range of the revenue-sharing fraction for the two players, and the maximum amount of monetary bargain space are obtained under explicit and implicit information, respectively. Numerical illustrations of the contracts for various scenarios are given to shed more insights. Keywords Dominance and bargaining • Nonlinear optimization • Supply chain coordination • Supply contracts

1 Introduction Revenue sharing mechanism has been applied extensively in various industries, such as internet service (e.g., He and Walrand 2005), airline (e.g., Zhang et al. 2010), and virtual enterprises (e.g., Chen and Chen 2006) – to name a few, as an efficient vehicle to achieve coordination, because it is relatively straightforward

J. Hou (*) Business School, Hohai University, Nanjing, Jiangsu 211100, China e-mail: [email protected] A.Z. Zeng School of Business, Worcester Polytechnic Institute, Worcester, MA 01609, USA e-mail: [email protected] T.-M. Choi and T.C. Edwin Cheng (eds.), Supply Chain Coordination under Uncertainty, International Handbooks on Information Systems, DOI 10.1007/978-3-642-19257-9_14, # Springer-Verlag Berlin Heidelberg 2011

347

348

J. Hou and A.Z. Zeng

for the decision makers to implement and manage the contract. The primary objective of the revenue-sharing contract is to align the two parties’ interests and actions by having the retailer share a portion of his/her revenue with the supplier. As a result, the supplier’s effort and willingness to collaborate should increase. Two desirable outcomes are expected from the revenue-sharing mechanism, namely higher profit level for the entire chain, and a “win-win” situation for each chain member. The classic problem of a revenue-sharing contract is how to determine the revenue-sharing fraction for better coordination outcomes. This contract parameter is determined under various decision-making configurations, one of which can be characterized by the power inequality of negotiation in the bargaining process. In a two-stage supply chain consisting of a single manufacturer (or supplier) and a single retailer, if the supplier has the ability to influence the retailer’s decision on revenue-sharing fraction, then he/she may receive larger increase in profit resulted from the coordination mechanism. On the other hand, if the retailer is dominant, then the revenue-sharing fraction may be set to satisfy the retailer’s requirements. The major contribution of this paper lies in the area where we obtain the key parameters of the revenue-sharing contract and the bargaining space of a singlesupplier-single-retailer supply chain with the consideration of a dominant player. The retailer’s profit depends upon the lead time that is affected by the supplier’s finite target inventory level. The contract requires the supplier to hold larger inventory level to achieve system optimization and also a “win–win” condition for two players. The dominant player (either the supplier or the retailer) in the bargaining process requires more increase in profit. In both situations, the ranges of the revenue-sharing fraction as well as the maximum monetary value that the two parties can bargain are obtained. The impacts of the explicit and implicit information about the supplier’s inventory holding cost on the decisions are also examined. As will be discussed in the literature review, the problem studied in this paper has not been fully addressed in the literature. Our numerical examples show that significant improvements can be accomplished by the proposed contract and the bargaining method. The remainder of the paper is organized as follows. Section 2 summarizes the literature related to revenue-sharing contract and the different ways of distributing the profit among the supply chain entities. In Sect. 3, we review the results from basic centralized and decentralized optimizations from our previous work, which will provide foundation for subsequent analysis. Section 4 examines the joint revenue-sharing and bargaining decisions between the two parties by taking into account of dominance and the kind of knowledge the retailer has about the supplier’s inventory holding cost structure. The ranges of the key contract parameter and monetary bargain space are derived and numerical examples are given. Finally, we provide concluding remarks and directions for future research in Sect. 5.

Bargaining in a Two-Stage Supply Chain Through Revenue-Sharing Contract

349

2 Literature Review Revenue-sharing contracts have attracted considerable attention. An extensive literature review can be found in Cachon and Lariviere (2000) and Yao et al. (2008). We herein focus on most recent examples of studies that have been published in the literature. Giannoccaro and Pontrandolfo (2004) propose a revenuesharing model that aims at coordinating a three-stage supply chain. The model increases the system efficiency as well as the profits of all the chain members by fine tuning the contract parameters. In analyzing a special two-stage supply chain where the revenue decreases with the lead time and increases with inventory, Gupta and Weerawat (2006) design a revenue-sharing contract to maximize the centralized revenue by choosing an appropriate inventory level. Chen et al. (2007) study the performance of the supply chain with one supplier and multiple buyers under deterministic price-sensitive customer demand. Yao et al. (2008) investigate a revenue-sharing contract for coordinating a supply chain comprising one manufacturer and two competing retailers who face stochastic demand before the selling season. Linh and Hong (2009) discuss how the revenue sharing fraction and the wholesale price are to be determined in revenue sharing contract in order to achieve channel coordination and a win–win outcome for a single retailer and a single wholesaler. Giannoccaro and Pontrandolfo (2009) model the negotiation process among the supply chain actors by adopting agent-based simulation, taking into account the contractual power and the collaboration among the SC actors. A number of researchers have recently demonstrated the effectiveness of revenue sharing contract in supply chain coordination by comparing or integrating it with other contract types. For example, Li and Hua (2008) and Li et al. (2009) have examined the coordination effectiveness of consignment contract with revenue sharing for decentralized supply chains. Bellantuono et al. (2009) present a model in which the supply chain partners participate in two different programs – a revenue sharing contract between the supplier and the retailer, and an advanced booking discount program offered by the retailer to the customers. Pan et al. (2010) discuss and compare the results of a wholesale price contract or a revenue-sharing contract under different channel power structures to check whether it is beneficial for manufacturers to use revenue-sharing contracts under different scenarios. Ouardighi and Kim (2010) compare the possible outcomes under a wholesale price contract and a revenue-sharing contract when studying a non-cooperative dynamic game in which a single supplier collaborates with two manufacturers on design quality improvements for their respective products. Lin et al. (2010) compare the revenue sharing contract with the insurance contract, under which the supplier shares the risk of overstock and under-stock with the retailer, improving the efficiency of the supply chain with a newsvendor-type product. In sum, the studies in this category do not consider revenue-sharing as a single coordination mechanism; rather as part of the supply chain collaboration methodology or an alternative to other contracts.

350

J. Hou and A.Z. Zeng

Bargaining and cooperation have been always playing a key role in profit allocation in a supply chain. For example, Jia and Yokoyama (2003) propose a scheme based on Game theory to decide the profit allocation of each independent power producers in the coalitions rationally and impartially. Guardiola et al. (2007) study the coordination of actions and the allocation of profit in supply chains under decentralized control in which a single supplier supplies several retailers with goods for replenishment of stocks. Nagarajan and Sosˇic´ (2008) use cooperative bargaining models to find allocations of the profit pie between supply chain partners. And the problem of how to split the additional profit among the supply chain entities in a revenue-sharing contract has been the subject of many recent researchers. In a study by Chauhan and Proth (2005) where the customer demand depends upon the retail price, a new approach is proposed to maximize the centralized profit by sharing the profit proportional to the risk among the partners. In the work of Jaber and Osman (2006), a simple profit-sharing contract is proposed in such a way that the profit is distributed proportionally to each partner’s investment amount. Rhee et al. (2010) propose a new way of generalizing contract mechanisms to multi-stage settings, where one supply chain entity takes the lead in negotiating a single contract with all other entities simultaneously. Two special cases are discussed – one in which all entities receive the same absolute increase in profit; and one in which all members receive the same relative increase in profit. In addition, Sucky (2006) considers the bargaining problem of a two-stage supply chain where the buyer has no access to the supplier’s complete information. To reduce the system-wide cost, the order quantity is treated as a variable and the coordination mechanism with the buyer being dominant is derived and compared with that under complete information. Inspired by the work of Rhee et al. (2010), our research assumes that either the supplier or the retailer is dominant in the bargaining process and requires more increase in profit. We derive the accepted range of revenue sharing fraction by both parties as well as the associated bargain space. In this paper we restrict our attention to a supply chain that consists of one supplier and one retailer, which are separate and independent organizations, actively seeking favorable opportunities to coordinate. We extend the study of Hou et al. (2009) by considering the situations under which one of the two supply chain members is dominant in the bargaining process. For both cases, we develop the key contract parameters and discuss the range of the monetary amount that can be shared between the two parties under explicit and implicit information about the supplier’s inventory cost, respectively.

3 The Basic Models We study the coordination issue between a supplier and a retailer in a two-stage supply chain that produces and sells one single product. The basic assumptions for this paper are identical to those made in our previous study (Hou et al. 2009) and are

Bargaining in a Two-Stage Supply Chain Through Revenue-Sharing Contract

351

briefly summarized here. The demand rate is known as l and the demand process is stationary and follows Poisson distribution. The supplier’s production cost is cs per unit and average unit inventory holding cost is h. The retailer’s unit cost is denoted as cr. Furthermore, the retailer’s unit profit of the final product, p, is assumed to be sensitive to the lead time. Similar to the study by Gupta and Weerawat (2006), we study the situation under which the supplier has the ability to influence the retailer’s average unit revenue p by setting his/her target inventory level b and by knowing the relationship between the lead time L and the target inventory level through the following expression: pðbÞ ¼ p0 bLðbÞ

(1)

where the parameter, b, is a scale factor, and p0 is the retailer’s largest possible unit revenue achieved in an ideal situation with the highest acceptable sales price by the end-users and the supplier’s shortest lead time. Both parameters ðb; p0 Þcan be estimated and hence can be assumed to be known. Since the retailer’s order lead time is determined by the time the supplier spends on production, transportation, and transaction, it is evident that the more inventory available at the supplier’s site, the shorter the lead time will be. Intuitively, the retailer’s average lead time is a decreasing function of the supplier’s target inventory level, that is, L0 ðbÞ < 0. The lead time is also limited by various factors besides the supplier’s target inventory level; for example, when the lead time is less influenced by the inventory, it is more determined by the other factors such as transportation time and order processing time. Hence, as the inventory level increases, the change rate of the lead time decreases. Therefore, the lead time, L(b), demonstrates the properties of a function that is decreasing but convex with respect to the inventory level (b), which means that L0 ðbÞ < 0 and L00 ðbÞ > 0. The assumed convexity of the lead-time function does simplify the subsequent analyses, but is also general enough to include many possible types of relationships between the inventory and lead time. Besides, we make an assumption that when there is no stock available at the supplier’s site, i.e., the lead time reaches its maximum, the customer will lower the acceptable price to an extent that the unit profit for the retailer becomes zero. As a result, the specific expression of L(b) is given as follows: LðbÞ ¼ lmax kbm ¼ p0 =b kbm ;

8 k > 0; 0 < m 1:

(2)

Where k is interpreted as a scale factor, and m is a known exponent. The values of both can be estimated based on sales history. In addition, lmax is the maximum lead time if there is no stock available when an order is placed, and LðbÞ > 0 holds for all values of b. In what follows, we first review the optimal planning parameters obtained from our previous study (Hou et al. 2009), which will be used as a basis for extensions in this paper.

352

3.1

J. Hou and A.Z. Zeng

The Centralized Planning Model

The goal of a centralized supply chain is to set the target inventory level so that the chain’s total expected profit, P0rþs , calculated in (3), is maximized. P0rþs ¼ lbkbm hb lðcs þ cr Þ

(3)

The first part of formula (3) is the revenue obtained by selling the final products. The second part is the inventory holding cost occurred at the supplier’s, while the last one indicates the supplier’s production cost and the retailer’s cost. It is easy to show that (3) is a concave function with respect to b and the supplier’s optimal inventory level is found as follows: b0

¼

h lbmk

1 m1

(4)

Note that the profit given at the above inventory quantity reaches to the highest point for the supply chain.

3.2

The Decentralized Profit-Sharing Model

In a decentralized supply chain, both players act independently and make decisions to maximize their respective profits. In this situation, the retailer determines a fraction (a) to share the sales revenue with the supplier, and then the supplier decides his/her target inventory level (b) based on the given revenue-sharing fraction. Denote (a*, b1*) as the optimal decisions and ðPr ; Ps Þ as the profits for the retailer and the supplier respectively, we summarize the results of this situation obtained by Hou et al. (2009) as follows: a ¼ m b1

Ps ðhja

¼

h lbm2 k

h ¼ mlbk lbm2 k

; b1 Þ

Pr ðhja

; b1 Þ

(5)

1 m1

m m1

< b0

h

(6)

h lbm2 k

h ¼ ð1 mÞlbk lbm2 k

m m1

1 m1

lcs

lcr

(7)

(8)

Bargaining in a Two-Stage Supply Chain Through Revenue-Sharing Contract

353

It is seen that the two methods provide somewhat different results. The objective now is to find an acceptable set of (a, b) to enhance the profitability of the supplier and the retailer. Note that a complete list of symbols and notations used throughout the paper is provided in Appendix 1.

4 The Bargaining Decision Under Dominance It is intuitive that a higher fraction of revenue offered by the retailer could motivate the supplier to hold a larger inventory, and as a result, a larger amount of revenue for the supply chain. Therefore, we want to see how they can work together to determine the revenue-sharing fraction so that the profits of both parties can increase to the levels they are able to achieve in a decentralized supply chain. The analysis will be performed in the following two situations (1) The supplier is the leader in the bargaining process, and we use the subscript “s” to label related notation, and (2) The retailer is the leader, and we use subscript “r” for all relevant symbols. Whoever is dominant in the supply chain requires larger increased profit from the new revenue sharing contract. Both situations are analyzed with explicit information and implicit information about the supplier’s inventory holding cost, h, respectively.

4.1 4.1.1

Supplier-Dominant Bargaining With Explicit Information

As the target inventory level of the supplier is set, which is b0 shown in (4), we only need to identify the new revenue-sharing fraction that will enable such an inventory quantity. Since increased inventory level causes higher inventory holding cost for the supplier but results in more revenue for the retailer, the dominant member (supplier) would require larger benefit from the coordination. Thus, we will first determine the new revenue-sharing fraction a in the presence of explicit information on the supplier’s inventory cost, h, and then discuss the range of the monetary amount that can be shared between the dominant supplier and the retailer. In a supplier-dominant supply chain, the range of a is given in the following statement: Proposition 4.1. In a supplier-dominant supply chain, to attract the two-stage supplier to hold a larger inventory level b0 and to achieve higher profits for both parties than those in the case of decentralized planning, the retailer’s new share of revenue, a, has the following range of values:

354

J. Hou and A.Z. Zeng

1 2m m 1 1 m m m1m m1m þ 2m1m þ 1 < as < 1 þ m1m m1m ; 2

8 0 < m < 1 (9)

Moreover, the range reaches the maximum when m ¼ 0.25. Proof. See Appendix 2. We refer to the right hand side of (9) as the upper limit of a, that is, 1

m

1m m1m ; aU s ¼1þm

(10)

and the left hand side of (9) as the lower limit, aLs ¼

1 2m m 1 m m1m m1m þ 2m1m þ 1 : 2

(11)

Now that the range of the revenue-sharing fraction is determined, we examine the range of monetary value that the retailer could share with the supplier. We denote such a monetary space as ð0; DPÞ, and give the values of the space in the following statement. Observation 4.1. In a supplier-dominant two-stage supply chain, there exist two possible scenarios when the monetary value for the two parties to share is found. The two scenarios are differentiated by a specific value of m* that is determined by the input parameters, ðl; b; k; h; cr cs Þ, as follows: Case (i): If 0 < m m* [except some values in m < m* when cr > cs, which fall into Case (ii)], then the range of the monetary amount that can be shared between the two parties is given by ð0; DPs1 Þ, where m DPs1 ¼ 0:5Ph ð1 mÞ 1 ð1 þ mÞm1m :

(12)

Case (ii): If m < m < 1 [plus those values of m < m* when cr > cs in Case (i)], there exists a revenue-sharing fraction, ah , where aLs < ah < aU s , at which the two parties’ new profits are identical: ah ¼

1 þ m lmðcs cr Þ : þ 2 2hb0

(13)

Therefore, a monetary quantity that allows the supplier’s new profit to be no less than the retailer’s is found as ð0; DPs2 Þ, where m

DPs2 ¼ Ph ðm 1Þðm1m 0:5Þ 0:5lðcs cr Þ

(14)

Note that in both (12) and (14), the factor, Ph , has the following expression:

h Ph ¼ lbk lbmk

m m1

(15)

Bargaining in a Two-Stage Supply Chain Through Revenue-Sharing Contract

355

Proof. See Appendix 3. It is seen from Observation 4.1 that retailer’s new revenue sharing fraction to attract the supplier to hold a higher inventory quantity falls into an interval, which is also true for the dollar amount. Hence, the final choice of the revenue-sharing fraction will be reached through bargaining between the two parties.

4.1.2

With Implicit Information

The supplier’s inventory cost, h, plays a critical role in decision making. In reality, the supplier may choose not to reveal the actual value of h to the retailer because he either considers it a piece of private information or has difficulty estimating the exact value. As such, the supplier may only tell the retailer a range of his cost structure in such a way that h1 < h < h2 . We call such a situation where the retailer has no specific value about the supplier’s cost structure as “decision-making under implicit information”. In this section, we will examine how to obtain the range of the revenue-sharing fraction and the impacts of the key input parameters on such a decision-making situation. Given the range of the supplier’s cost structure, h 2 ½h1 ; h2 , it is not difficult to show that the two parties’ profit functions are monotonically decreasing with the growth of the cost. Therefore, h1 and h2 represent the best and worst scenario, respectively, and we will only need to consider the new value of the contract parameter at these two limits. Suppose that the upper limit, h2, can be written as a function of the lower limit, that is, h2 ¼ ð1 þ dÞh1 ; where d > 0:

(16)

Table 1 reports two sets of the supplier’s inventory quantities based on the ; b results obtained from the previous sections: b 01 02 in centralized planning, and b11 ; b12 in decentralized coordination, as well as the profit functions of the two parties. Proposition 4.2. In a supplier-dominant two-stage supply chain, to coordinate with the supplier under the case of implicit information where the supplier provides an interval of the inventory cost, h1 h h2 , the retailer would select a revenuesharing fraction from the following range, aLs a aU s , where aLs ¼

1 2m m 1 m m1m m1m þ 2m1m þ 1 ; 2 1

m

1m m1m ; aU s ¼1þm

8 0<m<1

8 0 < m < 1:

However, the monetary bargain space to be negotiated between the two parties differs in the following way:

356

J. Hou and A.Z. Zeng

Table 1 Parameters in the gaming under implicit information Parameters hi, i ¼ 1, 2 a ¼ m a 1 1 m1 m1 b1 h1 h2 b11 ¼ ; b12 ¼ ; b11 > b12 2 2 lbm k lbm k 1 1 b0 h1 m1 h2 m1 b01 ¼ ; b02 ¼ ; b01 > b02 lbmk lbmk m m1 Retailer’s profit hi Pr ðhi ; a ; b1i Þ ¼ ð1 mÞlbk lcr lbm2 k m m1 hi Pr ðhi ; ai ; b0i Þ ¼ ð1 aÞlbk lcr lbmk m 1 m1 m1 Supplier’s profit hi hi Ps ðhi ; a ; b1i Þ ¼ mlbk hi ; lcs km2 bl km2 bl m 1 m1 m1 hi hi Ps ðhi ; ai ; b0i Þ ¼ albk hi lcs lbmk lbmk

Case (i): 0 < m m* [except some values of m < m* when cr > cs, which fall into Case (ii)]: As the supplier’s unit inventory holding cost range expands, which is captured by d (h2 h1 ¼ dh1 ), the gap between the maximum amount of income, ½1 GsDP , that the retailer would share with the supplier increases with d " ½1 GsDP ðdÞ

1 1 1þd

¼ As1

# m 1m

; where

m As1 ¼ 0:5ð1 mÞ 1 ð1 þ mÞm1m lbk

h1 lbmk

(17) m m1

(18)

Case (ii): m < m < 1 [plus some values of m < m* when cr > cs in Case (i)]: There exists a revenue-sharing fraction, ah , where aL < ah < aU , at which the two parties’ new profits are identical: ah ¼

1þm lmðcs cr Þ : þ 2 2hb0

(19)

Furthermore, as the supplier’s cost interval increases, the gap between the ½2 maximum amount of income, GsDP , that the retailer could share with the supplier increases with d " ½2 GsDP ðdÞ

¼ As2

1 1 1þd

# m 1m

; where

(20)

Bargaining in a Two-Stage Supply Chain Through Revenue-Sharing Contract

m

As2 ¼ ðm 1Þðm1m 0:5Þ lbk

h1 lbmk

m m1

:

357

(21)

Proof. See Appendix 4. As clearly stated in Proposition 4.2, it is interesting to see that even if the supplier provides an interval of the cost information rather than a specific value, the range of the fraction of revenue that the retailer will offer to encourage the retailer to hold larger quantity of inventory level remains the same; however, the amount of monetary value to be shared with the supplier varies as the range of the supplier’s cost value widens.

4.2

Retailer-Dominant Bargaining

The preceding section studies the key decision-making parameters when the supplier is dominant. In this section, we examine how the same parameters are determined in the opposite situation – the retailer is the dominator.

4.2.1

With Explicit Information

In this retailer-dominant supply chain in which the retailer has the explicit data about the inventory holding cost, h, we have found that the range of the revenuesharing fraction, a, can be described in the following proposition. Proposition 4.3. In a retailer-dominant two-stage supply chain, to attract the supplier to hold a larger inventory level b0 and to achieve higher profits for both parties than those in the case of decentralized planning, the retailer’s new share of revenue, a, satisfies the following range:

1 2m 2m m 1 m þ m1m m1m < ar < 0:5 m m1m m1m þ 2m1m þ 1 ; 8 0 < m < 1 (22)

Proof. See Appendix 2. Denote the right hand side of (22) as the upper limit of ar , that is, 2m m 1 L 1m m1m þ 2m1m þ 1 ; aU ¼ a ¼ 0:5 m m r s

(23)

and the left hand side of (22) as the lower limit, 1

2m

aLr ¼ m þ m1m m1m :

(24)

358

J. Hou and A.Z. Zeng

Comparing the conclusions in Proposition 4.3 with those in Proposition 4.1, we can see that the range of the revenue-sharing fraction remains unchanged; i.e., L U L aU r ar ¼ as as , and whoever is the leader, the range of the fraction reaches the maximum at m ¼ 0.25. The range of monetary value that the retailer could share with the supplier if the retailer is the leader is discussed in the following statement. Observation 4.2. In a retailer-dominant supply chain, two possible scenarios exist when finding the monetary amount for the two parties to share. The difference is dependent upon a specific range of (m1*, m2*) that is determined by the input parameters, ðl; b; k; h; cr cs Þ; as follows: Case (i): If m 2 = ðm1 ; m2 Þ and cr > cs , or cr cs , the range of the monetary amount that can be shared between the two parties is given by ð0; DPr1 Þ, where m DPr1 ¼ 0:5Ph ð1 mÞ 1 ð1 þ mÞm1m

(25)

Case (ii): If m 2 ðm1 ; m2 Þ and cr > cs , then there exists a revenue-sharing fraction, ah , where aLr < ah < aU r , at which the two parties’ new profits are identical. Therefore, a monetary quantity that allows the retailer’s new profit to be no less than the supplier’s is found as ð0; DPr2 Þ, where 1

2m

DPr2 ¼ Ph ð0:5 0:5m m1m þ m1m Þ þ 0:5lðcs cr Þ:

(26)

Proof. See Appendix 5. According to Observation 4.1 and 4.2, in the case where the supplier is dominant, only when cr–cs > d and m < m* can the supplier’s profit always be larger than the retailer’s (as shown in Appendix 3). This means that, to gain an advantage over the retailer, the supplier must lower his/her unit production cost (cs) to be less than the retailer’s unit cost (cr), and the supplier’s inventory level has minimal impact on the lead time, which is captured by the parameter, m. These two requirements are fairly stringent, and hence, it will be more difficult for the supplier to bargain. When the retailer is dominant, the condition under which the retailer’s profit is always higher than the supplier’s requires that only cr < cs is true. This means that the retailer only needs to ensure his/her unit cost (cr), lower than the supplier’s unit production cost (cs). This constraint is less stringent than that in the other case, and thus, it is easier for the retailer to gain advantage. The reason for this phenomenon is that, to achieve supply chain optimization, increased inventory level causes higher inventory holding cost for the supplier but results in higher revenue for the retailer, and thus makes it easier for the retailer to obtain higher profit than the supplier.

4.2.2

With Implicit Information

With the supplier’s holding cost information switching from a single value to an interval, the properties of the contract parameters in the retailer-dominant decision making are given in Proposition 4.4.

Bargaining in a Two-Stage Supply Chain Through Revenue-Sharing Contract

359

Proposition 4.4. In a retailer-dominant two-stage supply chain, to coordinate with the supplier under the case of implicit information where the supplier provides an interval of his inventory cost, h1 h h2 , the retailer would select a revenuesharing fraction from the following range, aLr a aU r , where 1

2m

aLr ¼ m þ m1m m1m ;

8 0<m<1

2m m 1 1m m1m þ 2m1m þ 1 ; aU r ¼ 0:5 m m

8 0<m<1

However, the monetary bargain space to be negotiated between the two parties differs in the following way: (i) When m 2 = ðm1 ; m2 Þ and cr > cs , or cr cs , and as the supplier’s average unit inventory holding cost range increases, the gap between the maximum amount ½1 of income, GrDP , that the retailer would share with the supplier increases with d " ½1 GrDP ðdÞ

where

¼ Ar1

1 1 1þd

m 1m

Ar1 ¼ 0:5ð1 mÞ 1 ð1 þ mÞm

# m 1m

;

h1 lbk lbmk

(27)

m m1

;

(28)

and (ii) When m 2 ðm1 ; m2 Þ and cr > cs , then there exists a revenue-sharing fraction, ah , where aLr < ah < aU r , at which the two parties’ new profits are identical: ah ¼

1 þ m lmðcs cr Þ þ 2 2hb0

Furthermore, as the supplier’s cost interval increases, the gap between the ½2 maximum amount of income, GrDP , that the retailer could share with the supplier increases with d " # m 1m 1 ½2 ; (29) GrDP ðdÞ ¼ Ar2 1 1þd where

m h1 m1 1 2m Ar2 ¼ 0:5 0:5m m1m þ m1m lbk : lbmk

Proof. See Appendix 4.

(30)

360

J. Hou and A.Z. Zeng

4.3

Numerical Examples

We provide two numerical examples to illustrate the range of a and the impact of the supplier’s holding cost range on the amount of revenue the retailer would share L U with the supplier. One Lexample shows the situation when ar < ah < as , and the U L U other considers ah 2 as ; as or ar ; ar . The values of the cost parameters, whose dimensions are all $/unit, for both cases are: b ¼ 2; l ¼ 500; k ¼ 0:5; h1 ¼ 0:5 Example 1. m ¼ 0.4, cr ¼ 6; cs ¼ 5. Based on these given parameters, we can easily derive values of a set of parameters when h1 ¼ 0.5 as follows: the L U as ; as ¼ ð0:6023; 0:6743Þ; aLr ; aU a ¼ 0:4; r ¼ ð0:5302; 0:6023Þ; b0 ¼ 21; 715; b1 ¼ 4; 716; Pr ðh1 ; a ; b1 Þ ¼ $5; 842; and Ps ðh1 ; a ; b1 Þ ¼ $1; 037: Furthermore, we have found that ah ¼ 0:6908; which exceeds the larger upper limit 0.6743, implying that the two members’ profits will never be the same within the sharing-fraction range. We then have calculated the two parties’ new profits, Pr ðh1 ; a; b0 Þ and Ps ðh1 ; a; b0 Þ, against the range of a (from 0.5302 to 0.6743), and display the results in Fig. 1. The profit values in the case of decentralized planning are also shown for benchmarking purpose. Note that the amount of capital that can be shared or bargained, DPs , for the dominant supplier and DPr for the dominant retailer are both about $1,954, which are also labeled in Fig. 1. 12000

10000

Pr (h1,a, b1*) 8000

P

Pr (h1,a *, b1* )

6000

DPs =

4000

Ps (h1,a,b*0 )

$1,954 DPr = $1,954

2000

Ps (h1,a *, b1* )

0 0.53 L

ar

0.56

0.6 U

0.64 L

ar, as a

Fig. 1 Example 1 – profits of the two parties: m ¼ 0.4; k ¼ 0.5; cr ¼ 6; cs ¼ 5

0.67 U

as

Bargaining in a Two-Stage Supply Chain Through Revenue-Sharing Contract

361

Fig. 2 The gap between the maximum amount of income for h1 < h < h2. (m ¼ 0.4, cr ¼ 6; cs ¼ 5)

With h1 fixed, we vary the range of h by increasing d from 0 to 1 at an increment of 0.2, and then calculate the maximum amount of revenue in dollars that could be bargained between the two parties. The results are shown in Fig. 2, which implies that as the upper limit of the supplier’s cost increases, the gap between the maximum amounts of income increases. Furthermore, the values of the gaps between supplier-dominant and retailer-dominant situations are identical; in particular, the gap is about $289,000 when h2 increases to twice as much as of h1 (i.e., d ¼ 1). 25; cs ¼ 5. Similarly, we have: a ¼ 0:45; aLs ; aU Example 2. m ¼ 0.45, s ¼ Lcr ¼ ð0:6463; 0:7138Þ; ar ; aU b0 ¼ 66; 684; b1 ¼ 15; 613; r ¼ ð0:5788; 0:6463Þ; Pr ðh1 ; a ; b1 Þ ¼ $8; 704; and Ps ðh1 ; a ; b1 Þ ¼ $7; 042: Unlike the previous example, ah ¼ 0:6575 is within aLs ; aU s ¼ ð0:6463; 0:71383Þ: The results are depicted in Fig. 3. Clearly, the amount of monetary capital that can be shared or bargained for the dominant supplier, DPs ¼ $4; 174; is smaller than for the dominant retailer, DPr ¼ $5; 003: Figure 4 shows the gap between the maximum amounts of income for this example. Unlike the previous case, this example sees a dramatic difference between the value of the gaps for supplier-dominant bargaining and retailer-dominant bargaining (About $764,000 when h2 increases to 200% of h1). This implies that the larger the bargain space (DP) is, the faster the gap between the maximum amounts of income ðGDP ðdÞÞ increases with the range of the inventory holding cost information ðdÞ. As such, the supplier would want to provide his cost information as specifically and exactly as possible in order to receive more sharing from the retailer.

362

J. Hou and A.Z. Zeng

Fig. 3 Example 2 – the profits of the two parties: m ¼ 0.45; k ¼ 0.5; cr ¼ 25; cs ¼ 5

Fig. 4 The gap between the maximum amount of income for h1 < h < h2 (m ¼ 0.45, cr ¼ 25; cs ¼ 5)

5 Concluding Remarks This paper studies the bargaining problem in a supplier–retailer supply chain based on revenue sharing. We assume that the retailer’s unit revenue is sensitive to the lead time, which is affected by the supplier’s target inventory level. A revenue-sharing

Bargaining in a Two-Stage Supply Chain Through Revenue-Sharing Contract

363

based coordination mechanism is constructed to select the supplier’s target inventory level and the revenue-sharing fraction to maximize the entire supply chain profit and to achieve a “win–win” condition. We obtain the range of the monetary amount that can be shared between the two parties under the situation where either the supplier or the retailer is dominant in the bargaining process. The impact of the supplier’s cost structure is also examined. This research leads to some subtle and important implications that can guide practices. First of all, the range of the revenue-sharing fraction that can be accepted by both parties remains the same in supplier-dominant and retailer-dominant situations. Secondly, in addition to the retailer’s unit cost (cr) and the supplier’s unit production cost (cs), the monetary amount that can be shared between the two parties depends heavily on the input parameter, m, the exponent associated with the supplier’s inventory level (b), which affects the supplier lead time, which in turn affects the profits of the two parties. Thirdly, in a retailer-dominant situation, it is easy for the retailer to obtain higher profit than the supplier; on the other hand, if the supplier is dominant, then it is hard for the supplier to gain such benefit. Finally, the format of the supplier’s cost structure (a single value versus an interval) does not affect the range of the revenue-sharing fraction that the retailer would use to coordinate the supplier. Furthermore, the larger the bargain space is, the faster the gap between the maximum amounts of capital increases with the range of the cost information. Several directions for future research can stem from our paper. First, given the optimal range of revenue-sharing fraction, how to distribute the additional profit among the supply chain entities would be an interesting issue. Second, the situation where the retailer chooses not to reveal his/her actual revenue to the counterpart is a problem that the supplier may encounter in the bargaining process. Finally, it will be worthwhile to investigate the supplier’s inventory decision if the supplier has a capacity constraint, as well as to consider the average lead time as a general function of the supplier’s inventory level. Acknowledgment This work is partially supported by the Scientific Research Foundation of Graduate School of Southeast University (YBJJ1003). We are also thankful for the helpful suggestions provided by the anonymous referees.

Appendix 1: The List of Symbols Used in the Paper l cs h cr L p

Demand rate (unit) Unit production cost of the supplier ($/unit) Average unit inventory holding cost of the supplier ($/unit) Unit cost of the retailer ($/unit) Lead time Final product’s unit profit of the retailer ($/unit)

364

p0 b b0 b1* P0rþs Pr Ps a* a aL aU ah ½ h 1 ; h2 DPs DPr GsDP

GrDP

J. Hou and A.Z. Zeng

The largest possible unit profit of the retailer ($/unit) Target inventory level of the supplier (unit) Supplier’s optimal target inventory level in a centralized supply chain (unit) Supplier’s optimal target inventory level in a decentralized supply chain (unit) Total expected profit of the supply chain Retailer’s profit function Supplier’s profit function Retailer’s optimal revenue sharing fraction in a decentralized supply chain New revenue-sharing fraction to attract the supplier to hold a larger inventory level (b0 ) and to achieve higher profits for both parties, where aL < a < a U Lower bound of the revenue-sharing fraction Upper bound of the revenue-sharing fraction Revenue-sharing fraction at which the two parties’ new profits are identical Range of the supplier’s inventory holding cost, where h2 ¼ ð1 þ dÞh1 and d > 0 Range of the monetary amount that can be shared between the two parties in a supplier-dominant supply chain Range of the monetary amount that can be shared between the two parties in a retailer-dominant supply chain Gap between the maximum amount of income that the retailer would share with the supplier in a supplier-dominant supply chain with implicit information Gap between the maximum amount of income that the retailer would share with the supplier in a retailer-dominant supply chain with implicit information

Appendix 2: This Appendix Contains the Proof for Proposition 4.1 and 4.3 If the retailer provides a higher share of revenue, a > a , to induce the supplier to hold an inventory level close to b0 , the value of the new revenue sharing fraction should satisfy the following requirements: Pr h; a; b0 > Pr h; a ; b1 ;

(31)

Bargaining in a Two-Stage Supply Chain Through Revenue-Sharing Contract

Ps h; a; b0 > Ps h; a ; b1 :

365

(32)

These two requirements ensure that the profits of both the two parties are increased so they would be willing to cooperate. If the dominant supplier gains more increase in profits, then Ps h; a; b0 Ps h; a ; b1 > Pr h; a; b0 Pr h; a ; b1

(33)

Else if the retailer is the leader, then Ps h; a; b0 Ps h; a ; b1 < Pr h; a; b0 Pr h; a ; b1

(34)

We first consider the case when the supplier is the leader. Since Pr ðh; a; b0 Þ

h ¼ ð1 aÞlbk lbmk

m m1

lcr ;

(35)

and Ps ðh; a; b0 Þ ¼ albk

h lbmk

m m1

1 m1 h h lcs lbmk

(36)

We see that the requirement in (31) implies that the fraction for revenue sharing, a, should take on the following range: 1 m a < 1 þ m1m m1m :

(37)

Similarly, we can derive another range of a based on the requirement in (32) as follows: 1 2m a > m þ m1m m1m :

(38)

And the range of a based on the requirement in (33) should be: 2m m 1 a > 0:5 m m1m m1m þ 2m1m þ 1

(39)

2m m 1 1 2m g1 ðmÞ ¼ 0:5 m m1m m1m þ 2m1m þ 1 m þ m1m m1m 1 m 2m m 1 and g2 ðmÞ ¼ 1 þ m1m m1m 0:5 m m1m m1m þ 2m1m þ 1 , it is easy m 2m to find that gðmÞ ¼ g1 ðmÞ ¼ g2 ðmÞ ¼ 0:5 1 m m1m þ m1m . Given the range Suppose

of m, 0 < m < 1, by plotting the value of g (m) to m (as seen in Fig. 5), we could

366

J. Hou and A.Z. Zeng m 0.01 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95

0.08 0.07 0.06 0.05

g(m)

0.04 0.03 0.02 0.01 0

m = 0.25 0

0.2

0.4

0.6

0.8

1

m

g (m) 0.0178 0.0490 0.0667 0.0753 0.0790 0.0797 0.0784 0.0757 0.0719 0.0675 0.0625 0.0571 0.0513 0.0453 0.0391 0.0327 0.0263 0.0198 0.0132 0.0066

Fig. 5 The plot of g(m) against m

prove that g1 ðmÞ ¼ g2 ðmÞ>0 holds for all values of m within (0, 1), and the function reaches the maximum when m ¼ 0.25. Thus, in this supplier-dominant supply chain, to entice the supplier to hold a larger inventory level b0 ; the retailer’s new share of revenue, a, which meets the requirements in (31), (32) and (33), should satisfy the following range: 2m m 1 1 m 0:5 m m1m m1m þ 2m1m þ 1 < a < 1 þ m1m m1m ;

8 0 < m < 1 (40)

Moreover, the range of a just equals g(m), which reaches the maximum (roughly about 0.0797) when m ¼ 0.25. If the retailer is the leader, since the proof is similar to that above, it is not repeated here. Hence, the proof for the proposition is complete.

Appendix 3: This Appendix Contains the Numerical Proof for Observation 4.1 The bargain space in terms of monetary value is the amount of capital shifted from the retailer to the supplier (as the supplier is the leader in the game, and thus has higher profit level). Hence, the bargain space, ð0; DPÞ, can be determined by the change of profit for the retailer as his share of revenue increases from aLs to aU s .

Bargaining in a Two-Stage Supply Chain Through Revenue-Sharing Contract

367

From Pr h; ah ; b0 ¼ Ps h; ah ; b0 , the fraction ah at which the two parties’ profits become identical can be derived as: ah ¼

1 þ m lmðcs cr Þ : þ 2 2hb0

(41)

We need to make sure that aL < ah < aU ; that is, to satisfy the following requirement, 1 þ m lmðc c Þ 1 2m m 1 1 m s r < 1 þ m1m m1m : m m1m m1m þ 2m1m þ 1 < þ 2 2 2hb0 (42) After some algebra, (42) indicates that the following relationship is required: f1 ðmÞ ¼

lmðcs cr Þ 1 2m m 1 m1m m1m þ 2m1m > 0 2hb0 2

f2 ðmÞ ¼

1 m lmðcs cr Þ 1 m >0 þ m1m m1m 2 2 2hb0

(43)

Since (43) does not offer a closed-form format of m, and 0 < m < 1, we rely on a numerical analysis again to see when the requirement of (43) can be met. The functions, f1(m) and f2(m), are plotted against m and the result are shown in Figs. 6–13 according to three different situations, with the basic parameters setting as l ¼ 500, k ¼ 0.5, b ¼ 2; h ¼ 1.

0.5

0.4

f1(m)

0.3

0.2

0.1

0

Fig. 6 The plot of f1(m) against m for cr ¼ cs

0

0.2

0.4

0.6

m

0.8

1

368

J. Hou and A.Z. Zeng 0.2 0.1

f2(m)

0 -0.1 -0.2 -0.3 -0.4 -0.5

0

0.2

0.4

0.6

0.8

1

m Fig. 7 The plot of f2(m) against m for cr ¼ cs

0.5

0

f1(m)

-0.5

-1

-1.5

-2

0

0.2

0.4

0.6

0.8

1

m Fig. 8 The plot of f1(m) against m for cr > cs (m* ¼ 0.36)

Case 1: cs ¼ cr As shown in Fig. 6, for all m within (0, 1), f1 ðmÞ > 0 holds; while it is seen clearly that when m > 0:5; the value of the function f2 ðmÞ is positive. Therefore, if the input parameter, m, is greater than 0.5 (but less than 1), it is possible for the two parties’ new profits to be identical when the retailer increases his share of revenue from aL to ah , but not yet to aU . Based on the above results, we see that there are two scenarios when the supplier and retailer are coordinating to improve their respective profit from that resulted in the decentralized planning situation (1) if 0 < m < 0.5, then the retailer’s profit is

Bargaining in a Two-Stage Supply Chain Through Revenue-Sharing Contract

369

2.5

2

f2(m)

1.5

1

0.5

0

0

0.2

0.4

0.6

0.8

1

m Fig. 9 The plot of f2(m) against m for cr > cs ðl ¼ 500; k ¼ 0:5; b ¼ 2; h ¼ 1; cs ¼ 5; cr ¼ 10Þ

0.1 0.05 0

f1(m)

-0.05 -0.1 -0.15 -0.2 -0.25 -0.3

0

0.2

0.4

0.6

0.8

1

m Fig. 10 The plot of f1(m) against m for cr > cs (m* ¼ 0.46)

always higher than the supplier’s at aL a aU ; ðiiÞ if 0:5 m < 1: then the two parties reach the same profit level when the revenue-sharing fraction, a ¼ ah ðaL < ah < aU Þ. We now calculate the bargain space for each scenario. In the first scenario where 0 < m < 0.5, the bargain space, DPs1, is the differ ence between the retailer’s profits at the following two points: aL ; b0 and aU ; b0 . Referring to (8) and (11), we can calculate the difference as follows:

370

J. Hou and A.Z. Zeng 0.35 0.3 0.25

f2(m)

0.2 0.15 0.1 0.05 0 -0.05

0

0.2

0.4

0.6

0.8

1

m Fig. 11 The plot of f2(m) against m for cr > cs ðl ¼ 500; k ¼ 0:5; b ¼ 2; h ¼ 1; cs ¼ 5; cr ¼ 6:5Þ

2.5

2

f1(m)

1.5

1

0.5

0

0

0.2

0.4

0.6

0.8

1

m Fig. 12 The plot of f1(m) against m for cr < cs (m* ¼ 0.52)

DPs1 ¼ Pr h; aL ; b0 Pr h; aU ; b0 m m m1 m1 (44) h h ¼ ð1 aL Þlbk ð1 aU Þlbk : lbmk lbmk m m1 h , and substituting aL in (11) to (44), we can be Denoting Ph ¼ lbk lbmk further simplify (44) to the following format:

Bargaining in a Two-Stage Supply Chain Through Revenue-Sharing Contract

371

0.5 0

f2(m)

-0.5 -1 -1.5 -2 -2.5

0

0.2

0.4

0.6

0.8

1

m Fig. 13 The plot of f2(m) against m for cr < cs ðl ¼ 500; k ¼ 0:5; b ¼ 2; h ¼ 1; cs ¼ 9; cr ¼ 5Þ

m DPs1 ¼ 0:5Ph ð1 mÞ 1 ð1 þ mÞm1m :

(45)

In the second scenario where 0:5 m < 1: the two parties’ profits approach to identical before the retailer’s revenue-sharing fraction reaches the upper limit. Since it is unfavorable for the supplier’s profit to be lower than the retailer’s, the monetary bargain space, DPs2 can be computed as DPs2 ¼ Pr h; ah ; b0 Pr h; aU ; b0 m m m1 m1 h h ð1 aU Þlbk ¼ ð1 ah Þlbk lbmk lbmk m

1

¼ Ph ð0:5 0:5m þ m1m m1m Þ 0:5lðcs cr Þ:

(46)

Case 2: cr > cs There is a specific m*, when m > m*, both f1 ðmÞ and f2 ðmÞ are positive. Then the monetary bargain space, DP s2 , is the difference between the retailer’s profits at the following two points: aL ; b0 and ah ; b0 , which can be computed as 1

m

DPs2 ¼ Ph ð0:5 0:5m þ m1m m1m Þ 0:5lðcs cr Þ

(47)

A. When cr – cs > d, where d is determined by other parameters of l; b; k; h, such m* is increasing with the distance between cr and cs, but not necessary larger or smaller than 0.5. For instance, if l ¼ 500, k ¼ 0:5, b ¼ 2, cs ¼ 5, h ¼ 1, then d is about 2, the values of m* are:

372

J. Hou and A.Z. Zeng

cr m*

6 0.11

7 0.25

10 0.36

15 0.42

20 0.45

25 0.47

And in this case when m < m*, the value of the function positive but f1 ðmÞ is f2 ðmÞis 2m m 1 lmðcs cr Þ 1 1m m1m þ2m1m þ1 < negative; i.e., the relationship of 1þm þ < mm 2hb 2 2 0

m

1

1þm1m m1m holds. That means the retailer’s profit is always smaller than supplier’s. Therefore, the bargain space L is the difference U between the retailer’s a ;b a ;b0 : DPs1 ¼0:5Ph ð1mÞ and profits at the following two points: 0 m 1ð1þmÞm1m . B. When cr – cs < d, m* is near 0.5 as the distance between cr and cs is really small. Using the same basic parameters above, m* is 0.47 for cr ¼ 6, and 0.46 for cr ¼ 6.5. But when m < m*, f2 ðmÞ f1 ðmÞ is negative for some values of m, and can be also positive for other values If nega within this range. m tive, then the bargain space is DPs1 ¼ Ph 12 ð1 mÞ 1 ð1 þ mÞm1m ; otherwise, 1 m DPs2 ¼ Ph ð0:5 0:5m þ m1m m1m Þ 0:5lðcs cr Þ. * Case 3: cr < cs . There is a specific m 0.5; its value is increasing slowly with the value of cs – cr. For instance, l ¼ 500, k ¼ 0:5, b ¼ 2, h ¼ 1, cr ¼ 5 the values of m* are: cs m*

6 0.5

7 0.51

8 0.51

9 0.52

10 0.52

11 0.53

13 0.53

15 0.54

20 0.54

When m < m*, the value of the function f2 ðmÞ is negative, while f1 ðmÞ 1 2m m 1 is positive. That is the relationship of m m1m m1m þ 2m2m þ 1 < 1 þ 2 1 m l mðCs Cr Þ þ holds. The retailer’s profit is always larger than m1m m1m < 1þm 2 2hb 0

supplier’s. Similar to m < 0.5, the bargain space is: case 1 when m DPs1 ¼ Ph 12 ð1 mÞ 1 ð1 þ mÞm1m . When m > m*, both f2 ðmÞ and f1 ðmÞ

lmðcs cr Þ are positive, then at ah ¼ 1þm , the two parties reach the same profit 2hb0 2 þ level. The monetary bargain space, DP2 , is the difference between the retailer’s profits at the following two points: aL ; b0 and ah ; b0 , which can be computed as 1 m DPs2 ¼ Ph ð0:5 0:5m þ m1m m1m Þ 0:5lðcs cr Þ. The proof for this observation is then complete.

Appendix 4: This Appendix Shows the Proof for Proposition 4.2 and 4.4 Since the proof for deriving the range of the revenue-sharing fractions is similar to that in Hou et al. (2009) and Appendix 1, it is not repeated here. Rather, we examine the monetary value that the retailer can share with the supplier. The results are summarized in Table 2 for Proposition 4.2 and Table 3 for Observation 4.4, respectively. It is seen that given a range of h½h1 h h1 ð1 þ dÞ instead of

Bargaining in a Two-Stage Supply Chain Through Revenue-Sharing Contract

373

Table 2 The maximums of shared monetary amount and gaps when the supplier is dominant (2) m < m < 1 (1) 0 < m < m L U h1 DPs11 ¼ Pr h1 ; as ; b01 Pr h1 ; as ; b01 DPs21 ¼ Pr h2 ; ah ; b01 Pr h2 ; aU s ; b01 m m m1 m1 h1 h1 lðcs cr Þ F1 ðmÞ F2 ðmÞ ¼ lbk ¼ lbk lbmk lbmk 2 L U U h2 DPs12 ¼ Pr h2 ; as ; b02 Pr h2 ; as ; b02 DPs22 ¼ Pr h2 ; ah ; b02 Pr h2 ; as ; b02 m m h2 m1 h2 m1 lðcs cr Þ F1 ðmÞ F2 ðmÞ ¼ lbk ¼ lbk 2 lbmk lbmk Gap G½1 ðdÞ ¼ DPs11 DPs12 DP " # m 1m 1 ¼ DPs11 1 1þd 1 m F1 ðmÞ ¼ ð1 mÞ 1 ð1 þ mÞm1m 2

½2

GDP ðdÞ ¼ DPs21 DPs22 " # m 1m 1 ¼ DPs21 1 1þd 1 1 1 m F2 ðmÞ ¼ m þ m1m m1m 2 2

Table 3 The maximums of shared monetary amount and gaps when the retailer is dominant (2) m 2 ðm1 ; m2 Þ and cr >cs (1) m 2 = ðm1 ; m2 Þ and cr > cs , or cr cs L U h1 DPr11 ¼ Pr h1 ; ar ; b01 Pr h1 ; ar ; b01 DPr21 ¼ Pr h1 ; ah ; b01 Pr h1 ; aLr ; b01 m m m1 m1 h1 h1 lðcs cr Þ F3 ðmÞ F4 ðmÞ þ ¼ lbk ¼ lbk 2 lbmk lbmk h2 DPr12 ¼ Pr h2 ; aLr ; b02 Pr h2 ; aU DPr22 ¼ Pr h2 ; ah ; b02 Pr h2 ; aLr ; b02 r ; b02 m m m1 m1 h2 h2 lðcs cr Þ F3 ðmÞ F4 ðmÞ þ ¼ lbk ¼ lbk lbmk lbmk 2

Gap G½1 ðdÞ ¼ DPr11 DPr12 DP " # m 1m 1 ¼ DPr11 1 1þd m F3 ðmÞ ¼ 0:5ð1 mÞ 1 ð1 þ mÞm1m

½2

GDP ðdÞ ¼ DPr21 DPr22 " # m 1m 1 ¼ DPr21 1 1þd 1 2m F4 ðmÞ ¼ 0:5 0:5m m1m þ m1m

one value, the gap between the shared monetary values at the two limits is a function of d. Let f ðdÞ ¼ 1

1 1þd

m 1m

:

(48)

As Hou et al. (2009) has proved, f ðdÞ increases as the range of h widens (i.e., h2 becomes bigger). The proof for Proposition 4.2 and 4.4 is then complete.

374

J. Hou and A.Z. Zeng

Appendix 5: This Appendix Shows the Numerical Proof for Observation 4.2 Similar to Appendix 3, from Pr h; ah ; b0 ¼ Ps h; ah ; b0 , the fraction ah at which the two parties’ profits become identical can be derived as: ah ¼

1 þ m lmðcs cr Þ þ : 2 2hb0

(49)

We need to make sure that

1 þ m lmðc c Þ 1 1 2m 2m m 1 s r < þ m m1m m1m þ 2m1m þ 1 : m þ m1m m1m < 2 2hb0 2 (50) After some algebra, (50) indicates that the following relationship is required: lmðcs cr Þ 1 1 1 2m 1m 1m >0 mþm m f1 ðmÞ ¼ 2hb0 2 2 lmðc c Þ 1 2m m 1 s r f2 ðmÞ ¼ >0 (51) m1m m1m þ 2m1m 2 2hb0

Since (51) does not offer a closed-form format of m, and 0 < m < 1, we rely on a numerical analysis again to see when the requirement of (51) can be met. The functions, f1(m) and f2(m), are plotted against m and the result are shown in Figs. 14–19 according to three different situations, with the basic parameters setting as l ¼ 500; k ¼ 0.5, b ¼ 2; h ¼ 1. 0.5

0.4

f1(m)

0.3

0.2

0.1

0

0

0.2

0.4

0.6

m Fig. 14 The plot of f1(m) against m for cr ¼ cs

0.8

1

Bargaining in a Two-Stage Supply Chain Through Revenue-Sharing Contract

375

0

-0.1

f2(m)

-0.2

-0.3

-0.4

-0.5

0

0.2

0.4

0.6

0.8

1

0.6

0.8

1

m Fig. 15 The plot of f2(m) against m for cr ¼ cs 0.5

0

f1(m)

-0.5

-1

-1.5

-2

0

0.2

0.4

m Fig. 16 The plot of f1(m) against m for cr > cs (m* ¼ 0.36)

Case 1: cs ¼ cr As shown in Fig. 14, for all m within (0, 1), f1 ðmÞ > 0 and f2 ðmÞ < 0 1 2m 2m m 1 holds; therefore, m þ m1m m1m Þ < 12 m m1m m1m þ 2m1m þ 1Þ < 1þm 2 þ lmðcs cr Þ L the retailer’s profit is always higher than the supplier’s at a a aU ; 2hb 0

and the bargain spaceDP1 , is the difference between the retailer’s profits at the following two points: aL ; b0 and aU ; b0 . DPr1 ¼ Pr h; aL ; b0 Pr h; aU ; b0 m m m1 m1 h h L U ð1 a Þlbk : (52) ¼ ð1 a Þlbk lbmk lbmk

376

J. Hou and A.Z. Zeng

Fig. 17 The plot of f2(m) against m for cr > cs. ðl ¼ 500; k ¼ 0:5; b ¼ 2; h ¼ 1; cs ¼ 5; cr ¼ 10Þ

2

1.5

f2(m)

1

0.5

0

-0.5

0

0.2

0.4

0.6

0.8

1

m

Fig. 18 The plot of f1(m) against m for cr < cs

2.5

2

f1(m)

1.5

1

0.5

0

0

0.2

0.4

0.6

0.8

1

m m m1 h Denoting Ph ¼ lbk lbmk , and substituting aL in (11) to (52), we can be further simplify (52) to the following format:

1 m DPr1 ¼ Ph ð1 mÞ 1 ð1 þ mÞm1m : 2

(53)

Case 2: cr > cs There is a specific range of (m1*, m2*), only when m is within this range, both f1 ðmÞ and f2 ðmÞ are positive, and the monetary bargain space DP1 is

Bargaining in a Two-Stage Supply Chain Through Revenue-Sharing Contract Fig. 19 The plot of f2(m) against m for cr < cs ðl ¼ 500; k ¼ 0:5; b ¼ 2; h ¼ 1; cs ¼ 9; cr ¼ 5Þ

377

0

-0.5

f2(m)

-1

-1.5

-2

-2.5

0

0.2

0.4

0.6

0.8

1

m

DPr2 ¼ Pr h; ah ; b0 Pr h; aL ; b0 m m m1 m1 h h L ¼ ð1 a Þlbk ð1 ah Þlbk lbmk lbmk 1 1 lðcs cr Þ 1 2m ¼ Ph m m1m þ m1m þ 2 2 2

(54)

Both values of m1* and m2* are increasing with the distance between cr and cs. And the interval between the two values is determined by other parameters of l; b; k; h. For instance, if l ¼ 500, k ¼ 0:5, b ¼ 2, cs ¼ 5, h ¼ 1, the range of (m1*, m2*) are: cr m*

6 (0.03, 0.11)

7 (0.21, 0.25)

10 (0.32, 0.36)

15 (0.38, 0.42)

20 (0.41, 0.45)

25 (0.43, 0.47)

But when m is outside this small range, f2 ðmÞm f1 ðmÞ is negative, and the bargain space is DPr1 ¼ 0:5Ph ð1 mÞ 1 ð1 þ mÞm1m . Case 3: cr < cs As seen from (51), the value of f1 ðmÞ increases with (cs – cr) and f2 ðmÞ decreases with (cs – cr), with other parameters unchanged. From Case 1 we know that, when cs ¼ cr, for all m within (0, 1), f1 ðmÞ > 0 and f2 ðmÞ < 0 holds; therefore, when cr < cs , we also have f1 ðmÞ > 0 and f2 ðmÞ < 0 for all m within (0, 1) (as shown in Figs. 18 and 19). Therefore, similarto case 1, the monetary bargain m space can be computed as DPr1 ¼ 0:5Ph ð1 mÞ 1 ð1 þ mÞm1m . The proof for this observation is then complete.

378

J. Hou and A.Z. Zeng

References Bellantuono N, Giannoccaro I, Pontrandolfo P, Tang CS (2009) The implications of joint adoption of revenue sharing and advance booking discount programs. Int J Prod Econ 121(2):383–394 Cachon G, Lariviere MA (2000) Supply chain coordination with revenue sharing contracts: strengths and limitations. Manage Sci 51(1):30–44 Chauhan SS, Proth JM (2005) Analysis of a supply chain partnership with revenue sharing. Int J Prod Econ 97(1):44–51 Chen J, Chen JF (2006) Study on revenue sharing contract in virtual enterprises. J Syst Sci Syst Eng 15(1):95–113 Chen K, Gao C, Wang Y (2007) Revenue-sharing contract to coordinate independent participants within the supply chain. J Syst Eng Electron 18(3):520–526 Giannoccaro I, Pontrandolfo P (2004) Supply chain coordination by revenue sharing contracts. Int J Prod Econ 89(2):131–139 Giannoccaro I, Pontrandolfo P (2009) Negotiation of the revenue sharing contract: an agent-based systems approach. Int J Prod Econ 122(2):558–566 Guardiola LA, Meca A, Timmer J (2007) Cooperation and profit allocation in distribution chains. Decis Support Syst 44(1):17–27 Gupta D, Weerawat W (2006) Supplier–manufacturer coordination in capacitated two-stage supply chains. Eur J Oper Res 175(1):67–89 He L, Walrand J (2005) Pricing and revenue sharing strategies for internet service providers. Paper presented in Proceedings of IEEE international conference on computer communications (INFOCOM), Miami, FL Hou J, Zeng AZ, Zhao L (2009) Achieving better coordination through revenue sharing and bargaining in a two-stage supply chain. Comput Ind Eng 57(1):383–394 Jaber MY, Osman IH (2006) Coordinating a two-level supply chain with delay in payments and profit sharing. Comput Ind Eng 50(4):385–400 Jia NX, Yokoyama R (2003) Profit allocation of independent power producers based on cooperative Game theory. Int J Electr Power Energy Syst 25(8):633–641 Li S, Hua Z (2008) A note on channel performance under consignment contract with revenue sharing. Eur J Oper Res 184(2):793–796 Li S, Hua Z, Huang L (2009) Supply chain coordination and decision making under consignment contract with revenue sharing. Int J Prod Econ 120(1):88–99 Lin Z, Cai C, Xu B (2010) Supply chain coordination with insurance contract. Eur J Oper Res 205 (2):339–345 Linh CT, Hong Y (2009) Channel coordination through a revenue sharing contract in a two-period newsboy problem. Eur J Oper Res 198(3):822–829 Nagarajan M, Sosˇic´ G (2008) Game-theoretic analysis of cooperation among supply chain agents: Review and extensions. Eur J Oper Res 187(3):719–745 Pan K, Lai KK, Leung SCH, Xiao D (2010) Revenue-sharing versus wholesale price mechanisms under different channel power structures. Eur J Oper Res 203(2):532–538 Ouardighi FE, Kim B (2010) Supply quality management with wholesale price and revenuesharing contracts under horizontal competition. Eur J Oper Res 206(2):329–340 Rhee B, Veen JAA, Venugopal V, Nalla VR (2010) A new revenue sharing mechanism for coordinating multi-stage supply chains. Oper Res Lett 38(4):296–301 Sucky E (2006) A bargaining model with implicit information for a single supplier–single buyer problem. Eur J Oper Res 171(2):516–535 Yao Z, Stephen CH, Leung KKL (2008) Manufacturer’s revenue-sharing contract and retail competition. Eur J Oper Res 186(2):637–651 Zhang A, Fu X, Yang HG (2010) Revenue sharing with multiple airlines and airports. Transp Res B Methodol 8–9(2):944–959

Should a Stackelberg-Dominated Supply-Chain Player Help Her Dominant Opponent to Obtain Better System-Parameter Knowledge? Jian-Cai Wang, Amy Hing Ling Lau, and Hon-Shiang Lau*

Abstract A manufacturer (Manu) supplies a product to a retailer (Reta). The uncertain knowledge of the dominant player (which may be either Manu or Reta) about a system parameter is represented by a subjective probability distribution. At the time when the dominant player is designing the supply or purchase contract, should the dominated player help the dominant player to improve his imperfect system-parameter knowledge? Can the dominant player induce the dominated player to share her superior knowledge by using (or by threatening to use) sophisticated “channelcoordinating” contract formats? It is likely that one would surmise from the literature that the answer to both questions is “yes”. However, this chapter shows that very often the correct answer is “no”. Specifically, for the basic cost and market parameters, we show that the dominated player is (1) always motivated to mislead the dominant player to have a biased mean value for his subjective distribution; and (2) motivated, over a wide range of likely conditions, to increase the variance of the dominant player’s subjective distribution. Moreover, the dominant player cannot narrow this range of confusion-encouraging conditions by using a more sophisticated contract format such as a “menu of contracts.” Our results highlight the need to develop arrangements that can actually motivate a dominated player to share knowledge honestly.

*

Authors contributed equally; names arranged in reverse alphabetical order

J.-C. Wang School of Business, University of Hong Kong, Pokfulam, Hong Kong and School of Management and Economics, Beijing Institute of Technology, Beijing, China e-mail: [email protected] A.H.L. Lau School of Business, University of Hong Kong, Pokfulam, Hong Kong e-mail: [email protected] H.-S. Lau* (*) Department of Management Sciences, City University of Hong Kong, Kowloon Tong, Hong Kong e-mail: [email protected] T.-M. Choi and T.C. Edwin Cheng (eds.), Supply Chain Coordination under Uncertainty, International Handbooks on Information Systems, DOI 10.1007/978-3-642-19257-9_15, # Springer-Verlag Berlin Heidelberg 2011

379

380

J.-C. Wang et al.

Keywords Supply chain contract design • Information sharing

1 Introduction 1.1

Problem Statement

In considering human/organizational interactions, two common notions appear to be intuitively plausible at first glance: 1. It is often “beneficial” to share knowledge and strive for a “bigger pie” for all. 2. There is often some way a player can benefit himself by hiding/distorting the information he is supposed to provide to the other player. Unfortunately, these two notions suggest opposite actions; i.e., sharing knowledge honestly versus hiding/distorting information. Much of the supply chain literature is motivated by the first notion. This chapter emphasizes the validity of the second notion, contrary to what one might surmise from the large supply chain literature on information sharing and channel coordination. Specifically, we consider a supply chain with an upstream “manufacturer” (a male called “Manu”) and a downstream “retailer” (a female called “Reta”). We will consider separately the situations where the dominant player is (1) Manu; and (2) Reta. To facilitate explanation, the first part of this chapter will concentrate on the case in which the dominant player is Manu. Manu will then specify the supply contract. Manu is uncertain about one of the system parameters (say, x), and perceive it as a random variable x~ with subjective probability distribution Fx(•). Reta knows x perfectly, and recognizes that Manu’s x~-knowledge will influence how Manu will specify the supply contract to be offered to Reta. Our question is: from Reta’s perspective, what are the ideal characteristics (or “quality”) of Manu’s x~-perception that would lead Manu to specify a contract that is most advantageous to Reta? The spirit of the current supply chain “movement” suggests that Reta should help to improve the quality of Manu’s x~-perception. In contrast, this chapter summarizes our research results (Wang et al. 2008, 2009) showing that, in most situations, the opposite is true – regardless of what contract format Manu would implement.

1.2 q p C(q)

Summary of Basic Symbols and Relationships The quantity supplied by Manu to Reta and sold by Reta to the retail market The unit retail price set by Reta A supply contract designed and offered by Manu to Reta, requiring Reta to pay Manu $C(q) if Reta wants Manu to supply her q units

Should a Stackelberg-Dominated Supply-Chain Player

m, c x~ PM , P R, P I PC PM, PR PMsub, PRsub

381

The unit variable cost of Manu and Reta, respectively and k (m þ c) A generic random variable with support [xmin, xmax], standard deviation sx, and coefficient of variation kx. x~’s mean is denoted by either mx or x (i.e., bold letter) The profit of, respectively, Manu, Reta, and the “integrated firm” Total channel profit, equals (PM + PR) Expected profit of, respectively, Manu and Reta The subsistence profit of, respectively, Manu and Reta

Reta incurs a unit retail-processing cost c and gets to set the unit retail price p and the purchase quantity q. Given a supply contract C(q) specified by Manu and the (p, q)-decision set by Reta, Manu’s and Reta’s profits are: For Manu : PM ¼ CðqÞ mq; For Reta : PR ¼ ðp cÞq CðqÞ:

(1)

Both players know that for any p Reta sets, the market demand is given by the linear demand curve: q ¼ a bp;

(2)

where parameters (a,b) reflect the basic market-demand characteristics. In this model, Reta’s decision variables are (p,q), and Manu’s decisions are (1) C(q)’s format; and (2) the numerical values of C(q)’s parameters. Besides m, Manu’s “environmental variables” (or “system parameters”) are {a, b, c}. Manu’s knowledge of one of these is imperfect, and perceives it as a random variable (i.e., a~, b~ or c~), with cumulative distribution function (cdf) Fa(•), Fb(•) or Fc(•), respectively.

2 Review and Overview Figure 1 depicts the decision/action sequence of our scenario. At time point A, the dominated player (“Reta” in the current scenario) anticipates that the systemparameter information s/he provides may influence how the dominant player (“Manu”) will specify the supply contract. At time point B the dominant player specifies the supply contract.

2.1

Positioning in the Literature

There exists now a huge literature on supply chain coordination and cooperation; see, e.g., Cachon (2003) and Chen (2003) for excellent reviews. Among others, the following two notions are likely to be learnt from this literature:

382

J.-C. Wang et al.

Reta contemplates the format of Manu’s Fx(•) that will be most beneficial to Reta.

Manu specifies C(q), using his knowledge Fx(•) on x. ˜

Reta orders q units from Manu.

Reta sets the retail price p and sells her stock.

time axis Point A

Point B

Point C

Fig. 1 Schematic diagram of action sequence showing time points A–C

Notion (A): At time point B depicted in Fig. 1, the dominant player with imperfect knowledge of system parameter (say, “a”) can use a series of increasingly sophisticated contract formats to increasingly improve his (and the supply chain’s) expected profit. See, e.g., Corbett et al. (2004), and Liu and C¸etinkaya (2009) about supply chain contract design under “stochastic” and “asymmetric” knowledge scenarios. Notion (B): At time point C depicted in Fig. 1 (after C(q) has been specified), if a player at one supply-chain echelon has better knowledge of the system’s environmental parameters than the player at the other echelon, the better-informed player can often improve the supply chain’s performance by sharing his/her superior information – i.e., the “bigger pie” notion (see, e.g., Lee et al., 2000, Wu and ¨ zer 2010). Note that in our stripped-down cost model Cheng 2008, and Liu and O defined by (1) and (2), by time point C it is too late for Reta to improve channel profit by sharing her superior information with Manu. Our paper extends the earlier related studies in the following aspects: 1. While the overwhelming majority of earlier related studies consider questions ¨ zer raised at time points B or C in Fig. 1 (see, e.g., Ha 2001; Lau and Lau 2005; O and Wei 2006), we consider a question raised at time point A. 2. We use the simplest possible two-echelon structure, summarized by (1) and (2). There is no manufacturing capacity consideration, production lead time, logistics cost, forecasting issue, knowledge transmission cost, etc. Thus, we have removed as much as possible those factors that are most readily identified as motivations for Reta to conceal her superior {a,b,c}-knowledge. The purpose is to minimize any likely confounding effects by factors not directly related to our questions. Among many others, Tirole (1988), Corbett et al. (2004), and Liu and C¸etinkaya (2009) also employ this stylized structure. Regarding Aspect (1) stated above, only a few other supply-chain contractdesign studies have also focused on time point A. For example, Li (2002) examined the following problem: At time point A, are the dominated players willing to sign an information-sharing contract with the dominant player before they learn the true information, noting that once information sharing is agreed, the private information obtained later must be revealed truthfully.

Should a Stackelberg-Dominated Supply-Chain Player

383

That is, although the contract-signing action is at time point A, the possible information acquisition and sharing occurs after time point A. Another example is Taylor and Xiao (2010), who investigated, from the dominant-player’s perspective, which format of Fx(•) would be optimal. In contrast, our study takes the perspective of the dominated player. We now reiterate the difference between our questions and the questions answered in most of the related earlier studies. Consider first the situation where Manu is the dominant player. The earlier studies showed that (1) Manu can use increasingly sophisticated contract formats to give himself increasingly higher expected profits; (2) under a specified supply contract (with specified contract parameter values), often Manu and the channel (and sometimes also Reta) will benefit if Manu becomes better informed about certain system parameters (such as a, b, or c). In contrast, our questions are: 1. Before Manu has finalized a supply contract, is Reta motivated to help Manu to become better informed about certain system parameters? 2. Can Manu motivate Reta to help him become better informed by telling Reta that he will use increasingly more sophisticated supply contract formats? This chapter will show that, under a wide range of plausible situations (hereafter “Situation A”), Reta prefers Manu to be more (rather than less) uncertain about the system’s parameter(s). Moreover, the range of “Situation A” cannot be narrowed significantly by using more sophisticated contract formats. Also, Reta is always motivated to bias Manu’s subjective distributions. For Reta, our results mean that, contrary to what she is likely to conclude from the current supply-chain literature, she should NOT share knowledge honestly but should mislead Manu before Manu finalizes his contract; we also showed what kind of Manu-misperception Reta should aim for. For Manu (and hence the researchers), our results mean that, again contrary to what one might conclude from the literature, the various sophisticated channel coordinating contract formats are unable to induce Reta to share information honestly. Our results therefore also establish the need to find new ways to motivate Reta to share knowledge. To facilitate understanding, the wording in the preceding two paragraphs is for the situation where Manu is the dominant player. For the situation where Reta is the dominant player, simply interchange the terms “Manu” and “Reta”.

2.2

Overview of the Chapter

Sections 3–5 will consider the dominant-Manu case. Section 6 outlines the dominantReta case. The main results are summarized in the concluding Sect. 7.

384

J.-C. Wang et al.

3 The Case of a Dominant Manu: Structure of the Problem 3.1

Supply Contract Formats Considered

We consider four C(q)-formats that Manu may impose (listed below in the order of increasing level of “sophistication”): 1. Price-only contract, label [w]. This is the simplest C(q) format: Manu charges Reta a wholesale price w for each unit she buys from Manu. 2. Franchise Fee contract, label [FF]. Manu requires Reta to pay a specified franchise fee FFM; he then supplies Reta the product at cost (i.e., m/unit). 3. Two-Parts Tariff contract, label [2P]. Manu requires Reta to pay a lump-sum fee L and also charges Reta a wholesale price w for each unit she buys from Manu. 4. Menu of Contracts, label [MC]. Assume for the time being that Manu knows deterministically all parameters except a, and [amin, amax] is the finite support of the subjective probability distribution of Manu’s perceived a˜. The format of Manu’s [MC] is then{[w(adec), L(adec)]|amin adec amax}; i.e., Manu informs Reta that if Reta declares the demand curve’s a-value to be adec, Manu will charge Reta a unit wholesale price w(adec) plus a lump-sum payment L(adec). That is, w(adec) and L(adec) are functions of adec. Among others, Myerson (1979) has shown that, for a given Manu’s initial state of stochastic a-knowledge, Manu can design a w(adec) and a L(adec) such that Reta is forced to reveal the real a-value as adec at time point C of Fig. 1; the resultant [MC] then gives Manu the highest expected profit he can get among all possible contract formats (specified at Point B). Similarly, if Manu knows deterministically all parameters except b or c, the format of Manu’s [MC] will be, respectively, {[w(bdec), L(bdec)]| bmin bdec bmax} or {[w(cdec), L(cdec)]|cmin cdec cmax}. The above four C(q)s are the most popularly considered contract formats. There are of course other formats besides these four. However, as will be justified in Sect. 4.5, considering these four is sufficient to support the conclusions we will be presenting in this chapter.

3.2

Characterizing Manu’s Subjective Distributions

Consider, for example, Manu’s subjective distribution on a˜, with cdf Fa(•), mean ma and standard deviation sa. We consider two aspects of the “quality” of Manu’s a˜-perception: its “bias” (ma a) and its “uncertainty” sa. Manu’s a˜-perception is “perfect” if both bias and uncertainty equal zero. Earlier related studies such as Ha (2001) and Corbett et al. (2004) had to restrict the imperfectly-known parameter’s subjective distribution to be uniform in order to obtain meaningful analytical results. We follow this approach and also obtain analytical results by assuming a uniform Fa(•). We then go one more step and

Should a Stackelberg-Dominated Supply-Chain Player

385

investigate the effects of assuming a more versatile distribution for Fa(•). The gamma distribution is chosen because it can take on a much wider range of coefficients of variations and skewnesses compared to (say) the popular exponential, Erlang or normal (which are special cases of the gamma). For our model, numerical results using a gamma Fa(•) reveal some important behavior unobservable under a uniform-Fa(•) assumption. Of course, the gamma numerical results also confirm the major uniform-based analytical results. In the following sections, numerical results under the gamma assumption will be presented first because they are easier to understand, analytical results for the uniform assumption are then used to provide further support.

3.3

Overview and Preview for the Dominant-Manu Case

Sections 4 and 5 consider, respectively, uncertainties in “a” and “c”. Since fairly similar results are obtained for all three parameters {a, b, c}, we present detailed results only for “a”, while results for “b” are completely omitted. Our results can be briefly summarized as follows: Regardless of what contract format Manu will use, Reta should always try to inflate mb and mc but deflate ma. Regarding the uncertainties, Reta does not want Manu’s (sa, sb, sc) to be too low, but also not too high. To emphasize, Reta’s preferences towards Manu’s m and s are quite different. Regarding the “error” (in m), Reta wants it to be as large as possible (as long as it is in the right direction). Regarding “uncertainty” s, Reta does not want Manu to be either too certain or too uncertain about his estimate. Our conclusions also mean that both Reta and Manu should behave in ways that are quite different from what one might surmise from the current supply chain literature; particularly, they are in stark contrast to the “all or nothing” result in Taylor and Xiao (2010).

3.4

Summary of Basic Benchmark Results

Detailed derivations of the results summarized in this subsection can be found in, e.g., Corbett et al. (2004) and Lau and Lau (2005). Under an “integrated firm” where Manu and Reta are merged into one entity, it is known that the optimal (p,q) decisions and the attainable channel profit are: pI ¼ ða þ bkÞ=ð2bÞ; qI ¼ ða bkÞ=2; and PI ¼ ða bkÞ2 =ð4bÞ; recall k ðm þ cÞ:

(3)

If Manu and Reta are two separate players, each with deterministic knowledge of all the parameters, then the dominant Manu knows that, for any w-value he declares

386

J.-C. Wang et al.

in a [w]-contract, The players’ profits and Reta’s responses (on retail-price and purchase-quantity) are: ðPM Þw ¼ ðw mÞ½a bðw þ cÞ=2; ðPR Þw ¼ ½a bðw þ cÞ2 =ð4bÞ; pw ¼ ða þ bc þ bwÞ=ð2bÞ; qw ¼ ða bc bwÞ=2:

) (4a)

Recognizing the above, Manu maximizes his profit by setting w ¼ ða þ bkÞ=ð2bÞ;

(4b)

leading to the following optimal profits for the players and the channel: PR ¼ ða bkÞ2 =ð16bÞ; PM ¼ ða bkÞ2 =ð8bÞ; PC ¼ PM þ PR ¼ 3ða bkÞ2 =ð8bÞ:

(4c)

Equations (3) and (4) show that PC* < PI*; i.e., [w] does not “coordinate the channel.” In contrast, it is known that, with deterministic parameter knowledge, either [FF] or [2P] enables Manu to not only coordinate the channel (i.e., achieve PC* ¼ PI*), but also acquire absolute power in deciding Reta’s share of ПI* (subject of course to the condition PR PRsub). In the deterministic knowledge context [MC] is irrelevant because it degenerates into [2P]. If Manu does not know all the parameters deterministically, it is known that no contract format enables Manu to achieve the same total channel profit as ПI*. However, in most “stochastic” or “asymmetric” knowledge scenarios, [w], [FF] and [2P] enable Manu to achieve progressively higher expected profit for himself. Ultimately, [MC] is the most powerful contract format for Manu – i.e., an optimized [MC] enables Manu to obtain the largest expected profit for himself. Relative to [w], we will refer to [FF], [2P] and [MC] collectively as “coordination encouraging” contract formats.

4 Dominant Manu is Uncertain About the Market Size a In this section, we will consider in Sect. 4.1 how Reta wants Manu to perceive “a” when both sides know that Manu will offer a price-only ([w]) contract. Then, in Sects. 4.2–4.4 we will consider how Reta wants Manu to perceive “a” when both sides know that Manu will offer, in turn, a franchise fee contract ([FF]), a two-part tariff contract ([2P]) and a menu of contract ([MC]). Under each contract, we first tabulate the numerical results for the situation where Manu’s a priori subjective knowledge of parameter “a” is gamma-distributed; this tabulation enables us to illustrate the main pattern of behavior we are emphasizing in this chapter. This pattern is then confirmed by analytical results we are able to derive for the situation

Should a Stackelberg-Dominated Supply-Chain Player

387

where Manu’s a priori subjective knowledge of parameter “a” is uniformdistributed. Our results consistently show that (1) Reta always prefers Manu to perceive (incorrectly) a deflated ma; moreover, over a wide range of plausible conditions, Reta prefers Manu to be more uncertain about a (i.e., higher sa); and (2) Manu cannot narrow or alter Reta’s range of confusion-preferring conditions by implementing (or threatening to implement) a more sophisticated contract format (e.g., [MC]) instead of a simpler one (e.g., [w]).

4.1

The Price-Only Contract [w]

4.1.1

Problem Statement: Manu’s Knowledge of “a” is Inferior to Reta’s

Manu perceives a˜ with subjective cdf Fa(•). Thus, after setting w, (4a) indicates that ~ M(perM) ¼ (w m)[a˜ b(w þ c)]/2 and Manu will perceive his own profit to be P ~ R(perM) ¼ [a˜ b(w þ c)]2/(4b). Manu will perceive Reta’s profit to be P recognizes that Reta will “play” only if Reta’s profit exceeds PRsub; i.e., Manu ~ R(perM) PRsub; or, equivalently, when a˜ b0 , perceives that Reta will play if P 0 where b ¼ b(w + c) þ √(4bPRsub) is the “cutoff value” (see Ha 2001 for a more ~ M(perM)] under a detailed explanation). Thus, Manu’s problem of maximizing E[P stochastically-perceived a˜ can be written as: ð amax fðw mÞ½a bðw þ cÞ=2g dFa ðaÞ; where b ¼ maxðamin ; b0 Þ: (5) max w

b

Thus, Manu will set the unit wholesale price at w*, where w* is the solution to (5). Then, Reta knows, from her perspective, that if Manu perceives a˜ with cdf F(a), her profit (as perceived by herself) is, PR(perR) ¼ [areal b(w* þ c)]2/(4b). This PR(perR)-expression shows that a higher PR(perR) is brought by a lower w*. Reta’s (and hence our) question is therefore: what kind of a Manu-perceived Fa(•) will lead to a lower w* – hence a higher PR(perR)? 4.1.2

Numerical Results

Table 1 presents the PR(perR)-values for different combinations of c-values and ka-values (or, equivalently, sa-values); recalling that kx x~’s coefficient of variation. Values of other parameters are set at: areal ¼ ma ¼ 5, b ¼ 1, ПRsub ¼ [ma b (c þ m)]2/24, and a˜ is gamma distributed. Without loss of generality, we set m ¼ 1 throughout this chapter. To obtain the PR(perR)-values, first solve (5) numerically for w*, then compute PR(perR) ¼ [areal b(w* þ c)]2/(4b). Table 1 shows that, for any given c-value (i.e., along each c column), PR(perR) decreases as sa increases in the lower (grayed) region where sa is “sufficiently large,” but PR(perR) increases as sa increases in the upper (non-grayed) region where sa is “sufficiently small.” A “Boundary B” separates the grayed and non-grayed regions (or “Situations”).

388

J.-C. Wang et al.

Table 1 ПR(perR) under a price-only contract ([w])

Gamma-distributed a˜, ma ¼ 5, b ¼ 1, and ПRsub ¼ (ma bk)2/(24b) 2.5

ΠR(perR)

2 [w] [FF] [2P] [MC]

1.5 1 0.5 0

2.5

3

3.5

4

4.5

5

5.5

6

6.5

7

7.5 ma

Fig. 2 PR(perR) under gamma-distributed a˜: sa ¼ 0.4, areal ¼ 5, b ¼ 1, c ¼ 1, and PRsub ¼ 0

In other words, over a significant range of plausible conditions, Reta is motivated to increase Manu’s uncertainty in a˜. We have repeated Table 1 computations using a grid system of different combinations of values of the parameters ma, b and ПRsub; their results confirm the pattern illustrated in Table 1 (the same verification approach has been used in all subsequent numerically illustrated patterns to be reported in this chapter). Table 1’s characteristics will be discussed again in greater detail in Sect. 4.1.4 after supporting analytical results are presented below. The “filled diamond” line in Fig. 2 illustrates, for a typical set of (sa, areal, b, c, ПRsub)-parameter values, how PR(perR) increases as ma decreases under a [w] contract. That is, Reta always prefers Manu to perceive (incorrectly) a deflated ma. 4.1.3

Analytical Proofs

The behavior depicted in Table 1 (and other effects) are derived analytically in Appendix 1 of Wang, Lau and Lau (hereafter “WLL”) (2008) for the case of a uniform F(a). The main results are summarized as Lemmas 1A to 1C. Note that one does not need to read these analytical results (and their counterparts in Sects. 4.2.3, 4.3.3 and 4.4.3) in order to follow the basic arguments of this chapter.

Should a Stackelberg-Dominated Supply-Chain Player

389

Corresponding to the slanting “Boundary B” in Table 1, WLL’s (2008) Appendix 1 shows that, for the case of a uniform F(a), there are four “regions” or “Situations,” separated by three boundaries defined as follows. First, the boundary functions saWA, saWB and saWC are derived in (A6), (A9) and (A10) of WLL’s (2008) Appendix A. For example, one boundary function is: pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ saWB ¼ fma bk 5 bPRsub þ

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 pﬃﬃﬃ ½ma bk bPRsub þ 8bPRsub g=ð4 3Þ : (6)

Second, these boundary functions delineate the following four Situations: 1. 2. 3. 4.

Situation Ia: when sa saWA Situation Ib: when saWA < sa saWB Situation II: when saWB < sa saWC Situation III: when sa > saWC

Lemma 1A (Manu’s optimal w-decision). Depending on the “Situation,” Manu’s w* is: Situation Ia (when sa saWA): w* ¼ [ma b(c m)]/(2b). pﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Situation Ib (when saWA < sa saWB): w ¼ ½ma 3sa bc 2 bPRsub =b: Situation II (when saWB < sa saWC):

w ¼

pﬃﬃﬃ 2ðma þ 3sa bcÞ þ bm

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃ 2 ½ma þ 3sa bk þ 12bPRsub =ð3bÞ:

Situation III (when sa > saWC): PR(perR) is less than PRsub, hence no trade occurs. Lemma 1B (Effect of sa on PR(perR)). Since (4) shows that PR(perR) increases as w decreases, one can obtain the following conclusions by simply observing the w*-expressions given in Lemma 1A above: In Situation Ia, w* and hence PR(perR) is constant w.r.t. sa. In Situation Ib, w* decreases and hence PR(perR) increases as sa increases. In Situation II, w* increases and hence PR(perR) decreases as sa increases. Lemma 1C (Effects of ma and PRsub on PR(perR)). Effect of ma: in all Situations, w* increases as ma increases; i.e., PR(perR) decreases as ma increases. Hence Reta is always motivated to mislead Manu into perceiving a smaller ma. Effect of PRsub: in all Situations, w* decreases and hence PR(perR) increases as PRsub increases. Hence Reta is always motivated to convince Manu to recognize an inflated PRsub.

4.1.4

Discussion

Although this chapter is not meant to consider PRsub, the effect of PRsub stated in Lemma 1C is worth noting. On one hand, its Lemma-1C effect appears intuitively reasonable once it is stated; on the other hand, earlier models incorporating PRsub

390

J.-C. Wang et al.

have always assumed that Manu knows and accepts PRsub as it is. Lemma 1C suggests that setting PRsub is an issue that warrants deeper investigations. We return now to our main issue: Manu’s is a˜-knowledge, which we quantify in two aspects: bias (ma a) and uncertainty sa. Reta’s Preference on ma, or a˜ ’s Bias A higher a-value implies a higher “base demand.” Thus, our Lemma-1C result means that Reta will always try to mislead Manu into perceiving as low a ma-value as possible. This is neither counter-intuitive nor intuitively obvious. It is not surprising that a dominated player will want the dominant player (something like a “boss”) to perceive the operating environment as “tougher” than it actually is. Reta’s Preference on sa, or a˜ ’s Level of Uncertainty In contrast to ma’s effects, we show here that sa’s (or, equivalently, ka’s) effects are quite counter-intuitive. Lemma 1B indicates that PR(perR) is a quasi-concave function of sa, attaining its maximum at saWB. This, of course, is the sa (or ka) effect illustrated graphically by Table 1, where the grayed area below Boundary B (the line “══”) corresponds to Situation II, and the white area above Boundary B corresponds to Situation Ib. It can be easily seen from the derivations in Appendix 1 that Situation Ia does not arise if a˜’s probability distribution has a long right-hand tail (as in a gamma distribution), hence Table 1 does not exhibit a Situation-Ia area – in contrast to Lemma-1’s uniform-distribution results. Under Situation Ib, Reta prefers Manu’s sa to be higher; i.e., instead of sharing her a-information, she is actually motivated to confuse Manu and muddy Manu’s a˜-knowledge. This is contrary to the increasingly popular supply chain notion of mutually beneficial information sharing. Nevertheless, Table 1 also depicts that the Situation-Ib area is always above the Situation-II area; i.e., Reta is motivated to increase sa when sa is “too low,” but to reduce sa when sa is “too high” – thus, this aspect of sa’s counter-intuitive effects does fit the intuitively attractive notion of “everything in moderation.” We now study how large the Situation-I area is relative to the Situation-II area. This is facilitated by Table 1’s numerical results. Consider first the case of a uniformly distributed a˜. Define kaBB as the ka-value of Boundary B. Assuming the simpler case of PRsub ¼ 0, (6) indicates that p kaBB ¼ ½1 ðbk=ma Þ=ð2 3Þ; i.e., kaBB should be less that [1/(2√3)], or 0.29. Table 1 depicts, for situations with non-zero PRsub and gamma-distributed (instead of uniform-distributed) a˜, kaBBvalues that are significantly less than 0.29. Thus, for the Table 1 column with c ¼ 1, kaBB 0.12. At this c-value, assuming that ma ¼ areal, (3) gives pI* ¼ (areal + bk)/(2b) ¼ (5 + 2)/2 ¼ 3.5, where k ¼ m+c ¼ 2. Hence the theoretical

Should a Stackelberg-Dominated Supply-Chain Player

391

optimal markup over cost is M ¼ (pI* k)/k ¼ 0.75, which is near the lower end of realistic M values, considering that this is the combined gross profit margin of both Manu and Reta. Thus, columns to the right of the “c ¼ 1” column in Table 1 represent less realistic conditions because they are not sufficiently profitable, whereas columns to the left of the “c ¼ 1” column in Table 1 represent increasingly profitable conditions. In other words, for most realistic combinations of system-parameter values, “Situation I” applies when ka is between 0 and (very roughly) 0.2. Thus, while Situation I is not entirely negligible, it is probably not as prevalent as Situation II. Note, however, that this conclusion will be contradicted in Sect. 5, where knowledge uncertainty in “c” (instead of “a”) is considered.

4.2 4.2.1

The Franchise Fee Contract [FF] Problem Statement: Manu’s Knowledge of “a” is Inferior to Reta’s

We stated in Sect. 3.1 that, under [FF], Manu charges Reta a lump-sum fee FFM but supplies her at cost; i.e., m/unit. If Manu perceives a˜, he then perceives Reta’s profit as, from (3), h i ~ R ðperMÞ ¼ ða~ bkÞ2 =ð4bÞ FFM : P (7a) Since Manu knows that Reta will “play” only if her profit is at least PRsub, ~ R(perM) ¼ PRsub” gives “b” (the “cut-off” a-value below which Reta solving “P “quits”) as pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ (7b) b ¼ bk þ 2 ½bðPRsub þ FFM Þ: Hence Manu’s problem is to set FFM to maximize ð amax FFM dFa ðaÞ:

(8)

maxðamin ;bÞ

From Reta’s perspective, if Manu charges FFM, her profit (as perceived by herself) is PR(perR) ¼ (areal bk)2/(4b) FFM. Thus, Reta prefers a Manuperceived Fa(•) that leads to a lower FFM, and hence a higher PR(perR).

4.2.2

Numerical Results

As counterpart to Table 1, Table 2 presents the PR(perR)-values for different combinations of c-values and ka-values (or, equivalently, sa-values) under [FF] when a˜ is gamma distributed. Values of the other parameters (i.e., ma, b, m and ПRsub) are as for Table 1. The sa-effects on PR(perR) depicted by Table 2 are very

392

J.-C. Wang et al.

similar to those depicted by Table 1. The position of “Boundary B” (marked “══”) remains largely unchanged when one moves from Table 1 to Table 2. That is, implementing [FF] instead [w] does not alter or narrow the range of conditions under which Reta prefers Manu’s sa to be larger.

4.2.3

Analytical Proofs

The behavior depicted in Table 2 (and other effects) are derived analytically in WLL’s (2008) Appendix B for the case of a uniformly-distributed a˜. The main results are summarized as Lemmas 2A to 2C; they parallel Lemmas 1A to 1C given in Sect. 4.1.3 for [w]. Lemma 2A (Manu’s optimal FFM decision). The boundary values saFB and saFC used below are defined in (B4) and (B5) of WLL’s (2008) Appendix B. Then, depending on the “Situation,” Manu’s FFM* is: Situation I (when sa saFB): FFM ¼ ðma

pﬃﬃﬃ 3sa bkÞ2 =ð4bÞ PRsub :

Situation II (when saFB < sa saFC): qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃ pﬃﬃﬃ 2 FFM ¼ fma þ 3sa bk þ ½ma þ 3sa bk þ 12bPRsub g2 =ð36bÞ PRsub :

(9)

(10)

Situation III (when sa > saFC): PR(perR) is less than PRsub, hence no trade occurs. Lemma 2B (Effect of sa on PR(perR)). Since Sect. 4.2.1 showed that PR(perR) increases as FFM decreases, one easily obtains the following conclusions by observing the FFM-expressions given in Lemma 2A above: In Situation I, FFM* decreases and hence PR(perR) increases as sa increases. In Situation II, FFM* increases and hence PR(perR) decreases as sa increases. Therefore, PR(perR) is a quasi-concave function of sa, with its maximum at saFB. Lemma 2C (Effects of ma and PRsub on PR(perR)). The effects of ma and PRsub on PR(perR) under [FF] are identical to those stated in Lemma 1C for the [w] contract. Table 2 ПR(perR) under a Franchise fee contract ([FF])

Gamma-distributed a˜, ma ¼ 5, b ¼ 1, and ПRsub ¼ (ma bk)2/(24b)

Should a Stackelberg-Dominated Supply-Chain Player

4.3 4.3.1

393

The Two-Part Tariff Contract [2P] Problem Statement: Manu’s Knowledge of “a” is Inferior to Reta’s

As stated in Sect. 3.1, under [2P], Manu charges Reta a lump-sum fee L on top of a unit wholesale price w. Similar to the arguments given in Sects. 4.1.1 and 4.2.1 for [w] and [FF], Manu’s problem under [2P] can be formulated as: ð amax ½a bðw þ cÞ þ L dFa ðaÞ; ðw mÞ (11) max w;L;b b 2 subject to

4.3.2

½b bðw þ cÞ2 L PRsub ; amin b amax : 4b

(12)

Numerical Results

As counterpart to Tables 1 and 2, Table 3 presents the PR(perR)-values for different combinations of c- and sa-values under [2P] when a˜ is gamma-distributed. Again, the sa-effects on PR(perR) depicted by Table 3 are very similar to those depicted by Tables 1 and 2, and the comments made in Sect. 4.2.2 for [FF] are also applicable here. 4.3.3

Analytical Proofs

The behavior depicted in Table 3 (and other effects) are derived analytically in WLL’s (2008) Appendix C for the case of a uniform F(a). The main results are summarized as Lemmas 3A to 3C; they parallel Lemmas 1A to 1C given in Sect. 4.1.3 for [w] and Lemmas 2A–2C given in Sect. 4.2.3 for [FF]. Similar to Lemma 1A, the Situation’s boundary functions saTA, saTB and saTC are derived in (C12), (C9) and (C13) of WLL’s (2008) Appendix C. Lemma 3A (Manu’s optimal [2P] decisions for w and L). Depending on the “Situation,” Manu’s optimal [2P] decisions for w and L are: Situation I (when sa min(saTA,saTB)): Table 3 ПR(perR) under a two-part tariff contract ([2P])

Gamma-distributed a˜, ma ¼ 5, b ¼ 1, and ПRsub ¼ (ma bk)2/(24b)

394

J.-C. Wang et al.

pﬃﬃﬃ pﬃﬃﬃ 3sa =b þ m; and L ¼ ðma 2 3sa bkÞ2 =ð4bÞ PRsub :

(13)

Situation IIa (when min(saTA,saTB) < sa saTB): pﬃﬃﬃ pﬃﬃﬃ w ¼ 3sa =b þ m; and L ¼ ðma 2 3sa bkÞ2 =ð4bÞ PRsub :

(14)

w ¼

Situation IIb (when saTB < sa saTC): qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃ pﬃﬃﬃ 2 w ¼ 4ðma þ 3sa bcÞ þ 11bm ðma þ 3sa bkÞ þ 60bPRsub =ð15bÞ; (15a) L ¼

pﬃﬃﬃ ma þ 3sa bk þ

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃ 2 ðma þ 3sa bkÞ þ 60bPRsub =ð100bÞ PRsub : (15b)

Situation III (when sa > saTC): PR(perR) is less than PRsub, hence no trade occurs. Lemma 3B (Effects of sa on PR(perR)). In Situation I, PR(perR) increases as sa increases. In Situation IIa and Situation IIb, PR(perR) decreases as sa increases. Therefore, PR(perR) is a quasi-concave function of sa, with its maximum at min (saTA, saTB). (Note that contrary to its counterparts Lemmas 1B and 2B, Lemma 3B cannot be obtained by merely observing the results given in Lemma 3A; it is derived in WLL’s Appendix 3). Lemma 3C (Effects of ma and PRsub on PR(perR)). The effects of ma and PRsub on PR(perR) under [2P] are identical to those stated in Lemmas 1C and 2C for [w] and [FF].

4.4 4.4.1

A Menu of Contract [MC] Brief Explanation of the Menu of Contracts ([MC]) Format

As explained in Myerson (1979) and Corbett et al. (2004), it is possible for Manu to specify functions w(adec) and L(adec) such that Reta is forced to declare the real a-value as adec, and that the resultant [MC] is the contract format that gives Manu the highest expected profit for a given state of stochastic a-knowledge (see Sect. 3.1).

4.4.2

Numerical Results

As counterpart to Tables 1–3, Table 4 presents the PR(perR)-values for different combinations of c- and sa-values under [MC] when a˜ is gamma distributed. Again, the sa-effects on PR(perR) depicted by Table 4 are very similar to those depicted by

Should a Stackelberg-Dominated Supply-Chain Player

395

Tables 1–3, and the comments made in Sects. 4.2.2 and 4.3.2 for [FF] and [2P] are also applicable here. 4.4.3

Analytical Proofs

The behavior depicted in Table 4 (and other effects) can be proven analytically for the case of a uniformly-distributed a˜. The substance of what amounts to “Lemma 4A” (i.e., the counterpart of Lemmas 1A, 2A and 3A) are detailed in WLL’s Appendix D. Lemmas 4B to 4C stated below are counterparts of the earlier Lemmas 1B/1C, 2B/2C and 3B/3C. Lemma 4B. The definitions of the following critical values saMA, saMB and saMC are given in (D14), (D12) and (D15) of WLL’s Appendix D. Then, depending on the “Situation,” the effects of sa on Reta’s PR(perR) are: Situation I (sa min(saMA,saMB)): PR(perR) increases as sa increases. Situation IIa (min(saMA,saMB) < sa saMB): PR(perR) decreases as sa increases. Situation IIb (saMB < sa saMC): PR(perR) decreases as sa increases. Situation III (sa > saMC): PR(perR) is less than PRsub, hence no trade occurs. Lemma 4C. PR(perR) decreases as ma increases and