International Handbooks on Information Systems
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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

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– 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

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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

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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

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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]

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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

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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]

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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]

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Part I

Introduction and Review

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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

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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

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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

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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; 0b:

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.

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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.

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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

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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).

□

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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,

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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 ; 00; 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 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.

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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)

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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 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.

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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

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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.

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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

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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.

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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

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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.

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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 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: 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

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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)

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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

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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.

□

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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

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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

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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

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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.

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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)

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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

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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

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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)

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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

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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.

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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

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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 .

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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

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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)

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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

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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

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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

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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

)

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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)

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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 .

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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

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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

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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.

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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

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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

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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

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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

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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.

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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

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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.

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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.

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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

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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.

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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,

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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.

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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)

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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

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.

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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.

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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

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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.

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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

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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

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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

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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

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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 :

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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

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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. □

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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

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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

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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: feryal.erhun@stanford.edu P. Keskinocak School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, GA, USA e-mail: pinar@isye.gatech.edu S. Tayur Tepper School of Business, Carnegie Mellon University, Pittsburgh, PA, USA e-mail: stayur@andrew.cmu.edu 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

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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

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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

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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

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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

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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

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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 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.

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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.

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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

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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:

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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.

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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

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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

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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

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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.

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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)

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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)

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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 .

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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.

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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: samar@skku.edu X. Su College of Business Administration, California State University Long Beach, 1250 Bellflower Blvd, Long Beach, CA 90840, USA e-mail: xsu@csulb.edu 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

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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

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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)

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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.

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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)

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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

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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)

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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.

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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

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2.87 k

Fig. 1 Optimal w and L for varying k

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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

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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)

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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.

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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

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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

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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

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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

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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.

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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

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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.

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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

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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 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

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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

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)

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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; f ðkÞ

The OEM’s profit is also monotonically decreasing in k.

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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

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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: santanu_snh@yahoo.com S.P. Sarmah (*) Department of Industrial Engineering and Management, Indian Institute of Technology, Kharagpur 721302, India e-mail: spsarmah@iem.iitkgp.ernet.in 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

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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.

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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.

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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

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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

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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.

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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 2bpi . 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).

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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 > = 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

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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. 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 g4 . 2

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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,

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C1. 1 Tr JpAD þ Det JpAD >0 C2. 1 þ Tr JpAD þ Det JpAD >0 C3. Det JpAD 1 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)

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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, (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@ =@xÞf 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 n0, 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 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

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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

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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 n0, 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.

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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)

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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.

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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),

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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),

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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):

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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

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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 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)

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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

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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

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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

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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)

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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

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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)

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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:

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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.

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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

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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.

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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=@l 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=@l 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=@l 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

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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).

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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

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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 @BPr@Q 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)

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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.

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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

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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

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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

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Part III

Channel Power, Bargaining and Coordination

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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: penguinhj@163.com A.Z. Zeng School of Business, Worcester Polytechnic Institute, Worcester, MA 01609, USA e-mail: azeng@wpi.edu 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

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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.

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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.

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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.

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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:

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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 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)

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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

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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 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

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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

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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:

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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

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0

0

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0.4

0.6

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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:

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0.5 0

f2(m)

-0.5 -1 -1.5 -2 -2.5

0

0.2

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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:

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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

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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.

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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

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m Fig. 14 The plot of f1(m) against m for cr ¼ cs

0.8

1

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0

-0.1

f2(m)

-0.2

-0.3

-0.4

-0.5

0

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1

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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

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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

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m

Fig. 18 The plot of f1(m) against m for cr < cs

2.5

2

f1(m)

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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

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f2(m)

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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.

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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: wangjc@business.hku.hk A.H.L. Lau School of Business, University of Hong Kong, Pokfulam, Hong Kong e-mail: ahlau@business.hku.hk H.-S. Lau* (*) Department of Management Sciences, City University of Hong Kong, Kowloon Tong, Hong Kong e-mail: mshslau@cityu.edu.hk 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

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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:

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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.

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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.

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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

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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

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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

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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”).

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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.

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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

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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

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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

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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)

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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)

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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

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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 increases as PRsub increases.

4.5

Summary and Discussion for the Scenario of Asymmetric a-Knowledge

Each of Sects. 4.1–4.4 showed that, given that Reta already knows the contract format Manu will use (be it [w], [FF], [2P] or [MC]), there is a “Situation-I” set of conditions under which Reta prefers Manu’s a-uncertainty to be higher when Manu is trying to determine the Manu-profit-maximizing parameter values of the supply contract. Furthermore, the expanse of Situation I remains largely unchanged for different contract formats. Since [w] ([MC]) is the simplest (most powerful) supply contract format a dominant Manu can impose, it is reasonably safe to assume that other plausible contract formats will also exhibit such a “Situation-I” behavior Table 4 PR ðperRÞ under a menu of contracts ([MC])

Gamma-distributed a˜, ma ¼ 5, b ¼ 1, and PRsub ¼ ðma bkÞ2 =ð24bÞ

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(we have actually confirmed this for a few other contract formats such as “quantity discount” and “(maximum) resale price maintenance”). Therefore, combining Sects. 4.1–4.4 enables us to conclude that, even if Reta does not know what contract format Manu will use, there is still a roughly constant set of conditions under which she prefers Manu to be less certain about a when Manu is formulating the supply contract. However, as explained in Sect. 4.1.4, these Situation-I conditions are probably not more prevalent than the Situation-II conditions (this conclusion will be contradicted in Sect. 5, where instead of “a,” Manu does not know “c” deterministically). However, Reta is always motivated to mislead Manu into believing an inflated value of PRsub but a deflated value of ma. The analytical derivations and numerical computations presented here have been done under the assumption that Reta knows a perfectly. However, it can be easily shown that the same conclusions are obtained if one assumes that Reta also knows a only as a random variable a˜0 (which is likely to be different from Manu’s perceived a˜). For example, under [w], (4) gives Reta’s expected profit as E[PR(perR)] ¼ E [a˜0 b(w* + c)]2/(4b), a moment’s reflection would reveal that a lower w* leads to a higher E[PR(perR)]. Combining this conclusion with Lemma 1A leads again to Lemma 1B. The same argument applies to Lemmas 2 and 3 (for [FF] and [2P]). [MC] presents greater difficulties, since now one has to derive a different set of [MC]’s [w(adec), L(adec)], given that Manu realizes that Reta also knows a only as a random variable. This is because, unlike [w], [FF] and [2P], Reta’s knowledge plays a role in Manu’s design of [MC]. In other words, extending our results, one can prove easily that, regardless of whether Reta herself is well informed of the parameter (say) a, a “Situation I” exists under [w], [FF] and [2P], where Reta prefers Manu to be less certain about a. While our analytical results cannot be similarly extended for [MC], it seems reasonable to conjecture that the same conclusion can be extended to [MC]. In other words, there exists a “Situation I” under which Reta prefers Manu’s sa to be larger, regardless of Reta’s own knowledge of a.

5 Dominant Manu is Uncertain About Reta’s Unit Cost c 5.1

Reta’s Preference on sc, or c’s Level of Uncertainty

We follow the same approach used in Sects. 3 and 4. First, as in a˜’s case, we are able to obtain analytical solutions (obtainable from the authors) for the case of uniformly-distributed c~ (i.e., the counterparts of Lemmas 1–4). We also obtained numerical solutions for the case of gamma-distributed c~, which not only reveal more characteristics than the uniformly-distributed-~ c analytical solutions, but are also much easier to understand. Since the c~-results here have much in common the ~ we will omit for brevity sake the [FF] numerical results. earlier results for a˜ and b, Thus, Tables 5–7 correspond to, respectively, Tables 1, 3 and 4 of Sect. 3 for [w],

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[2P] and [MC]. In the following paragraphs we point out the similarities to and emphasize the differences from the results presented earlier for a˜. The following similarities between Tables 5–7 and the earlier tables can be observed: 1. There is a Situation-I (Situation-II) area where kc is sufficiently low (high) and Reta prefers Manu to be more (less) uncertain about c~. The low/high-kc areas are graphically separated by a “Boundary B” (the line “══”) in the tabulated numerical data. 2. The position of Boundary B does not change substantially when Manu’s contract changes from the simplest format (i.e., [w]) to the most sophisticated and powerful (i.e., [MC]). The significant difference between this section’s c-uncertainty results (i.e., Tables 5–7) and the earlier a-uncertainty results (i.e., Tables 1–4) is in the k-magnitude at which Boundary B is located. In short, the Situation-I area is now much more prevalent than in the cases of uncertainties in a (and in b, detailed in WLL 2008). To elaborate, consider the Table 5 column with a ¼ 5. As explained in Sect. 4.1.4, we have, from (2), pI* ¼ [a + b(mc + m)]/(2b) ¼ (5 + 2)/2 ¼ 3.5, and the markup over cost is M ¼ (pI* k)/k ¼ 0.75. That is, the profitability here is the same as the column of c ¼ 1.0 in Table 1 discussed in Sect. 4.1.4. However, whereas Boundary B in Table 1’s “c ¼ 1” column occurs at around ka ¼ 0.13, Boundary B in Table 5’s “a ¼ 5” column occurs at around kc ¼ 0.5. Noting that the maximum possible k-value for a uniformly distributed random variable is 0.577, a Boundary-B location at kc ¼ 0.5 means that Situation I (the area above Boundary B) covers a very high proportion of likely uncertainty levels in Manu’s c~. Columns to the right of “a ¼ 5” in Table 5 represent low-profitability conditions that are less likely to occur. Moving to the left, the “a ¼ 10.0” column in Table 5 corresponds to a markup-over-cost (M) of 2 – an M-value not unrealistic high. However, at this M-level, Situation II does not arise until kc is as high as 0.95. Thus, ~ contrary to the conclusions reached in the preceding section for a˜ (and for b, detailed in WLL 2008), Table 5 shows that when Manu is uncertain about c, then

Table 5 ПR(perR) under a price-only contract ([w])

Gamma-distributed c~, mc ¼ 1, b ¼ 1, and ПRsub ¼ [a b(mc þ m)]2/(24b)

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Table 6 ПR(perR) under a two-part tariff ([2P])

Gamma-distributed c~, mc ¼ 1, b ¼ 1, and ПRsub ¼ [a b(mc þ m)]2/(24b)

Table 7 ПR(perR) under a menu of contracts ([MC])

Gamma-distributed c~, mc ¼ 1, b ¼ 1, and ПRsub ¼ [a b(mc þ m)]2/(24b)

in the majority of plausible conditions Reta is motivated to increase Manu’s c-uncertainty. Incidentally, Table 5 shows the necessity of considering both uniform and gamma distributions in this chapter; i.e., under some a-values “Situation II” can only arise when kc exceeds 0.577, thus Situation II cannot be adequately illustrated using only the uniform distribution. Therefore, to facilitate direct comparisons, Tables 1–7 are all compiled using gamma-distributed distributions. In Tables 6 and 7, Boundary B is slightly higher than its position in Table 5. However, it does not alter the conclusions reached in the preceding paragraph based on [w] and in the preceding sections based on a- and b-uncertainties – i.e.: 1. Over a range of plausible conditions (Situation I), Reta prefers Manu to be more uncertain about c. Also, Manu cannot significantly narrow or alter the range of Situation I by using a more sophisticated contract format (e.g., [MC]) instead of a simpler one (e.g., [w]). 2. Contrary to uncertainties in demand (i.e., sa and sb), under c-uncertainty Situation I becomes more prevalent than Situation II.

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399

Reta’s Preference on mc (or c’s bias) and PRsub

It can be easily proven that Reta will always try to mislead Manu into perceiving a higher mc (numerically illustrated in Fig. 3) and a higher PRsub. This effect matches intuition.

6 The Case of a Dominant Reta While WLL (2008) considered dominant-Manu scenarios, WLL (2009) considered the following dominant-Reta cost/profit model: For Manu : PM ¼ ðw mreal Þ q; Reta’s profit PR ¼ ðp wÞ q; where q ¼ a bp:

(16)

Referring to Fig. 1, the dominant Reta knows Manu’s mreal only stochastically as ~ and we investigate, from Manu’s perspective, what are the ideal characteristics m, ~ (or “quality”) of Reta’s m-perception that would lead Reta to specify a contract that is most advantageous to Manu. As in the preceding Sects. 3–5, we assume that Reta may implement any one of the four following contract-formats (1) price-only [rSU]; (2) [FF]; (3) [2P]; and (4) [MC]. Note that (1) [rSU] is [w]’s mirror image for the dominant-Reta scenario; and (2) a Reta-imposed [MC] involves more difficult calculations than a Manu-imposed [MC]. We developed procedures in WLL (2009) for determining, for each of the four contract formats, the parameter values that will optimize Reta’s expected profit. Using these optimizing procedures, we identified conditions under which Manu is motivated to share or distort asymmetrical information on the system parameters m and PMsub. These conditions are summarized as Points E–G in Sect. 7. 1.2 1 ΠR(perR)

0.8

[w] [FF] [2P] [MC]

0.6 0.4 0.2 0

mc 0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

Fig. 3 PR(perR) under gamma-distributed c~: sc ¼ 0.4, creal ¼ 1, a ¼ 5, b ¼ 1, and PRsub ¼ 0

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7 Concluding Summary We consider a very basic two-echelon supply chain system defined by the profit functions and demand curve given in (1), (2) and (16). At the initial stage (time point A in Fig. 1), the dominant player has not specified a supply contract, and has a stochastic perception of the value of one of the system’s parameters – i.e., one of {a,b,c} in the dominant-Manu case, or “m” in the dominant-Reta case. Sections 3–5 cover the dominant-Manu case, where we investigated how Reta would prefer Manu to perceive {a,b,c} and whether Reta is (or can be) motivated to improve the quality of Manu’s perception. We found that: (A) Regardless of how simple or sophisticated Reta’s contract format is, Reta is always motivated to mislead Manu into perceiving an inflated mc and an inflated PRsub. (B) Reta is always motivated to mislead Manu into perceiving a deflated ma-value but an inflated mb-value. (C) Reta is motivated to increase sx (where “x” is one of {a,b,c}) when sx is “too low,” but to reduce sx when sx is “too high” – i.e., “everything in moderation”. (D) Over a very substantial range of plausible and “desirable” (i.e., “profitable”) conditions, Reta is motivated to increase the uncertainty level (s) of Manu’s perception of {a,b,c}. Moreover, this range of confusion-preferring conditions cannot be narrowed by using (or threatening to use) more sophisticated contract formats such as a “menu of contracts” at time point B. Point A appears intuitively obvious after it is pointed out, even though it does not appear to have been explicitly recognized in the literature. Point B is neither intuitively obvious nor counter-intuitive. Points C and D are counter-intuitive and also appear to contradict some popularly held notions. Results (C) and (D) on Reta’s s-preference can be viewed from two directions. First, the current emphasis on supply chain coordination and collaboration might induce one to “guess” that a rational Reta would want to improve the quality (i.e., reduce the s) of Manu’s perception of (a,b,c), particularly when a coordination-encouraging contract is to be implemented. On the other hand, the existence of information rent might induce one to “guess” that Reta would not want to help Manu for free. Our results show that both guesses are half right (and half wrong). Thus, one can argue, coming from the “supply chain” angle, that it is counter-intuitive that the prospect of a coordination-encouraging contract has no effect on motivating Reta to reduce Manu’s s on (a,b,c). From the opposite direction, one might also argue that it is counter-intuitive that there is a range of conditions under which Reta would be interested in improving Manu’s s “for free.” Section 6 covers the dominant-Reta case, where we investigated how Reta would want Manu to perceive {m} and whether Reta is (or can be) motivated to improve the quality of Manu’s perception. Very similar to our dominant-Manu findings reported in Sects. 3–5, we found that:

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(E) Regardless of how simple or sophisticated Reta’s contract format is, Manu is always motivated to mislead Reta into perceiving an inflated mm and an inflated Manu’s subsistence profit level PMsub. (F) Manu is motivated to increase sm when sm is “too low,” but to reduce sm when sm is “too high” – i.e., “everything in moderation” (see Sect. 3.3). (G) Over a significant range of likely conditions, Manu is motivated to increase rather than reduce sm – i.e., to worsen rather than improve the degree of ~ certainty/confidence Reta feels about her initial m-perception before Reta designs her contracts. Manu is motivated to reduce the uncertainty of Reta’s ~ m-perception only in a relatively narrower range of conditions that can be roughly characterized as “low profitability.” Point E appears to be intuitively obvious after it is pointed out. Point F is not very counter-intuitive, but Point G is counter-intuitive and contradicts some popularly held notions.

References Cachon GP (2003) Supply chain coordination with contract. In: de Kok AG, Graves C (eds) Handbooks in operations research and management science, vol 11, Supply chain management: design, coordination, and operation. Elsevier, Amsterdam Chen FR (2003) Information sharing and supply chain coordination. In: de Kok AG, Graves C (eds) Handbooks in operations research and management science, vol 11, Supply chain management: design, coordination, and Operation. Elsevier, Amsterdam Corbett CJ, Zhou D, Tang CS (2004) Designing supply contracts: contract type and information asymmetry. Manage Sci 50:550–559 Ha AY (2001) Supplier-buyer contracting: asymmetric cost information and cutoff level policy for buyer participation. Nav Res Log 48:41–64 Lau AHL, Lau HS (2005) Some two-echelon supply-chain games: improving from deterministicsymmetric-information to stochastic-asymmetric-information. Eur J Oper Res 161:203–223 Lee HL, So KC, Tang CS (2000) The value of information sharing in a two-level supply chain. Manage Sci 46:626–643 Li L (2002) Information sharing in a supply chain with horizontal competition. Manage Sci 48:1196–1212 Liu XC, C ¸ etinkaya S (2009) Designing supply contracts in supplier vs buyer-driven channels: the impact of leadership, contract flexibility and information asymmetry. IIE Trans 41:687–701 ¨ zer O ¨ (2010) Channel incentives in sharing new product demand information and robust Liu H, O contracts. Eur J Oper Res 207(3):1341–1349 Myerson RB (1979) Incentive compatibility and the bargaining problem. Econometrica 47:61–73 ¨ zer O ¨ , Wei W (2006) Strategic commitments for an optimal capacity decision under asymmetric O forecast information. Manage Sci 52:1238–1257 Taylor TA, Xiao W (2010) Does a manufacturer benefit from selling to a better-forecasting retailer? Manage Sci 56(9):1584–1598 Tirole J (1988) The theory of industrial organization. MIT, Cambridge, MA Wang JC, Lau HS, Lau AHL (2008) How a retailer should manipulate a dominant manufacturere’s perception of market and cost parameters. Int J Prod Econ 116:43–60 Wang JC, Lau HS, Lau AHL (2009) When should a manufacturer share truthful manufacturing cost information with a dominant retailer? Eur J Oper Res 197:266–286 Wu YN, Cheng TCE (2008) The impact of information sharing in a multiple-echelon supply chain. Int J Prod Econ 115:1–11

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Supply Chain Coordination Under Demand Uncertainty Using Credit Option S. Kamal Chaharsooghi and Jafar Heydari

Abstract In this chapter, a coordination scheme based on the credit option for the simultaneous coordination of order quantity (Q) and reorder point (s) in a two-stage supply chain (SC) is developed. A decentralized SC including one buyer and one supplier is investigated. The buyer faces demand uncertainty and uses a continuous (s, Q) inventory model. It is shown that joint decision making for both s and Q is profitable for the whole SC. However, in centralized decision making the buyer always loses, whereas the supplier greatly profits. A credit option is proposed as a coordination scheme to encourage the buyer to accept the coordinated decisions. In the proposed model, the buyer can benefit from late payments that are subject to commitment to the jointly agreed s and Q. The lower and upper bounds for the credit period are calculated. The proposed scheme shares the benefits of coordinated decision making based on the bargaining power of each member. Numerical experiments showed that the proposed model can achieve channel coordination. Keywords Coordination • Credit option • Order quantity • Reorder point • Supply chain • Uncertainty

1 Introduction The supply chain concept is based on collaboration and coordination between all companies involved in producing, distributing, and delivering the right product at the right time for the consumers. From this viewpoint, all members involved in an SC must make their decisions in accordance with overall SC benefits. In the past, S.K. Chaharsooghi Industrial Engineering Department, Tarbiat Modares University, Tehran, Iran e-mail: skch@modares.ac.ir J. Heydari (*) Industrial Engineering Department, Shiraz University of Technology, Shiraz, Iran e-mail: JF.Heydari@gmail.com 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_16, # Springer-Verlag Berlin Heidelberg 2011

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companies competed with each other, whereas today, supply chains are racing against each other. The success of an SC depends on the willingness of the potential customers, which requires collaboration and coordination between all members in reducing the sale price, improving product quality, and delivering the right services to the customers. Due to the continuity of decisions in an SC, a wrong decision at one stage (e.g., a local optimum) can lead to additional costs for other members; thus, the SC can be greatly hindered in accomplishing its mission. Therefore, each SC member should make decisions regarding other members rather than only their own profitability. For example, the buyer (or retailer) has the authority to make inventory decisions, such as how much to order (order quantity) or when to order (reorder point). However, these decisions, in addition to influencing the buyer’s profitability, affect the supplier’s costs by changing its inventory-system inputs. Customers’ expectations are currently increasing; these include a high-quality product at the lowest price, quick delivery, and after-sale services. This increase of customer expectations, along with the many competitors in the market, has forced SC members to collaborate to meet the demands of customers at a higher level than their competitors. Thus, to increase customer loyalty and expand market share, each SC member should make appropriate decisions such that the overall SC costs are reduced and the customers’ expectations are met in the best possible manner. Additionally, uncertainties such as external factors outside of the SC members’ control can significantly affect the decisions made by SC partners (Chaharsooghi and Heydari 2010a). These uncertainties may include but are not limited to demand (quantity), supply (time), and price (changes). In facing uncertainties, there are two general strategies: first, if the uncertainties can be manipulated (by paying the relevant costs), shaper strategies are used, e.g., shaping market demand by offering a discount; second, if the uncertainty cannot be reshaped, it is then necessary to implement appropriate plans (e.g., increasing safety stocks) to face future uncertainties, known as adaptor strategies (Gupta and Maranas 2003). Member coordination is a key idea in the formation of the supply chain concept; in almost all definitions of the supply chain, coordination plays a critical role. Coordination in SC means that all decision makers in the SC make their decisions in alignment with other partners such that they globally optimize decision criteria. Sometimes, this alignment between decision makers requires members to set their decision variables far from their local optima; therefore, to encourage these members to accept the global optimum, a set of incentive schemes should be offered. Anything that is capable of encouraging an SC member to participate in the coordination model can be identified as an incentive scheme, e.g., quantity discounts, return policies (buy-back contracts), quantity flexibility, revenue sharing contracts, and credit options (delayed payment). The main focus of this chapter is on the credit option as an innovative scheme for facilitating SC coordination under demand uncertainty. The use of credit is very common in current business environments and plays an important role as a form of financing, particularly in developing countries (Sarmah et al. 2008). Although credit trading has been studied traditionally from the viewpoint of financing, from the operations-management viewpoint, using credit can reduces a company’s

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operating costs. In today’s business transactions, suppliers frequently allow credit for fixed time periods to encourage buyers to increase the size of their orders (Huang 2010). Allowing credit for a specific time period is advantageous for the buyer from several points of view. First, under a credit contract, the buyer does not have to pay the supplier immediately and therefore it is possible to pay the supplier from future earnings. Therefore, low-capital buyers can enter the business. Second, as the major portion of inventory holding costs is associated with investment depreciation, using credit therefore reduces these costs. In other words, delayed payment reduces the amount of capital invested in stock for the duration of the credit period (Huang 2010). Third, the unpaid balance can be invested during the credit period and create additional earnings. In general, before the end of the delay period, the retailer can sell the goods and accumulate revenue and earn interest (Huang 2007). In this chapter, we consider the benefit of credit from the second point of view. In this chapter, a two-stage, serially-connected SC is investigated. The downstream uses a continuous inventory-review system (s,Q). Based on this inventory-review system, the downstream places its order of size Q when its inventory level drops to the reorder point s. In the traditional mode, both Q and s are determined by the downstream so that minimize its costs. In the coordinated mode, unlike in the traditional mode, both downstream and upstream jointly determine these decision variables. Our analyses show that the simultaneous coordination of order quantity and reorder point is profitable for the supply chain as whole. Although the profit of the entire chain is increased, the buyer’s profitability is affected due to the deviation from its local optimum. To compensate for the downstream loss, an incentive scheme based on a credit option is proposed. According to the implemented credit option, the downstream can delay payments for purchased and delivered goods for a certain time, which is known as the credit period. Because a major part of inventory costs is due to investment depreciation on warehouses, a delay in payment can reduce these costs in the downstream site. The credit option is applied successfully as an innovative scheme to simultaneously coordinate both order quantity and reorder point in the supply chain. By implementing the credit option in the SC, the profitability of both members is increased over that before accepting the coordination model. Therefore, the participation of all members in the model is guaranteed. The lower and upper bounds for the credit period are determined based on the downstream and upstream profitabilities. Finally, the exact value of the credit period is calculated based on the bargaining power of each member. In this case, the profits of the coordination model are shared according to the bargaining power of the members. The main contribution of this chapter is the development of a model for the simultaneous optimization of order quantity and reorder point in a supply chain under demand uncertainty. We show that offering delayed payment can encourage the buyer to change its decisions to fall in line with the SC-optimal decisions. Delayed payment has been previously considered mostly for deterministic situations; in this study, we show that delayed payment is capable of coordinating an SC under a demand uncertainty. This chapter is organized as follows. In Sect. 2, a brief literature review is provided. SC model development in decentralized, centralized, and coordinated

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decision making is described in Sect. 3. Section 4 presents the numerical experiments. Finally conclusions are given in Sect. 5.

2 Literature Review Recently, SC coordination has attracted the attention of many researchers as well as practitioners around the world. Moving toward the development of supply chain management, the coordination issue has become a critical field of study. Various coordination models have been proposed in the literature. In this section, we focus on the recent studies in this area. Effective SC management requires the coordination of all SC members, including suppliers, manufacturers, distributors, and retailers, in delivering goods to the customers (Xu et al. 2001). Coordination mechanisms aim to find the optimum solution for the entire chain and then encourage chain members to accept this solution (Giannoccaro and Pontrandolfo 2004). Coordination can be considered in different functions of an SC such as inventory control, logistics, transportation, and forecasting, (Arshinder et al. 2008). Most of the research on SC coordination to date has considered the order quantity as the core parameter, while other parameters also exist that must be coordinated (Chaharsooghi and Heydari 2010a). In the centralized SC, there is one central planner who makes decisions and controls all members and therefore makes decisions that maximize the profit of the whole SC, whereas in decentralized, uncoordinated SC, each member maximizes its own profit (Chen et al. 2010). To coordinate a decentralized SC, various coordination contracts have been proposed in the literature, including quantity discounts (Tsai 2007; Shin and Benton 2007; Li and Liu 2006), buyback policies (Xiao et al. 2010; Ding and Chen 2008; Yao et al. 2008), revenue sharing (Van der Rhee et al. 2010; Hou et al. 2009; Li et al. 2009; Chauhan and Proth 2005), credit options (Chaharsooghi and Heydari 2010b; Jaber and Osman 2006; Sarker et al. 2000), and insurance contracts (Lin et al. 2010). The philosophy of all coordination contracts is the fair sharing of risk among partners. Cachon (2003) conducted a comprehensive literature review on SC-coordination contracts. In the following, the most popular coordination models are examined. Sharing downstream overstocking risk is the main concern of order-quantity coordination models. In a multi period setting, quantity-discount models – the most popular coordination models – have been developed. In quantity-discount models, the upstream encourages the downstream to change its order size by offering a quantity discount. Here, the problem is how to set the discount parameters such that the downstream has enough incentive to participate and, conversely, so that upstream profitability is not seriously affected. Various discount models have been introduced in the literature for various problem settings. For example, coordinating a three-level SC using quantity discounts has been investigated (Munson and Rosenblatt 2001; Jaber et al. 2010). A linear quantity-discount model to coordinate an SC including one manufacturer and multiple retailers after a demand disruption

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has been developed (Chen and Xiao 2009). A quantity-discount model in a twostage SC, in which the demand rate depends on the retailer’s stock level, has also been developed, and it was shown that this model could achieve channel coordination (Zhou et al. 2008). Return policies are also of interest to researchers in the SC-coordination field. In single-period inventory systems (or seasonal sales), unsold items at the retailer’s site at the end of the period pose a risk for the retailer. Return policies are adopted by upstream members to convince retailers to place their orders based on the wholly optimal solution. The effect of e-marketplaces on return polices in an SC when the returned product can be sold in the e-marketplace has been studied (Choi et al. 2004). Flexible return policies in a three-level SC have been proposed, and it was shown that this policy could achieve channel coordination (Ding and Chen 2008). The impact of price-sensitive demand on return polices has also been investigated and various situations were modeled (Yao et al. 2008). The effect of demand-information asymmetry on full return polices was investigated, and it was found that, under some conditions, it is possible that the upstream can lose profitability whereas the downstream always benefits from a return policy (Yue and Raghunathan 2007). Another coordination contract is the credit option or delayed payment, the main concern of this work. In trade, using credit or delayed payment means late payment for a purchased and delivered product. Credit trade has some advantages for the credit users (buyers) and is very common in business today. From the perspective of inventory control, one important advantage of the use of credit is the reduction of inventory-holding costs (Chaharsooghi and Heydari 2010b). A credit option was successfully applied for coordinating the base stock level and review period in a two-stage SC with periodic inventory model (Moses and Seshadri 2000). The optimal pricing and ordering policies with delayed payment, which are dependent on order size in a single-location inventory system, have been calculated, wherein it was assumed that a fixed and predetermined credit period is a function of order quantity and demand is a function of selling price (Shinn and Hwang 2003). The optimal cycle time of a single-location inventory model when delayed payment was allowed by the vendor and the credit period is not a decision variable has also been calculated (Chung et al. 2005); this model has been enhanced by optimizing the retailing price in addition to the cycle time (Teng et al. 2005). Subsequently, the model was further complemented to consider a partially permissible delay in payment when the order size is smaller than a predetermined quantity (Huang 2007). The above studies all considered the case of deterministic demand without shortages; Chung and Huang (2009) complemented the previous works by considering delayed payment in a single-location inventory system when shortages are allowed. This study was later extended to an economic production-quantity framework with a finite replenishment rate (Hu and Liu 2010). Most of the abovementioned studies in the field of delay in payments considered single-location inventory systems, whereas in supply chain management the problem is extended to multilocation inventory systems. Delay in payments has been successfully used to coordinate the order quantity in a deterministic environment

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(Jaber and Osman 2006). The optimal replenishment cycle and replenishment frequency in a two-stage SC under a credit option have been determined and the delay period calculated (Chen and Kang 2007). Sarmah et al. (2007) introduced a coordination model based on a credit option in a two-stage serial SC in which each party has its individual profit target. Afterwards, the coordination of singlemanufacturer and multiple heterogeneous buyers by considering transportation costs when demand is known and constant was investigated, and the coordinated production and replenishment cycle are determined (Sarmah et al. 2008). In another report, the effect of delayed payment, the inflation rate, and the depreciation rate on the inventory system when the supplier provides a permissible payment delay for the purchaser were discussed, and the optimal payment period and replenishment cycle were derived (Chang et al. 2009). Lead-time reduction in a two-stage supply chain with a permissible payment delay has been investigated, and the optimum inventory policy along with the optimum lead-time length has been calculated (Huang et al. 2010a). In the last study, the credit period was considered to be an input parameter of the system. The same authors further developed this work from a mathematical modeling viewpoint, in which the capital investment in reducing the order-processing time is a logarithmic function of ordering costs (Huang et al. 2010b). In this chapter, a credit-option model is proposed as an incentive scheme for coordinating one buyer and one supplier. In comparison with the previous works in the field of SC coordination and delay in payments contracts, this study can be distinguished in two aspects. First, the proposed model considers the coordination of two principal inventory decisions, i.e., order quantity and reorder point, simultaneously. Second, in the proposed model the credit option is investigated as an incentive scheme in the continuous inventory-review system in which the customer demand is stochastic. In the proposed model, the buyer uses an (s,Q) inventory system and the decision variables are order quantity and reorder point. By offering a credit period, the supplier encourages the buyer to select these decision variables such that the expected total SC costs are minimized. The coordinating parameter “Credit period” is determined such that both parties have sufficient incentive to participate in the model.

3 Model Development The investigated supply chain is a two-stage SC with one actor at each level, namely, buyer and supplier. The buyer faces the stochastic demand of the customers and uses a continuous inventory-review system (s,Q), in which an order of size Q is placed with the supplier when the buyer’s inventory level drops to the reorder point s. The placed order Q is delivered to the buyer after a stochastic lead time. In this model, the mean and variance of lead-time demand (LTD) is known. Both s and Q are buyer decision variables. The received customer orders must be responded to immediately; otherwise, the unfulfilled demand will be backlogged and must be realized as soon as possible.

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The supplier problem is how to replenish stocks such that is able to respond to the buyer’s orders. It is assumed that when a supplier places the replenishment order, the order is delivered immediately and instantaneously. In this situation, it has been proven that optimal size of a supplier order is an integer multiple of the buyer’s order quantity (Rosenblatt and Lee 1985). The following notation is used in this chapter: D Q1 Q2 h1 h2 A1 A2 B k m sL s

Expected demand per year Buyer’s order quantity (decision variable) Supplier’s order quantity, which is function of the buyer’s order quantity given by Q2 ¼ nQ1, where n is an integer (decision variable) Buyer’s unit inventory holding cost per year Supplier’s unit inventory holding cost per year Buyer’s ordering cost per replenishment Supplier’s ordering cost per replenishment Shortage cost per unit at buyer’s site Safety factor (decision variable) Expected lead time demand Standard deviation of demand during lead time Buyer’s reorder point, which is function of the safety factor k

It is assumed that the lead-time demand has a normal distribution. Therefore, for any given value of k, the reorder point s will be m + ksL (Silver et al. 1998), where ksL is the safety stock (SS).

3.1

Traditional Decision Making

In the traditional decision-making model, each SC member determines its authorized decision variables (k and Q1 for buyer and Q2 for supplier) by considering its own cost function such that it minimizes its own costs, regardless of other chain members. In this situation, each SC member should solve a traditional singlelocation inventory problem. In their textbook example of a single-location inventory model, Silver et al. (1998) calculate the optimized buyer’s order quantity and reorder point simultaneously. The expected buyer’s annual cost function, including expected annual ordering, holding and shortage costs, is calculated as follows: ðk; Q1 Þ ¼ A1 D=Q1 þ h1 ð0:5Q1 þ ksL Þ þ ðBDGu ðkÞsL Þ=Q1 TCTraditional 1

(1)

where Gu(k) is defined as: 1 ð

Gu ðkÞ ¼ k

1 u2 ðu kÞ pﬃﬃﬃﬃﬃﬃ e 2 du 2p

(2)

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The expected shortages per replenishment cycle are represented by Gu ðkÞsL . The buyer’s expected annual shortage costs can be calculated by multiplying the expected shortages per replenishment cycle by the expected number of replenishments per year, D/Q1, and the unit shortage cost B. Silver et al. (1998) propose an iterative procedure to find the optimum value of k and Q1 simultaneously. Meanwhile, the supplier solves its inventory problems by itself. As mentioned earlier, it has been proven that supplier cost will be minimized when Q2 is an integer multiple of Q1. The supplier expected annual-cost function, including expected annual ordering and holding costs, is: ðnÞ ¼ A2 D=ðnQ1 Þ þ 0:5h2 ðn 1ÞQ1 TCTraditional 2

(3)

In the real world, due to changing market behavior, the upstream members do not hold high inventory levels. It is assumed the supplier’s order quantity nQ1 should not exceed 1 year of customer demand, except in special cases (where Q1 is greater than D, in which case n is set at 1). Therefore, if D/Q1 is smaller than one then n ¼ 1; otherwise, the optimal value of n can be obtained by evaluating the integers in the interval [1, D/Q1]. In the traditional decision-making mode, buyer and supplier each make their decisions independently based on their own expected cost functions, i.e., (1) and (3), respectively.

3.2

Centralized Decision Making

In the centralized decision-making mode, there is one SC manager who controls all the businesses in the chain and makes decisions on Q1, k, and n based on the expected total profitability of the SC. The solution thus determined (i.e., the optimal values of the decision variables in the centralized mode) the globally optimum solution. The expected annual cost function (TC) for the whole chain is the sum of buyer and supplier expected cost functions (1) and (3): TCðk; Q1 ; nÞ ¼ ½nA1 þ A2 þ nBGu ðkÞsL D=ðnQ1 Þ þ 0:5½h1 þ ðn 1Þh2 Q1 þ h1 ksL

(4)

Considering the fact that the sum of convex functions must also be convex, to prove the convexity of TC with respect to Q1 and k, it is sufficient to prove the and TCTraditional . In the literature, it has been noted that convexity of TCTraditional 1 2 Traditional is a convex function of Q1 and k (Silver et al. 1998, p. 326). Also, TC1 is not a function of k and, therefore, this is sufficient to prove it is a TCTraditional 2 convex function of Q1. Because @ 2 TCTraditional =@Q21 ¼ 2A2 D=ðnQ31 Þ > 0, TCTraditional 2 2 is a convex function with respect to Q1 and k. Hence, TC is also a convex function with respect to Q1 and k.

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Using the first-order condition, the following equations are obtained: Q1 ¼ ½2DðnA1 þ A2 þ nBGu ðkÞsL Þ=ðn½h1 þ h2 ðn 1ÞÞ1=2 Fu ðkÞ ¼

1 ð

k

1 h1 Q 1 u2 pﬃﬃﬃﬃﬃﬃ :e 2 :du ¼ DB 2p

(5)

(6)

where Fu ðkÞ is the probability that a standard normal variable u takes on a value of k or larger or, correspondingly, the probability of a stock-out within the replenishment lead time. Also, similarly to the traditional decision-making mode, it is assumed that a supplier’s order quantity nQ1 should not exceed 1 year of customer demand except in special cases where Q1 > D. An iterative procedure that converges to the optimal solution is proposed as follows. Initialization: Set n as low as possible (n ¼ 1). Step 0: Set Gu ðkÞ ¼ 0. Step 1: Calculate Q1 using (5). Step 2: Calculate k using (6). Step 3: Repeat steps 1 and 2 iteratively until Q1 and k converge (the difference between two successive values reaches a sufficiently small value) then calculate TC using (4). Step 4: If n bD=Q1 c, then n ¼ n þ 1; go to Step 0. Step 5: Each combination of s, Q1, and n that results in a lower value of TC is the optimal solution (the obtained optimal values are denoted by the superscript “*”). In this procedure, the previous simpler procedure proposed by Silver et al. (1998) (corresponding to steps 0–3 of the proposed procedure) constitutes the middle of the newly proposed procedure. Due to the convex nature of the functions involved, convergence of the above procedure is ensured. Using the proposed procedure, the optimal values of decision variables will be obtained. The target point of the coordination model is to achieve this operational plan with a decentralized chain structure. To achieve channel coordination, the final values of the decision variables resulting from the coordination model must be equal to those of the centralized decision-making model. Indeed, coordination mechanisms should serve to create sufficient incentive for chain members to set their decision variables equal to the values obtained in the centralized model.

3.3

Coordinated Decision Making

Coordination mechanisms aim to encourage members to make decisions that are in line with those of the centralized SC, i.e., those of the globally optimum solution.

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In most cases, when the SC has a decentralized structure, setting the decision variables to optimal centralized values incurs losses for some members. In these cases, using an incentive scheme to fairly share the benefits of a coordination model is vital. At best, a coordination mechanism can increase SC profitability up to that of the centralized decision-making model; this is called channel coordination. Channel coordination is achieved if the expected total SC costs in the coordinated model are reduced to the centralized SC costs. Also, the coordination mechanism should be desirable for all members (Giannoccaro and Pontrandolfo 2004) to ensure that the model is applied. A model is desirable for a member when the application of the model does not reduce the member’s profit to less than that before applying the model. In this chapter, to coordinate both order quantity and reorder point simultaneously, a coordination mechanism based on a credit option is proposed. In this model, the buyer can take advantage of a payment delay subject to commitment to the jointly determined values of the decision variables. Based on the proposed model, the buyer must change its operational decisions regarding s and Q1 from its local optimum to the global optimum to access the benefit of delayed payment for purchased goods proposed by the supplier. Inventory holding cost is associated with cost factors such as leasing the warehouse space, staffing wages, overhead costs, and capital invested in stock. Although the cost of holding inventory is not fully due to capital invested in stock, this capital comprises the majority of inventory holding costs. The use of credit reduces the buyer’s inventory holding costs because it reduces the amount of capital invested in stock for the duration of the credit period (Huang et al. 2010a; Teng et al. 2005). Also, using a credit option has several other advantages for the buyer exploiting delayed payment which are not considered in this study; for instance, a credit user can accumulate sales revenue and earn interest by investing in external projects or depositing the earnings in an interest-bearing account during the credit period (Hu and Liu 2010; Chung et al. 2005). From the SC-coordination perspective, a credit option can convince the buyer to commit to the jointly agreed-upon replenishment rules by decreasing its inventory costs. By comparing the previously investigated traditional and centralized decisionmaking structures, it is possible to extract the operational coefficients for both buyer and supplier. By applying these operational coefficients to the local optimal solution (the values of the decision variables obtained in the traditional model), each party can extract the globally optimum solution. The buyer’s order-quantity coordinator coefficient can be defined as KQ ¼ Q1 =Q1 , where Q1 is the globally optimum value of the buyer’s order quantity that is obtained from the centralized decision-making model and Q1 is the local optimal value of the buyer’s order quantity calculated from the traditional decision-making model. Similarly, the buyer’s safety-factor coordinator coefficient is defined as Kk ¼ k =k and the supplier’s coordinator coefficient for the decision variable n is defined as Kn ¼ n =n. However, applying these defined coordinator coefficients is not possible without necessary arrangements. In fact, by applying the above coefficients, the buyer will

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lose due to deviation from its local optimum. To compensate for the incurred losses, the supplier proposes that the buyer purchases on credit. As mentioned earlier, using credit has several advantages for the buyer; their most important is reducing the expected inventory holding costs. As the majority of inventory holding cost is related to the capital invested in stock, this cost will be imposed only when the payment is cleared; therefore, delayed payment can reduce inventory holding costs. The portion of expected inventory holding costs due to capital invested in stock differs among various industries. For the model be applicable across different industries, we define this portion as a parameter in the proposed model. It is considered that b% of the expected inventory holding cost is related to the capital invested in stock and, therefore, the credit option can compensate these costs for the buyer in the Credit period (CP). From the mathematical point of view, during the credit period the buyer’s expected annual inventory holding costs are reduced by the coefficient (100 b)%. Figure 1 shows the buyer’s reduced inventory holding costs associated with the capital invested in stock after applying the credit option. Several replenishment cycles for both the buyer and supplier and their inventory positions by assuming n ¼ 3 are illustrated in Fig. 1. As shown here, the buyer places an order of size Q1 with the supplier when its inventory position drops to the Buyer Inventory Position

s µ

Q1

Q1

LT

LT

Q1

Q1

LT

LT

LT

Q1

SS=k.σL

CP

Supplier Inventory Position

CP

CP

First batch delivery

CP

Third batch delivery

Second batch delivery

CP

Time

Fifth batch delivery

Fourth batch delivery

Q1

Q1

2Q1

2Q1

Q1

Settle the bill for first batch Shipping first batch

Settle the bill for second batch

Shipping second batch

Q1

Settle the bill for third batch

Shipping third batch

Settle the bill for fourth batch

Shipping fourth batch

Settle the bill for fifth batch

Shipping fifth batch

Time

Shipping sixth batch

Fig. 1 Inventory positions for the buyer (top) and supplier (bottom) and sequence of events in several replenishment cycles

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reorder point s. The placed order is received just after the stochastic lead time LT. Due to the stochastic nature of lead time demand, a shortage may occur. After receiving a placed order, the waiting customers and new customers are served from the recently arrived batch of product (Q1) and revenue is earned. Without the credit option, the buyer must settle the bill at the time of receipt of each order. In the case where a credit option is proposed by the supplier, the buyer settles the payment at the end of the credit period CP. The amount of capital invested in stock for each unit of inventory during the credit period is zero. As b% of the inventory holding cost is associated with capital invested in stock, the cost of holding each unit during the credit period will be reduced. In Fig. 1, the area under the curve is divided into two parts. The hatched area represents the credit duration. In other words, at the beginning of each solid area in Fig. 1, the buyer pays the supplier for the previously delivered batch, CP time units ago. During the CP, the buyer earns revenue from selling the goods which have not yet been paid for. The inventory holding costs in the hatched area of Fig. 1 are reduced by the coefficient (100 b)% with respect to the solid area. Thus, the credit-option contract reduces the buyer’s inventory holding costs during the credit period. If this cost reduction compensates for the increased cost of changing from traditional to centralized decision making, then the buyer will participate. A longer credit period means a greater reduction in the buyer’s expected costs. From the buyer’s perspective, extending credit period is more advantageous. There is a lower bound for credit period, below which the participation is not beneficial for the buyer (when the extra savings from the delay in payment is less than extra cost imposed by accepting the centralized solution). Conversely, adjusting the credit period as low as reasonably possible is more beneficial for the supplier. In the credit period, payment has been not made to the supplier for product sold. Due to the time value of money, delaying the payments only up to a certain period is economical for the supplier. Therefore, from the supplier perspective, there is an upper bound for the credit period; when the delay time exceeds this upper bound, the participation will not be advantageous for the supplier. Figure 2, shows the concept of credit time interval and the preferences of both parties. As shown in Fig. 2, if the credit period becomes very low, the buyer will lose and will not participate, whereas if the credit period becomes very high the supplier will lose. Thus, there is an interval for the credit period between these lower and upper bounds; the appropriate CP lies within it. The bargaining power of members with

Low

Credit Period

High

Fig. 2 The interest of two parties in adjusting the credit period, CP

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each other is the measure for adjusting the coordinator variable CP. As all the inventory-system variables are determined for traditional and coordinated models, the only remaining decision variable is coordinator variable CP. Hereinafter, the Q1, k, and n denote locally optimal values of the decision variables in the traditional model and Q1 , k*, and n* denote globally optimal values of the decision variables in the centralized model, which are absolutely determined and known.

3.3.1

Condition for Buyer to Participate (Lower Bound for Credit Period)

In proposing the credit option to the buyer, the supplier intends to share the earned profits. In fact, the use of credit transfers part of the inventory costs to the supplier and therefore leads to a reduction in the buyer’s expected costs. As mentioned earlier, the buyer’s expected costs consist of three parts: expected ordering, holding inventory, and shortage costs. Using credit affects the second part; i.e., inventory holding costs. As in the traditional inventory model, it is possible to calculate the expected holding inventory costs before applying the credit option by calculating the area under the buyer’s inventory-level curve, which is ðQ1 =2Þ þ k:sL , where the first term is the average cycle inventory and the second term is the safety stock level. By applying the unit inventory-holding costs h1 to the derived value, the buyer’s expected annual inventory holding costs are calculated [see (1)]. To calculate the buyer’s expected annual inventory holding costs after applying the credit option, it is necessary to calculate the area under the buyer’s inventory-level curve based on Fig. 1. The hatched area in Fig. 1 is affected by the coefficient (1 b) while the solid area is not. As a result, the expected annual inventory holding costs will be: Inventory holding costs n ¼ h1 ðKQ Q1 DðCPÞÞ2 =2KQ Q1 þ ð1 bÞðCPÞðD D2 ðCPÞ=2KQ Q1 ÞðCPÞ þKk ksL ð1 bDðCPÞ=KQ Q1 Þ As can be seen in the above equation, the coordinator coefficients KQ and Kk are applied to the variables Q1 and k, respectively; note that Q1 and k in the above relation refer to the optimal values obtained from the traditional model. Finally, the buyer’s expected annual cost function after applying the credit option can be expressed as follows: TCCoordinated ðCPÞ 1

" D ðKQ Q1 DðCPÞÞ2 D2 ðCPÞ þ h1 þ ð1 bÞðD ÞðCPÞ ¼ A1 2KQ Q1 K Q Q1 2KQ Q1 D D þKk ksL ð1 bCP Þ þ ðBGu ðKk kÞsL Þ K Q Q1 KQ Q 1

(7)

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where the three terms denote the expected ordering cost, the expected inventory holding cost and the expected shortage cost, respectively. In the cost function (7), the buyer agrees to apply the coordinator coefficients KQ and Kk to the local optimum values of Q1 and k; in turn, the supplier accepts the delay in payments according to the credit period (delay period) CP. The only decision variable in (7) is CP. The buyer participates (i.e., applies the coordinator coefficients in the case of using credit) if and only if buyer costs do not rise after participation. In mathematical terms, TC after participation (7) must be smaller than the buyer’s expected cost in the traditional model (1). By comparing the buyer’s expected cost functions before and after the credit-option contract, a lower bound for CP can be calculated. Proposition 1. The minimum credit period that convinces the buyer to participate is: CPmin

1 ¼ ðKQ Q1 þ Kk ksL Þ D

rﬃﬃﬃﬃﬃﬃﬃ 2W D2

(8)

where W¼

ðKQ Q1 þ Kk ksL Þ2 1 h A1 Dð1 KQ Þ þ BDsL ½Gu ðKk kÞ Gu ðkÞKQ : 2 bh1 i Q1 2 ðKQ 1Þ þ h1 KQ kQ1 sL ðKk 1Þ (9) þh1 KQ 2

Proof. Set TCTraditional ðk; Q1 Þ TCCoordinated ðCPÞ; by simple mathematical opera1 1 tions it is verified that this inequality will be satisfied if and only if pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ CP D1 ðKQ Q1 þ Kk ksL Þ 2W=D2 . The right side of the resulting expression gives CPmin, which is the lowest value for the CP that encourages the buyer to participate. In fact, when CP is adjusted to CPmin, the expected total cost for the buyer after accepting the coordinator coefficients KQ and Kk is then equal to the expected total cost for buyer before accepting the coordination. If CP is adjusted to a value less than CPmin, the total cost imposed on the buyer after accepting the coordinator coefficients will then be greater than the traditional-model costs, and the buyer will therefore not participate in the plan.

3.3.2

Condition for Supplier to Participate (Upper Bound for Credit Period)

The supplier is the proposer of the credit option to the buyer. If the offer leads to disadvantages for the supplier, it will basically never occur. In this section, the costs imposed on the supplier by proposing credit are calculated and compared to the costs saved from the coordination model. When the supplier’s earned value is

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greater than the cost of convincing the buyer, supplier participation is then guaranteed. The difference between the supplier’s expected cost function in the coordinated and traditional models indicates the earned value of the supplier in moving from the traditional to coordinated model. In the coordinated model, in addition to the expected ordering and holding costs, the supplier is faced with another cost factor: the cost of convincing the buyer. The buyer’s convincing cost is related to the time value of money. The supplier receives payment CP units of time later; thus, with a known interest rate, it is easy to calculate this cost. Also note that all the costs by the supplier for convincing the buyer are delivered by the buyer; therefore, another simple method for calculating the convincing cost is the calculation of the hatched area in Fig. 1 over 1 year multiplied by the imposed cost coefficient bh1. Finally, the supplier’s expected annual cost function after proposing the credit option is: TCCoordinated ðCPÞ ¼ A2 D=ðKn KQ nQ1 Þ þ 0:5h2 KQ Q1 ðKn n 1Þ 2 þ bh1 DðCPÞ KQ Q1 0:5DðCPÞ þ Kk ksL =KQ Q1

(10)

where the first term is the expected ordering costs, the second term is the expected inventory holding costs and the third term is the expected cost of convincing the buyer. The supplier participates in the plan if and only if its cost after participation (10) is smaller than the cost before participation (3). By comparing the supplier’s expected cost functions before and after the credit-option contract, the following proposition establishes an upper bound for CP. Proposition 2. The maximum allowable credit period which is acceptable for the supplier is: CPmax

1 ¼ ðKQ Q1 þ Kk ksL Þ D

rﬃﬃﬃﬃﬃﬃﬃ 2M D2

(11)

where M¼

2DA2 ð1 KQ Kn Þ þ h2 KQ Kn nQ1 2 ð1 KQ n½1 Kn KQ Þ 2Kn nbh1 þ 0:5ðKQ Q1 þ Kk ksL Þ2

(12)

Proof. Set TCTraditional ðnÞ TCCoordinated ðCPÞ; by simple mathematical operations 2 2 it is verified that this inequality will be satisfied if and only if CP D1 ðKQ Q1 þ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Kk ksL Þ 2M=D2 . The right side of the achieved expression gives CPmax, the highest value for the CP that is acceptable for the supplier. When CP is adjusted to CPmax, the expected total cost to the supplier after proposing the credit option is then equal to its expected total cost in the traditional model. If the CP is adjusted to higher than CPmax, the total cost imposed on the supplier by proposing the credit option will be greater than in the traditional model, and the supplier will therefore refuse to participate.

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Equitable Credit Period

Channel coordination will be achieved by setting the credit period to a value between CPmin and CPmax. If CP is set to CPmin, then all the benefits are acquired by the supplier. In fact by setting CP ¼ CPmin, the buyer’s expected cost after coordination will be equal to the buyer’s expected cost in the traditional model. In this state, there is no change in profitability for the buyer by accepting the plan, and all the advantages are therefore gained by the supplier. Indeed, by setting the CP to CPmin, the buyer’s profitability will merely not be reduced. Conversely, if CP is set to CPmax, then all the benefits are gained by the buyer. The interval [CPmin, CPmax] is a continuous interval and therefore in selecting the credit period from this interval there are infinite options. But what point must be selected? Or, similarly, what is the criterion for setting CP? The answer to these questions is related to the power of each member versus the other. Recalling Fig. 2, the member with more power can set the CP. In the business environment this power is called “bargaining power”. The bargaining power is defined as the total advantage which one can obtain in a negotiation. In this section, the coordinator variable CP is determined based on the bargaining power of SC members. In this way, the share of benefits obtained by each member will be fairly distributed. We define a as the bargaining power of the buyer versus the supplier and therefore (1 a) is the bargaining power of the supplier versus the buyer. It is clear that a is between 0 and 1. By increasing the value of a to near 1, the bargaining power of the buyer increases. As a result, by increasing a, the expected earnings of the buyer are increased and vice versa. To calculate the credit period, it is necessary to calculate the extra earnings resulting from applying the proposed coordination model and share them between the two parties by adjusting the CP. The extra profit by applying the coordination scheme is: DTC ¼ TCðk; Q1 ; nÞ TCðk ; Q1 ; n Þ ¼ ½nA1 þ A2 þ nBGu ðkÞsL D=ðnQ1 Þ þ 0:5½h1 þ ðn 1Þh2 Q1 þ h1 ksL ½Kn nA1 þ A2 þ Kn nBGu ðKk kÞsL D=ðKn nKQ Q1 Þ 0:5½h1 þ ðKn n 1Þh2 KQ Q1 h1 Kk ksL

(13)

where Q1, k, and n are the (locally) optimal values of the decision variables in the traditional decision-making model, and Q1 , k*, n* are the optimal values of the decision variables in the centralized model, which are created by applying the coordinator coefficients KQ, Kk, and Kn, respectively, to the local optimum values. a According to the buyer–supplier bargaining power relationship, ðaþð1aÞÞ 100% of DTC should be obtained by the buyer and rest is retained by the supplier. From the mathematical point of view, the buyer’s expected cost after coordination must be smaller than the buyer’s expected cost in traditional mode by of the factor aDTC.

Supply Chain Coordination Under Demand Uncertainty Using Credit Option

TCTraditional ðk; Q1 Þ TCCoordinated ðk ; Q1 ; CPÞ ¼ aDTC 1 1

419

(14)

By substituting (1) and (7) into (14), the exact value of CP based on the buyer’s bargaining power a is found as follows: sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ# " 1 KQ Q1 aDTC Þ ðKQ Q1 þ Kk ksL Þ 2ðW CP ¼ D bh1

(15)

The channel coordination is achievable, If one value from (15) is obtained within the interval [CPmin, CPmax]. By applying the calculated CP, the supplier’s share of ð1aÞ the extra benefits will be ðaþð1aÞÞ 100%.

4 Numerical Experiments In this section, the performance of the introduced coordination scheme is measured by conducting a set of numerical examples. Also, the effect of uncertainties on the coordination model is examined by testing the model with various degrees of uncertainty. Table 1 shows the data set for the numerical experiments. After running the three models (traditional, centralized and coordinated) based on the Table 1 data set, the results shown in Table 2 were obtained. As shown in Table 2, the TC resulting from the coordinated decision-making model is equal to that of the centralized model. Therefore, the proposed coordination model can achieve channel coordination. Also, applying the model is desirable for both members due to the reduction of expected costs for each member with respect to the traditional model. By shifting from traditional to centralized decision making, the expected total SC cost can be reduced from the 885.89 to 814.03, an ~8.1% improvement. Nevertheless, this shift is not applicable due to the increase in the buyer’s expected costs (from 561.95 to 607.37). The credit-option scheme resolves this problem by fairly sharing the acquired benefits between members. As shown in Table 2, in the coordinated model, both members’ costs are reduced with respect to the traditional model and the total expected cost of the SC is also equal to the centralized model. The trend of the changes in the decision variables and the performance criteria were also tracked by increasing the uncertainty. The standard deviation of the leadtime demand (LTD) is the measure of uncertainty, and we can call it the degree of uncertainty. Figure 3 shows the total SC cost improvement resulting from the coordinated model with respect to traditional decision making with various degrees of uncertainty. As shown in Fig. 3, the proposed model can create an appropriate cost saving for an SC greater than 5% with all degrees of uncertainty. This improvement in total SC cost shows the ability of the coordination model for adjusting the decision

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Table 1 Numerical experiment data

Table 2 The results of running the numerical examples

Parameters D h1 h2 B A1 A2 m sL a b

Traditional decision making Decision variables Q1 123.48 s 77 n 1 CP (day) N/A

Performance criteria 561.95 TC1 TC2 323.94 TC 885.89

Values 500 4 3 5 50 80 80 20 0.6 0.4

Centralized Coordinated decision making decision making 193.57 69.93 1 N/A

193.57 69.93 1 CPmin ¼ 21.24 CPmax ¼ 65.22 CP ¼ 45.35

607.39 206.64 814.03

518.84 295.19 814.03

12

% Improvement

10 8 6 4 2 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 LTD standard deviation

Fig. 3 SC cost improvement of coordinated decision making compared to the traditional model

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200 Q1-Traditional

180

Q1-Coordinated Product unit

160

s-Traditional s-Coordinated

140 120 100 80 60

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 LTD standard deviation

Fig. 4 Coordinated buyer’s order quantity and reorder point versus traditional buyer’s order quantity and reorder point

variables. The proposed credit option encourages the buyer to decide on the variables Q1 and s jointly. Thus, the globally optimum solution can be achieved. Figure 4 illustrates the values of the buyer’s order quantity and reorder point before and after implementing the credit option. As shown here, the proposed model adjusts the buyer’s order quantity to a point higher than in traditional decision making while the reorder point is adjusted to less than in the traditional model. Note that in Fig. 4 the coordinated Q1 and s are equal to Q1 and s in centralized decision making. Figure 5 shows the buyer’s expected costs with the various degrees of uncertainty. As shown by the buyer’s expected cost curve, the cost in the coordinated model is lower than traditional and centralized models. As shown in Fig. 5, without conducting the credit option, the buyer’s expected cost increases (in going from the centralized to traditional model) and the buyer declines to participate. In fact, in coordinated decision making the buyer takes advantage of the delay in payment and therefore its cost is reduced compared to the centralized decision-making structure. Because the buyer suffers the uncertainties, as uncertainty increases the buyer’s expected costs also increase but the efficiency of the coordination model is not affected; i.e., channel coordination is achieved in all cases. Figure 6 shows the changes in the supplier’s expected costs versus the degree of uncertainty; also illustrated is a comparison between the three decision-making models from the supplier’s cost viewpoint. As shown here, the supplier’s expected cost increases in the coordinated model with respect to centralized decision making. The main cause of this increase is the proposal of the credit option to the buyer. Although the supplier’s expected cost in the coordinated model is greater than in the centralized model, it is always less than in the traditional model (see Fig. 6); therefore, the participation of the supplier is assured.

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Centralized Coordinated

Buyer's cost

600 550 500 450 400

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 LTD standard deviation

Fig. 5 Buyer’s expected costs versus degree of uncertainty (comparison between traditional, centralized and coordinated decision making)

400

350

Supplier's cost

300

250

200 Traditional 150

Centralized Coordinated

100

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 LTD standard deviation

Fig. 6 Supplier’s expected costs versus degree of uncertainty (comparison between traditional, centralized and coordinated decision making)

The change in the coordinator variable CP versus the degree of uncertainty is illustrated in Fig. 7. The credit period is reduced by increasing uncertainty. In summary, by increasing uncertainty, the expected costs of the supply chain will increase, but the proposed coordination model can reduce the cost as much as possible, and channel coordination is achievable by applying the model.

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100 CP-min

90

CPmax

Credit Period (day)

80

Fair CP

70 60 50 40 30 20 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

LTD standard deviation

Fig. 7 Credit periods with different degrees of uncertainty.

5 Concluding Remarks Supply chain coordination has two main aspects (1) what decisions need to be coordinated, and (2) how to coordinate them. In this chapter, a scheme based on a credit option for the simultaneous coordination of order quantity and reorder point in a two-stage SC under uncertainty was introduced. The presented scheme has the ability to achieve channel coordination. In this scheme, the buyer can use the advantage of delayed payment that is subject to commitment to the globally optimum decisions. Using a credit option, the supplier shares the extra benefits with the buyer. Surplus sharing is fairly distributed based on the members’ bargaining power. This study indicates that the simultaneous coordination of order quantity and reorder point is advantageous and the credit option can be used as a coordination scheme in achieving such advantages under demand uncertainty. The experimental results indicate the model effectiveness (achieving channel coordination) and suitability (dividing the benefits fairly). In addition, the numerical experiments show the advantages of the proposed model with various degrees of demand uncertainty. Although the credit option, or delay in payments, is a well-known mechanism both in financing and operations-management literature, its application under uncertainty has rarely been examined. This study shows the capability of the credit option as a supply chain coordination scheme under demand uncertainty. Compared to the existing literature in the field of SC coordination under uncertainty, this study can be viewed as an opening statement toward implementing the credit option as an SC-coordination scheme under uncertainty which is capable of aligning the objectives of the SC members with the supply chain goal. In comparison with the

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existing studies on the use of credit option in deterministic settings, this study enlarges the application area of credit option toward stochastic environments. The main limitation of the proposed model regards the structure of the investigated SC. The model is limited to a one-buyer, one-supplier configuration. The proposed model can be extended in future work by considering more SC tiers and a networked SC structure. In addition, all developments of the basic creditoption model in the deterministic environment, considering factors such as inflation, partial credit, and deteriorating products, can be investigated in future work.

References Arshinder K, Kanda A, Deshmukh SG (2008) Supply chain coordination: perspectives, empirical studies and research directions. Int J Prod Econ 115:316–335 Cachon GP (2003) Supply chain coordination with contract. In: Graves S, de Kok T (eds) Handbooks in operations research and management science. North-Holland, Amsterdam, pp 229–340 Chaharsooghi SK, Heydari J (2010a) Optimum coverage of uncertainties in a supply chain with an order size constraint. Int J Adv Manuf Technol 47:283–293 Chaharsooghi SK, Heydari J (2010b) Supply chain coordination for the joint determination of order quantity and reorder point using credit option. Eur J Oper Res 204(1):86–95 Chang CT, Wu SJ, Chen LC (2009) Optimal payment time with deteriorating items under inflation and permissible delay in payments. Int J Syst Sci 40(10):985–993 Chauhan SS, Proth JM (2005) Analysis of a supply chain partnership with revenue sharing. Int J Prod Econ 97(1):44–51 Chen LH, Kang FS (2007) Integrated vendor–buyer cooperative inventory models with variant permissible delay in payments. Eur J Oper Res 183:658–673 Chen K, Xiao T (2009) Demand disruption and coordination of the supply chain with a dominant retailer. Eur J Oper Res 197:225–234 Chen H, Chen Y, Chiu CH, Choi TM, Sethi S (2010) Coordination mechanism for the supply chain with lead time consideration and price-dependent demand. Eur J Oper Res 203:70–80 Choi TM, Li D, Yan H (2004) Optimal returns policy for supply chain with e-marketplace. Int J Prod Econ 88:205–227 Chung KJ, Huang CK (2009) An ordering policy with allowable shortage and permissible delay in payments. Appl Math Model 33:2518–2525 Chung KJ, Goyal SK, Huang YF (2005) The optimal inventory policies under permissible delay in payments depending on the ordering quantity. Int J Prod Econ 95:203–213 Ding D, Chen J (2008) Coordinating a three level supply chain with flexible return policies. Omega 36(5):865–876 Giannoccaro I, Pontrandolfo P (2004) Supply chain coordination by revenue sharing contracts. Int J Prod Econ 89:131–139 Gupta A, Maranas CD (2003) Managing demand uncertainty in supply chain planning. Comput Chem Eng 27:1219–1227 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 Hu F, Liu D (2010) Optimal replenishment policy for the EPQ model with permissible delay in payments and allowable shortages. Appl Math Model 34:3108–3117 Huang YF (2007) Economic order quantity under conditionally permissible delay in payments. Eur J Oper Res 176:911–924 Huang CK (2010) An integrated inventory model under conditions of order processing cost reduction and permissible delay in payments. Appl Math Model 34:1352–1359

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Huang CK, Tsai DM, Wu JC, Chung KJ (2010a) An integrated vendor–buyer inventory model with order-processing cost reduction and permissible delay in payments. Eur J Oper Res 202:473–478 Huang CK, Tsai DM, Wu JC, Chung KJ (2010b) An optimal integrated vendor–buyer inventory policy under conditions of order-processing time reduction and permissible delay in payments. Int J Prod Econ 128(1):445–451. doi:10.1016/j.ijpe.2010.08.001 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 Jaber MY, Bonney M, Guiffrida AL (2010) Coordinating a three level supply chain with learningbased continuous improvement. Int J Prod Econ 127(1):27–38 Li J, Liu L (2006) Supply chain coordination with quantity discount policy. Int J Prod Econ 101(1):89–98 Li S, Zhu 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 205:339–345 Moses M, Seshadri S (2000) Policy mechanisms for supply chain coordination. IIE Trans 32(3):245–262 Munson CL, Rosenblatt MJ (2001) Coordinating a three-level supply chain with quantity discounts. IIE Trans 33(5):371–384 Rosenblatt MJ, Lee HL (1985) Improving profitability with quantity discounts under fixed demand. IIE Trans 17:388–395 Sarker BR, Jamal AMM, Wang S (2000) Supply chain models for perishable products under inflation and permissible delay in payment. Comput Oper Res 27(1):59–75 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 182:1469–1478 Sarmah SP, Acharya D, Goyal SK (2008) Coordination of a single-manufacturer/multi-buyer supply chain with credit option. Int J Prod Econ 111:676–685 Shin H, Benton WC (2007) A quantity discount approach to supply chain coordination. Eur J Oper Res 180(2):601–616 Shinn SW, Hwang H (2003) Optimal pricing and ordering policies for retailers under order-sizedependent delay in payments. Comput Oper Res 30:35–50 Silver EA, Pyke DF, Peterson R (1998) Inventory management and production planning and scheduling, 3rd edn. Wiley, New York Teng JT, Chang CT, Goyal SK (2005) Optimal pricing and ordering policy under permissible delay in payments. Int J Prod Econ 97:121–129 Tsai JF (2007) An optimization approach for supply chain management models with quantity discount policy. Eur J Oper Res 177(2):982–994 Van der Rhee B, van der Veen JAA, Venugopal V, Nalla VR (2010) A new revenue sharing mechanism for coordinating multi-echelon supply chains. Oper Res Lett 38(4):296–301 Xiao T, Shi K, Yang D (2010) Coordination of a supply chain with consumer return under demand uncertainty. Int J Prod Econ 124(1):171–180 Xu K, Dong Y, Evers PT (2001) Towards better coordination of the supply chain. Transp Res E 37:35–54 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 187:275–282 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 Zhou YW, Min J, Goyal SK (2008) Supply-chain coordination under an inventory-level-dependent demand rate. Int J Prod Econ 113:518–527

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Supply Chain Coordination Under Consignment Contract with Revenue Sharing Sijie Li, Jia Shu, and Lindu Zhao

Abstract The balance of power between manufacturers and retailers is shifting, and consignment contract with revenue sharing has been widely applied in many industries, especially in on-line marketplaces. In this chapter we consider a supply chain with an upstream manufacturer and a downstream retailer where a singleperiod product is produced and sold. The manufacturer chooses the delivery quantity and the retail price, and the retailer sets the revenue shares. Utilizing Nash bargaining model, a cooperative game model is developed to implement profit sharing between the manufacturer and the retailer to achieve their cooperation. When the manufacturer and the retailer are assumed to be risk-neutral, under a very mild restriction on the demand distribution, the decentralized supply chain can be perfectly coordinated, and both players can earn more in the proposed cooperative setting. Furthermore, the impacts of supply chain system parameters on the optimal supply chain decisions and the supply chain performance are investigated in this chapter. Keywords Consignment contract with revenue game • Supply chain coordination • Uncertainty

sharing

•

Cooperative

S. Li (*) • L. Zhao Institute of Systems and Engineering, Southeast University, Nanjing, Jiangsu, People’s Republic of China e-mail: sjli@seu.edu.cn; ldzhao@seu.edu.cn J. Shu Department of Management Science and Engineering, School of Economics and Management, Southeast University, Nanjing, Jiangsu, People’s Republic of China e-mail: jshu@seu.edu.cn 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_17, # Springer-Verlag Berlin Heidelberg 2011

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1 Introduction In the traditional supply chain and distribution channel, the upstream seller, acting as the leader, has the ownership and price control of goods, and the downstream buyer often acts as the follower. The leader-follower scheme essentially implies that the supply chain member as the leader is dominant in the supply chain and distribution channel. The “dominant” implies that a supply chain member has the power of controlling/influencing another member’s actions and decisions. However, in the current market-oriented economy, the balance of power between manufacturers and retailers is shifting in industries as diverse as pharmaceuticals, consumer packaged goods, hardware, apparel, and furniture (Kumar 1996). Porter (1974) has confirmed the existence of retailer power in consumer goods industries, and defined it as the ability of a retailer to influence a manufacturer’s product differentiation. The supply chain with dominant retailer in the context of marketing and distribution channels has arguably been more pronounced in many academic researches and empirical analyses (e.g., Hua and Li 2008; Lau et al. 2008; Chen and Xiao 2009; Li et al. 2010). Consignment contract with revenue sharing has been widely applied in various industries, such as rental (Dana and Spier 2001; Mortimer 2004), retailing and auction (Wang et al. 2004b; G€ um€ us¸ et al. 2008) and procurement of industrial materials (Cachon 2001; Cachon and Lariviere 2001; Gerchak and Wang 2004). It is essentially a real realization of the power shifting in supply chain channel, and especially popular in on-line marketplaces, such as Amazon.com, Alibaba.com, eBay.com, etc. Under such a contract, ownership of the goods is retained by the supplier and price is also usually determined solely by the supplier. For each item sold, the retailer will deduct an agreed percentage from the selling price, remit the balance to the vendor and no money changes hands until the item is sold (Bolen 1978). Under the consignment contract with revenue sharing, the downstream retailer has more market power and acts as the leader in the supply chain channel, which means that the retailer is dominant over the supplier. In this chapter, we investigate the supply chain decisions and coordination under the consignment contract with revenue sharing. The centralized and decentralized supply chain structures are generally involved in supply chain organizations (Giannoccaro and Pontrandolfo 2004). In the centralized structure, the supply chain operates on the basis of centrally made decisions. In the decentralized structure, each firm makes its own decisions, based on its own knowledge, almost regardless of the rest of the supply chain (Bose and Pekny 2000). The coordination mechanism is an important issue in designing a contract for a decentralized supply chain. Cachon (2003) defined supply chain coordination as follows: A contract is said to coordinate the supply chain if the set of supply chain optimal actions is a Nash equilibrium, i.e., no firm has a profitable unilateral deviation from the set of supply chain optimal actions. This definition is concentrated on the decentralized supply chain since a centralized supply chain is perfectly coordinated. For a decentralized supply chain, if the decisions result in supply chain channel profit

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that is equal to the total profit achieved in a centralized supply chain, the decentralized supply chain is called perfect coordination or channel coordination (Bernstein and Federgruen 2003; Cachon 2004; Wang et al. 2004b). Many researches have addressed the supply chain coordination problem for the consignment contract with revenue sharing. These researches demonstrated that the decentralized supply chain cannot be perfectly coordinated under the consignment contract with revenue sharing. Cachon and Lariviere (2001) analyzed contracts of vendor management inventory (VMI) with revenue sharing, in which the manufacturer facing an uncertain demand offers various contracts to a component supplier. They demonstrated that the decentralized system provides less capacity than the integrated system, which means that the supply chain is not perfectly coordinated. Cachon (2003) studied coordination in a supply chain with one supplier and one retailer, and several different contract types were shown to coordinate the supply chain, such as buy back contracts, revenue sharing contracts, quantity flexibility contracts, sales rebate contracts and quantity discount contracts. They did not consider the consignment contract, and the supply chain is coordinated by an adjusting parameter l, which can loosely be interpreted as the retailer’s share of the supply chain’s profit. Wang et al. (2004b) modeled the decision making of the two firms in a supply chain as a non-cooperative game, and showed that the decentralized channel profit is always lower than the centralized channel profit under the proposed consignment contract. Li and Liu (2008) presented a coordination decision policy, in which a price discount policy is developed to allocate the expected increased profits between two sides of the supply chain; however the decentralized supply chain cannot be perfectly coordinated. Furthermore, many researches also showed that the consignment contract cannot perfectly coordinate the decentralized supply chain (e.g., Dong and Xu 2002; Choi et al. 2004; Cachon 2004). There are also some researches on perfectly coordinating the decentralized supply chain. Cachon (2001) proposed an approach of fixed transfer payments from the supplier to the retailer. Gerchak and Wang (2004) suggested a twoparameter contract (revenue-sharing plus surplus-subsidy) for an assembly system with multiple suppliers and one manufacturer. These coordination mechanisms can be regarded as an implementation of channel profits sharing by a transfer payment. In this chapter, we consider a two-echelon supply chain with an upstream manufacturer and a downstream retailer under the consignment contract with revenue sharing, and two specific demand function forms: additive and multiplicative demand cases (Petruzzi and Dada 1999; Wang et al. 2004b). The manufacturer produces a single-period product at a constant marginal cost. He has only one chance at production before the start of the selling season and sells his products through the retailer. The retailer does not pay the manufacturer upon receipt of the items but shares the sales revenue on units sold. The market demand for the singleperiod product is assumed to be price sensitive and uncertain. In this supply chain setting, the manufacturer chooses the delivery quantity and the retail price to be sold in the market, and the retailer sets the revenue shares.

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Utilizing Nash bargaining model (Nash 1950), we propose the cooperative game models to describe the relationship of payment bargaining between the manufacturer and the retailer and determine the optimal consignment contract with revenue sharing, i.e., the revenue sharing agreement attached with the equilibrium payment scheme. By the fair bargaining between the manufacturer and the retailer instead of adding the additional decision parameters or conditions, we explore the coordination mechanism for the decentralized supply chain under the consignment contract with revenue sharing. Compared with the existing researches on perfect coordination by Cachon (2001) and Gerchak and Wang (2004), the proposed coordination mechanism is more moderate and achievable by a Nash equilibrium contract. Following the measurement of the retailer’s dominance defined by Hua and Li (2008), we also investigate the retailer’s dominance in the two-echelon supply chain under the consignment contract with revenue sharing. Under a very mild restriction on the demand distribution function, we find that the cooperative game between the manufacturer and the retailer has unique Nash equilibrium solution, the manufacturer and the retailer can earn more by their cooperation and the decentralized supply chain is perfectly coordinated. By the theoretical and numerical analyses, we also explore supply chain system parameters (e.g., price-elasticity index, supply chain cost and demand uncertainty) how to affect the supply chain’s optimal decisions and the supply chain performance. This chapter is organized as follows. The assumptions and model are described in Sect. 2. In Sect. 3, the decisions of the centralized supply chain under the consignment contract with revenue sharing are characterized. The cooperative game models to coordinate the decentralized supply chain are provided in Sect. 4, and the impacts of supply chain system parameters on the supply chain performance are also discussed in this section. Detailed numerical results and managerial implications are contained in Sect. 5 and the last section concludes our work.

2 Model Assumptions and Descriptions We consider a two-echelon supply chain consisting of a risk-neutral manufacturer and a risk-neutral retailer, in which a single-period product is produced and sold in market with uncertain and price-sensitive (or price-dependent) demand. The price-sensitive demand is stochastic and can be modeled either in an additive or multiplicative fashion (Petruzzi and Dada 1999; Wang et al. 2004b). The manufacturer produces the single-period product at a constant marginal cost, and the product is sold at a retail price. The retailer incurs a constant cost at the retail stage. In economics and finance, marginal cost is the change in total cost that arises when the quantity produced changes by one unit, and it is typically increasing because of diminishing marginal productivity. Although the increasing marginal costs are common in the practical production deployment (Robert 1997; McAdams and Malone 2005), the constant marginal costs can appear, such as in the stage of

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proportional returns (Rowntree 1941). In literature, it is common that, without loss of generality, the marginal costs for both manufacturer and retailer are typically assumed to be constant (Lariviere and Porteus 2001; Li 2002; Wang et al. 2004b; Hua and Li 2008). Because the costs incurred by the manufacturer and the retailer are constant, the costs incurred by the manufacturer and the retailer can be regarded as a cost sharing agreement.

2.1

Notations and Assumptions

We first introduce the following notations and assumptions to develop the proposed model.

2.1.1

Notations

p D e f() F() h(x) g(x) cM cR c a q r y(p) z Pc(p,q) ðpc ; zc Þ Pd;M Pd;R Pe;M Pe;R

Product’s retail price per unit Price-sensitive demand, i.e., D(p) Random variable with a finite mean value of m and standard deviation value of s Probability density function (PDF) of e ¼ 1 FðÞ Cumulative distribution function (CDF) of e, FðÞ f(x)/[1 F(x)], the failure rate function of demand distribution ¼xh(x), the generalized failure rate function of demand distribution Manufacturer’s marginal production cost Retailer’s marginal cost at the retail stage for inventory handling, shelf-space usage, etc. cM + cR, the total supply chain cost per unit Retailer’s cost share that is incurred at the retail stage, and 1 a is manufacturer’s cost share that is incurred at the production stage Manufacturer’s production quantity for the single-period product Retailer’s revenue share for per unit sold Deterministic and decreasing function of p that captures the dependency between demand and price Stocking factor of inventory Total expected profit of supply chain for any chosen retail price p and production quantity q Supply chain decisions for the centralized supply chain Manufacturer’s expected profit functions in non-cooperative situation Retailer’s expected profit functions in non-cooperative situation Manufacturer’s expected profit functions in cooperative situation Retailer’s expected profit functions in cooperative situation

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ðpd ; zd ; rd Þ ðpe ; ze ; re Þ

Supply chain decisions at non-cooperation Supply chain decisions at cooperation

We add superscript “ values.

2.1.2

*

” to the relative variables to represent their optimal

Assumptions

1. The price-sensitive demand D has the functional forms of D(p) ¼ y(p)e in the multiplicative demand case and D(p) ¼ y(p) + e in the additive demand case. 2. The random variable e is supported on [A, B] with B A 0. 3. F() is strictly increasing, differentiable on [A, B], and F(A) ¼ 0, F(B) ¼ 1 (i.e., there is always some demand in market). 4. y(p) takes the form of y(p) ¼ apb (a > 0, b > 1) in the multiplicative case, and takes the form of y(p) ¼ a bp (a > 0, b > 0) in the additive case. The representations of y(p) are common in the literature, with the former representing an iso-price-elastic demand curve and the later representing a linear demand curve (Lau and Lau 2003; Khouja and Robbins 2005; Hua et al. 2006); see Petruzzi and Dada (1999) for an excellent review and extensions. In the formulation of y(p) ¼ apb, the parameter b is the price-elasticity index of (expected) demand. The larger the value of b, the more sensitive the demand is to a change in price. If the price-elasticity index is greater than 1, then a product is defined as price elastic; if the price-elasticity index is 1 or less, then a product is defined as inelastic. We focus on price-elastic products by assuming b > 1. In the linear demand case, the demand function is quite different from the iso-priceelasticity and multiplicative model. For example, it does not preserve the isoprice-elasticity property. Its price-elasticity index of the (expected) demand is given by bp/(a bp + m). The parameter b here is an indicator of the price “sensitivity” of demand and is closely related to the price-elasticity index. Specifically, the price-elasticity index is increasing in b at any price p. So, one can consider the parameter b as a surrogate of the price-elasticity index. 5. At the end of the selling season, the unsold units bear no salvage value or disposal cost and the unsatisfied demand incurs no loss-of-goodwill cost (i.e., shortage penalty). For the single-period or short life-cycle products, the assumptions of zero salvage value or holding cost and zero loss-of-goodwill cost are appropriate reflections of reality (Goto et al. 2004; Wang et al. 2004a). 6. The demand distribution has the strictly increasing failure rate (IFR) property: h0 (x) > 0, i.e., dh(x)/dx > 0. The IFR assumption is not restrictive because it captures most common distributions, such as the normal, uniform, as well as the gamma and Weibull families, subject to parameter restrictions (Barlow and Proschan 1975). According to Lariviere and Porteus (2001), IGFR (increasing generalized failure rate) is implied by IFR condition (there are IGFR distributions that are not IFR).

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433

The Model

In a decentralized supply chain, the manufacturer produces the product and then sells it to consumers through the retailer under a consignment contract. The consignment contract with revenue sharing adopted by the firms in the supply chain is the outcome of a bargaining process. The manufacturer and the retailer agree to negotiate to make a contract, and both firms accept the shares of the channel profit dictated by the negotiation process, which is subject to individual rationality. Under the consignment contract with revenue sharing, the retailer offers the manufacturer a revenue sharing contract which stipulates the product sold, he keeps r share of the revenue for per unit sold, and remits the rest, i.e., 1 r, to the manufacturer. In the non-cooperative situation, the two firms follow a Stackelberg (i.e., leaderfollower) game: The retailer, acting as the leader, offers the manufacturer a take-itor-leave-it contract, which specifies the percentage allocation of sales revenue between herself and the manufacturer. The manufacturer, acting as a follower, chooses how many units of the product to produce and the retail price. The manufacturer accepts the contract as long as he can earn a positive profit. According to the research of Wang et al. (2004b), we have the optimal decisions in the non-cooperative situation, which is depicted in Table 1. Firms will have market power as buyers, typically, when they have a dominant position in the market, and they use this power to extract favorable prices, terms and/or conditions from suppliers. There is few definition and measurement on the retailer dominance. Loewenstein et al. (2005) investigated the conversational dominance in negotiation by experimental and statistical method. The conversational dominance is measured by three basic measures: deception (misrepresentation of preferences and capabilities), extreme offers (offers outside the range specified for an issue), and quantitative arguments. Dukes et al. (2006) measured the bargaining power by the parameter in the generalized Nash bargaining model to Table 1 The optimal decisions for decentralized supply chain at non-cooperation Demand Iso-price-elastic Linear Condition F is IGFR F is IFR z

Þ ¼ ðb 1Þ½zd Lðzd Þ Fðz d bzd

ð1 rd Þ½a þ zd Lðzd Þ ð1 aÞbc ð1 rd Þ½a þ zd Lðzd Þ þ ð1 aÞbc

ð1 aÞc a þ zd Lðzd Þ þ 2ð1 rd Þ 2b 2 1 þ Fðzd Þ 1 þ rd aðb 2Þ þ 1 þ 2a ð1 aÞ ð1 aÞ rd ¼ r 1 Fðzd Þ 1 rd ba 2 2ðr aÞ½1 þ Fðzd Þ þ d ¼0 Iðrd ; zd Þ Note: Iðrd ; zd Þ ¼ ð1 rd Þ½1 Fðzd Þ2 ð1 rd Þf ðzd Þ½a þ zd Lðzd Þ ð1 aÞbcf ðzd Þ; a < rd < 1 ð1 aÞbc=ða þ AÞ

p

pd ¼

bczd 1a ðb 1Þ½zd Lðzd Þ 1 rd

Fðzd Þ ¼ pd ¼

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represent the manufacturer’s relative bargaining power. Canie¨ls and Gelderman (2007) observed the supplier dominance in the strategic quadrant of the Kraljic matrix, which is measured as the difference between supplier’s dependence and buyer’s dependence. In this chapter, we assume that the manufacturer and the retailer are devoted to pursue the long-range relationship, their bargaining and deal is repeated many times and both players must consider not only their short-term gains but also their longterm payoffs. Therefore, the expected profits are their suitable objectives in the business operations. The retailer’s dominance in a supply chain is defined as: d rm ¼

PR PM PR

ðif PR PM 0Þ;

(1)

in (1), drm denotes the retailer’s dominance over the manufacturer. PM and PR are the manufacturer’s and the retailer’s expected profits, respectively. The retailer’s dominance does not have a generalized description and definition in the existing researches, such as qualitative analysis and statistical hypothesis testing. The proposed measure of retailer’s dominance provides a clear and closed-form definition to describe the relative power or dominance between the manufacturer and the retailer. We employ cooperative game models in multiplicative and additive demand cases to show how to design an equilibrium contract and how to perfectly coordinate the decentralized supply chain through a cooperative bargaining process. The purpose of cooperation is to actually determine a channel profit allocation scheme between the manufacturer and the retailer. It should be noted that not all optimal profit sharing schemes are acceptable, neither the manufacturer nor the retailer would be willing to accept less profit at cooperation than at noncooperation. An optimal payment scheme ðPe;M ; Pe;R Þ at cooperation is called acceptable if DPe;M ¼ Pe;M Pd;M 0, and DPe;R ¼ Pe;R Pd;R 0, then the acceptable decision set Z can be defined as n o Z ¼ ðp; z; rÞjDPe;M ðp; z; rÞ ¼ Pe;M Pd;M 0; DPe;R ðp; z; rÞ ¼ Pe;R Pd;R 0 : (2) Equation (2) means that the optimal decisions ðpe ; ze ; re Þ belong to a point set, and the set of optimal decisions satisfies the individual rationality, which implies that both players earn more profits by their cooperation in the decentralized supply chain, i.e., DPe;M ðp; z; rÞ ¼ Pe;M Pd;M 0 and DPe;R ðp; z; rÞ ¼ Pe;R Pd;R 0. Suppose the manufacturer’s utility function of DPe;M is u1 ðÞ and the retailer’s utility function of DPe;R is u2 ðÞ. Then according to Nash bargaining model, the optimal bargaining payment scheme is obtained by solving the following problem: max u1 ðDPe;M Þ u2 ðDPe;R Þ:

ðp;z;rÞ2Z

(3)

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Based on the assumption that both the manufacturer and the retailer are riskneutral, the cooperative game model (3) can be formulized as max ðPe;M Pd;M ÞðPe;R Pd;R Þ:

ðp;z;rÞ2Z

(4)

We next study the manufacturer-retailer relationships and coordination decisions for the decentralized supply chain by conducting the proposed cooperative game model. We also investigate the impacts of demand uncertainty and supply chain cost share on the retailer’s dominance.

3 Decisions in Centralized Supply Chain We first characterize the optimal decision to the centralized supply chain, in which the retail price and the production quantity for the product are simultaneously chosen by a central decision-maker to maximize the total expected profit of supply chain, and this optimal decision has to be made before demand realization. For the single-period product problem, we have Pc ðp; qÞ ¼ cq þ pE½minfq; Dg:

3.1

(5)

The Iso-Price-Elastic Demand Case

For the iso-price-elastic demand model, DðpÞ ¼ yðpÞ e and yðpÞ ¼ apb , we then have Pc ðp; qÞ ¼ cq þ pE½minfq; yðpÞ eg:

(6)

Following Petruzzi and Dada (1999), we define z q=yðpÞ. This transformation of variables provides an alternative interpretation of the stocking decision: if z > e, then leftovers occur; if z < e, then shortages occur. Then, the problem of choosing a retail price p and a production quantity q is equivalent to choosing a retail price p and a stocking factor z, and (6) is equivalent to (7) Pc ðp; zÞ ¼ yðpÞfp½z LðzÞ czg; Rz where yðpÞ ¼ apb , and LðzÞ ¼ A ðz xÞf ðxÞdx. Wang et al. (2004b) provided the optimal decision ðpc ; zc Þ for the centralized supply chain with the iso-price-elastic demand: if F is IGFR, the centralized supply chain has unique optimal decision ðpc ; zc Þ, where zc is uniquely determined by Þ bczc c Þ ¼ ðb1Þ½zc Lðz ; pc ¼ ðb1Þ½z Lðz Fðz Þ : c bz c

c

c

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The Linear Demand Case

For the linear demand model, DðpÞ ¼ yðpÞ þ e and yðpÞ ¼ a bp. Since p c, it is assumed that a bc þ A 0 to ensure the non-negative demand when the product is priced at cost, i.e., p ¼ c. By substituting DðpÞ ¼ yðpÞ þ e and yðpÞ ¼ a bp into (5), the expected profit function of the centralized supply chain can be written as Pc ðp; qÞ ¼ cq þ pE½minfq; yðpÞ þ eg:

(8)

As the iso-price-elastic demand model, z q yðpÞ is defined as the stocking factor, then (8) can be rewritten as Pc ðp; zÞ ¼ ðp cÞða bpÞ þ p½z LðzÞ cz:

(9)

The following theorem is about the optimal decision ðpc ; zc Þ to the iso-priceelastic demand case. Theorem 1. For any fixed z 2 ½A; B, if 2h2 ðzÞ þ dhðzÞ=dz > 0, the centralized supply chain has a unique optimal decision ðpc ; zc Þ, where zc is uniquely deteraþz Lðz Þbc

mined by Fðzc Þ ¼ aþzc Lðzc Þþbc , and pc ðzc Þ ¼ c

c

aþzc Lðzc Þþbc . 2b

Proof. For any fixed z 2 ½A; B, it follows from (9) that @Pc ðp; zÞ ¼ a þ bc 2bp þ ½z LðzÞ ¼ 0; @p and @Pc ðp; zÞ ¼ p½1 FðzÞ c ¼ 0: @z By solving the above equations, we can obtain the feasible solution ðpc ; zc Þ aþzc Lðzc Þbc and pc ðzc Þ ¼ bcþaþz2bc Lðzc Þ . satisfying Fðzc Þ ¼ aþz c Lðzc Þþbc c Lðzc Þbc We next prove that zc is the unique solution to Fðzc Þ ¼ aþz aþzc Lðzc Þþbc . Let GðzÞ ¼ ½a þ z LðzÞ½1 FðzÞ bc½1 þ FðzÞ, then we have dGðzÞ 1 FðzÞ ; ¼ f ðzÞ a þ z LðzÞ þ bc dz hðzÞ 0 d2 GðzÞ f ðzÞ dGðzÞ f ðzÞ dhðzÞ 2 ½1 FðzÞ 2h ¼ ðzÞ þ : þ 2 dz2 f ðzÞ dz h ðzÞ dz d2 GðzÞ > 0, then It is obvious that, if 2h2 ðzÞ þ dhðzÞ dz dz2 dGðzÞ=dz¼0 < 0, which implies that GðzÞ itself is a unimodal function.

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Therefore zc is the unique solution since GðzÞ is continuous on ½A; B, GðAÞ ¼ a þ A bc > 0 and GðBÞ ¼ 2bc < 0. It follows from (9) that @ 2 Pc ðp; zÞ ¼ 2b; @p2

@ 2 Pc ðp; zÞ ¼ pf ðzÞ and @z2

@ 2 Pc ðp; zÞ ¼ 1 FðzÞ: @z@p

We then have 2 2 @ 2 Pc ðp; zÞ @ 2 Pc ðp; zÞ @ Pc ðp; zÞ ¼ 2bpf ðzÞ ½1 FðzÞ2 : @p2 @z2 @z@p

(10)

Equation (10) implies that ðpc ; zc Þ is the optimal solution to (9) if 2bpf ðzÞ ½1 FðzÞ2 > 0 at

ðpc ; zc Þ:

(11)

c Þþbc Substituting pc ¼ aþzc Lðz into (11), and we have 2b

ÞfFðz c Þ hðzc Þ½a þ zc Lðzc Þ þ bcg < 0: Fðz c Because GðzÞ is a unimodal function, GðAÞ > 0 and GðBÞ < 0, therefore c Þ hðzc Þ½a þ zc Lðzc Þ þ bc < 0, which means dGðzÞ=dz < 0 at z ¼ zc , and Fðz 2 that (11) can be met if 2h ðzÞ þ dhðzÞ=dz > 0. @ 2 Pc ðp;zÞ So we can have the following principal minors at ðp ; z Þ: < 0, 2 c c @p ðpc ;zc Þ 2

2 2 2 2 @ Pc ðp;zÞ @ Pc ðp;zÞ @ Pc ðp;zÞ @ Pc ðp;zÞ < 0, and > 0, then the Hessian 2 2 2 @z @p @z @z@p ðp ;z Þ c

ðpc ;zc Þ

c

matrix of Pc ðp; zÞ is negative definite at ðpc ; zc Þ Thus, if 2h2 ðzÞ þ dhðzÞ=dz > 0, there is a unique optimal contract ðpc ; zc Þ to the centralized supply chain. □ It is obvious that, if F is IFR, i.e., dhðzÞ=dz > 0, 2h2 ðzÞ þ dhðzÞ=dz > 0 for 8z 2 ½A; B. Theorem 1 indicates that it does not need any requirement on parameters other than the demand distribution itself to determine the optimal consignment contract with revenue sharing for the centralized supply chain. The following proposition describes how the optimal decision ðpc ; zc Þ changes with the supply chain system parameters b and c. Proposition 1. If 2h2 ðzÞ þ dhðzÞ=dz > 0, then (i) zc is decreasing in b, and is also decreasing in c. (ii) pc is decreasing in b, and its monotone property in c depends on b. Proof. (i) According to Theorem 1, we have Gðzc Þ ½a þ zc Lðzc Þ½1 Fðzc Þ bc½1 þ Fðzc Þ ¼ 0

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From the chain rule and differentiation of implicit functions, we have dzc c½1 þ Fðzc Þ ¼ : db @Gðzc Þ=@zc From the proof of Theorem 1, GðzÞ is a unimodal function, GðAÞ > 0 and GðBÞ < 0, and zc is the unique solution to Gðzc Þ ¼ 0, then @Gðzc Þ=@zc < 0. Therefore, if 2h2 ðzÞ þ dhðzÞ=dz > 0, dzc =db < 0, which means that zc is decreasing in b. For the same reason zc is also decreasing in c. (ii) From the chain rule and differentiation of implicit functions, we have dpc @pc @pc dzc ¼ þ : db @b @zc db Since pc ðzc Þ ¼

aþzc Lðzc Þþbc , 2b

we have

@pc a þ zc Lðzc Þ < 0 and ¼ @b 2b2

@pc 1 Fðzc Þ ¼ > 0: @zc 2b

Then dpc =db < 0, which means that pc is decreasing in b. From the chain rule and differentiation of implicit functions, we also have dpc @pc @pc dzc ¼ þ : dc @c @zc dc @p

@p

1Fðz Þ

dz

b½1þFðz Þ

c so It can be derived that @cc ¼ 12 , @zc ¼ 2b c and dcc ¼ @Gðz Þ=@z , c c c dpc Lðzc Þ dc ¼ 2 @Gðzc Þ=@zc , where Lðzc Þ ¼ Fðzc Þ þ bchðzc Þ 1. Because @Gðzc Þ=@zc < 0, the sign of dpc /dc isn determined by Lðz oc Þ.

1þFðzC Þ , Vc LðzC Þ

Since 1 FðzVc Þ ¼ bc aþz

1þFðz Þ

c Lðzc Þ ¼ bc hðzc Þ aþz Lðz Þ , which means c

c

aþz Lðz Þbc

that sign of Lðzc Þ is determined by zc . According to Fðzc Þ ¼ aþzc Lðzc Þþbc , zc is c c uniquely determined by b for any fixed c. □ Therefore, the monotone property of pc in c depends on b. Proposition 1 shows that, the price-elasticity index b affects the optimal stocking factor and the optimal retail price in the linear demand case as it does in the multiplicative demand case, but the effects of the total supply chain cost per unit c in the optimal stocking factor and the optimal retail price are different in the isoprice-elastic and the linear demand cases.

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4 Decentralized Supply Chain Coordination In this section, we investigate the coordination decisions for the decentralized supply chain in multiplicative and additive demand cases.

4.1

The Iso-Price-Elastic Demand Case

For the decentralized supply chain, we define the stocking factor of inventory as z q=yðpÞ. Hence the problem of choosing p, q and r is equivalent to choosing p, z and r. The manufacturer’s and the retailer’s expected profits can be written as Pe;M ðp; z; rÞ ¼ ð1 aÞcq þ ð1 rÞpE½minðq; DÞ

(12)

¼ yðpÞfð1 rÞp½z LðzÞ ð1 aÞczg; Pe;R ðp; z; rÞ ¼ acq þ rpE½minðq; DÞ ¼ yðpÞfrp½z LðzÞ aczg:

(13)

Substituting (12) and (13) into the cooperative game model (4), and we have the following conclusion from its first-order and second-order conditions. Theorem 2. For any fixed z 2 ½A; B, if F is IGFR, the cooperative game model (4) has a unique game equilibrium solution ðpe ; ze ; re Þ, where ze is uniquely determined by Þ ¼ ðb 1Þ½ze Lðze Þ ; Fðz e bze

pe ¼

(14)

bcze ; ðb 1Þ½ze Lðze Þ

(15)

and re ¼ ðb aÞb ½2ðb 1Þa þ 1 þ

ðb 1Þb1 ½ðb 2Þa þ 1 2bðb aÞb

:

(16)

Proof. By solving the first order conditions of cooperative game model (4) in the iso-price-elastic demand case, we have the feasible solutions ðpe ; ze ; re Þ, which are determined by the following equations, e Þ ¼ ðb 1Þ½ze Lðze Þ ; Fðz bze

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pe ¼

bcze ; ðb 1Þ½ze Lðze Þ

and re ¼ ðb aÞb ½2ðb 1Þa þ 1 þ

ðb 1Þb1 ½ðb 2Þa þ 1 2bðb aÞb

:

Following the proof of Theorem 1 in Wang et al. (2004b), there is a unique ze 2 e Þ ¼ ðb1Þ½ze Lðze Þ . Then, the feasible solution ðp ; z ; r Þ is ðA; BÞ satisfying Fðz e e e bze unique. From the second order conditions, the optimality conditions of the cooperative game model (4) are equivalent to (see, Gottfried and Weisman 1973): e Þ2 bcze f ðze Þ < 0; pe ½Fðz D ¼ Pe;M Pd;M ¼ Pe;R Pd;R > 0: From the proof of Theorem 1 in Wang et al. (2004b), 1 bze hðze Þ < 0 if F is IGFR. bcze e Þ Þ ¼ ðb1Þ½ze Lðz and pe ¼ ðb1Þ½z Lðz Since Fðz Þ , we then have pe Fðze Þ ¼ c; e bze e e thus, Þ2 bcz f ðz Þ ¼ cFðz Þ½1 bz hðz Þ < 0: pe ½Fðz e e e e e e Þ2 bcz f ðz Þ < 0 always holds if F is IGFR. It means that pe ½Fðz e e e From the optimal decision ðpd ; zd ; rd Þ and the unique feasible solution ðpe ; ze ; re Þ, we have D ¼ Pe;M Pd;M ¼ Pe;R Pd;R ¼ aczd ðpd Þb

ðb aÞb þ ðb 1Þb1 ½bða 2Þ þ 1 2ðb 1Þbþ1

:

Let gðaÞ ¼ ðb aÞb þ ðb 1Þb1 ½bða 2Þ þ 1, it can be verified that g0 ðaÞ ¼ b½ðb 1Þb1 ðb aÞb1 < 0, and lim gðaÞ ¼ 0. a!1

Because 0 < a < 1 and g0 ðaÞ < 0, so we have gðaÞ > 0. This implies that D > 0, and D ¼ Pe;M Pd;M ¼ Pe;R Pd;R > 0 holds. According to the above proofs, the unique feasible solution ðpe ; ze ; re Þ is optimal to the cooperative game model (4) if F is IGFR. □ From the proof of Theorem 2, the unique equilibrium solution ðpe ; ze ; re Þ always meets the acceptable decision condition Z when the demand distribution F is IGFR.

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This reveals that both the manufacturer and the retailer are better off at cooperation than at non-cooperation. Form the proof procedures of Theorem 2, if d½zhðzÞ=dz ¼ hðzÞ þ zdhðzÞ=dz > 0, the supply chain members can obtain more profits from their cooperation than those from non-cooperation, i.e., Pe;M > Pd;M , Pe;R > Pd;R ; and their incremental profits are equal, i.e., Pe;M Pd;M ¼ Pe;R Pd;R . The following proposition characterizes the optimal decision in the cooperative situation, its relationship to the optimal decision in the non-cooperative situation and to the optimal decision of the centralized supply chain. Proposition 2. In the decentralized supply chain with the iso-price-elastic demand model, (i) ze is independent of re , c, or a, and pe is independent of re or a. (ii) ze ¼ zc ¼ zd . (iii) pe ¼ pc < pd . (iv) re only depends on b and a, and re < rd . Proof. (i) From Theorem 2, we have ze is independent of re , c, or a, and pe is independent of re or a. (ii) Since there is a unique solution z to z bzFðzÞ þ ðb 1ÞLðzÞ ¼ 0, and compar e Þ d Þ c Þ Þ ¼ ðb1Þ½zc Lðz Þ ¼ ðb1Þ½zd Lðz e Þ ¼ ðb1Þ½ze Lðz with Fðz and Fðz , ing Fðz c d bzc bzd bze then ze ¼ zc ¼ zd . bcze bczc (iii) Since pe ¼ ðb1Þ½z Lðz Þ and pc ¼ ðb1Þ½z Lðz Þ , then pe ¼ pc . From Theorem e e c c 1 and Theorem 2 in Wang et al. (2004b), pc < pd . Then, pe ¼ pc < pd . (iv) It is obvious that re only depends on b and a. We have re

rd

" # 2ðb 1Þa þ 1 ðb 1Þb1 aðb 2Þ þ 1 ¼ 1 þ b1 2b ba 2bðb aÞ

0, which implies that b is 2ðbaÞbþ1 b

db ðb 1Þb1 ðb aÞ½aðb 2Þ þ 1 að1 aÞðb 1Þ b1 : ¼ þ log db ðb aÞ½aðb 2Þ þ 1 ba 2ðb aÞbþ1 að1aÞðb1Þ þ log Let lðbÞ ¼ ðbaÞ½aðb2Þþ1

b1 ba , we have

Supply Chain Coordination Under Consignment Contract with Revenue Sharing

l0 ðbÞ ¼

ð1 aÞ2 ðb 1Þðb aÞ2 ½ðb 2Þa þ 12

443

ðbÞ;

where ðbÞ ¼ ½bðb 5Þ þ 5a2 þ 2½bðb 1Þ 1a þ b. For ðbÞ, we have 0 ðbÞ ¼ 1 þ að4b þ 2ba 5a 2Þ > ð3a þ 1Þð1 aÞ > 0 for b > 1, hence, ðbÞ > ð1Þ ¼ a2 2a þ 1 > 0. Therefore we have l0 ðbÞ > 0 for b > 1. Since lim lðbÞ ¼ 0, lðbÞ < 0 for b > 1. b!1 Thus, db=db < 0, which implies that b is decreasing in b. (ii) From (20), we have 1 ðb 1Þb1 1 lim b ¼ þ > ; a!0 2bb 2 2 and 1 ðb 1Þ lim b ¼ þ ¼ 1: 2 2ðb 1Þ

a!1

(iii) From (20), we have b¼

Pe;R 1 ðb 1Þb1 ½ðb 2Þa þ 1 þ ¼ ; Pe;M þ Pe;R 2 2ðb aÞb

then 1 ð1 aÞ ¼ 1; lim b ¼ þ b!1 2 2ð1 aÞ and lim b ¼ 12 þ lim eb log½ba lim b1

b!1

b!1

b!1

n

ðb2Þaþ1 2ðbaÞ

o

ð1aÞ

¼ 12 þ e

a

2

> 12 .

□

Proposition 3 illustrates the behavior of the retailer’s profit share b. It is showed that the retailer can always extract more than 50% of the channel profit even if he does not incur any portion of the channel cost. From (20), the retailer’s and the manufacturer’s profit shares are not affected by demand uncertainty. Following the measurement of the retailer’s dominance defined in (1), at the isoprice-elastic demand case, the retailer’s dominance is as follows: drm ¼

2ðb 1Þb1 ½ðb 2Þa þ 1 ðb aÞb þ ðb 1Þb1 ½ðb 2Þa þ 1

:

(21)

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The following Proposition 4 describes the impacts of supply chain parameters b and a on the retailer’s dominance. It indicates that the retailer is always dominant over the manufacturer in the decentralized supply chain, and is the same as the meanings of Proposition 3, i.e., the retailer can always extract more than 50% of channel profit. Proposition 4. With the iso-price-elastic demand model, the retailer’s dominance d rm (i) is increasing in a for any given b > 1, and decreasing in b for any given a. 2 (ii) approaches 1þðb1Þ 1b b > 0 as a ! 0 and 1 as a ! 1. b a

(iii) approaches e 2ae + aea > 0 as b ! 1 and 1 as b ! 1. Proof. (i) From (21), we have ddrm 2ðb 1Þbþ2 ðb aÞb1 ½ðb 2Þa þ b ; ¼ 2 da fðb 1Þðb aÞb þ ðb 1Þb ½ðb 2Þa þ 1g and ddrm 2ðb 1Þbþ1 ðb aÞb ½aðb 2Þ þ 1lðbÞ ; ¼ 2 db fðb 1Þðb aÞb þ ðb 1Þb ½ðb 2Þa þ 1g where lðbÞ ¼

að1 aÞðb 1Þ b1 : þ log ðb aÞ½aðb 2Þ þ 1 ba

Since b > 1 and 0 < a < 1, ðb 2Þa þ b ¼ ðb 1Þa þ ðb aÞ > 0, then ddrm =da > 0. From the proof of Proposition 3, lðbÞ < 0 for 8b > 1, then dd rm =db < 0: (ii) From (21), we have lim d rm ¼

a!0

2ðb 1Þb1 bb

b1

þ ðb 1Þ

¼

2 1 þ ðb 1Þ1b bb

;

and lim d rm ¼ 1:

a!1

(iii) From (21), we have lim drm ¼ 1,

e

b!1 2aea + aea .

and

lim d rm ¼ lim

b!1

b!1 1þ

2 ðbaÞb ðb1Þb1 ½ðb2Þaþ1

¼

2 1þ lim

ðbaÞb

lim

ðb1Þ

b!1 ðb1Þb b!1 ðb2Þaþ1

¼ □

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Proposition 4 implies that the retailer dominance d rm is always greater than zero, that is to say that the retailer is dominant over the manufacturer in the decentralized supply chain under the consignment contract with revenue sharing. From (21), the retailer dominance is not affected by demand uncertainty, and is related to supply chain parameters b and a in the iso-price-elastic demand case.

4.2

The Linear Demand Case

In the linear demand case, the stocking factor of inventory is defined as z ¼ q y(p), then the problem of choosing p, q and r is equivalent to choosing p, z and r. The manufacturer’s and the retailer’s expected profit functions are Pe;M ðp; zÞ ¼ ð1 aÞcq þ ð1 rÞpE½minfq; Dg ¼ ð1 rÞp½a bp þ z LðzÞ ð1 aÞc½a bp þ z; Pe;R ðp; zÞ ¼ acq þ rpE½minfq; Dg ¼ rp½a bp þ z LðzÞ ac½a bp þ z:

(22)

(23)

Substituting (22) and (23) into the cooperative game model (4), and solving the optimal problem of cooperative game model (4), we have the following conclusion. Theorem 3. For any fixed z 2 ½A; B, if F is IFR, the cooperative game model (4) has a unique game equilibrium solution ðpe ; ze ; re Þ, where ze is uniquely determined by Fðze Þ ¼

a þ ze Lðze Þ bc ; a þ ze Lðze Þ þ bc

(24)

pe ðze Þ ¼

a þ ze Lðze Þ þ bc ; 2b

(25)

and Þ ð1 aÞð2r 1Þ Fðz Þ 2a 1 r a z z þ Lðz Þ Lðz Þ 1 Fðz e d e d e d d d : þ re ¼ þ Þ bc 2 Fðze Þ 2ð1 rd Þ 4 1 rd Fðz d (26) Proof. By solving the first order conditions of cooperative game model (4) in the linear demand case, we can have the feasible solutions, which are determined by the following equations:

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Fðze Þ ¼

a þ ze Lðze Þ bc ; a þ ze Lðze Þ þ bc

(27)

pe ðze Þ ¼

a þ ze Lðze Þ þ bc ; 2b

(28)

and e Þ ð1 aÞð2r 1Þ Fðz Þ 2a 1 r a z ze þ Lðz Þ Lðze Þ 1 Fðz d d d d d re ¼ þ : Þ þ 4 2 Fðze Þ 2ð1 rd Þ 1 rd bc Fðz d (29) According to Theorem 1, if 2h2 ðzÞ þ dhðzÞ=dz > 0, there is a unique optimal e Lðze Þbc solution ze 2 ðA; BÞ satisfying Fðze Þ ¼ aþz aþze Lðze Þþbc . Because zd and rd are unique, (27)–(29) determine a unique feasible solution ðpe ; ze ; re Þ if dhðzÞ=dz > 0. It is known that, if the Hessian matrix of cooperative game model is a negative definite matrix at ðpe ; ze ; re Þ, ðpe ; ze ; re Þ is the optimal solution to the cooperative game model (4). The Hessian matrix is a negative definite matrix at ðpe ; ze ; re Þ if the following two conditions are satisfied (see, Gottfried and Weisman 1973): ½1 Fðze Þ2 2bpe f ðze Þ < 0;

(30)

Pe;M Pd;M ¼ Pe;R Pd;R > 0:

(31)

Let Rðpe ; ze Þ ¼ 2bpe f ðze Þ ½1 Fðze Þ2 , (28) gives 1 Fðze Þ : Rðpe ; ze Þ ¼ f ðze Þ a þ ze Lðze Þ þ bc hðze Þ

(32)

From the proof of Theorem 1, we have GðzÞ ¼ ½a þ z LðzÞ½1 FðzÞ bc ½1 þ FðzÞ is a unimodal function if 2h2 ðzÞ þ dhðzÞ=dz > 0. dGðzÞ Sincen GðAÞ ¼ a þ A bc > 0 and GðBÞ ¼ 2bc < 0, o dz < 0, i.e., f ðze Þ a þ ze Lðze Þ þ bc

1Fðze Þ hðze Þ

z¼ze

< 0.

Therefore, Rðpe ; ze Þ > 0, and (30) holds if 2h2 ðzÞ þ dhðzÞ=dz > 0. From the first order optimality conditions, it can be derived that Pe;M Pd;M ¼ Pe;R Pd;R ¼

ðPe;M þ Pe;R Þ ðPd;M þ Pd;R Þ : 2

From Table 1, we have 1a ½a þ zd Lðzd Þ½1 Fðzd Þ gbc½1 þ Fðzd Þ ¼ 0, where g ¼ 1r > 1. d

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Let Yðzd Þ ¼ ½a þ zd Lðzd Þ½1 Fðzd Þ gbc½1 þ Fðzd Þ; then dYðzd Þ 1 Fðzd Þ ; ¼ f ðz Þ a þ z Lðz Þ þ gbc d d d dzd hðzd Þ and 0 d2 Yðzd Þ f ðzd Þ dYðzd Þ f ðzd Þ dhðzd Þ 2 ½1 Fðz : ¼ þ Þ 2h ðz Þ þ d d f ðzd Þ dzd h2 ðzd Þ dzd dzd 2 If 2h2 ðzÞ þ dhðzÞ dz > 0, then d2 Yðzd Þ f ðz Þ ¼ h2 ðzd Þ ½1 Fðzd Þ½2h2 ðzd Þ þ dhðzd Þ=dzd < 0, dz 2 d

dYðzd Þ=dzd ¼0

which

d

implies that Yðzd Þ is a unimodal function. 1a and g ¼ 1r Recall that a < rd < 1 ð1aÞbc , so we have YðAÞ ¼ a þ A aþA d

gbc > 0. Since Yðzd Þ is continuous on ½A; B, and YðBÞ ¼ 2gbc < 0, thus zd is the unique solution to Yðzd Þ ¼ 0 and @Yðzd Þ=@zd zd if g > 1. It can be directly observed that Pc , Pd;M þ Pd;R and Pe;M þ Pe;R have the same function form. According to Theorem 1, Pc achieves its maximum if and only if ðzc ; pc Þ ¼ ðzc ; pc Þ. It can also be observed from their expressions that ðze ; pe Þ ¼ ðzc ; pc Þ, thus ðze ; pe Þ is the unique optimal solution to Pe;M þ Pe;R . Because ze > zd , so we have Pe;M þ Pe;R > Pd;M þ Pd;R , and (31) holds. Therefore, the optimal solution ðpe ; ze ; re Þ determined by (24)–(26) is the Nash equilibrium solution to the cooperative game model (4) if dhðzÞ=dz > 0. □ In Theorem 3, zd and rd are the optimal stocking factor of inventory and the optimal revenue share of the decentralized supply chain in the linear demand case respectively, see Table 1. From the proof of Theorem 3, the unique equilibrium solution ðpe ; ze ; re Þ always meets the acceptable decision condition Z if the demand distribution F is IFR. This also reveals that both the manufacturer and the retailer are better off at cooperation than at non-cooperation in the linear demand case, i.e., Pe;M > Pd;M and Pe;R > Pd;R , which is the same as that in the iso-price-elastic demand case. Since the expressions of pe and ze are the same as those of pc and zc respectively, we have the following properties about pe and ze according to Proposition 1: if F is IFR, then ze is decreasing in b , and is also decreasing in c; pe is decreasing in b, and may be increasing, constant or decreasing in c. The following Proposition 5 characterizes the characters of the optimal decision in the cooperative situation, its relationship to the optimal decision in the noncooperative situation and to the optimal decision of the centralized supply chain.

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Proposition 5. In the decentralized supply chain with the linear demand model, (i) ze and pe do not depend on re or a, but depends on b and c; re depends on b, c and a. (ii) pe ¼ pc . (iii) ze ¼ zc > zd . Proof. (i)–(ii) They can be easily verified from the expressions of ðpd ; zd ; rd Þ, ðpe ; ze ; re Þ and ðpc ; zc Þ. (iii) The uniqueness of ze and zc gives ze ¼ zc . According to the proof of Theorem 3, we have ze > zd . Therefore ze ¼ zc > zd . □ It follows from Parts (ii) and (iii) of Proposition 5 that ðpe ; ze Þ ¼ ðpc ; zc Þ, which means that the decentralized supply chain with the linear demand model is perfectly coordinated under the consignment contract with revenue sharing ðpe ; ze ; re Þ. Since it is difficult to provide the closed forms of the manufacturer’s and the retailer’s expected profits, we cannot describe the impacts of parameters b, c and demand uncertainty s on supply chain decisions, b and drm analytically. We perform numerical studies to investigate the impacts.

5 Numerical Analysis To better illustrate the ideas in this chapter, we develop a numerical sample for the linear demand case. In the numerical studies, we assume that e ~ N(10,5), a ¼ 100, c ¼ 1.0. Given a ¼ 0.25, the optimal solutions can be obtained by applying Theorem 3 at a given b. The manufacturer’s and the retailer’s expected profits can then be computed numerically as well as the retailer’s profit share and dominance. When b varies from 1 to 80, the computational results are reported in Table 2. As shown in Propositions 3 and 4, we let a ¼ 0 and a ¼ 0.999 to analyze the behaviors of the retailer’s profit share b and dominance drm as a ! 0 and a ! 1, respectively. The computational results are reported in the following Tables 3 and 4. Results in Tables 2–4 show that, in the linear demand case, the revenue share r, the retail price p and the production quantity q is decreasing in b for any given a. Furthermore, the retailer’s profit share b and dominance drm are also decreasing in b for any given a, these observations are similar to Propositions 3 and 4 in the isoprice-elastic demand case. From Propositions 3 and 4, the retailer can always extract more than 50% of the channel profit in the iso-price-elastic demand case; however, in the linear demand case, the manufacturer can earn more than the retailers. For example, as shown in Table 3, when b > 36.07 (it corresponds to the italic and bold face), the retailer’s profit share is less than 0.5 and its dominance is negative. Table 4 shows that the retailer’s profit share and dominance approach 1 as a ! 1, which is similar to the

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Table 2 The optimal solutions in linear demand case, e ~ N(10,5), a ¼ 100, c ¼ 1.0, a ¼ 0.25 b

zd

1 13.518 5 11.247 10 10.057 15 9.255 20 8.616 25 8.064 30 7.566 40 6.656 50 5.792 80 2.804

rd pd 0.950 62.325 0.863 13.621 0.790 7.198 0.731 4.991 0.681 3.863 0.637 3.175 0.597 2.710 0.527 2.119 0.466 1.757 0.321 1.195

ze 20.491 16.804 14.835 13.520 12.490 11.618 10.849 9.491 8.262 4.402

re pe qe Pe;M Pe;R b drm 0.827 55.738 64.753 479.028 2,507.213 0.840 0.809 0.705 11.522 59.194 134.528 412.508 0.754 0.674 0.627 5.997 54.865 70.754 174.030 0.711 0.593 0.572 4.155 51.195 45.651 99.750 0.686 0.542 0.530 3.233 47.830 31.959 64.644 0.669 0.506 0.496 2.680 44.618 23.333 44.656 0.657 0.477 0.466 2.312 41.503 17.441 32.002 0.647 0.455 0.417 1.850 35.492 10.054 17.353 0.633 0.421 0.377 1.572 29.638 5.800 9.572 0.623 0.394 0.290 1.151 12.295 0.681 0.963 0.586 0.293

Table 3 The optimal solutions in linear demand case, e ~ N(10,5), a ¼ 100, c ¼ 1.0, a ¼ 0 b

zd

rd

pd

ze

re

pe

Pe;M

qe

Pe;R

b

drm

1.00 13.173 0.940 63.061 20.491 0.740 55.738 64.753 729.604 2,256.637 0.756 0.677 5.00 10.867 0.833 13.857 16.804 0.582 11.522 59.192 194.331 352.705 0.645 0.449 10.00 9.664 0.741 7.337 14.835 0.482 5.997 54.867 100.420 144.365 0.590 0.304 15.00 8.858 0.667 5.090 13.520 0.413 4.155 51.200 64.203 81.198 0.558 0.209 20.00 8.217 0.603 3.940 12.490 0.360 3.233 47.822 44.675 51.928 0.538 0.140 25.00 7.667 0.545 3.237 11.618 0.315 2.680 44.610 32.470 35.519 0.522 0.086 30.00 7.172 0.493 2.760 10.849 0.278 2.312 41.503 24.183 25.260 0.511 0.043 36.07 6.616 0.434 2.354 10.003 0.238 2.001 37.830 17.205 17.205 0.500 0.000 40.00 6.272 0.399 2.154 9.491 0.215 1.850 35.492 13.863 13.544 0.494 0.024 50.00 5.421 0.316 1.781 8.262 0.165 1.572 29.638 7.965 7.407 0.482 0.075 80.00 2.501 0.107 1.200 4.402 0.051 1.151 12.295 0.933 0.712 0.433 0.309

Table 4 The optimal solutions in linear demand case, e ~ N(10,5), a ¼ 100, c ¼ 1.0, a ¼ 0.999 b

zd

rd

pd

ze

re

pe

qe

Pe;M

Pe;R

b

d rm

15.00 20.00 25.00 30.00 40.00 50.00 80.00

13.459 12.449 11.590 10.827 9.478 8.253 4.398

0.999 0.999 0.999 0.999 0.999 0.999 0.999

4.163 3.238 2.684 2.314 1.851 1.573 1.151

13.520 12.490 11.618 10.849 9.491 8.262 4.402

0.999 0.999 0.999 0.999 0.999 0.999 0.999

4.155 3.233 2.680 2.312 1.850 1.572 1.151

51.200 47.822 44.610 41.503 35.492 29.638 12.295

0.048 0.052 0.044 0.036 0.022 0.013 0.001

145.353 96.551 67.945 49.407 27.385 15.359 1.644

1.00 1.00 1.00 1.00 1.00 1.00 1.00

1.00 1.00 1.00 1.00 1.00 1.00 1.00

results in Propositions 3 and 4. One issue should be illuminated in Table 4, when b is small (e.g., b 10), the manufacturer’s profit is negative at cooperation, and then the manufacturer and the retailer will not bargain to cooperate. For more detailed illustrations of impacts of price sensitivity b on the retailer’s profit share and dominance, we give another example that e ~ N(10,5), a ¼ 100, c ¼ 1.0 and a ¼ 0.5. The optimal solutions of the supply chain and the expected profits of the manufacturer and the retailer are then computed numerically as well as

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Table 5 The optimal solutions in linear demand case, e ~ N(10,5), a ¼ 100, c ¼ 1.0, a ¼ 0.5 b

zd

10.00 11.161 20.00 8.804 30.00 7.060 40.00 5.527 50.00 4.046 60.00 2.506

rd 0.838 0.760 0.702 0.656 0.617 0.583

pd 6.994 3.731 2.611 2.039 1.690 1.454

ze 17.809 14.036 11.381 9.167 7.140 5.137

re 0.760 0.690 0.643 0.608 0.580 0.557

pe 6.079 3.258 2.318 1.847 1.563 1.373

qe 57.015 48.878 41.852 35.293 28.972 22.761

Pe;M Pe;R 45.662 206.125 21.219 77.024 11.777 37.986 6.879 20.388 4.012 11.052 2.240 5.720

b 0.819 0.784 0.763 0.748 0.734 0.719

drm 0.778 0.725 0.690 0.663 0.637 0.608

Table 6 The optimal solutions in linear demand case, e ~ N(10,5), a ¼ 100, c ¼ 1.0, b ¼ 10.0 a

zd

0.01 0.05 0.10 0.15 0.20 0.25 0.50 0.75 0.99

9.678 9.735 9.810 9.888 9.970 10.057 10.586 11.424 14.112

rd 0.743 0.751 0.760 0.770 0.780 0.790 0.843 0.905 0.992

pd 7.332 7.312 7.285 7.258 7.229 7.198 7.019 6.758 6.118

ze 14.835 14.835 14.835 14.835 14.835 14.835 14.835 14.835 14.835

re 0.488 0.511 0.541 0.570 0.598 0.627 0.763 0.890 0.998

pe 5.997 5.997 5.997 5.997 5.997 5.997 5.997 5.997 5.997

qe 54.8650 54.8650 54.8650 54.8650 54.8650 54.8650 54.8650 54.8650 54.8650

Pe;M 99.192 94.315 88.293 82.357 76.510 70.754 43.491 19.222 0.168

Pe;R 145.592 150.469 156.491 162.427 168.275 174.030 201.293 225.562 244.616

b 0.595 0.615 0.639 0.664 0.687 0.711 0.822 0.921 0.999

drm 0.319 0.373 0.436 0.493 0.545 0.593 0.784 0.915 0.999

the retailer’s profit share and dominance. The computational results are reported in Table 5. Table 5 shows that the impacts of price sensitivity b on the manufacturer’s and the retailer’s expected profits, the retailer’s profit share and dominance are the same to those shown in Table 2. For analyzing the impacts of parameter a on the supply chain decisions, the retailer’s profit share and the retailer’s dominance, we next set b ¼ 10.0. When a varies from 0.01 to 0.99, the computational results are listed in Table 6. From Table 6, the revenue share r is increasing in a, however, for any given b > 0, the retailer’s cost share a has no impact on the production quantity q and the retail price p, i.e., the production quantity q and the retail price p are constant in a. As shown in Fig. 1, the retailer’s profit share b and dominance drm are increasing in a for any given b > 0. These observations are similar to Propositions 3 and 4 in the iso-price- elastic demand case. It can also be found that the retailer’s dominance is more sensitive in the price-elasticity index and the retailer’s cost share. To analyze the effects of demand uncertainty on the supply chain decisions and performance, we let b ¼ 10.0, a ¼ 0.25 and s varies from 5 to 50. Applying Theorem 3, we obtain the optimal solutions, the expected profits of supply chain members, the retailer’s profit share and dominance for any given demand uncertainty level s. Table 7 shows the computational results when the demand uncertainty varies.

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1.20 1.00 0.80 0.60 0.40 0.20 0.00

0.01

0.05

0.10

0.15

0.20

Retailer's profit share

0.25

0.50

0.75

0.99

Retailer's dominance

Fig. 1 The impacts of cost share on retailer’s profit share and dominance

Table 7 The optimal solutions in linear demand case, e ~ N(10,s), a ¼ 100, c ¼ 1.0, b ¼ 10.0, a ¼ 0.25 s

zd

5 10 15 20 25 30 35 40 45 50

10.057 10.563 11.478 12.749 14.335 16.201 18.317 20.658 23.203 25.934

rd 0.790 0.780 0.772 0.765 0.758 0.752 0.747 0.743 0.739 0.736

pd 7.198 7.146 7.139 7.153 7.187 7.238 7.305 7.386 7.480 7.586

ze 14.835 19.845 25.102 30.581 36.272 42.166 48.258 54.541 61.008 67.654

re 0.627 0.610 0.596 0.583 0.571 0.561 0.551 0.542 0.534 0.527

pe 5.997 6.156 6.369 6.591 6.819 7.052 7.290 7.533 7.782 8.036

qe 54.8650 58.2850 61.4120 64.6710 68.0820 71.6460 75.3580 79.2110 83.1880 87.2940

Pe;M 70.754 80.113 92.161 105.197 119.070 133.795 149.410 165.954 183.463 201.972

Pe;R 174.030 179.027 188.322 198.603 209.582 221.281 233.749 247.031 261.162 276.177

b 0.711 0.691 0.671 0.654 0.638 0.623 0.610 0.598 0.587 0.578

drm 0.593 0.553 0.511 0.470 0.432 0.395 0.361 0.328 0.298 0.269

Results in Table 7 show that, in the linear demand case, the production quantity and the retail price will increase in the demand uncertainty, and the retailer’s and the manufacturer’s expected profits is also increasing. These mean that the more demand uncertainty will lead to high expected profits for the manufacturer and the retailer because of the high risk in the market. As shown in Fig. 2, the retailer’s profit share b and dominance drm are decreasing in s for any given a > 0 and b > 0, which indicate that higher demand uncertainty will cut the retailer’s profit share and dominance in the decentralized supply chain. It is also can be found that the retailer’s dominance is more sensitive than the retailer’s profit share in the demand uncertainty. Comparing the optimal decisions at cooperation with those at non-cooperation in the above numerical analyses, it is found that the supply chain channel will supply more product quantity with lower retailer price in the market at cooperation. This means that the supply chain members can earn more profit by their cooperation and the consumers also obtain benefits from the supply chain cooperation.

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5

10

15

20

25

Retailer's profit share

30

35

40

45

50

Retailer's dominance

Fig. 2 The impacts of demand uncertainty on retailer’s profit share and dominance

6 Conclusion In this chapter, we investigate the coordination of a decentralized supply chain consisting of an upstream manufacturer and a downstream retailer, in which a single-period product is produced and sold in the retail market. The bargaining process between the supply chain members is governed by a consignment contract with revenue sharing. We present a cooperative model utilizing Nash bargaining model with two demand cases: the iso-price-elastic demand and the linear demand. In the proposed model, the retailer is the leader and the manufacturer is the follower in the non-cooperative game, and the two players are willing to cooperate if their profits at cooperation are no less than those at non-cooperation. They bargain and try to make an equilibrium agreement on the retail price, production quantity and revenue share. The research shows that the cooperative game between the manufacturer and the retailer has a unique equilibrium solution under a very mild restriction on the demand distribution. Under the equilibrium contract, the decentralized supply chain can be perfectly coordinated and both the manufacturer and the retailer are better off. It is found that the retailer’s profit share and dominance is increasing in retailer’s cost share for any given price elasticity, decreasing in price elasticity for any given retailer’s cost share. The retailer can always extract more than 50% of the channel profit even if he does not incur any portion of the channel cost in the multiplicative demand case; however, the manufacturer can earn more than the retailer by their cooperation in the linear demand case. Acknowledgements The research is supported by National Natural Science Foundation of China (No.71001024; No.70801014), Program for New Century Excellent Talents in University of China (NCET-09-0292) and PhD Programs Foundation of Ministry of Education of China (No.20100092120042).

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References Barlow RE, Proschan F (1975) Statistical theory of reliability and life testing probability models. Holt, Rinehart, and Winston, New York Bernstein F, Federgruen A (2003) Pricing and replenishment strategies in a distribution system with competing retailers. Oper Res 51(3):409–426 Bolen WH (1978) Contemporary retailing. Prentice-Hall, Englewood Cliffs, NJ Bose S, Pekny JF (2000) A model predictive framework for planning and scheduling problems: a case study of consumer goods supply chain. Comput Chem Eng 24:329–335 Cachon GP (2001) Stock wars: inventory competition in a two echelon supply chain. Oper Res 49(5):658–674 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 (2004) The allocation of inventory risk in supply chain: push, pull, and advancepurchase discount contracts. Manage Sci 50(2):222–238 Cachon GP, Lariviere MA (2001) Contracting to assure supply: how to share demand forecasts in a supply chain. Manage Sci 47(5):629–646 Canie¨ls MCJ, Gelderman CJ (2007) Power and interdependence in buyer supplier relationships: a purchasing portfolio approach. Ind Mark Manage 36:219–229 Chen KB, Xiao TJ (2009) Demand disruption and coordination of the supply chain with a dominant retailer. Eur J Oper Res 197(1):225–234 Choi KS, Dai JG, Song JS (2004) On measuring supplier performance under vendor-managedinventory programs in capacitated supply chains. Manuf Serv Oper Manage 6(1):53–72 Dana JD, Spier KE (2001) Revenue sharing and vertical control in the video rental industry. J Ind Econ 49(3):223–245 Dong Y, Xu KF (2002) A supply chain model of vendor managed inventory. Transp Res E 38(2):75–95 Dukes AJ, Gal-Or E, Srinivasan K (2006) Channel bargaining with retailer asymmetry. J Mark Res 43:84–97 Gerchak Y, Wang Y (2004) Revenue-sharing vs. wholesale-price contracts in assembly systems with random demand. Prod Oper Manage 13(1):23–33 Giannoccaro I, Pontrandolfo P (2004) Supply chain coordination by revenue sharing contracts. Int J Prod Econ 89(2):131–139 Goto JH, Lewis ME, Puterman ML (2004) Coffee, tea, or . . .?: a Markov decision process model for airline meal provisioning. Transp Sci 38(1):107–118 Gottfried BS, Weisman J (1973) Introduction to optimization theory. Prentice-Hall, Englewood Cliffs, NJ G€ um€us¸ M, Jewkes EM, Bookbinder JH (2008) Impact of consignment inventory and vendor managed inventory for a two-party supply chain. Int J Prod Econ 113(2):502–517 Hua ZS, Li SJ (2008) Impacts of demand uncertainty on retailer’s dominance and manufacturerretailer supply chain cooperation. Omega Int J Manage Sci 36(5):697–714 Hua ZS, Li SJ, Liang L (2006) Impact of demand uncertainty on supply chain cooperation of single-period products. Int J Prod Econ 100(2):268–284 Khouja M, Robbins SS (2005) Optimal pricing and quantity of products with two offerings. Eur J Oper Res 163(3):530–544 Kumar N (1996) The power of trust in manufacturer-retailer relationships. Harv Bus Rev 74(6):92–109 Lariviere MA, Porteus EL (2001) Selling to the newsvendor: an analysis of price-only contract. Manuf Serv Oper Manage 3(4):293–305 Lau AHL, Lau HS (2003) Effects of a demand-curves shape on the optimal solutions of a multiechelon inventory/pricing model. Eur J Oper Res 147(3):530–548

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Lau AHL, Lau HS, Wang JC (2008) How a dominant retailer might design a purchase contract for a newsvendor-type product with price-sensitive demand. Eur J Oper Res 190(2):443–458 Li L (2002) Information sharing in a supply chain with horizontal competition. Manage Sci 48(9):1196–1212 Li JL, Liu LW (2008) Supply chain coordination with manufacturer’s limited reserve capacity: an extended newsboy problem. Int J Prod Econ 112(2):860–868 Li J, Sheng ZH, Liu HM (2010) Multi-agent simulation for the dominant players’ behavior in supply chains. Simul Model Pract Theory 18(6):850–859 Loewenstein J, Morris MW, Chakravarti A, Thompson L, Kopelman S (2005) At a loss for words: dominating the conversation and the outcome in negotiation as a function of intricate arguments and communication media. Organ Behav Hum Dec 98:28–38 McAdams D, Malone TW (2005) Internal market for supply chain capacity allocation. Working Paper, MIT Sloan School of Management, Cambridge, MA Mortimer JH (2004) Vertical contracts in the video rental industry. Working paper, Harvard University, Cambridge, MA Nash JF (1950) The bargaining problem. Econometrica 18(2):155–162 Petruzzi NC, Dada M (1999) Pricing and the newsvendor problem: a review with extensions. Oper Res 47(2):183–194 Porter M (1974) Consumer behavior, retailer power and market performance in consumer goods industries. Rev Econ Stat 56:419–436 Robert D (1997) Durable-goods monopoly, increasing marginal cost and depreciation. Economica 64(253):137–154 Rowntree RH (1941) Note on constant marginal cost. Am Econ Rev 31(2):335–338 Wang HW, Guo M, Efstathiou J (2004a) A game-theoretical cooperative mechanism design for a two-echelon decentralized supply chain. Eur J Oper Res 157(2):372–388 Wang Y, Jiang L, Shen ZJ (2004b) Channel performance under consignment contract with revenue sharing. Manage Sci 50(1):34–47

Part IV

Technological Advancements and Applications in Supply Chain Coordination

.

DEAL: A Heuristic Approach for Collaborative Planning in Detailed Scheduling J. Benedikt Scheckenbach

Abstract Aroused by ongoing globalization, the division of labor steadily increases and future competition will likely take place between whole supply chains and not only between single companies. This compels suppliers and manufacturers to better coordinate their production plans in order to save production and holding costs. Despite this necessity, the willingness to share sensitive production data such as cost factors or resource availability has remained very limited. Usually the companies consider these data as vital to their business. However, today’s production planning models and solvers used in industry are of monolithic type and require full visibility of data in order to compute a solution. Hence, they cannot be used to tackle supply-chain-wide planning problems involving different companies. In recent years, “Collaborative Planning” as a joint decision making process under information asymmetry has received increased attention. The majority of present research in this field breaks with the monolithic approach but assumes that planning problems can be solved to optimality. On the contrary, industry struggles in operational business with large detailed scheduling problems that are only solvable by heuristics – violating this fundamental assumption. However, being computed only shortly before execution, badly aligned detailed schedules most obviously demand a coordinated solution. We propose a decentralized evolutionary algorithm (DEAL) for coordinating large-sized detailed schedules that does not demand the exchange of sensitive data but only transmits delivery dates and ordinal rankings. Experimental results prove that DEAL computes in the same time solutions of similar quality as monolithic heuristics are able to.

This work originated from a joint research project between the University of Hamburg, Institute for Logistics and Transport and the SAP AG Walldorf, Germany. Today, the author works as Supply Chain Consultant at Bayer Technology Services GmbH, Germany. J.B. Scheckenbach (*) Cranachstr. 16, 50733 Koeln, Germany e-mail: benedikt.scheckenbach@gmail.com 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_18, # Springer-Verlag Berlin Heidelberg 2011

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Keywords Collaborative planning • Detailed scheduling • Evolutionary algorithm • SAP APO

1 Introduction In the last decades, advanced planning systems (APS) have been put forward by industry and academia to solve supply chain (SC) planning problems. To cope with the complexity of the planning problems, APS are typically organized in a hierarchical manner. That is, on a mid-term level so-called master planning centrally calculates rough cut plans, generating enterprise-wide material quantity targets for subsequent production planning and detailed scheduling, which are then applied for each production site separately. Production planning is concerned with translating material quantity targets into concrete production activities, whereas detailed scheduling is about computing a good sequence and a feasible resource assignment for the activities. As detailed schedules per plant have to respect superordinated rough cut plans on enterprise level, APS inherently employ a hierarchical form of coordination. The problem with the above approach is that it is not applicable to companies that are interdependent on delivered items but legally separate organizations. A superordinated decision-making process fails because of the missing central entity and the reluctance of SC members to disclose sensitive data, such as production and holding cost or resource availability. As a remedy, successive planning strategies are employed today, a typical example is upstream planning. In upstream planning, the more powerful downstream entity calculates its production plan first, generating demand for subsequent upstream entities. Most of today’s “collaboration” systems, such as vendor-managed inventory, are actually implementations of successive planning strategies. From a mathematical perspective the interorganizational optimization problem is split into separate interdependent subproblems. The subproblems are then solved sequentially, whereas already-solved subproblems define the constraints for yet unsolved subproblems. For example, suppose a manufacturer setting up his production plan according to current demand and forecasts. By doing this, the manufacturer creates the demand for his suppliers that have to plan their production accordingly. However, successive planning pays no respect to the suppliers’ actual production capacities. Hence, successive planning strategies lead to suboptimal results, which often will result in wasted production capacity, larger safety stocks, decreased service levels or increased production costs. In recent years, academia has put forward advanced coordination mechanisms that try to attain the results of hierarchical coordination (or centralized planning) without requiring the parties to exchange sensitive data. Subsuming the different approaches, the term Collaborative Planning (CP) has emerged as a new research area. From a practical perspective, CP mechanisms have to cope with three difficulties: First, they should support complex operational planning problems. Second, the exchange of sensitive data must be avoided. Third, the mechanisms are required to be incentive compatible to prevent opportunistically acting partners

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from supplying systematically biased input that changes overall results to their advantage. It is very difficult to consider all three requirements to the fullest. This, and the still-prevailing paradigm of successive planning have hampered the introduction of CP mechanisms in practice. This work presents a decentralized evolutionary algorithm (DEAL) for aligning detailed schedules of several suppliers delivering to one manufacturer. In such a setting, production processes of the manufacturer cannot start until the suppliers have provided the necessary items. Being computed only shortly before execution, badly aligned detailed schedules most obviously demand a coordinated solution. Restricting to the limited scope of detailed scheduling allows to fulfill the above requirements to a degree acceptable for practical implementations. This work is organized as follows. Section 2 provides a brief summary of stateof-the-art CP approaches. In Sect. 3, the term “detailed scheduling” is further clarified by presenting the optimization problem underlying our further discussion. Section 4 introduces evolutionary algorithms. Section 5 presents the proprietary SAP metaheuristic to solve detailed scheduling problems – the so-called Production Planning and Detailed Scheduling (PP/DS) Optimizer. Building on this background, DEAL is presented in Section 6. Finally, we discuss experimental key results in Sect. 7 followed by concluding remarks in Sect. 8.

2 Literature Review on Collaborative Planning According to Stadtler (2007) Collaborative Planning (CP) can be defined as: . . . a joint decision making process for aligning plans of individual SC members with the aim of achieving coordination in light of information asymmetry.

Information asymmetry is a term stemming from game theory and describes the state where no SC member possesses all the information or preferences of other SC members. There exist several definitions for coordination, depending on the exact field of research. Roughly, literature can be divided into analytical and operational models. Analytical models employ very restrictive assumptions that allow a rigorous mathematical treatment. Here, coordination is often defined as optimal solution and Nash equilibrium. In contrast, the complexity of operational models might not even allow to numerically compute the optimal solution in reasonable time. Here, many authors speak of coordination, if the initial situation could be at least improved. The next two subsections provide a brief overview of the two approaches.

2.1

Analytical Models

Regarding analytical models, a large body of literature is concerned with multiechelon systems. Such a system includes several stages (e.g., representing supplier and manufacturer sites) to be coordinated.

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Static multi-echelon systems rely on restrictive assumptions, such as static demand in a single-item, single-production-stage scenario and aim at finding the optimal trade-off between the holding and fixed-order costs. The difficulty lies in the assumption that different costs accrue for each echelon. For example, large orders might be beneficial to suppliers, because of potential savings in order processing costs. However, a manufacturer might prefer more frequent, smaller orders due to high inventory cost (cf. Bhatnagar and Chandra 1993). If the manufacturer is the more powerful party, the supplier’s problem is to persuade the manufacturer to change his order policy by making concessions such as price discounts or compensation payments. Reviews of contributions assuming complete information are provided by Goyal and Gupta (1989) and Thomas and Salhi (1998). Papers that explicitly deal with incomplete information include Sucky (2006), Fransoo et al. (2001), and Corbett and de Groote (2000). Stochastic multi-echelon systems assume a stochastic demand distribution, generally with negligible fixed-order costs; and are concerned with retrieving the optimal safety stock for a given service level. For a single stage facing stochastic demand, the optimal safety stock is a quantile of the demand distribution. Research dates back to 1960, with the paper of Clark and Scarf (1960) who derive an exact algorithm for a serial multi-echelon system. Diks et al. (1996) and van Houtum et al. (1996) give an overview about relevant literature since then. Also contract theory focuses on the analytical investigation of the relationship between a supplier and a buyer of a good. Most of the literature in contract theory is about determining optimal contract parameters given the functional form of a contract, cf. Cachon (2003). As a main result, multi-echelon systems and contract theory provide the insight that lower system-wide costs can be achieved if some SC members provide compensation payments to create incentives for other members to accept a coordinated solution. Auction theory analyzes price-settling mechanisms under pure market conditions. There exist two possibilities for applying an auction mechanism to SC planning. First, the SC partners can be determined by an auction. A typical example is the choice of third-party logistics providers. However, it is questionable if market conditions really exist within an SC: Here, the choice for partners is usually a strategic one, binding a manufacturer to a supplier for a longer time period, cf. Stadtler (2007). Second, auction mechanisms can be employed for choosing the best interorganizational plan out of a set of alternatives. The typical proceeding is that the auctioneer grasps total visibility of resources and sells resource capacity to competing bidding agents, representing, for example, different company divisions. A survey can be found in Wellman and Walsh (2001). Most research focuses on incentive compatibility. In an incentive compatible mechanism, players fare best when they truthfully reveal any private information requested, i.e. truth-telling is a dominant strategy (cf. Myerson 1979). Incentive compatibility can be achieved by introducing compensation payments demanding each player to pay the opportunity cost that his presence introduces to all the other players. Auction mechanisms only focus on the problem of selecting the optimal

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production schedule but does not address the construction of candidate schedules under information asymmetry. A proposal on order quantities or delivery times can also be interpreted as a good offered from one SC member to the other. While each player knows his internal value of the good, he is assumed to have only limited information (in the form of a probability function) of his antagonist’s valuation. For such cases of information asymmetry, publications related to Bargaining theory analyze the efficiency of equilibrium strategies. Chatterjee and Samuelson (1983) provide such an analysis for a bilateral monopoly. Within a nonrecurring sealed-bid double auction, a buyer and a seller of an indivisible good simultaneously reveal their offers. If trading takes place (the buyer’s offer is higher than the seller’s claim) a settlement is computed according to an a priori known mechanism (e.g., by splitting the difference between both offers equally). Though the seller’s and buyer’s actual values of the good are assumed to be private knowledge, a probability distribution of likely values is supposed to be common knowledge. Being rational players, buyer and seller implement “best response strategies,” giving a price offer dependent on the own value and the counterpart’s probability distribution. These interlinked strategies lead to a Nash equilibrium. Related contributions have been put forward by Harsanyi and Selten (1972), Myerson and Satterthwaite (1983), Samuelson (1984), and Radoport and Fuller (1995). Bargaining theory provides interesting analytical insights into strategies leading to Nash equilibria and the related efficiency of the employed price-settlement mechanisms, i.e., the percentage of mutual beneficial agreements that can be actually attained by bargaining under the employed mechanism. However, most approaches assume the parties’ probability distributions of the value of the good to be public knowledge. It is questionable if this assumption can be justified for practical purposes. As auction theory, bargaining theory is not concerned how the good is created. In general, analytical models provide mathematical insight about optimal properties of coordination mechanisms, such as incentive compatibility. However, restrictive assumptions complicate a direct application to practical planning problems of an operational type.

2.2

Operational Models

In contrast to analytical models, operational models explicitly focus on generating production plans under information asymmetry. Using a metaheuristic, Fink (2004, 2006) investigates a supplier-manufacturer scenario where the sequence of delivery influences the quality of the related production plans. A central algorithm (assumed to be common knowledge) repeatedly proposes randomized delivery sequences to be accepted or rejected. The last jointly accepted delivery sequence is the final outcome of the coordination process. To sustain a fair outcome, both SC members are required to accept a certain percentage of the proposals, forcing them to also accept worse proposals.

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The internal acceptance probabilities are calculated according to a cooling scheme taken from Simulated Annealing. Though proposing purely randomized delivery sequences by a third party can be considered as a very fair approach, it might be regarded as inefficient in a real-world environment, where evaluating each proposal takes a considerable amount of time due to the size of the problems. An important aspect of CP mechanisms is security. Security can be defined by means of a trusted mediator (cf. Li and Atallah 2006). Assume an idealized mechanism, where each SC member supplies sensitive information to the trusted mediator, the mediator solves the global problem and reports back the individual solutions without disclosing private information to other SC members. A mechanism is said to be secure if any adversary interacting in the real mechanism can do no more harm than in the ideal scenario. The idea of security can be illustrated by a simple example. Assume three parties A, B, and C and each party holding a stock of items, e.g., A has three items, B has five items and C has two items in stock. Suppose that the parties aim at computing the sum of items without revealing how much items each single party has in stock. A secure sum algorithm can be set up as follows. Party A adds a random number, say 7, to its stock and transmits the sum of 10 to B. B adds its five items and transmits the sum of 15 to C. In turn, C adds 2 and transmits the sum of 17 to A. Eventually, A subtracts the initial random number and the correct sum of stock, i.e. ten items, has been computed. However, no party knows the exact number of items of any other party. Similar, though more complex, secure multiparty algorithms are available for other algebraic operations and different number of parties. These algorithms can be regarded as “one-way functions,” functions that are easy to evaluate but hard to invert (cf. Yao 1982). In simple terms, each protocol takes shares of the input and produces shares of the output (cf. Kerschbaum and Deitos 2008). Each SC member can encrypt his input with a private key, such that it cannot be decrypted by other parties without the private key, whereas the function can still be evaluated despite the encrypted values. Li and Atallah (2006) present a decentralized secure Simplex method for two parties. Atallah et al. (2003) present secure allocation protocols that prevent a disclosure of bidders’ sensitive information during auctions. Being a sophisticated encryption method, secure multiparty computation itself is not incentive compatible and does not prevent users from supplying systematically biased input. Operational planning problems are often formulated as linear problems (LP) or mixed-integer linear problems (MILP) and the solution is computed by an LP/ MILP-solver. Lagrangian relaxation (LR) is a common decomposition technique (cf. Conejo et al. 2006; Williams 1999) for problems having a sparse coefficient matrix, where most of the non-zero coefficients can be ordered into a block-angular structure. Consider the linear programming problem min

x1 ;x2 ;...;xn

n X j¼1

cj xj

(1)

DEAL: A Heuristic Approach for Collaborative Planning in Detailed Scheduling n X

s.t:

dij xj b fi

8i ¼ 1; . . . ; q

463

(2)

j¼1 n X

aij xj ¼ bi

8i ¼ 1; . . . ; m

(3)

j¼1

0 b xj

8j ¼ 1; . . . ; n;

(4)

where constraints (2) have a decomposable structure in n blocks. In the particular case of n ¼ 2, the problem can be written as 0 1 x½1 T T @ A; c½1 j c½2 min x½1 ;x½2 x½2 where the superindices in brackets refer to partitions, subject to 0

D½1 B B B B @ A½1

1 j 0 ½1 1 b C C x @ A b j D½2 C C ½2 A x ¼ j A½2

1 f ½1 B C C B B f ½2 C: C B @ A b 0

The problem can be decomposed by “dualizing” the coupling constraints 3. That is, adding them to the objective function. The problem 1–4 can be rewritten as " max

l1 ;l2 ;...;lm

s.t:

n X

min

x1 ;x2 ;...;xn

dij xj b fi

n X j¼1

cj xj

m X i¼1

li bi

n X

!# aij xj

j¼1

8i ¼ 1; . . . ; q

j¼1

0 b xj 0 b li

8j ¼ 1; . . . ; n; 8i ¼ 1; . . . m

where l1, l2, . . ., lm denote the Lagrangian multipliers. The idea of LR is to iteratively solve the inner minimization problem while updating the multipliers until the procedure has converged to a satisfying solution. Since the inner problem has a true block-angular constraint structure, it can be partitioned accordingly and solved separately. Subgradient procedures are frequently used to iteratively increase multiplier values in proportion to their constraint violation in the primal problem.

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J.B. Scheckenbach

Many SCM problems have a block-angular constraint matrix consisting of the intra-company constraints of each SC member, connected by inter-company material-flow-balance constraints. Subgradient methods only require realized material flows to update the multipliers, not the information of domain-internal production constraints. Hence, LR inherently supports scenarios with information asymmetry and is used by many authors to solve inter-company planning problems information asymmetry, cf. Kutanoglu and Wu (1999), Kutanoglu and Wu (2006), Ertogal and Wu (2000), Nie et al. (2006) and Walther et al. (2008). Multiplier values are dual information. Another possibility is to exchange primal information, target right-hand side (rhs) values for decision variables included in the coupling constraints, cf. Jung et al. (2005). Dudek and Stadtler (2005) decomposes the problem by fixing the material flow from a supplier and a manufacturer to certain values by setting the related rhs values. At the beginning, upstream planning is used to compute the initial “supply pattern.” Then the SC members alternatingly propose supply quantities and associated cost increases trying to compute reasonable proposals by approximating its antagonist’s cost changes within the local objective function. This is realized by penalizing the deviation from the currently confirmed supply quantities. The penalty coefficients are calculated on the basis of the antagonist’s cost reports. Thus, only effective proposals – that is, supply patterns with the highest trade-off of decrease of internal cost and increase of approximated “external” cost – are computed. After coordination, the worse-off player needs to be compensated by the player experiencing an improvement in its local objective function. Above approaches share the drawback that the adherence to imposed multipliers, rhs values or penalties cannot be monitored. An SC member can simply increase its local objective value by relaxing these bounds. Also the introduction of compensation payments does not necessarily ensure incentive compatibility. In a setting where the set of alternatives is already known to all players (as assumed for most analytical models), incentive compatibility in the sense of choosing the best interorganizational solution from a set of alternatives might be ensured by compensation payments. However, the situation is different if the reported values are also used for constructing new solutions. During construction, the future outcome of the search process is not determined yet. Thus, players are tempted to exaggerate their reported costs for receiving a higher compensation payment in order reap a good piece of the savings pie (as long they have the possibility for doing so). If an SC member reports exaggerated costs, its own objective gets a larger influence. At the end, those solutions will get explored that are, ceteris paribus, close to the locally optimal solution of the exaggerating domain obscuring the true global optimum with maximum saving. A possible solution to this problem is to decouple compensation payments from the coordination process. Albrecht (2010) suggests to compute a lumpsum (based on historical information) a priori to jointly constructing the solution. Under this premise the search process fosters truth telling of costs of intermediate solutions, since cheating obscures the true global optimum leading to an outcome where all partners are worse-off. For operational models of a certain structure, Albrecht

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(2010) proves that his mechanism is able to attain the global optimum within a finite number of proposal exchanges. Summarizing the above literature, coordination mechanism must support complex planning problems of operational type under information asymmetry, must not lead to a disclosure of sensitive data and must be incentive compatible. The difficulty lies in equally considering all the three aspects: For example, multi-echelon systems and contract theory support incentive compatibility but require very restrictive assumptions and simplified models. Secure multiparty computation is able to deal with complex models but is not incentive compatible. With the exception of Albrecht (2010), this critique also holds for coordination schemes relating to mathematical decomposition techniques. Moreover, although critical information is not explicitly exchanged, operational mechanisms relying on cost reporting provide opportunistically acting players the possibility to infer sensitive data by systematically probing other SC members. Thus, even if the mechanisms do not directly require the exchange of sensitive data, they are insecure in protecting these data. Practical requirements introduce further difficulties. If we assume planning problems to be NP-hard, the solution with maximum global saving is not guaranteed to be found in a reasonable amount of time – violating a fundamental assumption of many incentive compatible mechanisms. Moreover, in most approaches, the value of each generated alternative is measured as the difference to an initial solution. However, SC members are usually not of equal power in practice. A powerful party might be tempted to create bad initial settings for the weaker parties if there is the possibility to gain future compensation payments from the weaker parties’ “cost decrease” through coordination. For example, a (powerful) manufacturer could initially order items very early, resulting in huge overtime costs for the supplier and additional storage costs for the manufacturer, whereas storage costs are assumed to be less than overtime costs. Now, the coordination mechanism will search for a solution that avoids these costs. In the example, it is likely that such a solution will be found (since the costs have been generated by purpose) and that the coordinated solution will come with a higher cost decrease for the supplier than for the manufacturer. Hence, if savings are split “fairly”, the supplier will have to compensate the manufacturer. Thus, the manufacturer used his power to generate an additional margin at the expense of his supplier. Concluding, even if a mechanism was incentive compatible given a set of assumptions, its application in practice might suffer from a violation of some of these assumptions. Furthermore, real-world optimization usually requires considerable fine-tuning of parameters and data belonging to a company’s specific optimization problem. It is likely that companies would not accept a coordination mechanism that required a major change of the parameters or even abandoned their long-time tested and approved planning system. Moreover, companies often have specific planning problems that cannot be tackled by out-of-the-box solutions. In practice, the landscape of optimization tools is hence very heterogeneous and several software companies compete with different optimization methods with proprietary intrafirm developments. Finding a real-world setting with homogeneous optimization tools is unlikely. Hence, the coordination mechanism should rather treat the underlying

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solver as a black box. As will be discussed later, most models for production planning share commonalities in their structure. Our intention is to construct a coordination mechanism that dynamically changes the input data relating to common model structures but not the solver itself. Different solvers can then be supported by adapting the interface for changing the data.

3 The Resource-Constrained Project-Scheduling Problem The Resource-Constrained Project-Scheduling Problem (RCPSP) belongs to the class of NP-hard problems (cf. Blazewicz et al. 1983; Neumann et al. 2001) and represents a generalizations of different scheduling problems, such as job-shop or flow-shop problems. In our study, the RCPSP forms the underlying scheduling problem of each SC member. We present its basic problem definition below in order to provide the extension to collaborative planning in Sect. 6. We introduce the following definitions. The planning interval is defined as the 2 N0 consisting of several periods with a given discrete time interval ½0; . . . ; T period length (e.g., seconds), in which all activities start and end. An activity j refers to a single production activity (e.g., the assembly of parts). An activity has a start and a duration dj 0, defining the finish date fdj ¼ sdj + dj. date sdj 2 ½0; . . . ; T The set of all activities is referred to as J. Moreover, there exists a set R of renewable resources. Each resource r has a constant capacity of ar units that is renewed every period t ∈ T. When executed, an activity j places a constant resource usage ujr on resource r throughout its duration. The time for executing an activity is bounded by its release date rdj. The activities have to be executed in a predefined partial order, the so-called activity-precedence relation. For convenience, we introduce the sets Pj ¼ {i ∈ J | i is the immediate predecessor of j} and Sj ¼ {i ∈ J | i is the immediate successor of j}. An instance of the RCPSP is then given by • • • • •

A planning interval ½0; . . . ; T, A set of activities J, A set of renewable resources R, A precedence relation given by Pj or Sj for each activity j, respectively, The precedence and capacity constraints sdj r rdj sdj r fdi fdj ¼ sdj þ dj X j2Jjsdj b t b fdj

ujr b ar

8j 2 J

(5)

8j 2 J; i 2 Pj

(6)

8j 2 J

(7)

r2R 8t 2 ½0; . . . ; T;

• An objective function that shall be minimized or maximized.

(8)

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467

4 Evolutionary Algorithms Because of the extensive run times of mixed integer approaches and their complex implementation, many researchers have turned to metaheuristics. The term metaheuristic usually describes a generic optimization principle that is widely applicable to many different problem domains. Very often, the basic idea behind a metaheuristic is inspired by nature. Evolutionary algorithms (EAs) are stochastic iterative optimization heuristics inspired by natural evolution. Starting with a set of candidate solutions (population), in each iteration (generation), promising solutions are selected as potential parents (mating selection), and new solutions (individuals) are constructed by mixing information from the parents (crossover) and slightly modifying them (mutation). The resulting offspring are then inserted into the population, replacing some old or less fit solutions (environmental selection). By continually selecting good solutions for reproduction and then creating new solutions based on the knowledge represented in the selected individuals, the solutions “evolve” and become better and better adapted to the problem to be solved, just like in nature, where the individuals become better and better adapted to their environment through the means of evolution. In line with Deb (2001), an EA can be outlined as follows, t 0 initialize population: P(t) evaluate P(t) repeat select mating pool: M(t) s(P(t)) c(M(t)) construct offspring: M0 (t) evaluate P(t) \ M0 (t) update: P(t + 1) u(P(t) \ M0 (t)) t t+1 until terminated, with t denoting the generation counter, P(t) the population at generation t, and s, c and u representing the different genetic operators for selection, construction and update. For a more detailed introduction to EAs, the reader is referred to Eiben and Smith (2003).

5 The SAP Production Planning and Detailed Scheduling Optimizer SAP offers the PP/DS Optimizer as proprietary tool to solve RCPSPs of extended complexity, including multiple modes, setup times, precedence relations with maximum time constrains, varying capacity profiles and different objectives. Here, the

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J.B. Scheckenbach

above problem definition shall suffice. As objective we consider lateness minimization. In practice, soft-constrained due dates per activity ddj are often used to model customer preferences. In a simple way the objective can be put mathematically as Minimize

X

max fdj ddj ; 0 :

j2J

Due to the combinatorial complexity of the RCPSP, approaches for computing exact solutions (cf. Manne 1960; Blazewicz et al. 1996; Herroelen et al. 1998; Brucker and Knust 2000; Klein 2000; Neumann et al. 2001; Damay et al. 2007) are only applicable to small-sized problems of minor practical relevance. In line with the research of Hartmann and Kolisch (2000); Hartmann (2002); Alcaraz and Maroto (2001); Kolisch and Hartmann (2006), the PP/DS Optimizer employs an EA based on an activity sequence encoding and a serial schedule generation schemes (SSGS). An SGSS consists of |J| stages in each of which an activity is scheduled at the earliest precedence- and resource-feasible time. SSGS can be used to construct active schedules. Informally spoken, an active schedule is a schedule where no activity can be started earlier without delaying the start of another activity. It has been proved that for regular performance measures, such as lateness minimization, the optimal schedule lies within the set of all active schedules (cf. Sprecher et al. 1995; Neumann et al. 2001). Employing different SGSS as decoding functions, the PP/DS Optimizer uses several mutation and crossover operators to find the best sequence of activities. In other words, the search is not carried out in the space of all possible schedules (spanned by all start dates sdj not violating precedence and capacity constraints), but in the space of all sequences; using SGSS to evaluate the fitness of candidate sequences. Being a metaheuristic, the PP/DS Optimizer does not guarantee to find the optimal schedule. Thinking about compensation payments, it is hard to relate lateness to real monetary terms (cf. Stadtler 2005, p. 585). Especially on an operational level, machine-, workforce- and material-related costs are usually considered as sunk costs from an accountancy perspective. Using lateness costs, we primarily aim at seeking schedules that make optimal use of the available resources. As a detailed discussion of the PP/DS Optimizer would be beyond the scope of this paper, the interested reader is referred to Engelmann (1998) and Scheckenbach (2010). Earliness as a non-regular performance measure precludes the use of SSGS; the optimal schedule no longer lies within the set of all active schedules. However, from a practical point of view, the minimization of earliness is important to reduce holding costs. The strategy is then to modify the schedule constructed by SSGS in a subsequent right-alignment step. We define a right-aligned schedule as a schedule where no activity can be finished later without advancing some other activity, or violating the constraints, or increasing the objective function. Right-alignment considers activities from “right” to “left” (relating to their positions in a Gantt chart) and shifts each activity to the latest possible date. Also SAP provides a rightalignment algorithm.

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6 DEAL: A Decentral Evolutionary Algorithm This section presents a decentral evolutionary algorithm (DEAL) for aligning detailed schedules of n suppliers and one manufacturer. The algorithm works decentrally in the sense that a top-level evolutionary algorithm iteratively changes local problem instances that are solved locally and separately by the PP/DS Optimizer of each SC member. By doing this, each SC members evolves a population of candidate schedules. The top-level metaheuristic controls the composition of local schedules to global feasible schedules and the evolution of local populations, without demanding the exchange of sensitive data. We start by formulating the interorganizational problem in Subsection 6.1. Subsections 6.2 and 6.3 are about constructing candidate solutions. A solution relates to a composition of local schedules, feasible from an SC-wide perspective. In Subsection 6.4 we address the evaluation of solutions, the selection of the mating pool and the update of the population. Subsection 6.5 presents means to further speed up the search process.

6.1

The Interorganizational Problem

We assume each SC member having its unique set of resources. We further assume the activities between the different SC members to be connected by precedence constraints. That is, some activities of the manufacturer can only be started after the suppliers’ preceding activities have been finished, neglecting transportation times. Moreover, we assume that there is no cyclic dependency between the SC members. Thats is, neither suppliers nor manufacturer deliver to other suppliers taking part in coordination. Nevertheless, suppliers and manufacturer might delivery to external sinks and might be supplied by external sources, not participating in coordination. We extend our previous RCPSP formulation by defining • ddj as static, non-varying due date of activity j relating to external sinks, • rdj as static, non-varying release date of activity j relating to external sources, • e as index of an SC member (agreeing on the notation that the manufacturer is e ¼ 0 and suppliers are e > 0) and E as the set of all SC members, • Je J as the set of activities j of SC member e, Je, • Re as the set of resources r of e, • Pej as the set of predecessors within the problem of SC member e, • Sej as set of successors within the problem of SC member e, activities directly depending • JU0 as the set of manufacturer’s upstream-related on the suppliers’ delivery, i.e. JU0 ¼ j 2 J 0 j9i 2 J e ^ i 2 Pej with e 2 Enf0g , activities directly influencing • JDe as the set of supplier e’s downstream-related the manufacturer’s problem, i.e. JDe ¼ j 2 J e j9i 2 J 0 ^ i 2 S0j .

470

J.B. Scheckenbach

We assume the set of all jobs to be distributed between the SC members e, i.e. 8e 2 E : J e J; [ J e ¼ J; e2E

8e; f 2 E; e 6¼ f : J e \ J f ¼ ;: The same is true for resources. The intra-company RCPSPs of supplier and manufacturer interlink as follows. Finish dates of suppliers and static, nonvarying, release dates rdj determine the manufacturer’s release dates; expressed by including the following equations in the manufacturer’s RCPSP formulation. ( ) rdj ¼ max

max

k2Pej ;e2Enf0g

ðfdk Þ; rdj

8j 2 JU0

(9)

A supplier’s due dates of downstream-relating activities are defined by ( ddj ¼ min

min ðpdk Þ; ddj k2S0j

) 8j 2 JDe ; e 2 Enf0g;

(10)

where pdk denote proposed dates, send by the manufacturer to the supplier during coordination.

6.2

Initialization

During initialization, the manufacturer proposes dates that “myopically” optimize his intradomain problem. To do this, all release dates rdj are first set to their static, non-varying values rdj . Based on this relaxed problem, the manufacturer applies the PP/DS Optimizer for a predefined amount of time, resulting in a solution consisting of start dates sdj and finish dates fdj, j ∈ J0. The proposed dates pdj are then derived from the manufacturer’s related right-aligned schedule. Each supplier receives those proposed dates relating to his scheduling problem. The suppliers translate the proposed dates into downstream-related due dates using Equations 10, apply the PP/DS Optimizer to solve their own scheduling problems and report the realized finish dates back to the manufacturer. Applying (9), the manufacturer in turn converts these finish dates into upstream-related release dates and recalculates his schedule once again using the PP/DS Optimizer. The outcome is a feasible, though suboptimal, SC-wide schedule. After the above procedure, remaining candidates are computed according to the principles discussed in the next section, until the initial population has reached its size. Note that the strategy “propose everything as early as possible” does not provide an initial advantage to the manufacturer, since the suppliers’ capacity is limited

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(the RCPSP does not provide an option for overtime). Rather than absolute values, relative trade-offs between the proposed dates communicate the manufacturer’s preferences.

6.3

Construction

The construction of a new candidate solution is done analogously to initialization with the difference that the manufacturer tries to compute promising proposed dates based on the current population. To sustain an evolutionary process, the manufacturer should only change the most conflicting dates, otherwise construction would be similar to initialization. Different operators have been studied on the basis of this general idea. In the following, we will highlight a few examples, a detailed discussion can be found in Scheckenbach (2010). Proposed dates can be calculated by propagating the manufacturer’s due dates to upstream-related activities by means of backward passes. A backward pass propagates the upper bounds from the end of the horizon backward starting with pddn ¼ ddn : for j ¼ jJ 0 j; . . . ; 1; pddj ¼ min ddj ; min pddh dh jh 2 Sj where the manufacturers activities are assumed to be numbered according to a topological sorting (i.e. i < j, if i ∈ Pj, and pddj denotes the propagated due date. Proposed dates are then calculated as pdj ¼ pddj dj ; 8j 2 JU0 . Dates calculated by backward passes communicate the manufacturer’s minimal requirements (neglecting capacity constraints) to the suppliers. For most coordination problems, this strategy is too simple, since bottlenecks arise because of limited capacity and suboptimal activity sequences. An alternative is to generate an intermediate problem instance i by partially relaxing release dates in the manufacturer’s problem. By applying the PP/DS Optimizer to the intermediate instance, the manufacturer searches for better sequences of activities. The proposed dates are then derived from the start dates of the related right-aligned schedule. One possibility to calculate the set of release dates to relax is to take the history of coordination into account. With respect to an existing solution p, we define ðpÞ ðpÞ ðpÞ ðpÞ the amount of correction of activity j 2 JU0 as Dj ¼ rdj pdj , whereas rdj denotes the release date that results from the suppliers counterproposals ðpÞ (cf. Equation 9) to the proposed date pdj during constructing of solution p. The ðpÞ amount of correction Dj measures the suppliers’ ability to fulfill the manuðpÞ facturer’s proposals. Note that Dj can also become negative if a supplier exceeds the manufacturer’s requirements. Intuitively, a further relaxation of release dates of activities with a large positive amount of correction is not promising, as at least one of the suppliers is apparently not able to fulfill the resulting proposed dates. Instead ðpÞ we restrict the critical set to those activities with a low Dj . For the first a percent of

472

J.B. Scheckenbach ðpÞ

ðiÞ

activities with low Dj we set rdj ¼ rdj and for the remaining 1 a percent ðpÞ ðiÞ ðpÞ with high Dj we set rdj ¼ sdj . Lateness to ultimate customers is then reduced by applying the PP/DS Optimizer to the relaxed problem instance. The proposed dates are derived from the start dates of the subsequently right-aligned schedule, whereas right-alignment is not allowed to increase lateness, as discussed in Sect. 5. Another possibility is to shift weakly connected components. Imagine an undirected graph, where activities define the nodes and precedence relations the edges. Usually, this graph can be decomposed into several disjunct weakly connected components. Relaxing release dates of all activities belonging to the same component might have a larger effect on the lateness criterion than doing so for arbitrary selected activities. For each component C J, the maximum difference between finish date and propagated due date is computed as ðpÞ ðpÞ mdC ¼ max fdj pddj : j2C

We might aim at relaxing the release dates of the “most-delayed” activities contained in the first ba nC c components with largest maximum difference, where nC is the total number of components and a a further tuning parameter (the symbol b:c denotes the floor function). Crossover uses the information of several existing candidates to derive new proposed dates. For example, linear crossover can be used to compute proposed dates as ðpÞ

ðqÞ

rdj þ rdj ; 8j 2 JU0 ; pdj ¼ 2

(11)

where q and p denote two existing solutions from the mating pool. Instead of the former release dates, also proposed dates might be combined. Concluding, constructing a solution always takes the preferences of all SC members into account. First, the manufacturer proposes dates to the suppliers representing his most demanded corrections to an existing schedule. Second, the suppliers incorporate the manufacturer’s preferences into their objective function by adapting their downstream-related due dates, cf. Equation 10. However, the suppliers are not forced to comply to the manufacturer’s proposal and the SC-wide solution is based on their counterproposal. In particular, suppliers might not agree to delay any external customer. This goal can be expressed by giving the related delay a larger weight in the lateness objective before the start of coordination.

6.4

Evaluation, Selection and Update

Evaluating a candidate set of solutions is critical for advancing in the right search direction. In our approach, evaluating refers to the task of ranking the different SC-wide schedules in order to remove worst ones from the population and for

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selecting the mating pool. Following the above discussion this ranking scheme must support incentive compatibility, at least from a practical perspective. If compensation payments are not taken into account, we refer to a coordination mechanism as practically incentive compatible if it does not put any SC member worse than in the initial, uncoordinated setting. However, such a deterioration is not necessarily measured by the objective function. Costs of a supplier can be decomposed into coordination-related and domain-related costs. Coordination-related costs are penalty costs resulting from the violation of due dates proposed by the manufacturer. Since these costs mainly anticipate the manufacturer’s decision problem, they cannot be claimed by the supplier in an accountancy sense.1 Hence, in addition to the objective function of the local optimization method, a supplier might have another evaluation function representing his preferences regarding the manufacturer’s proposals. A simple, practical evaluation function distinguishes between acceptable and unacceptable solutions. Unacceptable solutions result from a setting of due dates that leave a supplier’s local optimization method too little freedom to produce acceptable results. We define ue ðpÞ ¼

0; if the proposal p is acceptable, z 2 N; if the proposal p is not acceptable:

Integer z denotes a ranking of unacceptable solutions. The smaller z, the better the related solution from a supplier’s perspective. For each solution, the suppliers submit their ranking to the manufacturer. After having computed his ranking in a similar manner, the manufacturer can combine the single ratings by non-dominated sorting. Solution p is said to dominate q, if 8e 2 E : ue ðpÞ b ue ðqÞ and 9d 2 E : ue ðpÞ < ue ðqÞ: Non-dominated sorting clusters the population into subsets dominating each other. This allows to construct a global ranking g, i.e. individuals that are not dominated by other individuals get a g ¼ 1, individuals with g ¼ 2 are only dominated by individuals with g ¼ 1 and so on. Having this information, the manufacturer decides which solutions to select in the mating pool and which to remove from the population. Although he is free in his single decisions, the ultimate solution to be implemented is required to be accepted by all suppliers, i.e. 8e 2 Enf0g : ue ðpÞ ¼ 0. In each generation, m offspring are created out of a population of l individuals. An offspring is created by either applying a mutation or crossover operator. An individual to be mutated is chosen by tournament selection, cf. Deb (2001). That is, two individuals are randomly drawn from the parent generation, and the one with better ranking wins the 1

Nevertheless, coordination-related costs are in direct relation to domain-related costs. For example, setting a due date earlier might result in a solution with higher lateness of other orders of external clients.

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tournament. The two individuals required for crossover are determined analogously by two tournaments. After constructing the offspring, the manufacturer sorts all individuals according to their global ranking. The manufacturer removes the worst m individuals from the population (environmental selection) and transmits this information to the suppliers who update their local populations accordingly. The above ranking has three advantages. First, the transmission of ordinal rankings disguises objective values considered as sensitive by the suppliers. The manufacturer has only limited possibilities for probing the suppliers to infer sensitive data. Although he knows that a solution is regarded by a supplier as better or worse than another solution, he does not know how much better or worse it is. Second, the dimensionality is reduced. The search process is not driven in dimensions were solutions are already acceptable for some suppliers (having a ranking of zero) but a multiobjective Goal Programming approach, as discussed in Deb (2001), is implemented instead. Third, and most important, suppliers have only limited possibilities for cheating. No suppliers can influence the trade-off between the local objectives by exaggerating his values, as such a trade-off does not exist in non-dominated sorting. Of course, suppliers can label actually acceptable solutions as unacceptable – but that is something they already can do today by reporting that present delivery dates cannot be maintained. In general, we assume that suppliers try to comply to delivery dates to the extent possible, as their status of being a reliable partner depends on it. Thus, we are tempted to label DEAL as a practically incentive compatible mechanism (however, not in a mathematical sense).

6.5

Means to Speed Up the Search Process

This section discusses means to further speed up the search process. First, the above coordination mechanism is characterized by periodic idle times. Suppliers wait for new manufacturer’s proposals to counter. Similarly, the manufacturer waits for counterproposal before the construction of a solution is continued. Today, computational power is a relatively cheap expense. As the single solutions are independent of each other, the coordination can be parallelized, allowing the manufacturer to compute several proposals in parallel, without waiting for counterproposals. Note that in such an environment the term generation becomes elusive – the focus is instead on a high workload in the systems. In contrast to classic EAs, individuals have different states. Intuitively, selection, construction and update operators can only be applied to the set of “complete” individuals. Second, a “warm reboot” was established. In a standard PP/DS run, different heuristics compute the initial activity sequences (e.g., the activities are sorted according to their propagated due dates). As DEAL holds a population of schedules, information of previous local PP/DS runs are available, however. Hence, we added the sequence of each schedule in the population to the set of initial standard sequences. Depending on the chosen due or release dates of the new offspring

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this information might not prove useful and is sorted out during environmental selection of the PP/DS run. However, if the problem instances vary only slightly, using existing activity sequences avoids a computation “from scratch” and gives good results already at the beginning of the PP/DS run. Third, a self-adaption of operators for generating proposals can be set up. We regard the decision of selecting an operator as a dynamically changing optimization problem. For each operator o, we define a value to. The probability for choosing operator o out of the set of all operators O is computed as t Po : to o2O

After an operator has been applied, all values are decreased by a factor d ∈ (0;1), to d, 8o ∈ O. If an operator has been applied successfully (the child c is i.e. to better globally ranked than the parent p, its value increases by the relative ranking improvement: to

to þ

gðpÞ gðcÞ : gðpÞ

The value to can be interpreted as pheromone value in an ant-colonization optimization approach, cf. Merkle et al. (2002). Fourth, redundant computations are avoided. After deriving proposed dates for offspring c, the following redundancy checks apply: If another individual p in the manufacturer’s population exists, that is accepted by all suppliers (ne ðpÞ ¼ 0; 8e 2 ðpÞ ðcÞ Enf0g) with rdj b pdj ; 8j 2 JU0 , the manufacturer discards solution c immediately and does not transmit the proposed dates to the suppliers. After an individual has passed the first check, a second check is carried out by each supplier. For supplier e, an individual c is redundant, if there exists a local candidate schedule ðpÞ ðcÞ p with ne(p) ¼ 0 and ddj b ddj ; 8j 2 JDe . In such a case, the supplier skips the local PP/DS run and counterproposes again the release dates of individual p. Each supplier stores the information how manufacturer proposals relate to local schedules in a table. Obviously, this table has to be looked up for computing the correct ranks and needs to be adapted if the manufacturer transmits which individuals are to be deleted.

7 Experimental Key Results The description of DEAL and the SAP PP/DS Optimizer in the above sections were very limited due to the scope of this contribution. With this background a detailed discussion of experimental results would be a futile endeavor. Hence, this section

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aims at a high-level presentation of key results only. More details are provided in Scheckenbach (2010). Two test data generators were used. The first one models the common procedure in practice. Given periodic demand patterns, available Bill-of-Materials, related resource capacity requirements and fixed lot sizes, a material requirements planning (MRP) algorithm calculates number and length of required production activities, precedence constraints and resource utilization. Usually the MRP run generates several lots to satisfy total demand of a finished good. As solely one Bill-ofMaterial per finished good is available, the generated activities show a repetitive structure akin to flow shop problems. The second test data generator is based on the work of Kolisch et al. (1992, 1995). Adding activity by activity, an algorithm randomly constructs an activity network and also randomly chooses the length, the resource requirements and the due dates of each activity. Restricting these decisions to be within certain bounds results in a test instance with predefined complexity measures. Having created a test instance by using one of the above generators, it is divided into the interconnected subinstances belonging to the different SC members. The division is carried out on the basis of a predefined assignment of resources to SC members. By respecting this assignment already during the generation of a test instance we assure that no cyclic dependency results between the different SC members (as assumed in Sect. 6.1). Independent of the chosen test data generator, the above procedure results in one fractioned test instance for the coordination mechanism and one integral benchmark instance to be “centrally” solved by a single PP/DS Optimizer. Several runs for different test instances with up to 10,000 activities have been carried out systematically to increase statistical confidence. Although the coordination mechanism has to cope with information asymmetry, neither a deterioration in solution quality nor run time could be observed with regard to the “central” solution. Several factors contributed to this result. First, proposed dates imply certain tradeoffs in the local optimization problem. Rather than the dates’ absolute values, these tradeoffs steer the local optimization methods. Hence, a crude alignment of delivery dates by DEAL seems to be sufficient – the exact setting of delivery dates is done subsequently by the local optimization methods themselves. Second, DEAL can be regarded as a decomposition method working problems that exhibit specific properties. For example, there are no cyclic dependencies between suppliers and manufacturer. It can be argued that DEAL makes use of these problem structures but not the generic PP/DS Optimizer. Third, the “warm reboot” property carries over information between several solutions. Avoiding an initialization from scratch allows to shorten runtimes of the locally applied PP/DS Optimizer without deteriorating the solution quality. In turn, shorter local runtimes allow more proposals to be exchanged. Due to the combinatorial complexity of the RCPSP, a frequent exchange of proposals is advantageous. Moreover it was observed that the warm reboot gets more effective as the process converges to a good setting of proposed or release dates, as previous activity sequences better fit to an offspring problem instance. Thus, the warm reboot

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supports the seamless shift from the strategy “try as much proposed dates as possible” at the beginning of coordination to the strategy “generate the best schedules for a fixed setting of proposed dates” towards the end of coordination. Fourth, even for larger problem instances a good coordinated solution was found in reasonable time. Although complexity increases, larger problem instances usually allow more freedom of rescheduling, i.e., a crude alignment of proposed dates suffices. Moreover, larger problem instances apparently come with a greater probability that large parts of previous activity sequences can be used for the warm reboot. Fifth, it could be observed that larger population sizes lead to better results. This is a common property of EAs: Larger populations have a better chance of overcoming local optima, at the cost of a slower convergence, however. By computing and evaluating proposals in parallel, DEAL allows larger population sizes without prolonging runtime. To further clarify the test methodology, Fig. 1 exemplary shows the results for a medium-sized test instance, also including multiple modes and setup times. Two suppliers deliver to one manufacturer, but also have other customers that should not be harmed by coordination.2 The convergence of the centrally applied PP/DS Optimizer is shown by the solid graph.3 The graph shows the average delay of 700000

central, monolithic solution sequential coordination parallel coordination

sum of no-priority lateness

650000 600000 550000 500000 450000 400000 350000 300000 250000

1000

2000

3000 4000 5000 runtime in seconds

6000

7000

Fig. 1 Exemplary results of a test instance with more than 2,700 activities in total, two suppliers and one manufacturer. Shown is the lateness to the manufacturer’s ultimate customers of central and coordination solutions, average of 10 runs each

2 Cf. Scheckenbach (2010), par. 8.2.2.1 and 8.2.2.4 for a detailed description of the underlying business problem. 3 The initial 500 s are required for problem generation and initialization.

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deliveries from the manufacturer to ultimate customers of ten different runs and of each run’s best-found solution at a specific point in time. It can be seen that the fitness of the initial population is gradually improved each generation, although the speed of improvement declines. The dashed graphs show the result of sequential and parallel coordination, respectively. An additional idle time can be observed required for the construction of the initial coordinated solution. For computing and evaluating a proposal, each partner runs his PP/DS Optimizer for 50 s. It should be highlighted that the graphs do not show the convergence of a local PP/DS run within these 50 s, but only the results of coordination, which explains the rather step-wise convergence. In order to compute acceptable solutions both suppliers prioritize the lateness to external customers in their local objective (cf. Sect. 6.3). Additionally the double-ranking scheme of Sect. 6.4 was implemented. Hence, improving acceptability can lead to a temporary increase in the lateness to the manufacturer’s customers (only the latter figure is drawn). It is worth mentioning, that all approaches led to acceptable solutions in the end of all runs, although the initial situation was not always accepted by the suppliers. Finally and most surprisingly, it can be seen that the parallel coordination (also employing a larger population size) does not only show better performance than the sequential coordination, but was also able to beat the central solution. It seems that the DEAL framework as a top-level metaheuristic was able to introduce new information that steered the overall search process in the right manner. Adjusting the due dates of the suppliers’ RCPSP instances by an evolutionary process and repeatedly solving these sub problems finally led to the good results. One might argue that the above results do not prove the effectiveness of DEAL but only the ineffectiveness of the PP/DS Optimizer. This is not true. First of all, it has to be mentioned that, for NP-hard planning problems, an optimal solution can usually not be computed in reasonable time. Moreover, one has to keep in mind that the PP/DS Optimizer was constructed to tackle RCPSPs of general type. For evaluating DEAL, we only consider a subset of all possible scheduling problems: problems, that exhibit regions that are only interdependent in a noncyclic manner. DEAL can be regarded as a specialized heuristic working on such problem structures. Hence, by decomposing the central scheduling problem, more detailed knowledge over the subproblems becomes available. According to the No Free Lunch Theorem of computer science, we have to expect that the results of a specialized heuristic are better than those of a general one.

8 Concluding Remarks When designing a coordination mechanism, three fundamental requirements have to be considered: Complex planning problems have to be supported, sensitive data must not be disclosed and incentive compatibility has to be guaranteed. To ensure the latter, most present approaches rely on compensation payments, payments made in an agreement by one or more parties to other parties to induce them to join the

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agreement. Though this idea seems to be logical at first sight, compensation payments are not without problems. As argued, such payments may – if not properly designed within the mechanism – provide possibilities for cheating and probing the other partner to infer sensitive information instead of preventing them from doing so. The presented heuristic restricts to generating ultimate solutions that are acceptable to all SC members without compensation payments. Naturally, this decision prunes the search space, possibly making the global optimum unattainable. However, it is questionable if this is considered as a drawback from a practical point of view. Usually, compensation payments are hardly deducible from control costs of objective functions and for many practical problems the optimal solution is usually not attained anyway. Practical acceptability adds further requirements: The mechanism should handle multiple domains, different optimization engines and different optimization models. It should be scalable to large problems and able to handle degraded, suboptimal solutions. Moreover, due to the limited time before escalation, short-term scheduling most urgently demands coordination. We believe EAs to be the logical choice under such conditions. They are not restricted to a particular type of problem requiring only a black-box fitness computation and are, as a population-based approach, easy to parallelize. Essentially, the idea is to adjust the data of underlying planning problems by an evolutionary process, using local optimization engines for decoding. As only the local data, but not the related optimization models are changed, different optimization engines can be connected to our framework. With regard to detailed scheduling, most practically used heuristics build up on release and due dates. If the possibility of a warm reboot is given, many proposals can be exchanged as well. For evaluating different proposals, we disguised sensitive data and limited possibilities for cheating by transmitting only ordinal rankings (reflecting a proposal’s quality) between the partners. Concluding, we claim that DEAL fulfills the three requirements complexity, security and incentive compatibility – at least from a practical perspective.

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Kolisch R, Sprecher A, Drexl A (1995) Characterization and generation of a general class of resource-constrained project scheduling problems. Manage Sci 41:1693–1703 Kutanoglu E, Wu SD (1999) On combinatorial auction and lagrangean relaxation for distributed resource scheduling. IIE Trans 31:813–826 Kutanoglu E, Wu SD (2006) Incentive compatible, collaborative production scheduling with simple communication among distributed agents. Int J Prod Res 44(3):421–446 Li J, Atallah M (2006) Secure and private collaborative linear programming. In: International conference on collaborative computing: networking, applications and worksharing Manne AS (1960) On the job-shop scheduling problem. Oper Res 8(2):219–223 Merkle D, Middendorf M, Schmeck H (2002) Ant colony optimization for resource constrained project scheduling. IEEE Trans Evol Comput 6(4):333–346 Myerson RB (1979) Incentive compatibility. Econometrica 47(1):61–73 Myerson RB, Satterthwaite MA (1983) Efficient mechanisms for bilateral trading. J Econ Theory 29:265–281 Neumann K, Schwindt C, Zimmermann J (2001) Project scheduling with time windows and scarce resources. Springer, Heidelberg Nie L, Xu X, Zhan D (2006) Collaborative planning in supply chains by lagrangian relaxation. In: Proceedings of the First International multi-symposiums on computer and computational science Radoport A, Fuller MA (1995) Bidding strategies in a bilateral monopoly with two-sided incomplete information. J Math Psychol 39:179–196 Samuelson W (1984) Bargaining under asymmetric information. Econometrica 52(4):995–1005 Scheckenbach B (2010) Collaborative planning in detailed scheduling. Ph.D thesis, University of Hamburg Sprecher A, Kolisch R, Drexl A (1995) Semi-active, active and non-delay schedules for the resource-constrained project-scheduling problem. Eur J Oper Res 80:94–102 Stadtler H (2005) Supply chain management and advanced planning – basics, overview and challenges. Eur J Oper Res 163:575–588 Stadtler H (2007) A framework for collaborative planning and state-of-the-art. OR Spectrum 31:5–30 Sucky E (2006) A bargaining model with asymmetric information for a single supplier-single buyer problem. Eur J Oper Res 171:516–533 Thomas PR, Salhi S (1998) A tabu search algorithm for the resource constrained project scheduling problem. J Heuristics 4:123–139 van Houtum GJ, Inderfurth K, Zijm WHM (1996) Materials coordination in stochastic multiechelon systems. Eur J Oper Res 95:1–23 Walther G, Schmid E, Spengler TS (2008) Negotiation based coordination in product recovery. J Prod Econ 111(2):334–350 Wellman MP, Walsh WE (2001) Auction protocols for decentralized scheduling. Game Econ Behav 35:271–303 Williams HP (1999) Model building in mathematical programming, 4th edn. Wiley, New York Yao A (1982) Protocols for secure computation. In: Proceedings of the 23rd Annual IEEE Symposium on Foundations of Computer Science

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Inventory Record Inaccuracy, RFID Technology Adoption and Supply Chain Coordination H. Sebastian Heese

Abstract Most retailers suffer from substantial discrepancies between inventory quantities recorded in the system and stocks truly available to customers. Promising full inventory transparency, RFID technology has often been suggested as a remedy to this problem. We consider inventory record inaccuracy in a supply chain model, where a Stackelberg manufacturer sets the wholesale price and a retailer determines how much to stock for sale to customers. We first analyze the impact of inventory record inaccuracy on optimal stocking decisions and profits. Contrasting optimal decisions in a decentralized supply chain with those in an integrated supply chain, we find that inventory record inaccuracy exacerbates the inefficiencies resulting from double marginalization in decentralized supply chains. Assuming that RFID technology can eliminate the problem of inventory record inaccuracy, we determine the cost thresholds at which RFID adoption becomes profitable. We show that a decentralized supply chain benefits more from RFID technology, such that RFID adoption improves supply chain coordination. Keywords Inventory • Record inaccuracy • RFID • Supply chain coordination

1 Introduction The discrepancy between inventory records and the amount of product effectively available for sale to customers presents a key problem in retail operations. Based on extensive empirical studies, Raman et al. (2001) report they found 65% of inventory This chapter is based on the article “Inventory record inaccuracy, double marginalization and RFID adoption” by H.S. Heese, published 2007 in Production and Operations Management (volume 16, issue 5, pages 542–553). H.S. Heese Kelley School of Business, Indiana University, 1309 East Tenth Street, Bloomington, IN 47405, USA e-mail: hheese@indiana.edu 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_19, # Springer-Verlag Berlin Heidelberg 2011

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records to be erroneous, i.e., the recorded inventory quantity did not match the quantity present at the store. Physical inventory deviated from inventory records by 35% on average, and about 16% of the product physically present at the store was not readily available to customers due to SKU misplacement. Transaction or scanning errors present yet another source of inventory record inaccuracies – all too often cashiers at the checkout use the number key to aggregate products with different flavors but the same price. Raman et al. (2001) conclude that “most retailers cannot, with any degree of precision, identify the number of units of a given item available at a store.” Since even the most sophisticated management system is handicapped if working with flawed data, the problem of lacking inventory record accuracy is often referred to as the missing link in retail execution and it has been estimated that the resulting lost sales and inventory costs reduce profits by more than 10% (Raman et al. 2001; Alexander et al. 2002). In this chapter we do not differentiate between the different sources of discrepancies between inventory records (system inventory or planned inventory) and inventory truly available for sale to customers (shelf inventory or available inventory). Since from the retailer’s perspective, the common underlying problem is the uncertainty of the inventory record, we will also use the term inventory uncertainty to refer to this problem. Among other developments in information technology, Radio-Frequency Identification (RFID) technology is being discussed as a powerful means to solve the problem of inventory uncertainty, among a plethora of other problems in supply chain execution. Enabling virtually full inventory transparency (both location and quantity) at any time, RFID technology, employed at the item-level, has indeed the potential to vastly improve retail execution (cf. McFarlane and Sheffi 2003). In recent years, several major retail chains have strongly promoted – or even mandated – RFID adoption by their suppliers, mostly at the pallet level. While RFID technology is suggested to enable substantial efficiency gains at the different stages of a supply chain, the associated costs are by no means negligible. Besides fixed costs related to the purchase and implementation of the necessary infrastructure, at this point especially the substantial cost of RFID tags seems to prohibit widespread use at the item level. Even if tag-costs decreased substantially, it is unlikely that item-level RFID adoption would be financially profitable at every retailer and for all products – it would likely start with more expensive items (cf. Want 2004). An optimal adoption decision needs to balance the value of full inventory transparency with the costs of RFID. In making this trade-off, it is important to distinguish the different incentives (benefits and costs) at the various stages of the supply chain. For example, adoption may be difficult if the retailer reaps most of the benefits, while the manufacturer bears most of the costs. More generally asked, how does the adoption decision of a decentralized supply chain differ from that of an integrated supply chain? We use the classic supply chain model of a single manufacturer, who as Stackelberg leader determines the wholesale price at which a product is sold to a retailer, who in turn determines how much to stock for sale to consumers (at an exogenous retail price). In making this stocking decision, the retailer faces

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uncertainty in customer demand as well as in inventory records. The latter source of uncertainty can be eliminated through RFID employment. We analyze the impact of inventory record inaccuracies on supply chain performance, and we investigate how double marginalization in a decentralized supply chain affects the different stages’ incentives to adopt RFID. While the possible implications of RFID technology on the supply chain are manifold, we focus on the improvements that result from having accurate inventory information. (With RFID all system inventory is available for sale to customers.) Poor execution at the retail level has been identified as the main driver of out-of-stock situations, which are a key problem in retailing (Andersen Consulting 1996; Gruen et al. 2002). While the benefit of such accuracy to the retailer is immediate, it is less clear, how RFID adoption affects the manufacturer, and the supply chain as a whole. Specifically, we address the following research questions: 1. How does inventory uncertainty affect stocking decisions compared to the standard case with accurate inventory records? 2. How does inventory uncertainty affect profits in a decentralized supply chain as compared to an integrated supply chain? 3. When do the manufacturer and the retailer individually benefit from RFID technology? 4. How does the RFID adoption decision in a decentralized supply chain differ from that of an integrated supply chain? We review the related literature in Sect. 2, and we introduce our base model in Sect. 3. Section 4 contains our model analysis and main results; we discuss the impact of inventory record inaccuracy on supply chain decisions and performance in an integrated and a decentralized supply chain (Sects. 4.1 and 4.2, respectively), before we contrast RFID adoption decisions for these two cases in Sect. 4.3. In Sect. 5, we conclude with a discussion of our results, limitations of our model, and suggestions for future research. All proofs are relegated to Appendix 1.

2 Literature Review The problem of inaccurate inventory records and misplaced SKUs at the retail level is widely discussed (e.g., Alexander et al. 2002; Raman et al. 2001; DeHoratius and Raman 2008). While most classic inventory models are based on the assumption of accurate inventory information, a subset of that research area considers the problem of inventory uncertainty under the term of yield uncertainty. An extensive review of this research stream is provided in Yano and Lee (1995). There is some very recent work that investigates optimal inventory management explicitly under inaccurate inventory records. Camdereli and Swaminathan (2010) consider the case where a fixed and known proportion of the retailer’s order quantity becomes unavailable for sale due to misplacement at the retailer. All misplaced products are recovered and salvaged

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at the end of the period. They analyze the impact of such proportional loss on the retailer’s optimal order quantity. They compare the performance of a decentralized supply chain with that of a vertically-integrated supply chain and investigate how coordination can be achieved by means of buy-back and revenue-sharing contracts. DeHoratius et al. (2008) propose the maintenance of a probabilistic inventory record instead of the commonly used point estimate to account for the presence of inventory record inaccuracy. Sources of such inaccuracies are modeled through an additional random variable called invisible demand. They suggest a simple Bayesian procedure to periodically update the inventory record. In K€ok and Shang (2007), the phenomenon of inventory inaccuracy is represented through random errors that change the physical inventory level at the end of each period. These errors keep accumulating, until the inventory record is updated by means of a costly inspection. They propose a near-optimal joint inventory inspection and replenishment heuristic and find that order quantities should increase as the number of periods since the last inspection increases, to accommodate for the added uncertainty. The inspection frequency should be higher for items with high value and larger error variance relative to demand variance. While explicitly addressing the impact of inventory record inaccuracy on inventory management, none of the above articles quantifies the value of inventory accuracy or the problem of RFID adoption. ¨ zer (2007) argue that most of the existing Dutta et al. (2007) and Lee and O estimates of the value of RFID are at best educated guesses, often lacking any comprehensible basis. Since most of these estimates are overly optimistic, the ¨ zer (2007) suggest that results of implementation are often frustrating. Lee and O this discrepancy between wild value propositions and sobering results has created a credibility gap and they call for the academic community to provide solid models to enable more realistic estimates of the RFID value. Atali et al. (2009) consider a single stage inventory control problem, where inventory records are inaccurate due to three additional demand streams: shrinkage, misplacement, and transaction errors. By prioritizing the different demand streams, they derive upper and lower bounds on the optimal solution as well as a simple heuristic. They compare the costs under inventory record inaccuracy with those under perfect transparency to obtain what they call the value of inventory visibility, which represents an upper bound on the benefits of RFID technology. Sahin (2004) considers a single-stage inventory system with inventory record inaccuracy. Considering different possible sources of such inaccuracies as well as different mathematical forms to represent these relations, Sahin investigates the consequences of inaccurate inventory records and thereby provides a quantifiable value for Auto ID technology. Rekik et al. (2008, 2009) study product loss through misplacement or theft. They demonstrate the importance of explicitly considering this problem in the stocking decision and derive threshold cost values, at which RFID adoption would become cost-effective. Karaer and Lee (2007) study the value inventory visibility in a manufacturer’s reverse channel. Their findings suggest that RFID technology might enable

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substantial benefits if the product return flow is highly volatile, if the duration of the reverse channel process is long, and if a significant portion of the returned products needs to be reworked. While these four articles explore the problem of inventory control under inaccurate inventory information and explicitly consider the role of RFID technology in mitigating the problem of inventory inaccuracy, their focus lies on the value of inventory visibility at a single stage or firm. In contrast, our work specifically focuses on the consequences of inaccurate inventory records in the presence of double marginalization in a decentralized supply chain, contrasting RFID adoption incentives of the individual supply chain stages with that of the integrated firm. Kang and Gershwin (2005) use simulation to study the consequences of inventory record inaccuracies in a setting where the inventory management process is automated. They find that even small undetected losses can lead to severe disruptions and stock-outs, especially in lean systems. They suggest several approaches to mitigate this problem, including the adoption of Auto-ID technology. Even though Auto-ID technology in their model enables full inventory transparency, their results suggest that simple heuristics that compensate for the stock loss can achieve near-optimal performance – at much lower cost. Fleisch and Tellkamp (2005) simulate the consequences of inventory record discrepancies in a three-stage supply chain. They find that an elimination of such inaccuracies, which could for example be achieved by RFID technology, can substantially reduce supply chain cost and stock-outs. While the simulation models of Kang and Gershwin (2005) and Fleisch and Tellkamp (2005) present a valuable first step in analyzing the value of inventory visibility in a supply chain, they assume a vertically integrated supply chain and hence cannot derive any insights with respect to the value of RFID adoption in a decentralized supply chain. Using simulation to estimate of the value of RFID technology in a four-stage supply chain under different degrees of collaboration, Sari (2010) finds that RFID adoption to be most beneficial in environments with longer lead times and limited demand uncertainty. Rekik et al. (2005) consider a supply chain with one manufacturer and one retailer both in the presence of inventory inaccuracies and under full inventory transparency due to RFID implementation. Comparing these two settings, they derive the value of RFID technology and they determine tag cost threshold values for profitable adoption. Similar to Camdereli and Swaminathan (2010), Rekik et al. (2005) also investigate the inefficiencies due to double marginalization in a decentralized supply chain and derive coordinating buy-back contracts. Gaukler et al. (2007) investigate the impact of RFID technology on a vertically-integrated supply chain vis-a`-vis a decentralized supply chain with one manufacturer and one retailer. They capture possible discrepancies between shelf inventory (available to satisfy customer demand) and backroom inventory through a parameter y that represents the efficiency of the retailer’s replenishment process. More specifically, they define y as the conditional probability that, given ample backroom inventory, a customer will find the product available on the retail shelf. If true demand is normally distributed, so is the effective demand that can be satisfied from the shelf.

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Assuming that neither the manufacturer nor the retailer include y in their decision making and that losing sales is the only penalty for empty shelves, the problem can be formulated as a standard newsvendor problem. Gaukler et al. (2007) analyze the benefits of full replenishment efficiency ðy ¼ 1Þ in a vertically-integrated supply chain and derive some insights into the threshold cost, at which RFID adoption would become profitable. While Rekik et al. (2005) and Gaukler et al. (2007) address problems that are similar to the one investigated in this article, their works differ in several important aspects. Most importantly, their research assumes that the impact of inventory record inaccuracy on the order yield is always negative, i.e., the retailer’s inventory available for sale is always less than ordered. However, it is reasonable to believe that inventory misplacements might eventually be corrected, implying available inventory that is larger than ordered in a specific period. Similarly, scanning errors will have a negative effect on the inventory records for some products and a positive for others (e.g., if scanning a vanilla yogurt as a strawberry yogurt). In such settings, RFID technology promises two potential benefits. On the one hand, it could help reduce the problem of shrinkage. On the other hand, RFID improves inventory transparency, reducing the uncertainty associated with inaccurate inventory records. We believe that these are two different value propositions, each of them important enough to merit attention. However, under the assumption that the source of inventory record inaccuracy always reduces order yield, the impact of inventory record inaccuracy can only be analyzed in the presence of such product losses, which are likely to have a confounding effect in estimating the value of RFID technology in achieving inventory transparency. The problem of uncertainty in inventory records cannot even be considered in a model where such inaccuracies are deterministic and known (as in Rekik et al. 2005). In our model, inventory record inaccuracies can be both positive or negative. Specifically, we capture this discrepancy between system inventory and available inventory by assuming that the ratio of the two follows a random variable. While this setup allows us to model shrinkage by assuming a mean below unity, we show that even in the absence of such losses, the mere uncertainty with respect to available inventory has a substantial impact on the supply chain. Gaukler et al. (2007) further assume that the retailer acted as if she was not aware of the inventory inaccuracy problem. This approach greatly simplifies the mathematical analysis, reducing it to a standard newsvendor problem. In our model, the retailer is conscious of the uncertainty with respect to inventory and she adjusts her orders accordingly, such that her execution problems also affect the manufacturer. As a consequence, in our setting both supply chain stages have an incentive to consider the costly adoption of RFID technology. Finally, the focus of Gaukler et al. (2007) is on the impact of RFID adoption on supply chain performance, rather than the adoption decision, while the question of how the adoption decision in a decentralized supply chain differs from that of an integrated supply chain is of central concern in our work.

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3 The Model In this section, we introduce our base model with its underlying assumptions. While we present the model for the integrated supply chain (which is analyzed in Sect. 4.1), the adjustments for the decentralized scenario (Sect. 4.2) are straightforward. A table with an overview of the notation is provided in Appendix 2. Generally, our model is based on the familiar newsvendor model. Given a starting inventory xn in period n, the decision maker needs to determine the optimal stocking quantity Qn in order to maximize expected (discounted) infinite-horizon profits. Demands in the different periods Dn are independent random variables with the identical support on ½0; D, density fD ðÞ and mean mD . A product is produced at cost c and offered for sale at a retail price r > c. We assume the firm has no market power and hence is a price-taker (i.e., the retail price r is exogenous). We do not explicitly consider penalty costs to capture the negative implications of shortages. Including such costs in our model would be straightforward and would not affect any of our insights. The key difference between our model and the standard newsvendor model with accurate inventory records is that the quantity available for sale to customers (shelf inventory) might differ from the planned stocking quantity. We assume that supply is reliable, so that such differences occur due to execution inefficiencies at the retailer, e.g., shrinkage, scanning errors, or SKU misplacement (cp. Sect. 1). Whereas some products might be lost forever due to theft, others might be physically present at the retailer (stored or lost in the backroom or on the wrong shelf), but not readily available for sale. However, in some periods these units can be made available to customers – since record inaccuracies are certainly not planned, such recoveries might come as a surprise and hence might actually increase available inventory beyond the planned stocking level. As a consequence, besides demand uncertainty, the decision maker faces uncertainty with respect to the validity of the inventory records. For example, let Q denote the number of units on the inventory record (and supposedly in the store) and let QA denote the number of units actually available for sales to the customers (on the right shelf). If inventory is misplaced, QA < Q. If those units are then found and placed at the right spot, it might also occur that QA > Q. On average, the available inventory should not be larger than the planned inventory, but it can be smaller, if there is shrinkage. Noting the obvious similarity to the problem of inventory management under yield uncertainty, we build on the model of Inderfurth (2004) and model the discrepancy between planned and available inventory in any given period through a stochastic multiplier Yn on ½0; Y with density fY ðÞ and mean mY 1. The random variable Y represents the ratio of available to recorded (planned) inventory, so on average there is no shrinkage if mY ¼ 1, while there is some loss if mY < 1. To achieve mathematical tractability in an environment complicated by uncertainty in inventory records, we assume that the ending inventory in each period is no larger than the desired stocking quantity of the next period (i.e., xn Qn ).

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Following the seminal work of Veinott (1965), this assumption has frequently been used in the inventory literature to achieve tractability in multi-period settings with non-perishable inventory. This assumption is always satisfied in traditional inventory models with accurate inventory records, if cost parameters are stationary and demand is stochastically non-decreasing, since optimal stocking levels are non-decreasing over time and leftover inventory can never exceed the intended stocking quantity. However, this assumption is non-trivial in our setting, where stochastic order yield might lead to leftover inventory in excess of the intended stocking quantity. With the assumption that left-over inventory in any given period is less than the optimal stocking quantity of the following period, consecutive periods can be decoupled by assuming that there is a salvage value s on end-of-period leftover inventory, which is equal to the (discounted) unit cost of the following period minus a potential per-unit per-period holding cost. Hence, under this assumption, consecutive periods are independent and can be analyzed separately, so that the analysis of a multi-period inventory model with inventory holding cost can be reduced to a single-period analysis with the appropriate salvage value and zero starting inventory, without any loss of generality. While we base our following analysis on a single-period model, we ask the reader to keep in mind that our results and insights apply to any period of a multi-period horizon, and thus also for the corresponding multi-period problem. Let pðQÞ denote the supply chain’s expected profit if ordering Q.1 The finite To obtain a continuously differentiable support of fD ðÞ implies a discontinuity at D. representation of the problem, we distinguish two cases depending on whether the Y. 2 order quantity Q is smaller (case A) or larger (case B) than D= D Case A : Q Y

(1)

0YQ 1 ð pA ðQÞ ¼ cQ þ @ ðrD þ sðYQ DÞÞfD ðDÞdDAfY ðYÞdY ðY

ðY

0

B þ @ 0

D Case B : Q > Y 1

0

ðD

0

1

C rYQfD ðDÞdDAfY ðYÞdY

YQ

(2)

Analyzing a single-period model with zero starting inventory, we will use the terms order and stocking quantity interchangeably. 2 In the following, the cases where inequality conditions are satisfied with equality are assigned arbitrarily, without loss of generality.

Inventory Record Inaccuracy, RFID Technology Adoption

0YQ D=Q ð ð @

pB ðQÞ ¼ cQ þ 0

ðY þ D=Q D=Q ð

þ 0

491

1 ðrD þ sðYQ DÞÞfD ðDÞdDAfY ðYÞdY

0

0 D 1 ð @ ðrD þ sðYQ DÞÞfD ðDÞdDAfY ðYÞdY 0 B @

0

ðD

1 C rYQfD ðDÞdDAfY ðYÞdY

YQ

It is easy to confirm that the expected profit functions are strictly concave and to derive sufficient first-order conditions. However, analytical closed form expressions can only be obtained for special types of demand and yield distributions, for example the uniform and the exponential distribution (Inderfurth 2004). To obtain analytically tractable and crisp results, we assume that both yield and demand are uniformly distributed. Numerical studies based on normally distributed demand and yield largely confirmed our analytical results. With these assumptions, (1) and (2) can be simplified to the following expressions. Case A : Q

mD : mY

pA ðQÞ ¼ ðrmY cÞQ

ðr sÞmY 2 2 Q 3mD

Case B : Q >

mD : mY

pB ðQÞ ¼ ðr sÞmD ðc smY ÞQ

ðr sÞmD 2 1 Q 3mY

(3)

(4)

In the following section, we analyze how inventory record uncertainty affects stocking decisions and supply chain performance, focusing specifically on the impact of double marginalization in a decentralized supply chain. Assuming that the adoption of RFID technology implies additional per-unit costs, but leads to full inventory transparency (the problem then becomes a standard newsvendor problem), we derive closed form solutions for the RFID cost thresholds that make adoption profitable.

4 Analysis In this section, we investigate the consequences of inventory uncertainty and we contrast RFID adoption incentives of an integrated supply chain with those of a decentralized supply chain. We first determine the optimal decisions and supply chain profits for the integrated (Sect. 4.1) and for the decentralized supply chain

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(Sect. 4.2). In Sect. 4.3, we derive and contrast the RFID cost thresholds for the two scenarios, below which adoption would be profitable. In the following, superscripts are used to indicate the concerned supply chain stage (M: manufacturer, R: retailer, SC: supply chain), and subscripts (1–4) are used to distinguish between the following four cases (1) an integrated supply chain with inaccurate inventory records, (2) an integrated supply chain with RFID, (3) a decentralized supply chain with inaccurate inventory records, and (4) a decentralized supply chain with RFID.

4.1

The Effect of Inventory Record Inaccuracy in an Integrated Supply Chain

We first derive the optimal order decisions and resulting profits for an integrated supply chain that experiences inventory record inaccuracies. This supply chain faces average procurement/production cost of c=mY and, for notational conveY nience, we use the parameter a ¼ rc=m rs to denote its critical fractile. Proposition 1. The optimal order quantity for the integrated supply chain under 3a otherwise. D, if a 23 , and Q1B ¼ p1ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ D, inventory inaccuracy is Q1A ¼ 4m 2mY

Y

¼ Correspondingly, expected profits are qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ðrsÞðc=mY sÞ pSC , otherwise. 1B ¼ ðr sÞmD 2mD 3 pSC 1A

3ðrc=mY Þ2 4ðrsÞ

3ð1aÞ

mD , if a 23 , and

Reflecting the earlier distinction of two possible regimes [cp. cases A and B in (1) and (2)], we see that the rule for determining the optimal order quantity under inventory uncertainty differs depending on the specific value of the critical fractile ðaÞ.3 While the optimal order rule for small service levels (case A) has a similar structure as in the standard newsvendor problem, it is very different for larger critical fractiles (case B). Shrinkage (if mY < 1) generally has two conflicting effects on the optimal order quantity. On the one hand, one could expect shrinkage to increase the optimal order quantity to compensate for the expected loss in product. On the other hand, shrinkage increases the expected per unit production costs, with a decreasing effect on the target service level and hence the optimal order quantity. The net outcome of these two effects can be both positive or negative, depending on the specific values of the different cost parameters (see Lemma 1 in Appendix 1). For sufficiently high salvage value ðs > r=4Þ, both order quantities strictly increase in mY . If the salvage value is lower, Q1A is increasing-decreasing in mY , and Q1B is either decreasingincreasing (if s > c=2) or even strictly decreasing in mY ðs < c=2Þ. 3 While the specific value of two-thirds is characteristic of the uniform distribution, a threshold based stocking policy is likely for all demand distributions with finite support (see earlier discussion of the two cases). We observed similar threshold based stocking policies in numerical experiments for normally distributed demand and order yield.

Inventory Record Inaccuracy, RFID Technology Adoption 140

493

Q1B

120 100 80 60

Q1A

a = 2/3

40 20 0 20%

µY

100%

Fig. 1 Optimal order quantity of the integrated supply chain

Figure 1 illustrates the optimal order decisions and the contrast between the different stocking rules Q1A and Q1B for D ¼ 100, c ¼ 10, s ¼ 8, and r ¼ 50. Shrinkage lowers the critical fractile under inventory uncertainty, and increasing mY from 20 to 100% raises a from 0 to approximately 95%. Figure 1 demonstrates how the integrated supply chain’s optimal order quantity is raised for higher critical fractiles. Compared to the optimal order quantity for lower critical fractiles, the optimal order quantity for higher critical fractiles is always inflated to buffer against the comparatively high underage costs that might occur in the event of extremely low order yield (Q1B > Q1A , if a > 2=3). Lemma 1 in Appendix 1 proves that the validity of this observation is not limited to this specific numeric example. In this case, the threshold value a ¼ 2=3 corresponds to an average ratio between available inventory and system inventory of about mY ¼ 46%. It is interesting to analyze the optimal order quantity for the case without shrinkage (i.e., mY ¼ 1) and to compare it to the optimal order quantity Q^ for the corresponding newsvendor problem without inventory uncertainty (i.e., Y ¼ 1). It is well-known that Q^ ¼ aD and without shrinkage, deviations from this order quantity can only be due to the inherent inventory uncertainty. The results of Lemma 1 (see Appendix 1) show that without shrinkage ðmY ¼ 1Þ there is a threshold value 2=3 < a < 1, such that the optimal order quantity under inventory uncertainty is larger than the optimal order to the corresponding problem without inventory uncertainty, if and only if a > a, i.e., if the critical fractile is sufficiently large.4 For the specific example presented in Fig. 1, it can easily be verified that the optimal order quantity under inventory uncertainty is always higher than the optimal order quantity for the standard newsvendor problem without inventory uncertainty (Q^ ¼ aD increases from 0 to 95). Employing RFID technology has two effects on supply chain performance. On the one hand, this technology enables better (we assume perfect) inventory 4 This threshold value solves 12a2 ð1 aÞ ¼ 1, so a 89:6%. This threshold result has been mentioned in Inderfurth (2005).

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transparency, such that all inventory is available for sale.5 It can be easily confirmed that such accuracy increases expected supply chain profits. On the other hand, there are costs associated with RFID technology. In practice, the cost of item-level RFID is largely driven by the cost of the tags. Correspondingly we assume that employment of RFID increases the per-unit cost by a fixed amount t.6 Assuming full inventory transparency under RFID, the supply chain in this scenario (subscript 2) faces the standard newsvendor problem (cf. Porteus 2002) and expected profits are ðQ

ðD

pRFID ðQÞ ¼ ðc þ tÞQ þ ðrD þ sðQ DÞÞfD ðDÞdD þ rQfD ðDÞdD 0

(5)

Q

Proposition 2 provides the optimal order quantity and expected profits for the integrated supply chain under RFID employment. Proposition 2. The optimal order of the integrated supply chain with RFID 2 yields maximum expected profit pSC ¼ ðrctÞ mD . D employment Q2 ¼ rct 2 rs rs Since inventory transparency increases expected profits, there must be a threshold value for the RFID costs, below which adoption is profitable. Before analyzing these cost thresholds (in Sect. 4.3), in the following section (Sect. 4.2) we derive the optimal decisions for a decentralized supply chain.

4.2

The Effect of Inventory Record Inaccuracy in a Decentralized Supply Chain

As in the previous section, we first study the optimal decisions and resulting supply chain performance for the case of inaccurate inventory records, and then analyze the setting with RFID technology. The previous section studied the decisions in an integrated supply chain; here we assume that the Stackelberg manufacturer sets a wholesale price, anticipating the retailer’s corresponding optimal stocking decision. While the manufacturer is not directly affected by the inventory record inaccuracy, he is so indirectly through the retailer’s adjustments in her order decision. In a setting were the retailer does not consider the inventory uncertainty in making her order decisions (as in Gaukler et al. 2007), the only reason for her to adjust her orders under RFID would be a change in the per-item cost. However, if the retailer 5

An implication of this assumption is that RFID adoption eliminates shrinkage. While this effect can be an important driver of RFID adoption, much of the following discussion focuses on the potential benefit of RFID technology in reducing inventory uncertainty, assuming there is no shrinkage ðmY ¼ 1Þ. However, unless noted otherwise, all results are also valid for the case with shrinkage ðmY < 1Þ. 6 To avoid trivial cases we assume t < r c.

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is aware of the inventory uncertainty problem and adjusts the order quantity accordingly, the impact of inventory uncertainty (or RFID adoption) on the manufacturer benefits is not immediately clear. Proposition 3 describes the equilibrium wholesale price and order decisions as well as resulting expected profits for the decentralized supply chain with inventory uncertainty. Proposition 3. In a decentralized supply chain with inventory inaccuracies, the Y and the retailer orders manufacturer charges a wholesale price w3 ¼ cþrm 2 2 3a D. Manufacturer, retailer and supply chain profits are pM ¼ 3mD ðrc=mY Þ , Q3 ¼ pR3

¼

8mY 3mD ðrc=mY Þ2 16ðrsÞ

3

and

pSC 3

¼

9mD ðrc=mY Þ2 16ðrsÞ

8ðrsÞ

, respectively.

Interestingly, we find that in the decentralized supply chain with endogenous wholesale pricing, the optimal order decision no longer follows the two different rules encountered in the previous section, where we distinguished between regimes A and B depending on the critical fractile a. While for high critical fractiles the retailer would still order according to a different function (similar to Q1B in Proposition 1), with the double marginalization (Spengler 1950) under endogenous wholesale pricing she simply never faces such high critical fractiles. Even if the supply chain as a whole had a higher critical fractile (a > 2=3), the manufacturer’s mark-up above cost always reduces the retailer’s critical fractile sufficiently to induce an order following the rule for regime A. It can easily be seen that, for a 2=3, double marginalization in the decentralized supply chain reduces the order quantity to 50% of the quantity that maximizes integrated supply chain profits (Q1A ). We know from Proposition 1 that the integrated supply chain follows a different stocking rule (Q1B ) for higher critical fractiles and it can be shown that Q1B > Q1A for a > 2=3 (see Lemma 1 in Appendix 1). Hence, under inventory uncertainty, the integrated supply chain inflates the order quantity for high critical fractiles (compared to case A). For a > 2=3, the decentralized supply chain keeps following the same rule (case A) and hence orders even less than half of what the integrated supply chain orders. As a result, we find that for products with high critical fractiles inventory inaccuracies exacerbate the problem of double marginalization in a decentralized supply chain. We now explore the benefits of RFID employment on the decentralized supply chain with double marginalization. Similar to Proposition 2, Proposition 4 provides the equilibrium decisions and resulting performance measures after RFID adoption for the decentralized supply chain. Proposition 4. In a decentralized supply chain with RFID technology, the manufacturer charges a wholesale price w4 ¼ rþcþt and the retailer orders Q4 ¼ 2 2 1 rct D. Manufacturer, retailer and supply chain profits are pM ¼ ðrctÞ mD , 2

pR4

rs Þ2 ¼ ðrct 4ðrsÞ

4

mD and

pSC 4

¼

3ðrctÞ2 4ðrsÞ

2ðrsÞ

mD , respectively.

Recall that the costs of RFID technology are borne by the manufacturer, but that he can, with endogenous wholesale pricing, pass some of these costs on to the

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retailer. Comparing the wholesale prices before and after RFID adoption, we see YÞ , so the manufacturer’s surcharge on the wholesale price that w4 w3 ¼ tþrð1m 2 increases with the severity of the retailer’s pre-RFID shrinkage problem (it decreases in mY ). If there is no shrinkage at the retailer (i.e., if mY ¼ 1), the costs of RFID technology are split evenly. Consistent with the earlier observation, we find that the decentralized supply chain again orders exactly half of what the integrated supply chain would order. However, since there is only one stocking rule for the case with inventory transparency, here the impact of double marginalization on the optimal order quantity is consistent for all values of a. The fact that RFID adoption restores this proportionality of order quantities for high critical fractiles hints at the coordinating effect of RFID adoption. In the following section we derive and analyze the adoption thresholds for the integrated and the decentralized supply chain, and we show that the inventory transparency following RFID adoption might indeed improve supply chain coordination.

4.3

Double Marginalization and RFID Adoption

Proposition 5 provides the RFID cost threshold below which the integrated supply chain would find RFID adoption profitable. Proposition 5. An integrated supply chain would adopt RFID technology pﬃﬃ for t < tA ¼ ðr cÞ 23 ðr c=mY Þ if a 23 , and for t < tB ¼ ðr cÞ rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ qﬃﬃﬃﬃﬃﬃﬃ ðr sÞ 1 2 1a 3 , otherwise. As expected, we see that the benefit of RFID adoption differs depending on the balance of underage and overage costs, captured by the supply chain’s critical fractile a. It can be easily verified that both cost thresholds are positive, confirming the value of inventory transparency. The following Proposition 6 provides the analogue to Proposition 5 for the decentralized supply chain. Proposition 6. In a decentralized supply chain where the manufacturer set the wholesale price as a Stackelberg leader, both the retailer and the manufacturer are better off with RFID technology if and only if t < tA . Interestingly, we find that with endogenous wholesale pricing, the retailer’s and the manufacturer’s RFID adoption decisions are perfectly aligned, i.e., the retailer is better off under RFID adoption if and only if the manufacturer is better off. As a consequence, the (leading) manufacturer’s adoption decision is also optimal in terms of total supply chain profits. Surprisingly, Proposition 6 also shows that, if the critical fractile is relatively low ða 2=3Þ, the cost threshold of the decentralized supply chain is identical to that of the integrated supply chain. Consequently, the double marginalization in the

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decentralized supply chain in this case does not distort the evaluation of RFID. However, adoption decisions are different for higher critical fractiles ða > 2=3Þ, as described in the following Proposition 7, which provides one of the key insights of this chapter. Proposition 7. For a > 2=3, the cost threshold for profitable RFID adoption is strictly larger in a decentralized supply chain than in an integrated supply chain ðtA > tB Þ. No matter if the RFID adoption is undertaken by the manufacturer or mandated by the retailer, as long as the manufacturer is able to adjust wholesale prices as Stackelberg leader, for products with a sufficiently high critical fractile, the decentralized supply chain has a larger incentive to adopt RFID than the integrated supply chain, as it extracts a higher relative benefit from RFID technology. As a consequence, the performance gap due to double marginalization is decreased and RFID adoption improves supply chain performance. Figure 2 illustrates the difference between the two cost thresholds and contrasts the RFID adoption decisions of an integrated and a decentralized supply chain. For this graph, we use the same parameter values as in Fig. 1 (D ¼ 100, c ¼ 10, s ¼ 8, and r ¼ 50). Shrinkage in our model increases the expected procurement cost and thus has a significant impact on the supply chain’s critical fractile. For this specific example, the critical fractile a increases from 0 to approximately 95% as the average ratio of available-to-planned inventory mY increases from 20 to 100%. For settings with substantial shrinkage and relatively smaller critical fractiles under inventory uncertainty (a 2=3 for mY < 46%), we see that the incentives of the integrated and the decentralized supply chain are perfectly aligned ðtA ¼ tB Þ. The two threshold values are the same, even if the product would have a high critical fractile in the absence of shrinkage. However, for higher critical fractiles under inventory record inaccuracy, the decentralized supply chain extracts a higher value from RFID and finds RFID 40

8

35

tA

7 30

a=2/3

6

25 20

5 tA = tB

4

tA > tB

15

3 80%

10 5 0 20%

µY

Fig. 2 Cost thresholds for profitable RFID adoption

tB

100%

µY

100%

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H.S. Heese

adoption profitable at a higher per-unit cost compared to the cost threshold of the integrated supply chain ðtA > tB Þ. In settings with very high shrinkage, the value of RFID as a tool to prevent such losses is substantial (for this specific numeric example we have tA ¼ tB ¼ 40 for mY ¼ 20%). In order to better illustrate the value of RFID in reducing the uncertainty in inventory records, in the small frame of Fig. 2, we contrast the threshold values for higher values of mY (from 80 to 100%). We see that the relative difference between the two thresholds might be substantial for such settings with minor shrinkage problems (higher mY ) – arguably the case in most real life instances. In the absence of shrinkage ðmY ¼ 1Þ, the integrated supply chain in this example would invest up to tB ¼ 3:67 per unit into RFID technology, whereas the decentralized supply chain would find RFID adoption profitable up to a significantly higher cost threshold of tA ¼ 5:36. Why does the decentralized supply chain gain proportionally the same from RFID adoption for relatively low critical fractiles ða 2=3Þ, but more if the critical fractile is higher ða > 2=3Þ? As discussed before, if the critical fractile is relatively low, both the decentralized and the integrated supply chain increase their orders by the same fraction ðQ3 =Q1A ¼ Q4 =Q2 ¼ 1=2Þ, and both supply chains make the same RFID adoption decision (Proposition 6). For higher critical fractiles however, the optimal order quantity for the integrated supply chain is proportionally larger under inventory inaccuracies (Q1B > Q1A , see Lemma 1 in Appendix 1). While both the integrated and the decentralized supply chain make the same proportional adjustments to their optimal order quantities when employing RFID in settings with relatively low optimal service levels, we see that with inaccurate inventory information for higher critical fractiles, the integrated supply chain inflates the optimal order quantity (cp. Fig. 1). Since the order quantities are proportionally the same for both supply chain structures with inventory transparency, the change after RFID adoption is proportionally larger for the decentralized supply chain, counteracting the problem of double marginalization. Note that improved supply chain coordination in our model is a result of individually rational RFID adoption decisions, and coordination was not an explicit objective at the outset. Clearly the performance of the decentralized supply chain could be further improved, for example by means of buy-back contracts. However, the focus of our research lies on RFID adoption decisions under inventory record inaccuracy. The general problem of supply chain coordination has been addressed by a wide body of research (see Cachon (2003) for a review).

5 Conclusion The discrepancy between inventory records and the amount of product effectively available for sale to customers presents a key problem in retail operations. Promising full transparency, RFID technology is often proposed as a remedy to this problem. However, RFID technology is not free of cost, so an optimal

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adoption decision needs to quantify the achievable benefits and balance them against these costs. We consider a simple supply chain model, where a Stackelberg manufacturer sets the wholesale price and a retailer determines how much to stock for sale to customers. Besides demand uncertainty, the retailer faces uncertainty with respect to her inventory records, i.e., she is uncertain by how much available stock differs from the planned stocking quantity. We first analyze how inventory record uncertainty affects optimal stocking decisions in an integrated supply chain. We find that in the presence of such inaccuracies, the optimal stocking rule can follow two different types, depending on the balance between underage and overage costs. For relatively low critical fractiles, the optimal stocking rule resembles the solution to the standard newsvendor problem without inventory inaccuracies. However, for higher critical fractiles, we find that the optimal stocking quantity for the integrated supply chain is increased to reduce the risk of costly stock-outs in the event that available inventory turns out to be much lower than system inventory. We then determine the optimal wholesale price and stocking decisions in the decentralized supply chain. We show that with endogenous wholesale pricing by a Stackelberg manufacturer, the manufacturer’s and the retailer’s RFID adoption decisions are perfectly aligned, i.e., the manufacturer is better off with RFID technology if and only if the retailer is better off. As a consequence, the manufacturer’s optimal RFID adoption decision is also optimal in terms of total supply chain profits. Interestingly, we find that with endogenous wholesale pricing, the critical fractile to the retailer is never sufficiently high to warrant an stocking quantity increase as encountered for the integrated supply chain. In the absence of inventory record uncertainty, both supply chains resort to stocking rules of the same structure and our findings are consistent with the classic result that the stocking quantity of the decentralized supply chain with double marginalization is half of that of the integrated supply chain. We show that this proportionality of stocking quantities is maintained under inventory record uncertainty, as long as the (integrated) supply chain’s critical fractile is sufficiently low. However, for higher critical fractiles the (integrated) supply chain’s optimal stocking quantity is increased. Since the decentralized supply chain with double marginalization fails to adjust stocking quantities sufficiently to maintain proportionality, we find that for relatively high critical fractiles, the double marginalization in a decentralized supply chain may exacerbate the negative consequences of inventory record inaccuracy. Analyzing the cost thresholds for profitable RFID adoption, interestingly we find that for low critical fractiles the incentives for RFID adoption in the decentralized supply chain are perfectly aligned with those in the integrated supply chain, such that both supply chains would find RFID technology attractive at the same cost. However, for products with higher critical fractiles, the relative benefit of RFID is larger for the decentralized supply chain and consequently adoption would become profitable at a strictly higher cost. The rationale behind this surprising finding lies in the decentralized supply chain’s failure to sufficiently increase stocking quantities

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under inventory record inaccuracy. Since RFID adoption eliminates the need for such stocking quantity inflation and restores the proportionality of stocking quantities of the two supply chains, it mitigates the consequences of double marginalization and hence improves coordination of the decentralized supply chain.

Appendix 1: Additional Results and Proofs Lemma 1. (a) Q1A is increasing in mY for 0 < a 2=3, if s > r=4, and increasing-decreasing otherwise. (b) Q1B is increasing in mY (for 2=3 < a < 1), if s > r=4, decreasing if s < c=2, and decreasing-increasing otherwise. ^ and Q1B > Q^ , a > a, where 2=3 < a < 1. (c) If mY ¼ 1, then Q1A < Q, (d) If a > 2=3, then Q1B > Q1A . Proof of Lemma 1. 2c 2 3c 1A Part (a): @Q @mY > 0 , mY < r ; a 3 , mY rþ2s . Q1A is increasing in mY for 3c 2c r 0 < a 2=3, if rþ2s < r , s > 4 , and increasing-decreasing otherwise. c 2 3c 1B Part (b): @Q @mY > 0 , mY > 2s ; a > 3 , mY > rþ2s . Q1B is increasing in mY on 3c 2=3 < a < 1, if 2sc < rþ2s , s > 4r , decreasing if 2sc > 1 , s < 2c , and decreasingincreasing otherwise. 1 ﬃ < a , 12a2 ð1 aÞ > 1. Part (c) Q1A < Q^ by inspection; Q1B < Q^ , pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 3ð1aÞ The left side of this inequality is decreasing in a(for a > 2=3). At a ¼ 2=3, it is equal to 16=9 > 1, and it is equal to zero at a ¼ 1. Hence there exists a 2 23 ; 1 , where the inequality is satisfied with equality. 3mD a 1 9a2 D ﬃ> Part (d) Q1B >Q1A , pmﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2mY , 3ð1aÞ > 4 (since both sides of the mY 3ð1aÞ □ inequality are positive) ,14ð3aþ1Þ ð23aÞ2 >0,a6¼23. Proof of Proposition 1. The optimal order quantities are derived in Inderfurth (2004). The resulting expected profits can be obtained by substituting these quantities into (3) and (4). □ Proof of Proposition 2. The optimal order quantity Q2 is the well-known critical fractile solution to the classic newsvendor model. The expected profits follow from □ substituting Q2 into the expected cost function (5) and simplification. Proof of Proposition 3. The retailer’s optimal order quantities are as given in 3 =mY Proposition 1, but with a critical fractile of a3 ¼ rwrs instead of a. The M manufacturer’s profit equals p3 ¼ ðw3 cÞQ3 . Two cases need to be distinguished 3 ¼ ðr þ 2sÞ m3Y to denote the depending on the resulting value of a3 . Use w

Inventory Record Inaccuracy, RFID Technology Adoption

501

3 (i.e., a3 2=3), a3 ¼ 2=3. For w3 w @ 2 pM @pM 3mD rw3A =mY 3mD 3A ðw3A Þ 3A ðw3A Þ ¼ ðw3A cÞQ3A ¼ ðw3A cÞ 2m < 0; @w ¼ ; @w3A 2 ¼ ðrsÞm rs 3A Y Y cþrmY 3 , a3 jw3A < 2=3 , 3ðc mY sÞ þ ðr sÞmY > 0. w3A > w For 0 ! w3A ¼ 2 ; qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ @pM ðw Þ mD 3B M rs 3B 3 (i.e., a3 > 2=3), p3B ¼ ðw3B cÞQ3B ¼ ðw3B cÞ m 3ðw3B =m sÞ; @w w3 < w ¼ 3B Y Y wholesale

price

at

which

pM 3A

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

mD cþw3B 2smY 1 3ð1a3 Þ 2w3B 2smY mY M pM 3A a3 ¼2=3 ¼ p3B a3 ¼2=3

3 (both cases are equivalent at w 3 ). Since > 0 ! w3B ¼ w 3 , the optimal wholesale price is w3 ¼ w3A . and w3A > w

Substituting this optimal wholesale price gives the expressions in the Proposition.□ Proof of Proposition 4. The retailers optimal order quantity Q4 is the standard critical fractile solution to the newsvendor problem in (5) – the critical fractile 4 The manufacturer’s profit equals pM is rw 4 ¼ ðw4 ðc þ tÞÞQ4 ¼ rs . rw @ 2 pM @pM 4mD rþcþt 4 ðw4 Þ 4 ðw4 Þ 4 ðw4 ðc þ tÞÞ2mD rs ; @w4 ¼ 0 ! w4 ¼ 2 . @w4 2 ¼ ðpþrsÞ < 0; Substitution of the optimal wholesale price and simplification gives the expressions in the Proposition. □ 2

2

ðrctÞ 3ðrc=mY Þ SC Proof of Proposition 5. If a 23 : pSC 2 > p1A , rs mD > 4ðrsÞ mD p ﬃﬃ ﬃ 2 2 , 4ðr c tÞ > 3ðr c=mY Þ , 2ðr c tÞ > 3ðr c=mY Þ (both sides are pﬃﬃ Þ2 SC positive) , t < ðr cÞ 23 ðr c=mY Þ; If a > 23 : pSC , ðrct 2 > p1B rs mD > qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Y sÞ ðr sÞmD 2mD ðrsÞðc=m 3 qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ qﬃﬃﬃﬃﬃﬃ qﬃﬃﬃﬃﬃﬃ rct2 12=3 1a 1a , rs > 1 2 3 1 2 3 > 1 2 ¼ 1=3 > 0 3

, rct rs >

rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ qﬃﬃﬃﬃﬃﬃﬃ 12

1a 3

, t < ðr cÞ ðr sÞ

rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ qﬃﬃﬃﬃﬃﬃﬃ 12

1a 3 .

□

M Proof of Proposition 6. The inequalities pR4 > pR3 and pM 4 > p3 (hence by definition SC SC also p4 > p3 ) can both easily be transformed to 4ðr c tÞ2 > 3ðr c=mY Þ2 , which is the same inequality as in the proof of Proposition 5 for case A. □ pﬃﬃ Proof of Proposition 7. tA > tB , ðr cÞ 23 ðr c=mY Þ > ðr cÞ ðr sÞ rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ qﬃﬃﬃﬃﬃﬃﬃ qﬃﬃﬃﬃﬃﬃﬃ pﬃﬃ qﬃﬃﬃﬃﬃﬃ 3 1a 1a a , 1 2 > 3 a2 (both sides of the > 1 2 3 , 1 2 1a 2 3 q3ﬃﬃﬃﬃﬃﬃ 4 inequality are positive for a > 2=3) , 1 34 a2 > 2 1a 3 (both sides of the inequal 3 3 2 9 4 4 9 ð2 þ aÞ a 23 > ity are positive for a > 2=3) , 1 2 a þ 16 a > 3 43 a , 16

0 , a > 23 :

□

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H.S. Heese

Appendix 2: Overview of Notation Symbol c r s t Y U½0; Y mY D U½0; D mD w Q p

a

Description Unit cost Unit revenue Unit salvage value RFID tag cost (per unit) Yield (random variable) Mean yield Demand (random variable) Mean demand Unit wholesale price (decision variable) Order quantity (decision variable) Expected profit Superscript (M ¼ manufacturer; R ¼ retailer; SC ¼ supply chain) Critical fractile for the supply chain

References Alexander K, Birkhofer G, Gramling K, Kleinberger H, Leng S, Moogimane D, Woods M (2002) Focus on retail: applying Auto-ID to improve product availability at the retail shelf. White paper, Auto-ID Center, MIT, Cambridge Anderson Consulting (1996) Where to look for incremental sales gains: the retail problem of outof-stock merchandise. The Coca-Cola Retailing Research Council, Atlanta ¨ zer O ¨ (2009) If the inventory manager knew: value of visibility and RFID under Atali A, Lee HL, O imperfect inventory information. Working paper, Stanford University, Stanford, CA Cachon GP (2003) Supply chain coordination with contracts. In: Graves S, de Kok T (eds) Handbooks in OR/MS: supply chain management: design, coordination and operation. North-Holland, Amsterdam, pp 229–239 Camdereli AZ, Swaminathan JM (2010) Misplaced inventory and radio-frequency identification (RFID) technology: information and coordination. Prod Oper Manage 19(1):1–18 DeHoratius N, Raman A (2008) Inventory record inaccuracy: an empirical analysis. Manage Sci 54(4):627–641 DeHoratius N, Mersereau A, Schrage L (2008) Retail inventory management when records are inaccurate. Manuf Serv Oper Manage 10(2):257–277 Dutta A, Lee HL, Whang S (2007) RFID and operations management: technology, value and incentives. Prod Oper Manage 16(5):646–655 Fleisch E, Tellkamp C (2005) Inventory inaccuracy and supply chain performance: a simulation study of a retail supply chain. Int J Prod Econ 95(3):373–385 Gaukler GM, Seifert RW, Hausman WH (2007) Item-level RFID in the retail supply chain. Prod Oper Manage 16(1):65–76 Gruen TW, Corsten DS, Bharadwaj S (2002) Retail out-of-stocks: a worldwide examination of extent, causes, and consumer responses. The Grocery Manufacturers of America, Washington, DC Inderfurth K (2004) Analytical solution for a single-period production-inventory problem with uniformly distributed yield and demand. Cent Eur J Oper Res 12:117–127

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Inderfurth K (2005) Incorporating demand and yield uncertainty in advanced MRP systems. In: Lasch R, Janker CG (eds) Logistik management – innovative logistikkozepte. Deutscher Universitaets-Verlag, Wiesbaden Kang Y, Gershwin SB (2005) Information inaccuracy and inventory systems: stock loss and stockout. IIE Trans 37(9):843–859 ¨ , Lee HL (2007) Managing the reverse channel with RFID-enabled negative demand Karaer O information. Prod Oper Manage 16(5):625–645 K€ ok AG, Shang KH (2007) Inspection and replenishment policies for systems with inventory record inaccuracy. Manuf Serv Oper Manage 9(2):185–205 ¨ zer O ¨ (2007) Unlocking the value of RFID. Prod Oper Manage 16(1):40–64 Lee HL, O McFarlane D, Sheffi Y (2003) The impact of automatic identification on supply chain operations. Int J Logistics Manage 14(1):1–17 Porteus EL (2002) Foundations of stochastic inventory theory. Stanford University Press, Stanford, CA Raman A, DeHoratius N, Ton Z (2001) Execution: the missing link in retail operations. Calif Manage Rev 43(3):136–152 Rekik Y, Jemai Z, Sahin E, Dallery Y (2005) Involving the performance of retail stores subject to execution errors: coordination versus Auto-ID technology. Technical report, LGI – Ecole Centrale Paris, Paris Rekik Y, Sahin E, Dallery Y (2008) Analysis of the impact of the RFID technology on reducing product misplacement errors at retail stores. Int J Prod Econ 112(1):264–278 Rekik Y, Sahin E, Dallery Y (2009) Inventory inaccuracy in retail stores due to theft: an analysis of the benefits of RFID. Int J Prod Econ 118(1):189–198 Sahin E (2004) A qualitative and quantitative analysis of the impact of the auto ID technology on the performance of supply chains. PhD thesis, LGI – Ecole Centrale Paris, Paris Sari K (2010) Exploring the impacts of radio frequency identification (RFID) technology on supply chain performance. Eur J Oper Res 207(1):174–183 Spengler JJ (1950) Vertical integration and antitrust policy. J Polit Econ 58(4):347–352 Veinott AF (1965) Optimal policy for a multi-product, dynamic, nonstationary inventory problem. Manage Sci 12(3):206–222 Want R (2004) RFID: a key to automating everything. Sci Am 290(1):56–65 Yano CA, Lee HL (1995) Lot sizing with random yields: a review. Oper Res 43(2):311–334

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Possibilistic Mixed Integer Linear Programming Approach for Production Allocation and Distribution Supply Chain Network Problem in the Consumer Goods Industry Bilge Bilgen

Abstract In the consumer goods industry there is an ongoing trend towards an increased product variety and shorter replenishment cycle times. Hence, manufacturers seek a better coordination of production and distribution activities. Our study is motivated by the production-distribution problem encountered by a soft-drink company operating in consumer goods industry. The problem is to determine the optimal allocation of products and routing decisions for a multiechelon supply chain to minimize the total supply chain cost comprising of production, setup, inventory and distribution costs. A mixed integer linear programming (MILP) model is proposed to describe the optimization problem. However, a real supply chain operates in a highly dynamic and uncertain environment. The ambiguity of cost parameters is considered in the objective function of the model. The proposed approach uses the strategy of minimizing the most possible cost, maximizing the possibility of obtaining lower cost, and minimizing the risk of obtaining higher cost. Zimmermann’s fuzzy multi objective programming method is then applied for achieving an overall satisfactory compromise solution. The applicability of the proposed model is illustrated through a case study in consumer goods industry.

1 Introduction A supply chain system is comprised of all organizations that are involved in transforming raw materials to a final product. Supply chain management has received considerable attention from academicians and practitioners during the last several decades. In today’s world fast economic changes and the increasing pressure of market competition lead firms to focus on integrated supply chains. The coordination and integration of the production (supply), inventory, and distribution B. Bilgen Department of Industrial Engineering, Dokuz Eylul University, 35160 Izmir, Turkey e-mail: bilge.bilgen@deu.edu.tr 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_20, # Springer-Verlag Berlin Heidelberg 2011

505

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B. Bilgen

(demand) operations is widely perceived to be a route to obtain a competitive advantage. Arshinder et al. (2008) present a survey and classification of studies on supply chain coordination found in the literature. In the consumer goods industry the focus in production planning and scheduling is shifting from the management of plant-specific operations to a holistic view of the entire supply chain comprising value adding functions like purchasing, manufacturing and distribution. A closer coordination of production and distribution activities is required in order to avoid excessive inventories at the manufacturers’ warehouses. While traditionally minimizing production costs has been considered as the major objective, attention has shifted towards faster replenishment and improved logistical performance. Thus, finished product inventories are merely regarded as buffers between the manufacturing and the distribution stage of the supply chain. As a result, distribution costs have to be included in the overall objective function (Bilgen and Guenther 2010). The dynamic and complex nature of supply chain imposes a high degree of uncertainty in supply chain planning decisions and significantly influences the overall performance and effectiveness of the configuration and coordination of supply chain network (Klibi et al. 2010). The significance of accounting for uncertainty has prompted the researchers to address uncertain parameters in supply chain planning. Several authors have analyzed the sources of uncertainty present in a supply chain. In a recent review Peidro et al. (2009a) have presented an analysis of the literature on supply chain planning under uncertainty conditions by adopting quantitative approaches. In their review, some of the strengths and weaknesses of the approaches currently used have been pointed. Interested reader could refer to excellent reviews of Klibi et al. (2010) and Peidro et al. (2009a) regarding the review of supply chain network design problems under uncertainty, and review of quantitative models for supply chain planning under uncertainty, respectively. In this chapter, the optimal design and operation of a multi-product, multiperiod, multi-echelon, supply chain consisting of multiple manufacturers, multiple production lines and multiple distribution centers is considered. The problem is to assign products to the production lines, and to determine the routes to be traveled to coordinate the production and transportation routing operations so that the customer demand, capacity constraints, production, and inventory constraints are all satisfied, while the resulting cost (i.e. the sum of the production, inventory, setup, and transportation costs) over a given planning horizon is minimized. The problem is formulated as a MILP model. Because of price fluctuations in a dynamic market, assigning crisp values for parameters is no longer appropriate for dealing ambiguous decision problems. Possibility distribution offers an effectual alternative for proceeding with inherent ambiguous phenomena in determining cost parameters. Therefore, in this study a possibilistic MILP model is developed. The purpose of this study is twofold: first to develop more relatively sophisticated MILP model able simultaneously to form production and distribution network in the consumer goods industry, secondly to demonstrate the usefulness and significance of the fuzzy programming through a possibilistic programming approach.

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The remainder of this chapter is organized as follows. The relevant literature is reviewed in Sect. 2. In Sect. 3 the key characteristics of the problem are outlined, and the model is described in detail. The proposed possibilistic mixed integer linear programming is elaborated in Sect. 4. The results of a numerical investigation which show the practical applicability of the developed possibilistic optimization model is presented in Sect. 5. Finally, concluding remarks and future directions for further studies are stated in Sect. 6.

2 Literature Review The efficient coordination of production and distribution systems becomes a challenging problem as companies move towards higher collaborative and competitive environments. In the academic literature, integrated production and distribution planning problem has been the subject of many studies during the last decade. For general in-depth description of the field of production and distribution planning in supply chain management, the reader is referred to Ereng€uc¸ et al. (1999), Sarmiento and Nagi (1999), Bilgen and Ozkarahan (2004), Stadler (2005), Chen and Vairaktarakis (2005) Arshinder et al. (2008), Peidro et al. (2010a), Chen (2010). Research effort on the supply chain coordination issues have been overwhelming and many fruitful research results have obtained. e.g. Chandra and Fisher (1994), Fumero and Vercellis (1999), Dhaenens-Flipo and Finke (2001), de Matta and Miller (2004), Park (2005), Lei et al. (2006), Cordeau et al. (2006), Eksioglu et al. (2007), Stadtler and Kilger (2008), Rizk et al. (2008), Elhedhli and Gzara (2008), Tsiakis and Papageorgiou (2008), and Bilgen and Guenther (2010). The models stated above assume the parameters that influence the design decisions to be deterministic. However, much of decision making in real world takes place in an environment where the objectives, constraints, or parameters are not known precisely. Several approaches take into account sources of uncertainty are arising in the area of supply chain management. These approaches can be roughly classified into analytic approaches, and simulation-based approaches (Guillen et al. 2005). On the other hand another classification is done by Peidro et al. (2009a) as analytic models, models based on artificial intelligence (AI), simulation based models and hybrid models. A number of researchers have proposed stochastic supply chain management models that are closer to real situations. Most research has modeled the supply chain uncertainty (e.g., uncertain demand) by probability distribution that is usually predicted from historical data. However, whenever statistical data is unreliable or even unavailable, stochastic models may not be the best choice. Fuzzy set theory may provide an alternative approach for dealing with the supply chain uncertainty (Lai and Hwang 1992a). Fuzzy set theory was proposed by Zadeh and has been found extensive applications in various fields such as operations research, management science, control theory and artificial intelligence. Fuzzy mathematical programming (FMP) is one of the most popular decision making approaches based on fuzzy set theory. Fuzzy sets theory has been

508

B. Bilgen

implemented in mathematical programming since 1970 when Bellman and Zadeh (1970) introduced the basic concepts of fuzzy goals, fuzzy constraints, and fuzzy decisions. Many of the developments in the area of FMP are based on this seminal paper. A detailed discussion of the FMP procedures can be found in (Lai and Hwang 1992a; Zimmermann 1996). In the context of FMP, two research directions that are being pursued are flexible programming, and possibilistic programming. Flexibility is modeled by fuzzy sets and may reflect the fact that constraints and goals are linguistically formulated. Some applications of flexible programming can be found in Tsai et al. (1997), Miller et al. (1997), and Mula et al. (2006). On the other hand there is uncertainty, corresponding to an objective variability in the model parameters (randomness), or a lack of knowledge of the parameter values (epistemic uncertainty). Epistemic uncertainty is concerned with ill-known parameters modeled by fuzzy intervals in the setting of possibility theory (Mula et al. 2007). Different applications of possibilistic programming approach can also be found in the literature. Hsu and Wang (2001) develop a possibilistic linear programming model to determine appropriate strategies regarding the safety stock levels for assembly materials, regulating dealers’ forecast demands, and number of key machines. Wang and Liang (2005) present a novel interactive possibilistic programming approach for solving the multi-product aggregate planning problem with imprecise forecast demand, related operating costs, and capacity. In a more recent work, Sakalli et al. (2010) develop a possibilistic aggregate production planning (APP) model for blending problem in a brass factory. In the proposed model, the Lai and Hwang’s fuzzy ranking concept is relaxed by using either-or constraints. The important works concerning application of possibilistic programming in SCM are presented by Wang and Shu (2005, 2007). Other applications of possibilistic programming in production and distribution planning problems can be found in Chen and Chang (2006), Mula et al. (2007), Liang (2008), Torabi and Hassani (2008a), Liang and Cheng (2009), Mula et al. € (2010), Kabak and Ulengin (2011). In recent years there is a significant growth in the study of production and distribution models using the FMP approach. In a pioneering study, Sakawa et al. (2001) have presented fuzzy programming for the production and transportation planning in the light of obscure estimation of parameters. Chen and Chang (2006) simultaneously handle multi-product, multi-echelon, and multi-period supply chain model with fuzzy parameters. And they propose a solution procedure that is able to calculate the fuzzy objective value of the fuzzy supply chain model. Subsequent research on FMP include by Aliev et al. (2007). They develop a fuzzy integrated multi-product, multi-period production and distribution planning model within supply chain. The model is formulated in terms of fuzzy programming and solution is provided by Genetic Algorithm (GA). Torabi and Hassani (2008a) develop a novel multi-objective possibilistic MILP model for a supply chain master planning problem consisting of multiple suppliers, one manufacturer and multiple

Possibilistic Mixed Integer Linear Programming Approach for Production Allocation

509

distribution centers. Their model integrates the procurement, production and distribution plans considering various conflicting objectives simultaneously as well as the imprecise nature of some impractical parameters such as market demands, cost/time coefficients, and capacity levels. In another study they develop a more comprehensive multi-objective supply chain master planning framework for a multi-echelon supply chain by extending their previous work. They propose a FGP model which is solved through a novel auxiliary crisp formulation (Torabi and Hassani 2008b). Liang (2008) presents an interactive fuzzy multi-objective linear programming (FMOLP) model for solving integrated production and transportation problem with multiple fuzzy goals in fuzzy environments. Liang and Cheng (2009) proposes a FMOLP model to simultaneously minimize total costs and total delivery time with reference to inventory levels, available machine capacity and labor levels at each source, as well as market demand and available warehouse space at each destination, and the constraint on total budget. More recently, Peidro et al. (2009b, 2010b) adopt two well-known fuzzy programming approaches to solve the tactical supply chain planning problem arises in the automobile industry. The fuzzy model integrally handles all the epidemistic uncertainty sources identified in tactical supply chain planning problem. The proposed model in their papers integrates procurement, production, and distribution planning activities into a multi-echelon, multi-product, multi-level and multi-period supply chain network. Bilgen (2010) apply flexible programming approach on the integrated production and distribution planning problem within consumer goods industry. The effects of different fuzzy operators on the model under different demand granularity levels are investigated. Jolai et al. (2010) consider a multi-objective, multi-product, and multi-period collaborative production and distribution planning model using Fuzzy Goal Programming (FGP) approach. A simple genetic algorithm, a particle swarm optimization (PSO) and hybrid GA are developed to solve the problem. Kabak and € Ulengin (2011) propose a possibilistic linear programming model to make strategic resource planning decisions in the SC context. In its current form, the proposed model is expected to provide an important guide to SC managers in preparing their strategic plans, taking into account the fuzziness of long-term plans. Table 1 summarizes the relevant studies. In this chapter, we present a novel model consisting of multiple production lines, multiple plants, and multiple distribution centers considering imprecise nature of the integrated production and distribution planning problem. Our study also extends this literature in terms of the model scope. It enhances the above mentioned studies by considering detailed distribution routing, minor and major setup time and costs, and assignments of products to production lines. Its main advantage lies in its ability to simultaneously coordinate production allocation and transportation operations of the entire planning horizon. The main contribution of this chapter is the integration of production allocation and distribution supply chain network problem through a possibilistic programming approach, accompanied by experiments on real data in consumer goods industry.

S S

M M M M

S

S M S

S

S

Chen and Chang (2006) Aliev et al. (2007)

Torabi and Hassani (2008a) Torabi and Hassani (2008b) Liang (2008) Liang and Cheng (2009)

Peidro et al. (2009b)

Mula et al. (2010) Jolai et al. (2010) Bilgen (2010)

€ Kabak and Ulengin (2011)

Proposed research

M

M

M M M

M

M M S S

M M

S

M

M

M M M

M

M M S M

M M

M

Number of products

M

S

M M M

M

M M S M

M M

S

Number of periods

T

S/T

T T T

T

T T S T

T T

T

Decision level

þ

þ

þ þ

þ

þ

þ

þ þ þ

þ þ

þ

þ

þ

Objective Constraints function

Source of uncertainty

S strategic, T tactical, O operational, S single, M multi, V vagueness, A ambiguity, H hypothetic

S

Sakawa et al. (2001)

Objective (S vs. M)

Number of echelons

Problem characteristics

Table 1 Literature on integrated production and distribution planning models using FMP

A A

þ þ

A V V

þ þ

A

þ

A A A A V A

þ þ

þ þ

V

Ambiguity vs. Parameters vagueness

Zimmermann’s approach Zadeh’s extentions principle Genetic algorithm Possibilistic programming FGP FMOLP FMOLP Chanas and Verdegay method Possibilistic programming FGP, GA, PSO Flexible programming Possibilistic programming Possibilistic programming

Solution approach

þ

þ

þ H þ

þ

H H þ þ

H þ

þ

Industrial application

510 B. Bilgen

Possibilistic Mixed Integer Linear Programming Approach for Production Allocation

511

3 Problem Description and Model Formulation Our study is motivated by the production-distribution problem encountered by a soft-drink company, which has to decide routinely the best way of delivering a set of orders to its customers over a multi-day planning horizon. Often it is not economical to install production equipment at each plant for the entire portfolio of products due to the high investment costs. Therefore, dedicated lines for the production of a specific range of products are established at each plant. Each individual plant has two production lines. No plant can produce the whole product range but just a part of the product range. Each production line at each individual plant can be viewed as a single stage process capable of producing several product groups. Setup costs are incurred at each plant whenever a production line changes production to a different product group. Each plant has an attached distribution center (DC) which serves as a buffer for the local production, and storage for products, which cannot be produced at the corresponding plant. Each DC has to be able to deliver the whole range of products. Products which cannot be produced in the corresponding plant must be delivered to a DC from another plant or another DC. Delivery takes place by means of homogenous vehicles with limited capacity. The distribution of goods from the plants to the DCs occurs in one of two ways: on a straight-and-back basis, i.e. there is only one DC on a given delivery vehicle’s path, which is the case if the customer requirements constitute a full truck load (FTL), or in a route involving multiple DCs on the route have individual requirements less than a truckload (LTL). For LTL multiple customers are served in a single route. In this case, transportation costs only depend on the transportation distance, not on the specific load. Usually, optimization models for production and distribution planning are based on the simplifying assumption that transportation costs are linear with respect to quantity and distance. This assumption, however, is not rectified in real transportation systems since most often different modes of transportation can be used. Particularly, in the consumer goods industry road transport is the most preferred mode in the distribution of finished goods due to its high flexibility (Guenther and Seiler 2009). The problem is to assign products to the production lines, and to determine the routes to be traveled to coordinate the production and transportation routing operations so that the customer demand, capacity constraints, production, and inventory constraints are all satisfied, while the resulting cost (i.e. the sum of the production, inventory, setup, and transportation costs) over a given planning horizon is minimized.

3.1

Problem Assumptions and Notation

The following considerations further define and delimit the problem: • The supply network consists of several plants which deliver the final products to various distribution centers.

512

B. Bilgen

• Each plant comprises several not necessarily identical production lines. Each line produces a given range of products. Multiple assignments of products to the lines are allowed. • For each product inventory balances at distribution centers are updated on a daily basis according to the production output from the various lines at the plants, the inbound and outbound transportation quantities, and the given external demand. • All vehicles used in LTL transportation are assumed to be identical. • No specific handling capacities and costs at distribution centers are considered. • Transportation activities are carried out within a single day. Nevertheless, lead times for long-distance transportation can be modeled simply by offsetting the time index of the respective decision variables. • Each vehicle can only travel according to pre-defined route with fixed operational cost. It is assumed that a vehicle can pick up products from a plant in a travel. Each vehicle is allowed to discharge cargo at most two distribution centers.

3.2

Model Formulation

The MILP model for the considered problem proposed by Bilgen (2010) is adopted as the basis of this work. It reflects major issues arising in the consumer goods industry, e.g. major and minor setups, daily demand assignments, use of different transportation modes etc. Nevertheless, its main features are also relevant for other types of industries. The sets, indices, parameters and variables used to formulate the problem mathematically are described below.

3.2.1

Notation

Sets I J Ji L P Lp IL W T Ji Rp Rw

Set of products (i ¼ 1,2,. . .,I) Set of product groups (j ¼ 1,2,. . .,J) Set of products that belong to group j Set of production lines (l ¼ 1,2. . .,L) Set of plants (p ¼ 1,2,. . .,P) Set of production lines at plant p Set of products that can be processed on line l Set of distribution centers (DC) (w ¼ 1,2,. . .,W) Set of time periods Set of product groups including product p Set of routes that begin with plant p Set of routes including distribution center w

Possibilistic Mixed Integer Linear Programming Approach for Production Allocation

513

Parameters PCapl ECapl diwt ail ai V tsil TSjl ~ l EC C~prod il C~SMin il C~SMaj

Capacity of production line l Extra capacity of production line l External demand of product i at DC w in period t Time consumed to produce product i on line l Factor for converting quantities of product i into unit loads, e.g. pallets Loading capacity of a vehicle Minor setup time of product i on line l Major setup time of product group j on line l Extra cost for using production line l Processing cost of product i on production line l

C~Inven iw C~LTL

Inventory holding cost of product i at DC w in time period t Transportation cost per vehicle on route r

Minor setup cost of product i on production line l Major setup cost of product group j on production line l

jl

r

Decision variables xilt elt yirwt qipwt Iiwt nrt zilt ujlt

Production volume for product i produced on production line l in time period t Extra capacity used on production line l in time period t Quantity of product i delivered to warehouse w via route r in time period t Quantity of product i shipped from plant p to DC w in time period t Inventory level of product i at DC w at the end of time period t The number of vehicle used on route r in time period t. (identical vehicles are used) 1, if product i is setup on line l in period t 0, otherwise 1, if product group j is setup on line l in period t 0, otherwise

3.2.2

Objective Function

Minimize XXX i2I l2L t2T

XXX i

w

t

od xilt þ C~Pr il

XXX

zilt þ C~SMin il

i2I l2L t2T

C~Inven iw Iiwt þ

XX

XXX j2J l2L t2T

ujlt þ C~SMaj jl

XX r

t

C~LTL r nrt

ðCapÞ EC~l elt

l2L t2T

(1)

514

B. Bilgen

The first term in objective function defines the production costs, the second and the third terms represent the minor and major setup costs for products, and product groups, respectively. Finally last three terms represent transportation ~ l , C~prod , cost of the system, and the inventory costs, and extra capacity costs. EC il SMin ~SMaj ~Inven ~LTL ~ Cil , Cjl , Ciw , Cr are imprecise coefficients.

3.2.3

Model Constraints

subject to X

ail xilt þ

X

i

tsil zilt þ

X

i

TSjl ujlt PCapl þ elt

8l; t

(2)

j

Constraints (2) are the time capacity constraints. Capacity is the upper bound on the total time that can be consumed to produce products. It specifies that the time used for processing (manufacturing) on a line cannot exceed the capacity of that line in time period t. ail xilt ðPCapl þ ECapl Þzilt

8i; l; t

(3)

Constraints (3) enforce the production quantity i on line l to zero, if no corresponding setup operation is performed (i.e. zilt ¼ 0). elt ECapl

8l; t

(4)

Constraints (4) guarantee that the extra capacity used for production line l and time period t cannot exceed maximum available extra capacity for that production line. X zilt Mujlt 8j; l; t (5) i2Ji

Constraints (5) ensure that product i belong to product group j can only be set, if the line is set up for the product group j. XX X X yirwt þ qipwt ¼ xilt 8p; i; t (6) r2Rp

w

w

l2Lp

Total output quantities achieved from producing product i at the production lines in plant p must be equal to the shipping quantities on the routes starting from plant p and including DC w plus the shipping quantities to the DCs which are supplied from plant p. That is, it ensures the availability of the product i at plant p in time period t. XX yirwt ai Vnrt 8r; t (7) i

w2Wr

Possibilistic Mixed Integer Linear Programming Approach for Production Allocation

515

Constraints (7) guarantee that, the total quantity of products (converted into unit loads) to be transported to the various DCs w 2 Wr included in route r determines the number of vehicles nrt required for that route in time period t. Note that variable nrt is defined as an integer number of identical vehicles each having a transportation capacity of Vunit loads. X X Iiwt1 þ yirwt þ qipwt diwt Iiwt ¼ 0 8i; w; t with Iiw0 ¼ given r2RðwÞ

p

(8) The daily demands of products must be satisfied. Constraints (8) ensure the inventory flow balance at DCs, and require each DC to have enough supply (from either inventory and/or the quantity arrived in that period) to meet the demand. That is, the inventory of product i at distribution center w at the end of time period t is determined by the ending inventory of the previous time period, the quantities received, and the external demand to be satisfied on the respective time period. xilt ; yirwt ; qipwt ; 8i; j; l; p; w; r; t

I iwt 0;

nrt 0;

and integer,

zilt ;

ujlt 2 f0; 1g

(9)

Finally, constraints (9) are integrality constraints of integer and binary variables, and non-negativity constraints of continuous variables. In real-life situation for a production and distribution planning problem, many input information related to the process are not known with certainty. Fuzzy and/or imprecise natures of the problem cannot be described adequately by the conventional approach. Fortunately, possibility distribution offers an effectual alternative for proceeding with inherent ambiguous phonemia in determining environmental coefficients and related parameters. In our model, cost coefficients are all imprecise. Therefore a possibilistic programming model is developed to determine the optimum production allocation and distribution decisions. Based on Lai and Hwang’s (1992b) approach, a possibilistic linear programming model is transformed into crisp multiobjective programming models. Finally, Zimmermann’s fuzzy programming method (1978) is applied to obtain composite single objective. In this study, production, setup, transportation, and inventory costs are represented by triangular possibility distributions. The parameters of a triangular possibility distribution are given as the optimistic, the most possible, and the pessimistic values, which were estimated by decision maker.

4 Solution Methodology 4.1

Converting Imprecise Cost Coefficients to Crisp Numbers

There are many studies to tackle with the imprecise cost coefficients in the objective function in the literature. The first method for getting a compromise solution was

516

B. Bilgen

proposed by Tanaka et al. (1984). They adapt a weighted average as a substitute for the fuzzy objective with a crisp compromise objective. An a-Pareto optimal solution proposed by Sakawa and Yano (1989) restricts the fuzzy coefficients to a-level sets. A similar concept is the b-possibility efficient solution of Luhandjula (1987). He obtained a single objective semi-infinite linear programming problem, and provided a cutting plane method to solve this semi-infinite problem. Rommelganger et al. (1989) proposed a multiple objective linear programming by using r a-level sets and establishing membership functions of the upper and lower bound for each a-level set to solve linear programming problem with triangular fuzzy costs. In contrast to the approach of Rommelganger et al. (1989) has restricted their solution method to one a-level. They used decomposition theory, and considered a convex set with extreme points defined by the lower and upper bound of the n a-level sets of the fuzzy coefficients. On the other hand, Lai and Hwang (1992b) converted the fuzzy objective with a triangular possibility distribution into three crisp objectives. Following the Lai and Hwang’s (1992b) approach, the approach developed here minimizes cm, maximizes (cm co), and minimizes (cp cm), rather than simultaneously minimizing cm, cp, co. Figure 1 represents a triangular possibility distribution of imprecise numbers. Geometrically, this imprecise objective is fully defined by three corner points (cm,1), (cp, 0) ,(co,0) in Fig. 1. Using Lai and Hwang’s approach (1992b), we substitute minimizing cm, maximizing (cm co), and minimizing (cp cm). That is the approach used in this work involves minimizing the most possible value of the imprecise costs, cm, maximizing the possibility of lower costs (cm co), and minimizing the risk of obtaining higher cost (cp cm). The three replaced objective functions can be minimized by pushing the three prominent points towards left. In this way, our problem can be transformed into a multi-objective linear programming as follows:

π~ Ci 1

Fig. 1 The triangular possibility distribution for cost coefficients

~ Ci(o)

~ (m) Ci

~ (p) Ci

Possibilistic Mixed Integer Linear Programming Approach for Production Allocation

XXX

z1 ¼

Min

XX

mðPr odÞ C~il xilt þ

i2I l2L t2T

þ

XXX

þ

t

r

z2 ¼

Max

XXX

mðSMinÞ C~il zilt þ

C~rmðLTLÞ nrt

þ

moðPr odÞ xilt þ C~il

þ

t

r

Min z3 ¼

XXX

w

XX r

t

moðCapÞ

elt

XXX

moðSMajÞ C~jl ujlt

j2J l2L t2T

w

i pmðPr odÞ C~il xilt þ

moðInvenÞ Iiwt C~iw

t

XX

EC~l

pmðCapÞ

elt

l2L t2T pmðSMinÞ C~il zilt þ

i2I l2L t2T

þ

EC~l

XXX

C~rmoðLTLÞ nrt þ

XXX

mðInvenÞ Iiwt C~iw

t

XX

moðSMinÞ C~il zilt þ

i2I l2L t2T

þ

mðSMajÞ C~jl ujlt

l2L t2T

i2I l2L t2T

XX

elt

j2J l2L t2T

i

XXX

XXX

XXX

i2I l2L t2T

þ

mðCapÞ

l2L t2T

i2I l2L t2T

XX

EC~l

517

C~rpmðLTLÞ nrt þ

XXX

pmðSMajÞ C~jl ujlt

(10)

j2J l2L t2T

XXX i

w

pmðInvenÞ Iiwt C~iw

t

To solve, many multi-objective linear programming techniques can be used, such as utility theory, goal programming, and so on. Lai and Hwang (1992b) suggested using Zimmermann’s (1978) fuzzy programming method to convert the auxiliary MOLP model into an equivalent single goal LP problem. Initially, the positive ideal solutions (PIS) and negative ideal solutions (NIS) of the three objective functions should be obtained to construct the linear membership functions of the objectives (Lai and Hwang 1992b). For each objective function, the corresponding linear membership function is computed as:

mz1 ¼

8 > < 1NIS z

if

9 z1 =

8 > : 1 : NIS ; 0 if z1 >z1 0 9 8 if z3 3 ;> = < 1NIS z3 z3 NIS ; : 3 3 0 if z3 >zNIS 3

if if

9 z2 >zPIS 2 ;> =

PIS zNIS 2

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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]

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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

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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]

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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]

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Part I

Introduction and Review

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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

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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

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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

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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.

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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

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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; 0b:

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.

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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

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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.

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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 ; 00; 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 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.

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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

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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)

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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 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.

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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

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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.

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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.

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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

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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

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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.

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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

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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.

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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 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: 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

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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.

□

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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

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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

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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

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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).

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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.

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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)

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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

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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

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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)

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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

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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.

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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

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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 .

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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

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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)

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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

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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

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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

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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

)

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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)

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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 .

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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

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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

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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.

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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

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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

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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

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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

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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.

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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

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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.

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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.

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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

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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.

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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,

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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.

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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)

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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

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.

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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.

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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

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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.

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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

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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

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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

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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

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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 :

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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

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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. □

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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

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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

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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: feryal.erhun@stanford.edu P. Keskinocak School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, GA, USA e-mail: pinar@isye.gatech.edu S. Tayur Tepper School of Business, Carnegie Mellon University, Pittsburgh, PA, USA e-mail: stayur@andrew.cmu.edu 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

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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

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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

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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

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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

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n=1

1

a 2

2

9a 16

←⎯ ⎯⎯

3a 8

3

75a 128

←⎯ ⎯ ⎯ 5

15a 32

←⎯ ⎯ ⎯ 3

5a 16

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1225a 2048

⎯ ←⎯ ⎯ 7

525a 1024

←⎯ ⎯⎯ ⎛ 5⎞

105a 256

…

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Prices

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n=4 …

n=3

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n=2

n=4 …

a 8b

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3a 16b

a 12b

⎯⎯ ⎯ → 5

5a ⎯⎯ ⎯→ ⎛×3 ⎞ ⎜ ⎟ 48b ⎝ 2⎠

5a 32b

a 16b

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7a 96b

⎯⎯ ⎯ → 5

35a 384b

…

…

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⎛× ⎞ ⎜ ⎟ ⎝ 6⎠

⎛× ⎞ ⎜ ⎟ ⎝ 4⎠

11a 32b ⎯⎯ ⎯ → 3 ⎛× ⎞ ⎜ ⎟ ⎝ 2⎠

35a 256

93a 256b …

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5a 16b

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⎛× ⎞ ⎜ ⎟ ⎝ 2⎠

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⎛× ⎞ ⎜ ⎟ ⎝ 2⎠

…

⎛× ⎞ ⎜ ⎟ ⎝ 4⎠

…

⎛×3 ⎞ ⎜ ⎟ ⎝ 2⎠

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a 4b

QN

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

100 90 80

% of Profits

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DM Loss 25.00 9.77

5 6.06

7 4.39

9

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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

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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 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.

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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.

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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:

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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.

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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

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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

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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

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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.

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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)

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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 .

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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.

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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: samar@skku.edu X. Su College of Business Administration, California State University Long Beach, 1250 Bellflower Blvd, Long Beach, CA 90840, USA e-mail: xsu@csulb.edu 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

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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

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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)

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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.

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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)

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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

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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)

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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.

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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

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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)

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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.

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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

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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

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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

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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

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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.

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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

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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.

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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

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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 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

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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

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)

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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; f ðkÞ

The OEM’s profit is also monotonically decreasing in k.

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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

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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: santanu_snh@yahoo.com S.P. Sarmah (*) Department of Industrial Engineering and Management, Indian Institute of Technology, Kharagpur 721302, India e-mail: spsarmah@iem.iitkgp.ernet.in 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

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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.

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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.

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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

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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

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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.

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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 2bpi . 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).

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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 > = 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

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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. 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 g4 . 2

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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,

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C1. 1 Tr JpAD þ Det JpAD >0 C2. 1 þ Tr JpAD þ Det JpAD >0 C3. Det JpAD 1 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)

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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, (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@ =@xÞf 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 n0, 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 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

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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

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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 n0, 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.

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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)

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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

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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

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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),

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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

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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),

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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):

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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

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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 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

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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)

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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

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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.

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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

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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)

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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

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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)

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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:

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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.

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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

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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.

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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=@l 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=@l 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=@l 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

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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).

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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

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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 @BPr@Q 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)

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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.

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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

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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

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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

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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: penguinhj@163.com A.Z. Zeng School of Business, Worcester Polytechnic Institute, Worcester, MA 01609, USA e-mail: azeng@wpi.edu 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

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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.

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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.

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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

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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.

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3.1

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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)

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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:

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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 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)

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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 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:

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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.

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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

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m Fig. 14 The plot of f1(m) against m for cr ¼ cs

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0

-0.1

f2(m)

-0.2

-0.3

-0.4

-0.5

0

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0.6

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0.6

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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

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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

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0

-0.5

0

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0.4

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m

Fig. 18 The plot of f1(m) against m for cr < cs

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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

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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.

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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: wangjc@business.hku.hk A.H.L. Lau School of Business, University of Hong Kong, Pokfulam, Hong Kong e-mail: ahlau@business.hku.hk H.-S. Lau* (*) Department of Management Sciences, City University of Hong Kong, Kowloon Tong, Hong Kong e-mail: mshslau@cityu.edu.hk 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

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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:

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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.

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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.

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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

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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

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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

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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”).

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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.

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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

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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

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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

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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)

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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)

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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

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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 increases as PRsub increases.

4.5

Summary and Discussion for the Scenario of Asymmetric a-Knowledge

Each of Sects. 4.1–4.4 showed that, given that Reta already knows the contract format Manu will use (be it [w], [FF], [2P] or [MC]), there is a “Situation-I” set of conditions under which Reta prefers Manu’s a-uncertainty to be higher when Manu is trying to determine the Manu-profit-maximizing parameter values of the supply contract. Furthermore, the expanse of Situation I remains largely unchanged for different contract formats. Since [w] ([MC]) is the simplest (most powerful) supply contract format a dominant Manu can impose, it is reasonably safe to assume that other plausible contract formats will also exhibit such a “Situation-I” behavior Table 4 PR ðperRÞ under a menu of contracts ([MC])

Gamma-distributed a˜, ma ¼ 5, b ¼ 1, and PRsub ¼ ðma bkÞ2 =ð24bÞ

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(we have actually confirmed this for a few other contract formats such as “quantity discount” and “(maximum) resale price maintenance”). Therefore, combining Sects. 4.1–4.4 enables us to conclude that, even if Reta does not know what contract format Manu will use, there is still a roughly constant set of conditions under which she prefers Manu to be less certain about a when Manu is formulating the supply contract. However, as explained in Sect. 4.1.4, these Situation-I conditions are probably not more prevalent than the Situation-II conditions (this conclusion will be contradicted in Sect. 5, where instead of “a,” Manu does not know “c” deterministically). However, Reta is always motivated to mislead Manu into believing an inflated value of PRsub but a deflated value of ma. The analytical derivations and numerical computations presented here have been done under the assumption that Reta knows a perfectly. However, it can be easily shown that the same conclusions are obtained if one assumes that Reta also knows a only as a random variable a˜0 (which is likely to be different from Manu’s perceived a˜). For example, under [w], (4) gives Reta’s expected profit as E[PR(perR)] ¼ E [a˜0 b(w* + c)]2/(4b), a moment’s reflection would reveal that a lower w* leads to a higher E[PR(perR)]. Combining this conclusion with Lemma 1A leads again to Lemma 1B. The same argument applies to Lemmas 2 and 3 (for [FF] and [2P]). [MC] presents greater difficulties, since now one has to derive a different set of [MC]’s [w(adec), L(adec)], given that Manu realizes that Reta also knows a only as a random variable. This is because, unlike [w], [FF] and [2P], Reta’s knowledge plays a role in Manu’s design of [MC]. In other words, extending our results, one can prove easily that, regardless of whether Reta herself is well informed of the parameter (say) a, a “Situation I” exists under [w], [FF] and [2P], where Reta prefers Manu to be less certain about a. While our analytical results cannot be similarly extended for [MC], it seems reasonable to conjecture that the same conclusion can be extended to [MC]. In other words, there exists a “Situation I” under which Reta prefers Manu’s sa to be larger, regardless of Reta’s own knowledge of a.

5 Dominant Manu is Uncertain About Reta’s Unit Cost c 5.1

Reta’s Preference on sc, or c’s Level of Uncertainty

We follow the same approach used in Sects. 3 and 4. First, as in a˜’s case, we are able to obtain analytical solutions (obtainable from the authors) for the case of uniformly-distributed c~ (i.e., the counterparts of Lemmas 1–4). We also obtained numerical solutions for the case of gamma-distributed c~, which not only reveal more characteristics than the uniformly-distributed-~ c analytical solutions, but are also much easier to understand. Since the c~-results here have much in common the ~ we will omit for brevity sake the [FF] numerical results. earlier results for a˜ and b, Thus, Tables 5–7 correspond to, respectively, Tables 1, 3 and 4 of Sect. 3 for [w],

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[2P] and [MC]. In the following paragraphs we point out the similarities to and emphasize the differences from the results presented earlier for a˜. The following similarities between Tables 5–7 and the earlier tables can be observed: 1. There is a Situation-I (Situation-II) area where kc is sufficiently low (high) and Reta prefers Manu to be more (less) uncertain about c~. The low/high-kc areas are graphically separated by a “Boundary B” (the line “══”) in the tabulated numerical data. 2. The position of Boundary B does not change substantially when Manu’s contract changes from the simplest format (i.e., [w]) to the most sophisticated and powerful (i.e., [MC]). The significant difference between this section’s c-uncertainty results (i.e., Tables 5–7) and the earlier a-uncertainty results (i.e., Tables 1–4) is in the k-magnitude at which Boundary B is located. In short, the Situation-I area is now much more prevalent than in the cases of uncertainties in a (and in b, detailed in WLL 2008). To elaborate, consider the Table 5 column with a ¼ 5. As explained in Sect. 4.1.4, we have, from (2), pI* ¼ [a + b(mc + m)]/(2b) ¼ (5 + 2)/2 ¼ 3.5, and the markup over cost is M ¼ (pI* k)/k ¼ 0.75. That is, the profitability here is the same as the column of c ¼ 1.0 in Table 1 discussed in Sect. 4.1.4. However, whereas Boundary B in Table 1’s “c ¼ 1” column occurs at around ka ¼ 0.13, Boundary B in Table 5’s “a ¼ 5” column occurs at around kc ¼ 0.5. Noting that the maximum possible k-value for a uniformly distributed random variable is 0.577, a Boundary-B location at kc ¼ 0.5 means that Situation I (the area above Boundary B) covers a very high proportion of likely uncertainty levels in Manu’s c~. Columns to the right of “a ¼ 5” in Table 5 represent low-profitability conditions that are less likely to occur. Moving to the left, the “a ¼ 10.0” column in Table 5 corresponds to a markup-over-cost (M) of 2 – an M-value not unrealistic high. However, at this M-level, Situation II does not arise until kc is as high as 0.95. Thus, ~ contrary to the conclusions reached in the preceding section for a˜ (and for b, detailed in WLL 2008), Table 5 shows that when Manu is uncertain about c, then

Table 5 ПR(perR) under a price-only contract ([w])

Gamma-distributed c~, mc ¼ 1, b ¼ 1, and ПRsub ¼ [a b(mc þ m)]2/(24b)

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Table 6 ПR(perR) under a two-part tariff ([2P])

Gamma-distributed c~, mc ¼ 1, b ¼ 1, and ПRsub ¼ [a b(mc þ m)]2/(24b)

Table 7 ПR(perR) under a menu of contracts ([MC])

Gamma-distributed c~, mc ¼ 1, b ¼ 1, and ПRsub ¼ [a b(mc þ m)]2/(24b)

in the majority of plausible conditions Reta is motivated to increase Manu’s c-uncertainty. Incidentally, Table 5 shows the necessity of considering both uniform and gamma distributions in this chapter; i.e., under some a-values “Situation II” can only arise when kc exceeds 0.577, thus Situation II cannot be adequately illustrated using only the uniform distribution. Therefore, to facilitate direct comparisons, Tables 1–7 are all compiled using gamma-distributed distributions. In Tables 6 and 7, Boundary B is slightly higher than its position in Table 5. However, it does not alter the conclusions reached in the preceding paragraph based on [w] and in the preceding sections based on a- and b-uncertainties – i.e.: 1. Over a range of plausible conditions (Situation I), Reta prefers Manu to be more uncertain about c. Also, Manu cannot significantly narrow or alter the range of Situation I by using a more sophisticated contract format (e.g., [MC]) instead of a simpler one (e.g., [w]). 2. Contrary to uncertainties in demand (i.e., sa and sb), under c-uncertainty Situation I becomes more prevalent than Situation II.

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399

Reta’s Preference on mc (or c’s bias) and PRsub

It can be easily proven that Reta will always try to mislead Manu into perceiving a higher mc (numerically illustrated in Fig. 3) and a higher PRsub. This effect matches intuition.

6 The Case of a Dominant Reta While WLL (2008) considered dominant-Manu scenarios, WLL (2009) considered the following dominant-Reta cost/profit model: For Manu : PM ¼ ðw mreal Þ q; Reta’s profit PR ¼ ðp wÞ q; where q ¼ a bp:

(16)

Referring to Fig. 1, the dominant Reta knows Manu’s mreal only stochastically as ~ and we investigate, from Manu’s perspective, what are the ideal characteristics m, ~ (or “quality”) of Reta’s m-perception that would lead Reta to specify a contract that is most advantageous to Manu. As in the preceding Sects. 3–5, we assume that Reta may implement any one of the four following contract-formats (1) price-only [rSU]; (2) [FF]; (3) [2P]; and (4) [MC]. Note that (1) [rSU] is [w]’s mirror image for the dominant-Reta scenario; and (2) a Reta-imposed [MC] involves more difficult calculations than a Manu-imposed [MC]. We developed procedures in WLL (2009) for determining, for each of the four contract formats, the parameter values that will optimize Reta’s expected profit. Using these optimizing procedures, we identified conditions under which Manu is motivated to share or distort asymmetrical information on the system parameters m and PMsub. These conditions are summarized as Points E–G in Sect. 7. 1.2 1 ΠR(perR)

0.8

[w] [FF] [2P] [MC]

0.6 0.4 0.2 0

mc 0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

Fig. 3 PR(perR) under gamma-distributed c~: sc ¼ 0.4, creal ¼ 1, a ¼ 5, b ¼ 1, and PRsub ¼ 0

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7 Concluding Summary We consider a very basic two-echelon supply chain system defined by the profit functions and demand curve given in (1), (2) and (16). At the initial stage (time point A in Fig. 1), the dominant player has not specified a supply contract, and has a stochastic perception of the value of one of the system’s parameters – i.e., one of {a,b,c} in the dominant-Manu case, or “m” in the dominant-Reta case. Sections 3–5 cover the dominant-Manu case, where we investigated how Reta would prefer Manu to perceive {a,b,c} and whether Reta is (or can be) motivated to improve the quality of Manu’s perception. We found that: (A) Regardless of how simple or sophisticated Reta’s contract format is, Reta is always motivated to mislead Manu into perceiving an inflated mc and an inflated PRsub. (B) Reta is always motivated to mislead Manu into perceiving a deflated ma-value but an inflated mb-value. (C) Reta is motivated to increase sx (where “x” is one of {a,b,c}) when sx is “too low,” but to reduce sx when sx is “too high” – i.e., “everything in moderation”. (D) Over a very substantial range of plausible and “desirable” (i.e., “profitable”) conditions, Reta is motivated to increase the uncertainty level (s) of Manu’s perception of {a,b,c}. Moreover, this range of confusion-preferring conditions cannot be narrowed by using (or threatening to use) more sophisticated contract formats such as a “menu of contracts” at time point B. Point A appears intuitively obvious after it is pointed out, even though it does not appear to have been explicitly recognized in the literature. Point B is neither intuitively obvious nor counter-intuitive. Points C and D are counter-intuitive and also appear to contradict some popularly held notions. Results (C) and (D) on Reta’s s-preference can be viewed from two directions. First, the current emphasis on supply chain coordination and collaboration might induce one to “guess” that a rational Reta would want to improve the quality (i.e., reduce the s) of Manu’s perception of (a,b,c), particularly when a coordination-encouraging contract is to be implemented. On the other hand, the existence of information rent might induce one to “guess” that Reta would not want to help Manu for free. Our results show that both guesses are half right (and half wrong). Thus, one can argue, coming from the “supply chain” angle, that it is counter-intuitive that the prospect of a coordination-encouraging contract has no effect on motivating Reta to reduce Manu’s s on (a,b,c). From the opposite direction, one might also argue that it is counter-intuitive that there is a range of conditions under which Reta would be interested in improving Manu’s s “for free.” Section 6 covers the dominant-Reta case, where we investigated how Reta would want Manu to perceive {m} and whether Reta is (or can be) motivated to improve the quality of Manu’s perception. Very similar to our dominant-Manu findings reported in Sects. 3–5, we found that:

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(E) Regardless of how simple or sophisticated Reta’s contract format is, Manu is always motivated to mislead Reta into perceiving an inflated mm and an inflated Manu’s subsistence profit level PMsub. (F) Manu is motivated to increase sm when sm is “too low,” but to reduce sm when sm is “too high” – i.e., “everything in moderation” (see Sect. 3.3). (G) Over a significant range of likely conditions, Manu is motivated to increase rather than reduce sm – i.e., to worsen rather than improve the degree of ~ certainty/confidence Reta feels about her initial m-perception before Reta designs her contracts. Manu is motivated to reduce the uncertainty of Reta’s ~ m-perception only in a relatively narrower range of conditions that can be roughly characterized as “low profitability.” Point E appears to be intuitively obvious after it is pointed out. Point F is not very counter-intuitive, but Point G is counter-intuitive and contradicts some popularly held notions.

References Cachon GP (2003) Supply chain coordination with contract. In: de Kok AG, Graves C (eds) Handbooks in operations research and management science, vol 11, Supply chain management: design, coordination, and operation. Elsevier, Amsterdam Chen FR (2003) Information sharing and supply chain coordination. In: de Kok AG, Graves C (eds) Handbooks in operations research and management science, vol 11, Supply chain management: design, coordination, and Operation. Elsevier, Amsterdam Corbett CJ, Zhou D, Tang CS (2004) Designing supply contracts: contract type and information asymmetry. Manage Sci 50:550–559 Ha AY (2001) Supplier-buyer contracting: asymmetric cost information and cutoff level policy for buyer participation. Nav Res Log 48:41–64 Lau AHL, Lau HS (2005) Some two-echelon supply-chain games: improving from deterministicsymmetric-information to stochastic-asymmetric-information. Eur J Oper Res 161:203–223 Lee HL, So KC, Tang CS (2000) The value of information sharing in a two-level supply chain. Manage Sci 46:626–643 Li L (2002) Information sharing in a supply chain with horizontal competition. Manage Sci 48:1196–1212 Liu XC, C ¸ etinkaya S (2009) Designing supply contracts in supplier vs buyer-driven channels: the impact of leadership, contract flexibility and information asymmetry. IIE Trans 41:687–701 ¨ zer O ¨ (2010) Channel incentives in sharing new product demand information and robust Liu H, O contracts. Eur J Oper Res 207(3):1341–1349 Myerson RB (1979) Incentive compatibility and the bargaining problem. Econometrica 47:61–73 ¨ zer O ¨ , Wei W (2006) Strategic commitments for an optimal capacity decision under asymmetric O forecast information. Manage Sci 52:1238–1257 Taylor TA, Xiao W (2010) Does a manufacturer benefit from selling to a better-forecasting retailer? Manage Sci 56(9):1584–1598 Tirole J (1988) The theory of industrial organization. MIT, Cambridge, MA Wang JC, Lau HS, Lau AHL (2008) How a retailer should manipulate a dominant manufacturere’s perception of market and cost parameters. Int J Prod Econ 116:43–60 Wang JC, Lau HS, Lau AHL (2009) When should a manufacturer share truthful manufacturing cost information with a dominant retailer? Eur J Oper Res 197:266–286 Wu YN, Cheng TCE (2008) The impact of information sharing in a multiple-echelon supply chain. Int J Prod Econ 115:1–11

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Supply Chain Coordination Under Demand Uncertainty Using Credit Option S. Kamal Chaharsooghi and Jafar Heydari

Abstract In this chapter, a coordination scheme based on the credit option for the simultaneous coordination of order quantity (Q) and reorder point (s) in a two-stage supply chain (SC) is developed. A decentralized SC including one buyer and one supplier is investigated. The buyer faces demand uncertainty and uses a continuous (s, Q) inventory model. It is shown that joint decision making for both s and Q is profitable for the whole SC. However, in centralized decision making the buyer always loses, whereas the supplier greatly profits. A credit option is proposed as a coordination scheme to encourage the buyer to accept the coordinated decisions. In the proposed model, the buyer can benefit from late payments that are subject to commitment to the jointly agreed s and Q. The lower and upper bounds for the credit period are calculated. The proposed scheme shares the benefits of coordinated decision making based on the bargaining power of each member. Numerical experiments showed that the proposed model can achieve channel coordination. Keywords Coordination • Credit option • Order quantity • Reorder point • Supply chain • Uncertainty

1 Introduction The supply chain concept is based on collaboration and coordination between all companies involved in producing, distributing, and delivering the right product at the right time for the consumers. From this viewpoint, all members involved in an SC must make their decisions in accordance with overall SC benefits. In the past, S.K. Chaharsooghi Industrial Engineering Department, Tarbiat Modares University, Tehran, Iran e-mail: skch@modares.ac.ir J. Heydari (*) Industrial Engineering Department, Shiraz University of Technology, Shiraz, Iran e-mail: JF.Heydari@gmail.com 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_16, # Springer-Verlag Berlin Heidelberg 2011

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companies competed with each other, whereas today, supply chains are racing against each other. The success of an SC depends on the willingness of the potential customers, which requires collaboration and coordination between all members in reducing the sale price, improving product quality, and delivering the right services to the customers. Due to the continuity of decisions in an SC, a wrong decision at one stage (e.g., a local optimum) can lead to additional costs for other members; thus, the SC can be greatly hindered in accomplishing its mission. Therefore, each SC member should make decisions regarding other members rather than only their own profitability. For example, the buyer (or retailer) has the authority to make inventory decisions, such as how much to order (order quantity) or when to order (reorder point). However, these decisions, in addition to influencing the buyer’s profitability, affect the supplier’s costs by changing its inventory-system inputs. Customers’ expectations are currently increasing; these include a high-quality product at the lowest price, quick delivery, and after-sale services. This increase of customer expectations, along with the many competitors in the market, has forced SC members to collaborate to meet the demands of customers at a higher level than their competitors. Thus, to increase customer loyalty and expand market share, each SC member should make appropriate decisions such that the overall SC costs are reduced and the customers’ expectations are met in the best possible manner. Additionally, uncertainties such as external factors outside of the SC members’ control can significantly affect the decisions made by SC partners (Chaharsooghi and Heydari 2010a). These uncertainties may include but are not limited to demand (quantity), supply (time), and price (changes). In facing uncertainties, there are two general strategies: first, if the uncertainties can be manipulated (by paying the relevant costs), shaper strategies are used, e.g., shaping market demand by offering a discount; second, if the uncertainty cannot be reshaped, it is then necessary to implement appropriate plans (e.g., increasing safety stocks) to face future uncertainties, known as adaptor strategies (Gupta and Maranas 2003). Member coordination is a key idea in the formation of the supply chain concept; in almost all definitions of the supply chain, coordination plays a critical role. Coordination in SC means that all decision makers in the SC make their decisions in alignment with other partners such that they globally optimize decision criteria. Sometimes, this alignment between decision makers requires members to set their decision variables far from their local optima; therefore, to encourage these members to accept the global optimum, a set of incentive schemes should be offered. Anything that is capable of encouraging an SC member to participate in the coordination model can be identified as an incentive scheme, e.g., quantity discounts, return policies (buy-back contracts), quantity flexibility, revenue sharing contracts, and credit options (delayed payment). The main focus of this chapter is on the credit option as an innovative scheme for facilitating SC coordination under demand uncertainty. The use of credit is very common in current business environments and plays an important role as a form of financing, particularly in developing countries (Sarmah et al. 2008). Although credit trading has been studied traditionally from the viewpoint of financing, from the operations-management viewpoint, using credit can reduces a company’s

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operating costs. In today’s business transactions, suppliers frequently allow credit for fixed time periods to encourage buyers to increase the size of their orders (Huang 2010). Allowing credit for a specific time period is advantageous for the buyer from several points of view. First, under a credit contract, the buyer does not have to pay the supplier immediately and therefore it is possible to pay the supplier from future earnings. Therefore, low-capital buyers can enter the business. Second, as the major portion of inventory holding costs is associated with investment depreciation, using credit therefore reduces these costs. In other words, delayed payment reduces the amount of capital invested in stock for the duration of the credit period (Huang 2010). Third, the unpaid balance can be invested during the credit period and create additional earnings. In general, before the end of the delay period, the retailer can sell the goods and accumulate revenue and earn interest (Huang 2007). In this chapter, we consider the benefit of credit from the second point of view. In this chapter, a two-stage, serially-connected SC is investigated. The downstream uses a continuous inventory-review system (s,Q). Based on this inventory-review system, the downstream places its order of size Q when its inventory level drops to the reorder point s. In the traditional mode, both Q and s are determined by the downstream so that minimize its costs. In the coordinated mode, unlike in the traditional mode, both downstream and upstream jointly determine these decision variables. Our analyses show that the simultaneous coordination of order quantity and reorder point is profitable for the supply chain as whole. Although the profit of the entire chain is increased, the buyer’s profitability is affected due to the deviation from its local optimum. To compensate for the downstream loss, an incentive scheme based on a credit option is proposed. According to the implemented credit option, the downstream can delay payments for purchased and delivered goods for a certain time, which is known as the credit period. Because a major part of inventory costs is due to investment depreciation on warehouses, a delay in payment can reduce these costs in the downstream site. The credit option is applied successfully as an innovative scheme to simultaneously coordinate both order quantity and reorder point in the supply chain. By implementing the credit option in the SC, the profitability of both members is increased over that before accepting the coordination model. Therefore, the participation of all members in the model is guaranteed. The lower and upper bounds for the credit period are determined based on the downstream and upstream profitabilities. Finally, the exact value of the credit period is calculated based on the bargaining power of each member. In this case, the profits of the coordination model are shared according to the bargaining power of the members. The main contribution of this chapter is the development of a model for the simultaneous optimization of order quantity and reorder point in a supply chain under demand uncertainty. We show that offering delayed payment can encourage the buyer to change its decisions to fall in line with the SC-optimal decisions. Delayed payment has been previously considered mostly for deterministic situations; in this study, we show that delayed payment is capable of coordinating an SC under a demand uncertainty. This chapter is organized as follows. In Sect. 2, a brief literature review is provided. SC model development in decentralized, centralized, and coordinated

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decision making is described in Sect. 3. Section 4 presents the numerical experiments. Finally conclusions are given in Sect. 5.

2 Literature Review Recently, SC coordination has attracted the attention of many researchers as well as practitioners around the world. Moving toward the development of supply chain management, the coordination issue has become a critical field of study. Various coordination models have been proposed in the literature. In this section, we focus on the recent studies in this area. Effective SC management requires the coordination of all SC members, including suppliers, manufacturers, distributors, and retailers, in delivering goods to the customers (Xu et al. 2001). Coordination mechanisms aim to find the optimum solution for the entire chain and then encourage chain members to accept this solution (Giannoccaro and Pontrandolfo 2004). Coordination can be considered in different functions of an SC such as inventory control, logistics, transportation, and forecasting, (Arshinder et al. 2008). Most of the research on SC coordination to date has considered the order quantity as the core parameter, while other parameters also exist that must be coordinated (Chaharsooghi and Heydari 2010a). In the centralized SC, there is one central planner who makes decisions and controls all members and therefore makes decisions that maximize the profit of the whole SC, whereas in decentralized, uncoordinated SC, each member maximizes its own profit (Chen et al. 2010). To coordinate a decentralized SC, various coordination contracts have been proposed in the literature, including quantity discounts (Tsai 2007; Shin and Benton 2007; Li and Liu 2006), buyback policies (Xiao et al. 2010; Ding and Chen 2008; Yao et al. 2008), revenue sharing (Van der Rhee et al. 2010; Hou et al. 2009; Li et al. 2009; Chauhan and Proth 2005), credit options (Chaharsooghi and Heydari 2010b; Jaber and Osman 2006; Sarker et al. 2000), and insurance contracts (Lin et al. 2010). The philosophy of all coordination contracts is the fair sharing of risk among partners. Cachon (2003) conducted a comprehensive literature review on SC-coordination contracts. In the following, the most popular coordination models are examined. Sharing downstream overstocking risk is the main concern of order-quantity coordination models. In a multi period setting, quantity-discount models – the most popular coordination models – have been developed. In quantity-discount models, the upstream encourages the downstream to change its order size by offering a quantity discount. Here, the problem is how to set the discount parameters such that the downstream has enough incentive to participate and, conversely, so that upstream profitability is not seriously affected. Various discount models have been introduced in the literature for various problem settings. For example, coordinating a three-level SC using quantity discounts has been investigated (Munson and Rosenblatt 2001; Jaber et al. 2010). A linear quantity-discount model to coordinate an SC including one manufacturer and multiple retailers after a demand disruption

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has been developed (Chen and Xiao 2009). A quantity-discount model in a twostage SC, in which the demand rate depends on the retailer’s stock level, has also been developed, and it was shown that this model could achieve channel coordination (Zhou et al. 2008). Return policies are also of interest to researchers in the SC-coordination field. In single-period inventory systems (or seasonal sales), unsold items at the retailer’s site at the end of the period pose a risk for the retailer. Return policies are adopted by upstream members to convince retailers to place their orders based on the wholly optimal solution. The effect of e-marketplaces on return polices in an SC when the returned product can be sold in the e-marketplace has been studied (Choi et al. 2004). Flexible return policies in a three-level SC have been proposed, and it was shown that this policy could achieve channel coordination (Ding and Chen 2008). The impact of price-sensitive demand on return polices has also been investigated and various situations were modeled (Yao et al. 2008). The effect of demand-information asymmetry on full return polices was investigated, and it was found that, under some conditions, it is possible that the upstream can lose profitability whereas the downstream always benefits from a return policy (Yue and Raghunathan 2007). Another coordination contract is the credit option or delayed payment, the main concern of this work. In trade, using credit or delayed payment means late payment for a purchased and delivered product. Credit trade has some advantages for the credit users (buyers) and is very common in business today. From the perspective of inventory control, one important advantage of the use of credit is the reduction of inventory-holding costs (Chaharsooghi and Heydari 2010b). A credit option was successfully applied for coordinating the base stock level and review period in a two-stage SC with periodic inventory model (Moses and Seshadri 2000). The optimal pricing and ordering policies with delayed payment, which are dependent on order size in a single-location inventory system, have been calculated, wherein it was assumed that a fixed and predetermined credit period is a function of order quantity and demand is a function of selling price (Shinn and Hwang 2003). The optimal cycle time of a single-location inventory model when delayed payment was allowed by the vendor and the credit period is not a decision variable has also been calculated (Chung et al. 2005); this model has been enhanced by optimizing the retailing price in addition to the cycle time (Teng et al. 2005). Subsequently, the model was further complemented to consider a partially permissible delay in payment when the order size is smaller than a predetermined quantity (Huang 2007). The above studies all considered the case of deterministic demand without shortages; Chung and Huang (2009) complemented the previous works by considering delayed payment in a single-location inventory system when shortages are allowed. This study was later extended to an economic production-quantity framework with a finite replenishment rate (Hu and Liu 2010). Most of the abovementioned studies in the field of delay in payments considered single-location inventory systems, whereas in supply chain management the problem is extended to multilocation inventory systems. Delay in payments has been successfully used to coordinate the order quantity in a deterministic environment

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(Jaber and Osman 2006). The optimal replenishment cycle and replenishment frequency in a two-stage SC under a credit option have been determined and the delay period calculated (Chen and Kang 2007). Sarmah et al. (2007) introduced a coordination model based on a credit option in a two-stage serial SC in which each party has its individual profit target. Afterwards, the coordination of singlemanufacturer and multiple heterogeneous buyers by considering transportation costs when demand is known and constant was investigated, and the coordinated production and replenishment cycle are determined (Sarmah et al. 2008). In another report, the effect of delayed payment, the inflation rate, and the depreciation rate on the inventory system when the supplier provides a permissible payment delay for the purchaser were discussed, and the optimal payment period and replenishment cycle were derived (Chang et al. 2009). Lead-time reduction in a two-stage supply chain with a permissible payment delay has been investigated, and the optimum inventory policy along with the optimum lead-time length has been calculated (Huang et al. 2010a). In the last study, the credit period was considered to be an input parameter of the system. The same authors further developed this work from a mathematical modeling viewpoint, in which the capital investment in reducing the order-processing time is a logarithmic function of ordering costs (Huang et al. 2010b). In this chapter, a credit-option model is proposed as an incentive scheme for coordinating one buyer and one supplier. In comparison with the previous works in the field of SC coordination and delay in payments contracts, this study can be distinguished in two aspects. First, the proposed model considers the coordination of two principal inventory decisions, i.e., order quantity and reorder point, simultaneously. Second, in the proposed model the credit option is investigated as an incentive scheme in the continuous inventory-review system in which the customer demand is stochastic. In the proposed model, the buyer uses an (s,Q) inventory system and the decision variables are order quantity and reorder point. By offering a credit period, the supplier encourages the buyer to select these decision variables such that the expected total SC costs are minimized. The coordinating parameter “Credit period” is determined such that both parties have sufficient incentive to participate in the model.

3 Model Development The investigated supply chain is a two-stage SC with one actor at each level, namely, buyer and supplier. The buyer faces the stochastic demand of the customers and uses a continuous inventory-review system (s,Q), in which an order of size Q is placed with the supplier when the buyer’s inventory level drops to the reorder point s. The placed order Q is delivered to the buyer after a stochastic lead time. In this model, the mean and variance of lead-time demand (LTD) is known. Both s and Q are buyer decision variables. The received customer orders must be responded to immediately; otherwise, the unfulfilled demand will be backlogged and must be realized as soon as possible.

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The supplier problem is how to replenish stocks such that is able to respond to the buyer’s orders. It is assumed that when a supplier places the replenishment order, the order is delivered immediately and instantaneously. In this situation, it has been proven that optimal size of a supplier order is an integer multiple of the buyer’s order quantity (Rosenblatt and Lee 1985). The following notation is used in this chapter: D Q1 Q2 h1 h2 A1 A2 B k m sL s

Expected demand per year Buyer’s order quantity (decision variable) Supplier’s order quantity, which is function of the buyer’s order quantity given by Q2 ¼ nQ1, where n is an integer (decision variable) Buyer’s unit inventory holding cost per year Supplier’s unit inventory holding cost per year Buyer’s ordering cost per replenishment Supplier’s ordering cost per replenishment Shortage cost per unit at buyer’s site Safety factor (decision variable) Expected lead time demand Standard deviation of demand during lead time Buyer’s reorder point, which is function of the safety factor k

It is assumed that the lead-time demand has a normal distribution. Therefore, for any given value of k, the reorder point s will be m + ksL (Silver et al. 1998), where ksL is the safety stock (SS).

3.1

Traditional Decision Making

In the traditional decision-making model, each SC member determines its authorized decision variables (k and Q1 for buyer and Q2 for supplier) by considering its own cost function such that it minimizes its own costs, regardless of other chain members. In this situation, each SC member should solve a traditional singlelocation inventory problem. In their textbook example of a single-location inventory model, Silver et al. (1998) calculate the optimized buyer’s order quantity and reorder point simultaneously. The expected buyer’s annual cost function, including expected annual ordering, holding and shortage costs, is calculated as follows: ðk; Q1 Þ ¼ A1 D=Q1 þ h1 ð0:5Q1 þ ksL Þ þ ðBDGu ðkÞsL Þ=Q1 TCTraditional 1

(1)

where Gu(k) is defined as: 1 ð

Gu ðkÞ ¼ k

1 u2 ðu kÞ pﬃﬃﬃﬃﬃﬃ e 2 du 2p

(2)

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The expected shortages per replenishment cycle are represented by Gu ðkÞsL . The buyer’s expected annual shortage costs can be calculated by multiplying the expected shortages per replenishment cycle by the expected number of replenishments per year, D/Q1, and the unit shortage cost B. Silver et al. (1998) propose an iterative procedure to find the optimum value of k and Q1 simultaneously. Meanwhile, the supplier solves its inventory problems by itself. As mentioned earlier, it has been proven that supplier cost will be minimized when Q2 is an integer multiple of Q1. The supplier expected annual-cost function, including expected annual ordering and holding costs, is: ðnÞ ¼ A2 D=ðnQ1 Þ þ 0:5h2 ðn 1ÞQ1 TCTraditional 2

(3)

In the real world, due to changing market behavior, the upstream members do not hold high inventory levels. It is assumed the supplier’s order quantity nQ1 should not exceed 1 year of customer demand, except in special cases (where Q1 is greater than D, in which case n is set at 1). Therefore, if D/Q1 is smaller than one then n ¼ 1; otherwise, the optimal value of n can be obtained by evaluating the integers in the interval [1, D/Q1]. In the traditional decision-making mode, buyer and supplier each make their decisions independently based on their own expected cost functions, i.e., (1) and (3), respectively.

3.2

Centralized Decision Making

In the centralized decision-making mode, there is one SC manager who controls all the businesses in the chain and makes decisions on Q1, k, and n based on the expected total profitability of the SC. The solution thus determined (i.e., the optimal values of the decision variables in the centralized mode) the globally optimum solution. The expected annual cost function (TC) for the whole chain is the sum of buyer and supplier expected cost functions (1) and (3): TCðk; Q1 ; nÞ ¼ ½nA1 þ A2 þ nBGu ðkÞsL D=ðnQ1 Þ þ 0:5½h1 þ ðn 1Þh2 Q1 þ h1 ksL

(4)

Considering the fact that the sum of convex functions must also be convex, to prove the convexity of TC with respect to Q1 and k, it is sufficient to prove the and TCTraditional . In the literature, it has been noted that convexity of TCTraditional 1 2 Traditional is a convex function of Q1 and k (Silver et al. 1998, p. 326). Also, TC1 is not a function of k and, therefore, this is sufficient to prove it is a TCTraditional 2 convex function of Q1. Because @ 2 TCTraditional =@Q21 ¼ 2A2 D=ðnQ31 Þ > 0, TCTraditional 2 2 is a convex function with respect to Q1 and k. Hence, TC is also a convex function with respect to Q1 and k.

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Using the first-order condition, the following equations are obtained: Q1 ¼ ½2DðnA1 þ A2 þ nBGu ðkÞsL Þ=ðn½h1 þ h2 ðn 1ÞÞ1=2 Fu ðkÞ ¼

1 ð

k

1 h1 Q 1 u2 pﬃﬃﬃﬃﬃﬃ :e 2 :du ¼ DB 2p

(5)

(6)

where Fu ðkÞ is the probability that a standard normal variable u takes on a value of k or larger or, correspondingly, the probability of a stock-out within the replenishment lead time. Also, similarly to the traditional decision-making mode, it is assumed that a supplier’s order quantity nQ1 should not exceed 1 year of customer demand except in special cases where Q1 > D. An iterative procedure that converges to the optimal solution is proposed as follows. Initialization: Set n as low as possible (n ¼ 1). Step 0: Set Gu ðkÞ ¼ 0. Step 1: Calculate Q1 using (5). Step 2: Calculate k using (6). Step 3: Repeat steps 1 and 2 iteratively until Q1 and k converge (the difference between two successive values reaches a sufficiently small value) then calculate TC using (4). Step 4: If n bD=Q1 c, then n ¼ n þ 1; go to Step 0. Step 5: Each combination of s, Q1, and n that results in a lower value of TC is the optimal solution (the obtained optimal values are denoted by the superscript “*”). In this procedure, the previous simpler procedure proposed by Silver et al. (1998) (corresponding to steps 0–3 of the proposed procedure) constitutes the middle of the newly proposed procedure. Due to the convex nature of the functions involved, convergence of the above procedure is ensured. Using the proposed procedure, the optimal values of decision variables will be obtained. The target point of the coordination model is to achieve this operational plan with a decentralized chain structure. To achieve channel coordination, the final values of the decision variables resulting from the coordination model must be equal to those of the centralized decision-making model. Indeed, coordination mechanisms should serve to create sufficient incentive for chain members to set their decision variables equal to the values obtained in the centralized model.

3.3

Coordinated Decision Making

Coordination mechanisms aim to encourage members to make decisions that are in line with those of the centralized SC, i.e., those of the globally optimum solution.

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In most cases, when the SC has a decentralized structure, setting the decision variables to optimal centralized values incurs losses for some members. In these cases, using an incentive scheme to fairly share the benefits of a coordination model is vital. At best, a coordination mechanism can increase SC profitability up to that of the centralized decision-making model; this is called channel coordination. Channel coordination is achieved if the expected total SC costs in the coordinated model are reduced to the centralized SC costs. Also, the coordination mechanism should be desirable for all members (Giannoccaro and Pontrandolfo 2004) to ensure that the model is applied. A model is desirable for a member when the application of the model does not reduce the member’s profit to less than that before applying the model. In this chapter, to coordinate both order quantity and reorder point simultaneously, a coordination mechanism based on a credit option is proposed. In this model, the buyer can take advantage of a payment delay subject to commitment to the jointly determined values of the decision variables. Based on the proposed model, the buyer must change its operational decisions regarding s and Q1 from its local optimum to the global optimum to access the benefit of delayed payment for purchased goods proposed by the supplier. Inventory holding cost is associated with cost factors such as leasing the warehouse space, staffing wages, overhead costs, and capital invested in stock. Although the cost of holding inventory is not fully due to capital invested in stock, this capital comprises the majority of inventory holding costs. The use of credit reduces the buyer’s inventory holding costs because it reduces the amount of capital invested in stock for the duration of the credit period (Huang et al. 2010a; Teng et al. 2005). Also, using a credit option has several other advantages for the buyer exploiting delayed payment which are not considered in this study; for instance, a credit user can accumulate sales revenue and earn interest by investing in external projects or depositing the earnings in an interest-bearing account during the credit period (Hu and Liu 2010; Chung et al. 2005). From the SC-coordination perspective, a credit option can convince the buyer to commit to the jointly agreed-upon replenishment rules by decreasing its inventory costs. By comparing the previously investigated traditional and centralized decisionmaking structures, it is possible to extract the operational coefficients for both buyer and supplier. By applying these operational coefficients to the local optimal solution (the values of the decision variables obtained in the traditional model), each party can extract the globally optimum solution. The buyer’s order-quantity coordinator coefficient can be defined as KQ ¼ Q1 =Q1 , where Q1 is the globally optimum value of the buyer’s order quantity that is obtained from the centralized decision-making model and Q1 is the local optimal value of the buyer’s order quantity calculated from the traditional decision-making model. Similarly, the buyer’s safety-factor coordinator coefficient is defined as Kk ¼ k =k and the supplier’s coordinator coefficient for the decision variable n is defined as Kn ¼ n =n. However, applying these defined coordinator coefficients is not possible without necessary arrangements. In fact, by applying the above coefficients, the buyer will

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lose due to deviation from its local optimum. To compensate for the incurred losses, the supplier proposes that the buyer purchases on credit. As mentioned earlier, using credit has several advantages for the buyer; their most important is reducing the expected inventory holding costs. As the majority of inventory holding cost is related to the capital invested in stock, this cost will be imposed only when the payment is cleared; therefore, delayed payment can reduce inventory holding costs. The portion of expected inventory holding costs due to capital invested in stock differs among various industries. For the model be applicable across different industries, we define this portion as a parameter in the proposed model. It is considered that b% of the expected inventory holding cost is related to the capital invested in stock and, therefore, the credit option can compensate these costs for the buyer in the Credit period (CP). From the mathematical point of view, during the credit period the buyer’s expected annual inventory holding costs are reduced by the coefficient (100 b)%. Figure 1 shows the buyer’s reduced inventory holding costs associated with the capital invested in stock after applying the credit option. Several replenishment cycles for both the buyer and supplier and their inventory positions by assuming n ¼ 3 are illustrated in Fig. 1. As shown here, the buyer places an order of size Q1 with the supplier when its inventory position drops to the Buyer Inventory Position

s µ

Q1

Q1

LT

LT

Q1

Q1

LT

LT

LT

Q1

SS=k.σL

CP

Supplier Inventory Position

CP

CP

First batch delivery

CP

Third batch delivery

Second batch delivery

CP

Time

Fifth batch delivery

Fourth batch delivery

Q1

Q1

2Q1

2Q1

Q1

Settle the bill for first batch Shipping first batch

Settle the bill for second batch

Shipping second batch

Q1

Settle the bill for third batch

Shipping third batch

Settle the bill for fourth batch

Shipping fourth batch

Settle the bill for fifth batch

Shipping fifth batch

Time

Shipping sixth batch

Fig. 1 Inventory positions for the buyer (top) and supplier (bottom) and sequence of events in several replenishment cycles

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reorder point s. The placed order is received just after the stochastic lead time LT. Due to the stochastic nature of lead time demand, a shortage may occur. After receiving a placed order, the waiting customers and new customers are served from the recently arrived batch of product (Q1) and revenue is earned. Without the credit option, the buyer must settle the bill at the time of receipt of each order. In the case where a credit option is proposed by the supplier, the buyer settles the payment at the end of the credit period CP. The amount of capital invested in stock for each unit of inventory during the credit period is zero. As b% of the inventory holding cost is associated with capital invested in stock, the cost of holding each unit during the credit period will be reduced. In Fig. 1, the area under the curve is divided into two parts. The hatched area represents the credit duration. In other words, at the beginning of each solid area in Fig. 1, the buyer pays the supplier for the previously delivered batch, CP time units ago. During the CP, the buyer earns revenue from selling the goods which have not yet been paid for. The inventory holding costs in the hatched area of Fig. 1 are reduced by the coefficient (100 b)% with respect to the solid area. Thus, the credit-option contract reduces the buyer’s inventory holding costs during the credit period. If this cost reduction compensates for the increased cost of changing from traditional to centralized decision making, then the buyer will participate. A longer credit period means a greater reduction in the buyer’s expected costs. From the buyer’s perspective, extending credit period is more advantageous. There is a lower bound for credit period, below which the participation is not beneficial for the buyer (when the extra savings from the delay in payment is less than extra cost imposed by accepting the centralized solution). Conversely, adjusting the credit period as low as reasonably possible is more beneficial for the supplier. In the credit period, payment has been not made to the supplier for product sold. Due to the time value of money, delaying the payments only up to a certain period is economical for the supplier. Therefore, from the supplier perspective, there is an upper bound for the credit period; when the delay time exceeds this upper bound, the participation will not be advantageous for the supplier. Figure 2, shows the concept of credit time interval and the preferences of both parties. As shown in Fig. 2, if the credit period becomes very low, the buyer will lose and will not participate, whereas if the credit period becomes very high the supplier will lose. Thus, there is an interval for the credit period between these lower and upper bounds; the appropriate CP lies within it. The bargaining power of members with

Low

Credit Period

High

Fig. 2 The interest of two parties in adjusting the credit period, CP

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each other is the measure for adjusting the coordinator variable CP. As all the inventory-system variables are determined for traditional and coordinated models, the only remaining decision variable is coordinator variable CP. Hereinafter, the Q1, k, and n denote locally optimal values of the decision variables in the traditional model and Q1 , k*, and n* denote globally optimal values of the decision variables in the centralized model, which are absolutely determined and known.

3.3.1

Condition for Buyer to Participate (Lower Bound for Credit Period)

In proposing the credit option to the buyer, the supplier intends to share the earned profits. In fact, the use of credit transfers part of the inventory costs to the supplier and therefore leads to a reduction in the buyer’s expected costs. As mentioned earlier, the buyer’s expected costs consist of three parts: expected ordering, holding inventory, and shortage costs. Using credit affects the second part; i.e., inventory holding costs. As in the traditional inventory model, it is possible to calculate the expected holding inventory costs before applying the credit option by calculating the area under the buyer’s inventory-level curve, which is ðQ1 =2Þ þ k:sL , where the first term is the average cycle inventory and the second term is the safety stock level. By applying the unit inventory-holding costs h1 to the derived value, the buyer’s expected annual inventory holding costs are calculated [see (1)]. To calculate the buyer’s expected annual inventory holding costs after applying the credit option, it is necessary to calculate the area under the buyer’s inventory-level curve based on Fig. 1. The hatched area in Fig. 1 is affected by the coefficient (1 b) while the solid area is not. As a result, the expected annual inventory holding costs will be: Inventory holding costs n ¼ h1 ðKQ Q1 DðCPÞÞ2 =2KQ Q1 þ ð1 bÞðCPÞðD D2 ðCPÞ=2KQ Q1 ÞðCPÞ þKk ksL ð1 bDðCPÞ=KQ Q1 Þ As can be seen in the above equation, the coordinator coefficients KQ and Kk are applied to the variables Q1 and k, respectively; note that Q1 and k in the above relation refer to the optimal values obtained from the traditional model. Finally, the buyer’s expected annual cost function after applying the credit option can be expressed as follows: TCCoordinated ðCPÞ 1

" D ðKQ Q1 DðCPÞÞ2 D2 ðCPÞ þ h1 þ ð1 bÞðD ÞðCPÞ ¼ A1 2KQ Q1 K Q Q1 2KQ Q1 D D þKk ksL ð1 bCP Þ þ ðBGu ðKk kÞsL Þ K Q Q1 KQ Q 1

(7)

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where the three terms denote the expected ordering cost, the expected inventory holding cost and the expected shortage cost, respectively. In the cost function (7), the buyer agrees to apply the coordinator coefficients KQ and Kk to the local optimum values of Q1 and k; in turn, the supplier accepts the delay in payments according to the credit period (delay period) CP. The only decision variable in (7) is CP. The buyer participates (i.e., applies the coordinator coefficients in the case of using credit) if and only if buyer costs do not rise after participation. In mathematical terms, TC after participation (7) must be smaller than the buyer’s expected cost in the traditional model (1). By comparing the buyer’s expected cost functions before and after the credit-option contract, a lower bound for CP can be calculated. Proposition 1. The minimum credit period that convinces the buyer to participate is: CPmin

1 ¼ ðKQ Q1 þ Kk ksL Þ D

rﬃﬃﬃﬃﬃﬃﬃ 2W D2

(8)

where W¼

ðKQ Q1 þ Kk ksL Þ2 1 h A1 Dð1 KQ Þ þ BDsL ½Gu ðKk kÞ Gu ðkÞKQ : 2 bh1 i Q1 2 ðKQ 1Þ þ h1 KQ kQ1 sL ðKk 1Þ (9) þh1 KQ 2

Proof. Set TCTraditional ðk; Q1 Þ TCCoordinated ðCPÞ; by simple mathematical opera1 1 tions it is verified that this inequality will be satisfied if and only if pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ CP D1 ðKQ Q1 þ Kk ksL Þ 2W=D2 . The right side of the resulting expression gives CPmin, which is the lowest value for the CP that encourages the buyer to participate. In fact, when CP is adjusted to CPmin, the expected total cost for the buyer after accepting the coordinator coefficients KQ and Kk is then equal to the expected total cost for buyer before accepting the coordination. If CP is adjusted to a value less than CPmin, the total cost imposed on the buyer after accepting the coordinator coefficients will then be greater than the traditional-model costs, and the buyer will therefore not participate in the plan.

3.3.2

Condition for Supplier to Participate (Upper Bound for Credit Period)

The supplier is the proposer of the credit option to the buyer. If the offer leads to disadvantages for the supplier, it will basically never occur. In this section, the costs imposed on the supplier by proposing credit are calculated and compared to the costs saved from the coordination model. When the supplier’s earned value is

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greater than the cost of convincing the buyer, supplier participation is then guaranteed. The difference between the supplier’s expected cost function in the coordinated and traditional models indicates the earned value of the supplier in moving from the traditional to coordinated model. In the coordinated model, in addition to the expected ordering and holding costs, the supplier is faced with another cost factor: the cost of convincing the buyer. The buyer’s convincing cost is related to the time value of money. The supplier receives payment CP units of time later; thus, with a known interest rate, it is easy to calculate this cost. Also note that all the costs by the supplier for convincing the buyer are delivered by the buyer; therefore, another simple method for calculating the convincing cost is the calculation of the hatched area in Fig. 1 over 1 year multiplied by the imposed cost coefficient bh1. Finally, the supplier’s expected annual cost function after proposing the credit option is: TCCoordinated ðCPÞ ¼ A2 D=ðKn KQ nQ1 Þ þ 0:5h2 KQ Q1 ðKn n 1Þ 2 þ bh1 DðCPÞ KQ Q1 0:5DðCPÞ þ Kk ksL =KQ Q1

(10)

where the first term is the expected ordering costs, the second term is the expected inventory holding costs and the third term is the expected cost of convincing the buyer. The supplier participates in the plan if and only if its cost after participation (10) is smaller than the cost before participation (3). By comparing the supplier’s expected cost functions before and after the credit-option contract, the following proposition establishes an upper bound for CP. Proposition 2. The maximum allowable credit period which is acceptable for the supplier is: CPmax

1 ¼ ðKQ Q1 þ Kk ksL Þ D

rﬃﬃﬃﬃﬃﬃﬃ 2M D2

(11)

where M¼

2DA2 ð1 KQ Kn Þ þ h2 KQ Kn nQ1 2 ð1 KQ n½1 Kn KQ Þ 2Kn nbh1 þ 0:5ðKQ Q1 þ Kk ksL Þ2

(12)

Proof. Set TCTraditional ðnÞ TCCoordinated ðCPÞ; by simple mathematical operations 2 2 it is verified that this inequality will be satisfied if and only if CP D1 ðKQ Q1 þ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Kk ksL Þ 2M=D2 . The right side of the achieved expression gives CPmax, the highest value for the CP that is acceptable for the supplier. When CP is adjusted to CPmax, the expected total cost to the supplier after proposing the credit option is then equal to its expected total cost in the traditional model. If the CP is adjusted to higher than CPmax, the total cost imposed on the supplier by proposing the credit option will be greater than in the traditional model, and the supplier will therefore refuse to participate.

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Equitable Credit Period

Channel coordination will be achieved by setting the credit period to a value between CPmin and CPmax. If CP is set to CPmin, then all the benefits are acquired by the supplier. In fact by setting CP ¼ CPmin, the buyer’s expected cost after coordination will be equal to the buyer’s expected cost in the traditional model. In this state, there is no change in profitability for the buyer by accepting the plan, and all the advantages are therefore gained by the supplier. Indeed, by setting the CP to CPmin, the buyer’s profitability will merely not be reduced. Conversely, if CP is set to CPmax, then all the benefits are gained by the buyer. The interval [CPmin, CPmax] is a continuous interval and therefore in selecting the credit period from this interval there are infinite options. But what point must be selected? Or, similarly, what is the criterion for setting CP? The answer to these questions is related to the power of each member versus the other. Recalling Fig. 2, the member with more power can set the CP. In the business environment this power is called “bargaining power”. The bargaining power is defined as the total advantage which one can obtain in a negotiation. In this section, the coordinator variable CP is determined based on the bargaining power of SC members. In this way, the share of benefits obtained by each member will be fairly distributed. We define a as the bargaining power of the buyer versus the supplier and therefore (1 a) is the bargaining power of the supplier versus the buyer. It is clear that a is between 0 and 1. By increasing the value of a to near 1, the bargaining power of the buyer increases. As a result, by increasing a, the expected earnings of the buyer are increased and vice versa. To calculate the credit period, it is necessary to calculate the extra earnings resulting from applying the proposed coordination model and share them between the two parties by adjusting the CP. The extra profit by applying the coordination scheme is: DTC ¼ TCðk; Q1 ; nÞ TCðk ; Q1 ; n Þ ¼ ½nA1 þ A2 þ nBGu ðkÞsL D=ðnQ1 Þ þ 0:5½h1 þ ðn 1Þh2 Q1 þ h1 ksL ½Kn nA1 þ A2 þ Kn nBGu ðKk kÞsL D=ðKn nKQ Q1 Þ 0:5½h1 þ ðKn n 1Þh2 KQ Q1 h1 Kk ksL

(13)

where Q1, k, and n are the (locally) optimal values of the decision variables in the traditional decision-making model, and Q1 , k*, n* are the optimal values of the decision variables in the centralized model, which are created by applying the coordinator coefficients KQ, Kk, and Kn, respectively, to the local optimum values. a According to the buyer–supplier bargaining power relationship, ðaþð1aÞÞ 100% of DTC should be obtained by the buyer and rest is retained by the supplier. From the mathematical point of view, the buyer’s expected cost after coordination must be smaller than the buyer’s expected cost in traditional mode by of the factor aDTC.

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TCTraditional ðk; Q1 Þ TCCoordinated ðk ; Q1 ; CPÞ ¼ aDTC 1 1

419

(14)

By substituting (1) and (7) into (14), the exact value of CP based on the buyer’s bargaining power a is found as follows: sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ# " 1 KQ Q1 aDTC Þ ðKQ Q1 þ Kk ksL Þ 2ðW CP ¼ D bh1

(15)

The channel coordination is achievable, If one value from (15) is obtained within the interval [CPmin, CPmax]. By applying the calculated CP, the supplier’s share of ð1aÞ the extra benefits will be ðaþð1aÞÞ 100%.

4 Numerical Experiments In this section, the performance of the introduced coordination scheme is measured by conducting a set of numerical examples. Also, the effect of uncertainties on the coordination model is examined by testing the model with various degrees of uncertainty. Table 1 shows the data set for the numerical experiments. After running the three models (traditional, centralized and coordinated) based on the Table 1 data set, the results shown in Table 2 were obtained. As shown in Table 2, the TC resulting from the coordinated decision-making model is equal to that of the centralized model. Therefore, the proposed coordination model can achieve channel coordination. Also, applying the model is desirable for both members due to the reduction of expected costs for each member with respect to the traditional model. By shifting from traditional to centralized decision making, the expected total SC cost can be reduced from the 885.89 to 814.03, an ~8.1% improvement. Nevertheless, this shift is not applicable due to the increase in the buyer’s expected costs (from 561.95 to 607.37). The credit-option scheme resolves this problem by fairly sharing the acquired benefits between members. As shown in Table 2, in the coordinated model, both members’ costs are reduced with respect to the traditional model and the total expected cost of the SC is also equal to the centralized model. The trend of the changes in the decision variables and the performance criteria were also tracked by increasing the uncertainty. The standard deviation of the leadtime demand (LTD) is the measure of uncertainty, and we can call it the degree of uncertainty. Figure 3 shows the total SC cost improvement resulting from the coordinated model with respect to traditional decision making with various degrees of uncertainty. As shown in Fig. 3, the proposed model can create an appropriate cost saving for an SC greater than 5% with all degrees of uncertainty. This improvement in total SC cost shows the ability of the coordination model for adjusting the decision

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Table 1 Numerical experiment data

Table 2 The results of running the numerical examples

Parameters D h1 h2 B A1 A2 m sL a b

Traditional decision making Decision variables Q1 123.48 s 77 n 1 CP (day) N/A

Performance criteria 561.95 TC1 TC2 323.94 TC 885.89

Values 500 4 3 5 50 80 80 20 0.6 0.4

Centralized Coordinated decision making decision making 193.57 69.93 1 N/A

193.57 69.93 1 CPmin ¼ 21.24 CPmax ¼ 65.22 CP ¼ 45.35

607.39 206.64 814.03

518.84 295.19 814.03

12

% Improvement

10 8 6 4 2 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 LTD standard deviation

Fig. 3 SC cost improvement of coordinated decision making compared to the traditional model

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200 Q1-Traditional

180

Q1-Coordinated Product unit

160

s-Traditional s-Coordinated

140 120 100 80 60

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 LTD standard deviation

Fig. 4 Coordinated buyer’s order quantity and reorder point versus traditional buyer’s order quantity and reorder point

variables. The proposed credit option encourages the buyer to decide on the variables Q1 and s jointly. Thus, the globally optimum solution can be achieved. Figure 4 illustrates the values of the buyer’s order quantity and reorder point before and after implementing the credit option. As shown here, the proposed model adjusts the buyer’s order quantity to a point higher than in traditional decision making while the reorder point is adjusted to less than in the traditional model. Note that in Fig. 4 the coordinated Q1 and s are equal to Q1 and s in centralized decision making. Figure 5 shows the buyer’s expected costs with the various degrees of uncertainty. As shown by the buyer’s expected cost curve, the cost in the coordinated model is lower than traditional and centralized models. As shown in Fig. 5, without conducting the credit option, the buyer’s expected cost increases (in going from the centralized to traditional model) and the buyer declines to participate. In fact, in coordinated decision making the buyer takes advantage of the delay in payment and therefore its cost is reduced compared to the centralized decision-making structure. Because the buyer suffers the uncertainties, as uncertainty increases the buyer’s expected costs also increase but the efficiency of the coordination model is not affected; i.e., channel coordination is achieved in all cases. Figure 6 shows the changes in the supplier’s expected costs versus the degree of uncertainty; also illustrated is a comparison between the three decision-making models from the supplier’s cost viewpoint. As shown here, the supplier’s expected cost increases in the coordinated model with respect to centralized decision making. The main cause of this increase is the proposal of the credit option to the buyer. Although the supplier’s expected cost in the coordinated model is greater than in the centralized model, it is always less than in the traditional model (see Fig. 6); therefore, the participation of the supplier is assured.

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Centralized Coordinated

Buyer's cost

600 550 500 450 400

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 LTD standard deviation

Fig. 5 Buyer’s expected costs versus degree of uncertainty (comparison between traditional, centralized and coordinated decision making)

400

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Supplier's cost

300

250

200 Traditional 150

Centralized Coordinated

100

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 LTD standard deviation

Fig. 6 Supplier’s expected costs versus degree of uncertainty (comparison between traditional, centralized and coordinated decision making)

The change in the coordinator variable CP versus the degree of uncertainty is illustrated in Fig. 7. The credit period is reduced by increasing uncertainty. In summary, by increasing uncertainty, the expected costs of the supply chain will increase, but the proposed coordination model can reduce the cost as much as possible, and channel coordination is achievable by applying the model.

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100 CP-min

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70 60 50 40 30 20 10 0

1

2

3

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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

LTD standard deviation

Fig. 7 Credit periods with different degrees of uncertainty.

5 Concluding Remarks Supply chain coordination has two main aspects (1) what decisions need to be coordinated, and (2) how to coordinate them. In this chapter, a scheme based on a credit option for the simultaneous coordination of order quantity and reorder point in a two-stage SC under uncertainty was introduced. The presented scheme has the ability to achieve channel coordination. In this scheme, the buyer can use the advantage of delayed payment that is subject to commitment to the globally optimum decisions. Using a credit option, the supplier shares the extra benefits with the buyer. Surplus sharing is fairly distributed based on the members’ bargaining power. This study indicates that the simultaneous coordination of order quantity and reorder point is advantageous and the credit option can be used as a coordination scheme in achieving such advantages under demand uncertainty. The experimental results indicate the model effectiveness (achieving channel coordination) and suitability (dividing the benefits fairly). In addition, the numerical experiments show the advantages of the proposed model with various degrees of demand uncertainty. Although the credit option, or delay in payments, is a well-known mechanism both in financing and operations-management literature, its application under uncertainty has rarely been examined. This study shows the capability of the credit option as a supply chain coordination scheme under demand uncertainty. Compared to the existing literature in the field of SC coordination under uncertainty, this study can be viewed as an opening statement toward implementing the credit option as an SC-coordination scheme under uncertainty which is capable of aligning the objectives of the SC members with the supply chain goal. In comparison with the

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existing studies on the use of credit option in deterministic settings, this study enlarges the application area of credit option toward stochastic environments. The main limitation of the proposed model regards the structure of the investigated SC. The model is limited to a one-buyer, one-supplier configuration. The proposed model can be extended in future work by considering more SC tiers and a networked SC structure. In addition, all developments of the basic creditoption model in the deterministic environment, considering factors such as inflation, partial credit, and deteriorating products, can be investigated in future work.

References Arshinder K, Kanda A, Deshmukh SG (2008) Supply chain coordination: perspectives, empirical studies and research directions. Int J Prod Econ 115:316–335 Cachon GP (2003) Supply chain coordination with contract. In: Graves S, de Kok T (eds) Handbooks in operations research and management science. North-Holland, Amsterdam, pp 229–340 Chaharsooghi SK, Heydari J (2010a) Optimum coverage of uncertainties in a supply chain with an order size constraint. Int J Adv Manuf Technol 47:283–293 Chaharsooghi SK, Heydari J (2010b) Supply chain coordination for the joint determination of order quantity and reorder point using credit option. Eur J Oper Res 204(1):86–95 Chang CT, Wu SJ, Chen LC (2009) Optimal payment time with deteriorating items under inflation and permissible delay in payments. Int J Syst Sci 40(10):985–993 Chauhan SS, Proth JM (2005) Analysis of a supply chain partnership with revenue sharing. Int J Prod Econ 97(1):44–51 Chen LH, Kang FS (2007) Integrated vendor–buyer cooperative inventory models with variant permissible delay in payments. Eur J Oper Res 183:658–673 Chen K, Xiao T (2009) Demand disruption and coordination of the supply chain with a dominant retailer. Eur J Oper Res 197:225–234 Chen H, Chen Y, Chiu CH, Choi TM, Sethi S (2010) Coordination mechanism for the supply chain with lead time consideration and price-dependent demand. Eur J Oper Res 203:70–80 Choi TM, Li D, Yan H (2004) Optimal returns policy for supply chain with e-marketplace. Int J Prod Econ 88:205–227 Chung KJ, Huang CK (2009) An ordering policy with allowable shortage and permissible delay in payments. Appl Math Model 33:2518–2525 Chung KJ, Goyal SK, Huang YF (2005) The optimal inventory policies under permissible delay in payments depending on the ordering quantity. Int J Prod Econ 95:203–213 Ding D, Chen J (2008) Coordinating a three level supply chain with flexible return policies. Omega 36(5):865–876 Giannoccaro I, Pontrandolfo P (2004) Supply chain coordination by revenue sharing contracts. Int J Prod Econ 89:131–139 Gupta A, Maranas CD (2003) Managing demand uncertainty in supply chain planning. Comput Chem Eng 27:1219–1227 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 Hu F, Liu D (2010) Optimal replenishment policy for the EPQ model with permissible delay in payments and allowable shortages. Appl Math Model 34:3108–3117 Huang YF (2007) Economic order quantity under conditionally permissible delay in payments. Eur J Oper Res 176:911–924 Huang CK (2010) An integrated inventory model under conditions of order processing cost reduction and permissible delay in payments. Appl Math Model 34:1352–1359

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Huang CK, Tsai DM, Wu JC, Chung KJ (2010a) An integrated vendor–buyer inventory model with order-processing cost reduction and permissible delay in payments. Eur J Oper Res 202:473–478 Huang CK, Tsai DM, Wu JC, Chung KJ (2010b) An optimal integrated vendor–buyer inventory policy under conditions of order-processing time reduction and permissible delay in payments. Int J Prod Econ 128(1):445–451. doi:10.1016/j.ijpe.2010.08.001 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 Jaber MY, Bonney M, Guiffrida AL (2010) Coordinating a three level supply chain with learningbased continuous improvement. Int J Prod Econ 127(1):27–38 Li J, Liu L (2006) Supply chain coordination with quantity discount policy. Int J Prod Econ 101(1):89–98 Li S, Zhu 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 205:339–345 Moses M, Seshadri S (2000) Policy mechanisms for supply chain coordination. IIE Trans 32(3):245–262 Munson CL, Rosenblatt MJ (2001) Coordinating a three-level supply chain with quantity discounts. IIE Trans 33(5):371–384 Rosenblatt MJ, Lee HL (1985) Improving profitability with quantity discounts under fixed demand. IIE Trans 17:388–395 Sarker BR, Jamal AMM, Wang S (2000) Supply chain models for perishable products under inflation and permissible delay in payment. Comput Oper Res 27(1):59–75 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 182:1469–1478 Sarmah SP, Acharya D, Goyal SK (2008) Coordination of a single-manufacturer/multi-buyer supply chain with credit option. Int J Prod Econ 111:676–685 Shin H, Benton WC (2007) A quantity discount approach to supply chain coordination. Eur J Oper Res 180(2):601–616 Shinn SW, Hwang H (2003) Optimal pricing and ordering policies for retailers under order-sizedependent delay in payments. Comput Oper Res 30:35–50 Silver EA, Pyke DF, Peterson R (1998) Inventory management and production planning and scheduling, 3rd edn. Wiley, New York Teng JT, Chang CT, Goyal SK (2005) Optimal pricing and ordering policy under permissible delay in payments. Int J Prod Econ 97:121–129 Tsai JF (2007) An optimization approach for supply chain management models with quantity discount policy. Eur J Oper Res 177(2):982–994 Van der Rhee B, van der Veen JAA, Venugopal V, Nalla VR (2010) A new revenue sharing mechanism for coordinating multi-echelon supply chains. Oper Res Lett 38(4):296–301 Xiao T, Shi K, Yang D (2010) Coordination of a supply chain with consumer return under demand uncertainty. Int J Prod Econ 124(1):171–180 Xu K, Dong Y, Evers PT (2001) Towards better coordination of the supply chain. Transp Res E 37:35–54 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 187:275–282 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 Zhou YW, Min J, Goyal SK (2008) Supply-chain coordination under an inventory-level-dependent demand rate. Int J Prod Econ 113:518–527

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Supply Chain Coordination Under Consignment Contract with Revenue Sharing Sijie Li, Jia Shu, and Lindu Zhao

Abstract The balance of power between manufacturers and retailers is shifting, and consignment contract with revenue sharing has been widely applied in many industries, especially in on-line marketplaces. In this chapter we consider a supply chain with an upstream manufacturer and a downstream retailer where a singleperiod product is produced and sold. The manufacturer chooses the delivery quantity and the retail price, and the retailer sets the revenue shares. Utilizing Nash bargaining model, a cooperative game model is developed to implement profit sharing between the manufacturer and the retailer to achieve their cooperation. When the manufacturer and the retailer are assumed to be risk-neutral, under a very mild restriction on the demand distribution, the decentralized supply chain can be perfectly coordinated, and both players can earn more in the proposed cooperative setting. Furthermore, the impacts of supply chain system parameters on the optimal supply chain decisions and the supply chain performance are investigated in this chapter. Keywords Consignment contract with revenue game • Supply chain coordination • Uncertainty

sharing

•

Cooperative

S. Li (*) • L. Zhao Institute of Systems and Engineering, Southeast University, Nanjing, Jiangsu, People’s Republic of China e-mail: sjli@seu.edu.cn; ldzhao@seu.edu.cn J. Shu Department of Management Science and Engineering, School of Economics and Management, Southeast University, Nanjing, Jiangsu, People’s Republic of China e-mail: jshu@seu.edu.cn 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_17, # Springer-Verlag Berlin Heidelberg 2011

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1 Introduction In the traditional supply chain and distribution channel, the upstream seller, acting as the leader, has the ownership and price control of goods, and the downstream buyer often acts as the follower. The leader-follower scheme essentially implies that the supply chain member as the leader is dominant in the supply chain and distribution channel. The “dominant” implies that a supply chain member has the power of controlling/influencing another member’s actions and decisions. However, in the current market-oriented economy, the balance of power between manufacturers and retailers is shifting in industries as diverse as pharmaceuticals, consumer packaged goods, hardware, apparel, and furniture (Kumar 1996). Porter (1974) has confirmed the existence of retailer power in consumer goods industries, and defined it as the ability of a retailer to influence a manufacturer’s product differentiation. The supply chain with dominant retailer in the context of marketing and distribution channels has arguably been more pronounced in many academic researches and empirical analyses (e.g., Hua and Li 2008; Lau et al. 2008; Chen and Xiao 2009; Li et al. 2010). Consignment contract with revenue sharing has been widely applied in various industries, such as rental (Dana and Spier 2001; Mortimer 2004), retailing and auction (Wang et al. 2004b; G€ um€ us¸ et al. 2008) and procurement of industrial materials (Cachon 2001; Cachon and Lariviere 2001; Gerchak and Wang 2004). It is essentially a real realization of the power shifting in supply chain channel, and especially popular in on-line marketplaces, such as Amazon.com, Alibaba.com, eBay.com, etc. Under such a contract, ownership of the goods is retained by the supplier and price is also usually determined solely by the supplier. For each item sold, the retailer will deduct an agreed percentage from the selling price, remit the balance to the vendor and no money changes hands until the item is sold (Bolen 1978). Under the consignment contract with revenue sharing, the downstream retailer has more market power and acts as the leader in the supply chain channel, which means that the retailer is dominant over the supplier. In this chapter, we investigate the supply chain decisions and coordination under the consignment contract with revenue sharing. The centralized and decentralized supply chain structures are generally involved in supply chain organizations (Giannoccaro and Pontrandolfo 2004). In the centralized structure, the supply chain operates on the basis of centrally made decisions. In the decentralized structure, each firm makes its own decisions, based on its own knowledge, almost regardless of the rest of the supply chain (Bose and Pekny 2000). The coordination mechanism is an important issue in designing a contract for a decentralized supply chain. Cachon (2003) defined supply chain coordination as follows: A contract is said to coordinate the supply chain if the set of supply chain optimal actions is a Nash equilibrium, i.e., no firm has a profitable unilateral deviation from the set of supply chain optimal actions. This definition is concentrated on the decentralized supply chain since a centralized supply chain is perfectly coordinated. For a decentralized supply chain, if the decisions result in supply chain channel profit

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that is equal to the total profit achieved in a centralized supply chain, the decentralized supply chain is called perfect coordination or channel coordination (Bernstein and Federgruen 2003; Cachon 2004; Wang et al. 2004b). Many researches have addressed the supply chain coordination problem for the consignment contract with revenue sharing. These researches demonstrated that the decentralized supply chain cannot be perfectly coordinated under the consignment contract with revenue sharing. Cachon and Lariviere (2001) analyzed contracts of vendor management inventory (VMI) with revenue sharing, in which the manufacturer facing an uncertain demand offers various contracts to a component supplier. They demonstrated that the decentralized system provides less capacity than the integrated system, which means that the supply chain is not perfectly coordinated. Cachon (2003) studied coordination in a supply chain with one supplier and one retailer, and several different contract types were shown to coordinate the supply chain, such as buy back contracts, revenue sharing contracts, quantity flexibility contracts, sales rebate contracts and quantity discount contracts. They did not consider the consignment contract, and the supply chain is coordinated by an adjusting parameter l, which can loosely be interpreted as the retailer’s share of the supply chain’s profit. Wang et al. (2004b) modeled the decision making of the two firms in a supply chain as a non-cooperative game, and showed that the decentralized channel profit is always lower than the centralized channel profit under the proposed consignment contract. Li and Liu (2008) presented a coordination decision policy, in which a price discount policy is developed to allocate the expected increased profits between two sides of the supply chain; however the decentralized supply chain cannot be perfectly coordinated. Furthermore, many researches also showed that the consignment contract cannot perfectly coordinate the decentralized supply chain (e.g., Dong and Xu 2002; Choi et al. 2004; Cachon 2004). There are also some researches on perfectly coordinating the decentralized supply chain. Cachon (2001) proposed an approach of fixed transfer payments from the supplier to the retailer. Gerchak and Wang (2004) suggested a twoparameter contract (revenue-sharing plus surplus-subsidy) for an assembly system with multiple suppliers and one manufacturer. These coordination mechanisms can be regarded as an implementation of channel profits sharing by a transfer payment. In this chapter, we consider a two-echelon supply chain with an upstream manufacturer and a downstream retailer under the consignment contract with revenue sharing, and two specific demand function forms: additive and multiplicative demand cases (Petruzzi and Dada 1999; Wang et al. 2004b). The manufacturer produces a single-period product at a constant marginal cost. He has only one chance at production before the start of the selling season and sells his products through the retailer. The retailer does not pay the manufacturer upon receipt of the items but shares the sales revenue on units sold. The market demand for the singleperiod product is assumed to be price sensitive and uncertain. In this supply chain setting, the manufacturer chooses the delivery quantity and the retail price to be sold in the market, and the retailer sets the revenue shares.

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Utilizing Nash bargaining model (Nash 1950), we propose the cooperative game models to describe the relationship of payment bargaining between the manufacturer and the retailer and determine the optimal consignment contract with revenue sharing, i.e., the revenue sharing agreement attached with the equilibrium payment scheme. By the fair bargaining between the manufacturer and the retailer instead of adding the additional decision parameters or conditions, we explore the coordination mechanism for the decentralized supply chain under the consignment contract with revenue sharing. Compared with the existing researches on perfect coordination by Cachon (2001) and Gerchak and Wang (2004), the proposed coordination mechanism is more moderate and achievable by a Nash equilibrium contract. Following the measurement of the retailer’s dominance defined by Hua and Li (2008), we also investigate the retailer’s dominance in the two-echelon supply chain under the consignment contract with revenue sharing. Under a very mild restriction on the demand distribution function, we find that the cooperative game between the manufacturer and the retailer has unique Nash equilibrium solution, the manufacturer and the retailer can earn more by their cooperation and the decentralized supply chain is perfectly coordinated. By the theoretical and numerical analyses, we also explore supply chain system parameters (e.g., price-elasticity index, supply chain cost and demand uncertainty) how to affect the supply chain’s optimal decisions and the supply chain performance. This chapter is organized as follows. The assumptions and model are described in Sect. 2. In Sect. 3, the decisions of the centralized supply chain under the consignment contract with revenue sharing are characterized. The cooperative game models to coordinate the decentralized supply chain are provided in Sect. 4, and the impacts of supply chain system parameters on the supply chain performance are also discussed in this section. Detailed numerical results and managerial implications are contained in Sect. 5 and the last section concludes our work.

2 Model Assumptions and Descriptions We consider a two-echelon supply chain consisting of a risk-neutral manufacturer and a risk-neutral retailer, in which a single-period product is produced and sold in market with uncertain and price-sensitive (or price-dependent) demand. The price-sensitive demand is stochastic and can be modeled either in an additive or multiplicative fashion (Petruzzi and Dada 1999; Wang et al. 2004b). The manufacturer produces the single-period product at a constant marginal cost, and the product is sold at a retail price. The retailer incurs a constant cost at the retail stage. In economics and finance, marginal cost is the change in total cost that arises when the quantity produced changes by one unit, and it is typically increasing because of diminishing marginal productivity. Although the increasing marginal costs are common in the practical production deployment (Robert 1997; McAdams and Malone 2005), the constant marginal costs can appear, such as in the stage of

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proportional returns (Rowntree 1941). In literature, it is common that, without loss of generality, the marginal costs for both manufacturer and retailer are typically assumed to be constant (Lariviere and Porteus 2001; Li 2002; Wang et al. 2004b; Hua and Li 2008). Because the costs incurred by the manufacturer and the retailer are constant, the costs incurred by the manufacturer and the retailer can be regarded as a cost sharing agreement.

2.1

Notations and Assumptions

We first introduce the following notations and assumptions to develop the proposed model.

2.1.1

Notations

p D e f() F() h(x) g(x) cM cR c a q r y(p) z Pc(p,q) ðpc ; zc Þ Pd;M Pd;R Pe;M Pe;R

Product’s retail price per unit Price-sensitive demand, i.e., D(p) Random variable with a finite mean value of m and standard deviation value of s Probability density function (PDF) of e ¼ 1 FðÞ Cumulative distribution function (CDF) of e, FðÞ f(x)/[1 F(x)], the failure rate function of demand distribution ¼xh(x), the generalized failure rate function of demand distribution Manufacturer’s marginal production cost Retailer’s marginal cost at the retail stage for inventory handling, shelf-space usage, etc. cM + cR, the total supply chain cost per unit Retailer’s cost share that is incurred at the retail stage, and 1 a is manufacturer’s cost share that is incurred at the production stage Manufacturer’s production quantity for the single-period product Retailer’s revenue share for per unit sold Deterministic and decreasing function of p that captures the dependency between demand and price Stocking factor of inventory Total expected profit of supply chain for any chosen retail price p and production quantity q Supply chain decisions for the centralized supply chain Manufacturer’s expected profit functions in non-cooperative situation Retailer’s expected profit functions in non-cooperative situation Manufacturer’s expected profit functions in cooperative situation Retailer’s expected profit functions in cooperative situation

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ðpd ; zd ; rd Þ ðpe ; ze ; re Þ

Supply chain decisions at non-cooperation Supply chain decisions at cooperation

We add superscript “ values.

2.1.2

*

” to the relative variables to represent their optimal

Assumptions

1. The price-sensitive demand D has the functional forms of D(p) ¼ y(p)e in the multiplicative demand case and D(p) ¼ y(p) + e in the additive demand case. 2. The random variable e is supported on [A, B] with B A 0. 3. F() is strictly increasing, differentiable on [A, B], and F(A) ¼ 0, F(B) ¼ 1 (i.e., there is always some demand in market). 4. y(p) takes the form of y(p) ¼ apb (a > 0, b > 1) in the multiplicative case, and takes the form of y(p) ¼ a bp (a > 0, b > 0) in the additive case. The representations of y(p) are common in the literature, with the former representing an iso-price-elastic demand curve and the later representing a linear demand curve (Lau and Lau 2003; Khouja and Robbins 2005; Hua et al. 2006); see Petruzzi and Dada (1999) for an excellent review and extensions. In the formulation of y(p) ¼ apb, the parameter b is the price-elasticity index of (expected) demand. The larger the value of b, the more sensitive the demand is to a change in price. If the price-elasticity index is greater than 1, then a product is defined as price elastic; if the price-elasticity index is 1 or less, then a product is defined as inelastic. We focus on price-elastic products by assuming b > 1. In the linear demand case, the demand function is quite different from the iso-priceelasticity and multiplicative model. For example, it does not preserve the isoprice-elasticity property. Its price-elasticity index of the (expected) demand is given by bp/(a bp + m). The parameter b here is an indicator of the price “sensitivity” of demand and is closely related to the price-elasticity index. Specifically, the price-elasticity index is increasing in b at any price p. So, one can consider the parameter b as a surrogate of the price-elasticity index. 5. At the end of the selling season, the unsold units bear no salvage value or disposal cost and the unsatisfied demand incurs no loss-of-goodwill cost (i.e., shortage penalty). For the single-period or short life-cycle products, the assumptions of zero salvage value or holding cost and zero loss-of-goodwill cost are appropriate reflections of reality (Goto et al. 2004; Wang et al. 2004a). 6. The demand distribution has the strictly increasing failure rate (IFR) property: h0 (x) > 0, i.e., dh(x)/dx > 0. The IFR assumption is not restrictive because it captures most common distributions, such as the normal, uniform, as well as the gamma and Weibull families, subject to parameter restrictions (Barlow and Proschan 1975). According to Lariviere and Porteus (2001), IGFR (increasing generalized failure rate) is implied by IFR condition (there are IGFR distributions that are not IFR).

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The Model

In a decentralized supply chain, the manufacturer produces the product and then sells it to consumers through the retailer under a consignment contract. The consignment contract with revenue sharing adopted by the firms in the supply chain is the outcome of a bargaining process. The manufacturer and the retailer agree to negotiate to make a contract, and both firms accept the shares of the channel profit dictated by the negotiation process, which is subject to individual rationality. Under the consignment contract with revenue sharing, the retailer offers the manufacturer a revenue sharing contract which stipulates the product sold, he keeps r share of the revenue for per unit sold, and remits the rest, i.e., 1 r, to the manufacturer. In the non-cooperative situation, the two firms follow a Stackelberg (i.e., leaderfollower) game: The retailer, acting as the leader, offers the manufacturer a take-itor-leave-it contract, which specifies the percentage allocation of sales revenue between herself and the manufacturer. The manufacturer, acting as a follower, chooses how many units of the product to produce and the retail price. The manufacturer accepts the contract as long as he can earn a positive profit. According to the research of Wang et al. (2004b), we have the optimal decisions in the non-cooperative situation, which is depicted in Table 1. Firms will have market power as buyers, typically, when they have a dominant position in the market, and they use this power to extract favorable prices, terms and/or conditions from suppliers. There is few definition and measurement on the retailer dominance. Loewenstein et al. (2005) investigated the conversational dominance in negotiation by experimental and statistical method. The conversational dominance is measured by three basic measures: deception (misrepresentation of preferences and capabilities), extreme offers (offers outside the range specified for an issue), and quantitative arguments. Dukes et al. (2006) measured the bargaining power by the parameter in the generalized Nash bargaining model to Table 1 The optimal decisions for decentralized supply chain at non-cooperation Demand Iso-price-elastic Linear Condition F is IGFR F is IFR z

Þ ¼ ðb 1Þ½zd Lðzd Þ Fðz d bzd

ð1 rd Þ½a þ zd Lðzd Þ ð1 aÞbc ð1 rd Þ½a þ zd Lðzd Þ þ ð1 aÞbc

ð1 aÞc a þ zd Lðzd Þ þ 2ð1 rd Þ 2b 2 1 þ Fðzd Þ 1 þ rd aðb 2Þ þ 1 þ 2a ð1 aÞ ð1 aÞ rd ¼ r 1 Fðzd Þ 1 rd ba 2 2ðr aÞ½1 þ Fðzd Þ þ d ¼0 Iðrd ; zd Þ Note: Iðrd ; zd Þ ¼ ð1 rd Þ½1 Fðzd Þ2 ð1 rd Þf ðzd Þ½a þ zd Lðzd Þ ð1 aÞbcf ðzd Þ; a < rd < 1 ð1 aÞbc=ða þ AÞ

p

pd ¼

bczd 1a ðb 1Þ½zd Lðzd Þ 1 rd

Fðzd Þ ¼ pd ¼

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represent the manufacturer’s relative bargaining power. Canie¨ls and Gelderman (2007) observed the supplier dominance in the strategic quadrant of the Kraljic matrix, which is measured as the difference between supplier’s dependence and buyer’s dependence. In this chapter, we assume that the manufacturer and the retailer are devoted to pursue the long-range relationship, their bargaining and deal is repeated many times and both players must consider not only their short-term gains but also their longterm payoffs. Therefore, the expected profits are their suitable objectives in the business operations. The retailer’s dominance in a supply chain is defined as: d rm ¼

PR PM PR

ðif PR PM 0Þ;

(1)

in (1), drm denotes the retailer’s dominance over the manufacturer. PM and PR are the manufacturer’s and the retailer’s expected profits, respectively. The retailer’s dominance does not have a generalized description and definition in the existing researches, such as qualitative analysis and statistical hypothesis testing. The proposed measure of retailer’s dominance provides a clear and closed-form definition to describe the relative power or dominance between the manufacturer and the retailer. We employ cooperative game models in multiplicative and additive demand cases to show how to design an equilibrium contract and how to perfectly coordinate the decentralized supply chain through a cooperative bargaining process. The purpose of cooperation is to actually determine a channel profit allocation scheme between the manufacturer and the retailer. It should be noted that not all optimal profit sharing schemes are acceptable, neither the manufacturer nor the retailer would be willing to accept less profit at cooperation than at noncooperation. An optimal payment scheme ðPe;M ; Pe;R Þ at cooperation is called acceptable if DPe;M ¼ Pe;M Pd;M 0, and DPe;R ¼ Pe;R Pd;R 0, then the acceptable decision set Z can be defined as n o Z ¼ ðp; z; rÞjDPe;M ðp; z; rÞ ¼ Pe;M Pd;M 0; DPe;R ðp; z; rÞ ¼ Pe;R Pd;R 0 : (2) Equation (2) means that the optimal decisions ðpe ; ze ; re Þ belong to a point set, and the set of optimal decisions satisfies the individual rationality, which implies that both players earn more profits by their cooperation in the decentralized supply chain, i.e., DPe;M ðp; z; rÞ ¼ Pe;M Pd;M 0 and DPe;R ðp; z; rÞ ¼ Pe;R Pd;R 0. Suppose the manufacturer’s utility function of DPe;M is u1 ðÞ and the retailer’s utility function of DPe;R is u2 ðÞ. Then according to Nash bargaining model, the optimal bargaining payment scheme is obtained by solving the following problem: max u1 ðDPe;M Þ u2 ðDPe;R Þ:

ðp;z;rÞ2Z

(3)

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Based on the assumption that both the manufacturer and the retailer are riskneutral, the cooperative game model (3) can be formulized as max ðPe;M Pd;M ÞðPe;R Pd;R Þ:

ðp;z;rÞ2Z

(4)

We next study the manufacturer-retailer relationships and coordination decisions for the decentralized supply chain by conducting the proposed cooperative game model. We also investigate the impacts of demand uncertainty and supply chain cost share on the retailer’s dominance.

3 Decisions in Centralized Supply Chain We first characterize the optimal decision to the centralized supply chain, in which the retail price and the production quantity for the product are simultaneously chosen by a central decision-maker to maximize the total expected profit of supply chain, and this optimal decision has to be made before demand realization. For the single-period product problem, we have Pc ðp; qÞ ¼ cq þ pE½minfq; Dg:

3.1

(5)

The Iso-Price-Elastic Demand Case

For the iso-price-elastic demand model, DðpÞ ¼ yðpÞ e and yðpÞ ¼ apb , we then have Pc ðp; qÞ ¼ cq þ pE½minfq; yðpÞ eg:

(6)

Following Petruzzi and Dada (1999), we define z q=yðpÞ. This transformation of variables provides an alternative interpretation of the stocking decision: if z > e, then leftovers occur; if z < e, then shortages occur. Then, the problem of choosing a retail price p and a production quantity q is equivalent to choosing a retail price p and a stocking factor z, and (6) is equivalent to (7) Pc ðp; zÞ ¼ yðpÞfp½z LðzÞ czg; Rz where yðpÞ ¼ apb , and LðzÞ ¼ A ðz xÞf ðxÞdx. Wang et al. (2004b) provided the optimal decision ðpc ; zc Þ for the centralized supply chain with the iso-price-elastic demand: if F is IGFR, the centralized supply chain has unique optimal decision ðpc ; zc Þ, where zc is uniquely determined by Þ bczc c Þ ¼ ðb1Þ½zc Lðz ; pc ¼ ðb1Þ½z Lðz Fðz Þ : c bz c

c

c

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The Linear Demand Case

For the linear demand model, DðpÞ ¼ yðpÞ þ e and yðpÞ ¼ a bp. Since p c, it is assumed that a bc þ A 0 to ensure the non-negative demand when the product is priced at cost, i.e., p ¼ c. By substituting DðpÞ ¼ yðpÞ þ e and yðpÞ ¼ a bp into (5), the expected profit function of the centralized supply chain can be written as Pc ðp; qÞ ¼ cq þ pE½minfq; yðpÞ þ eg:

(8)

As the iso-price-elastic demand model, z q yðpÞ is defined as the stocking factor, then (8) can be rewritten as Pc ðp; zÞ ¼ ðp cÞða bpÞ þ p½z LðzÞ cz:

(9)

The following theorem is about the optimal decision ðpc ; zc Þ to the iso-priceelastic demand case. Theorem 1. For any fixed z 2 ½A; B, if 2h2 ðzÞ þ dhðzÞ=dz > 0, the centralized supply chain has a unique optimal decision ðpc ; zc Þ, where zc is uniquely deteraþz Lðz Þbc

mined by Fðzc Þ ¼ aþzc Lðzc Þþbc , and pc ðzc Þ ¼ c

c

aþzc Lðzc Þþbc . 2b

Proof. For any fixed z 2 ½A; B, it follows from (9) that @Pc ðp; zÞ ¼ a þ bc 2bp þ ½z LðzÞ ¼ 0; @p and @Pc ðp; zÞ ¼ p½1 FðzÞ c ¼ 0: @z By solving the above equations, we can obtain the feasible solution ðpc ; zc Þ aþzc Lðzc Þbc and pc ðzc Þ ¼ bcþaþz2bc Lðzc Þ . satisfying Fðzc Þ ¼ aþz c Lðzc Þþbc c Lðzc Þbc We next prove that zc is the unique solution to Fðzc Þ ¼ aþz aþzc Lðzc Þþbc . Let GðzÞ ¼ ½a þ z LðzÞ½1 FðzÞ bc½1 þ FðzÞ, then we have dGðzÞ 1 FðzÞ ; ¼ f ðzÞ a þ z LðzÞ þ bc dz hðzÞ 0 d2 GðzÞ f ðzÞ dGðzÞ f ðzÞ dhðzÞ 2 ½1 FðzÞ 2h ¼ ðzÞ þ : þ 2 dz2 f ðzÞ dz h ðzÞ dz d2 GðzÞ > 0, then It is obvious that, if 2h2 ðzÞ þ dhðzÞ dz dz2 dGðzÞ=dz¼0 < 0, which implies that GðzÞ itself is a unimodal function.

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Therefore zc is the unique solution since GðzÞ is continuous on ½A; B, GðAÞ ¼ a þ A bc > 0 and GðBÞ ¼ 2bc < 0. It follows from (9) that @ 2 Pc ðp; zÞ ¼ 2b; @p2

@ 2 Pc ðp; zÞ ¼ pf ðzÞ and @z2

@ 2 Pc ðp; zÞ ¼ 1 FðzÞ: @z@p

We then have 2 2 @ 2 Pc ðp; zÞ @ 2 Pc ðp; zÞ @ Pc ðp; zÞ ¼ 2bpf ðzÞ ½1 FðzÞ2 : @p2 @z2 @z@p

(10)

Equation (10) implies that ðpc ; zc Þ is the optimal solution to (9) if 2bpf ðzÞ ½1 FðzÞ2 > 0 at

ðpc ; zc Þ:

(11)

c Þþbc Substituting pc ¼ aþzc Lðz into (11), and we have 2b

ÞfFðz c Þ hðzc Þ½a þ zc Lðzc Þ þ bcg < 0: Fðz c Because GðzÞ is a unimodal function, GðAÞ > 0 and GðBÞ < 0, therefore c Þ hðzc Þ½a þ zc Lðzc Þ þ bc < 0, which means dGðzÞ=dz < 0 at z ¼ zc , and Fðz 2 that (11) can be met if 2h ðzÞ þ dhðzÞ=dz > 0. @ 2 Pc ðp;zÞ So we can have the following principal minors at ðp ; z Þ: < 0, 2 c c @p ðpc ;zc Þ 2

2 2 2 2 @ Pc ðp;zÞ @ Pc ðp;zÞ @ Pc ðp;zÞ @ Pc ðp;zÞ < 0, and > 0, then the Hessian 2 2 2 @z @p @z @z@p ðp ;z Þ c

ðpc ;zc Þ

c

matrix of Pc ðp; zÞ is negative definite at ðpc ; zc Þ Thus, if 2h2 ðzÞ þ dhðzÞ=dz > 0, there is a unique optimal contract ðpc ; zc Þ to the centralized supply chain. □ It is obvious that, if F is IFR, i.e., dhðzÞ=dz > 0, 2h2 ðzÞ þ dhðzÞ=dz > 0 for 8z 2 ½A; B. Theorem 1 indicates that it does not need any requirement on parameters other than the demand distribution itself to determine the optimal consignment contract with revenue sharing for the centralized supply chain. The following proposition describes how the optimal decision ðpc ; zc Þ changes with the supply chain system parameters b and c. Proposition 1. If 2h2 ðzÞ þ dhðzÞ=dz > 0, then (i) zc is decreasing in b, and is also decreasing in c. (ii) pc is decreasing in b, and its monotone property in c depends on b. Proof. (i) According to Theorem 1, we have Gðzc Þ ½a þ zc Lðzc Þ½1 Fðzc Þ bc½1 þ Fðzc Þ ¼ 0

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From the chain rule and differentiation of implicit functions, we have dzc c½1 þ Fðzc Þ ¼ : db @Gðzc Þ=@zc From the proof of Theorem 1, GðzÞ is a unimodal function, GðAÞ > 0 and GðBÞ < 0, and zc is the unique solution to Gðzc Þ ¼ 0, then @Gðzc Þ=@zc < 0. Therefore, if 2h2 ðzÞ þ dhðzÞ=dz > 0, dzc =db < 0, which means that zc is decreasing in b. For the same reason zc is also decreasing in c. (ii) From the chain rule and differentiation of implicit functions, we have dpc @pc @pc dzc ¼ þ : db @b @zc db Since pc ðzc Þ ¼

aþzc Lðzc Þþbc , 2b

we have

@pc a þ zc Lðzc Þ < 0 and ¼ @b 2b2

@pc 1 Fðzc Þ ¼ > 0: @zc 2b

Then dpc =db < 0, which means that pc is decreasing in b. From the chain rule and differentiation of implicit functions, we also have dpc @pc @pc dzc ¼ þ : dc @c @zc dc @p

@p

1Fðz Þ

dz

b½1þFðz Þ

c so It can be derived that @cc ¼ 12 , @zc ¼ 2b c and dcc ¼ @Gðz Þ=@z , c c c dpc Lðzc Þ dc ¼ 2 @Gðzc Þ=@zc , where Lðzc Þ ¼ Fðzc Þ þ bchðzc Þ 1. Because @Gðzc Þ=@zc < 0, the sign of dpc /dc isn determined by Lðz oc Þ.

1þFðzC Þ , Vc LðzC Þ

Since 1 FðzVc Þ ¼ bc aþz

1þFðz Þ

c Lðzc Þ ¼ bc hðzc Þ aþz Lðz Þ , which means c

c

aþz Lðz Þbc

that sign of Lðzc Þ is determined by zc . According to Fðzc Þ ¼ aþzc Lðzc Þþbc , zc is c c uniquely determined by b for any fixed c. □ Therefore, the monotone property of pc in c depends on b. Proposition 1 shows that, the price-elasticity index b affects the optimal stocking factor and the optimal retail price in the linear demand case as it does in the multiplicative demand case, but the effects of the total supply chain cost per unit c in the optimal stocking factor and the optimal retail price are different in the isoprice-elastic and the linear demand cases.

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4 Decentralized Supply Chain Coordination In this section, we investigate the coordination decisions for the decentralized supply chain in multiplicative and additive demand cases.

4.1

The Iso-Price-Elastic Demand Case

For the decentralized supply chain, we define the stocking factor of inventory as z q=yðpÞ. Hence the problem of choosing p, q and r is equivalent to choosing p, z and r. The manufacturer’s and the retailer’s expected profits can be written as Pe;M ðp; z; rÞ ¼ ð1 aÞcq þ ð1 rÞpE½minðq; DÞ

(12)

¼ yðpÞfð1 rÞp½z LðzÞ ð1 aÞczg; Pe;R ðp; z; rÞ ¼ acq þ rpE½minðq; DÞ ¼ yðpÞfrp½z LðzÞ aczg:

(13)

Substituting (12) and (13) into the cooperative game model (4), and we have the following conclusion from its first-order and second-order conditions. Theorem 2. For any fixed z 2 ½A; B, if F is IGFR, the cooperative game model (4) has a unique game equilibrium solution ðpe ; ze ; re Þ, where ze is uniquely determined by Þ ¼ ðb 1Þ½ze Lðze Þ ; Fðz e bze

pe ¼

(14)

bcze ; ðb 1Þ½ze Lðze Þ

(15)

and re ¼ ðb aÞb ½2ðb 1Þa þ 1 þ

ðb 1Þb1 ½ðb 2Þa þ 1 2bðb aÞb

:

(16)

Proof. By solving the first order conditions of cooperative game model (4) in the iso-price-elastic demand case, we have the feasible solutions ðpe ; ze ; re Þ, which are determined by the following equations, e Þ ¼ ðb 1Þ½ze Lðze Þ ; Fðz bze

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pe ¼

bcze ; ðb 1Þ½ze Lðze Þ

and re ¼ ðb aÞb ½2ðb 1Þa þ 1 þ

ðb 1Þb1 ½ðb 2Þa þ 1 2bðb aÞb

:

Following the proof of Theorem 1 in Wang et al. (2004b), there is a unique ze 2 e Þ ¼ ðb1Þ½ze Lðze Þ . Then, the feasible solution ðp ; z ; r Þ is ðA; BÞ satisfying Fðz e e e bze unique. From the second order conditions, the optimality conditions of the cooperative game model (4) are equivalent to (see, Gottfried and Weisman 1973): e Þ2 bcze f ðze Þ < 0; pe ½Fðz D ¼ Pe;M Pd;M ¼ Pe;R Pd;R > 0: From the proof of Theorem 1 in Wang et al. (2004b), 1 bze hðze Þ < 0 if F is IGFR. bcze e Þ Þ ¼ ðb1Þ½ze Lðz and pe ¼ ðb1Þ½z Lðz Since Fðz Þ , we then have pe Fðze Þ ¼ c; e bze e e thus, Þ2 bcz f ðz Þ ¼ cFðz Þ½1 bz hðz Þ < 0: pe ½Fðz e e e e e e Þ2 bcz f ðz Þ < 0 always holds if F is IGFR. It means that pe ½Fðz e e e From the optimal decision ðpd ; zd ; rd Þ and the unique feasible solution ðpe ; ze ; re Þ, we have D ¼ Pe;M Pd;M ¼ Pe;R Pd;R ¼ aczd ðpd Þb

ðb aÞb þ ðb 1Þb1 ½bða 2Þ þ 1 2ðb 1Þbþ1

:

Let gðaÞ ¼ ðb aÞb þ ðb 1Þb1 ½bða 2Þ þ 1, it can be verified that g0 ðaÞ ¼ b½ðb 1Þb1 ðb aÞb1 < 0, and lim gðaÞ ¼ 0. a!1

Because 0 < a < 1 and g0 ðaÞ < 0, so we have gðaÞ > 0. This implies that D > 0, and D ¼ Pe;M Pd;M ¼ Pe;R Pd;R > 0 holds. According to the above proofs, the unique feasible solution ðpe ; ze ; re Þ is optimal to the cooperative game model (4) if F is IGFR. □ From the proof of Theorem 2, the unique equilibrium solution ðpe ; ze ; re Þ always meets the acceptable decision condition Z when the demand distribution F is IGFR.

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This reveals that both the manufacturer and the retailer are better off at cooperation than at non-cooperation. Form the proof procedures of Theorem 2, if d½zhðzÞ=dz ¼ hðzÞ þ zdhðzÞ=dz > 0, the supply chain members can obtain more profits from their cooperation than those from non-cooperation, i.e., Pe;M > Pd;M , Pe;R > Pd;R ; and their incremental profits are equal, i.e., Pe;M Pd;M ¼ Pe;R Pd;R . The following proposition characterizes the optimal decision in the cooperative situation, its relationship to the optimal decision in the non-cooperative situation and to the optimal decision of the centralized supply chain. Proposition 2. In the decentralized supply chain with the iso-price-elastic demand model, (i) ze is independent of re , c, or a, and pe is independent of re or a. (ii) ze ¼ zc ¼ zd . (iii) pe ¼ pc < pd . (iv) re only depends on b and a, and re < rd . Proof. (i) From Theorem 2, we have ze is independent of re , c, or a, and pe is independent of re or a. (ii) Since there is a unique solution z to z bzFðzÞ þ ðb 1ÞLðzÞ ¼ 0, and compar e Þ d Þ c Þ Þ ¼ ðb1Þ½zc Lðz Þ ¼ ðb1Þ½zd Lðz e Þ ¼ ðb1Þ½ze Lðz with Fðz and Fðz , ing Fðz c d bzc bzd bze then ze ¼ zc ¼ zd . bcze bczc (iii) Since pe ¼ ðb1Þ½z Lðz Þ and pc ¼ ðb1Þ½z Lðz Þ , then pe ¼ pc . From Theorem e e c c 1 and Theorem 2 in Wang et al. (2004b), pc < pd . Then, pe ¼ pc < pd . (iv) It is obvious that re only depends on b and a. We have re

rd

" # 2ðb 1Þa þ 1 ðb 1Þb1 aðb 2Þ þ 1 ¼ 1 þ b1 2b ba 2bðb aÞ

0, which implies that b is 2ðbaÞbþ1 b

db ðb 1Þb1 ðb aÞ½aðb 2Þ þ 1 að1 aÞðb 1Þ b1 : ¼ þ log db ðb aÞ½aðb 2Þ þ 1 ba 2ðb aÞbþ1 að1aÞðb1Þ þ log Let lðbÞ ¼ ðbaÞ½aðb2Þþ1

b1 ba , we have

Supply Chain Coordination Under Consignment Contract with Revenue Sharing

l0 ðbÞ ¼

ð1 aÞ2 ðb 1Þðb aÞ2 ½ðb 2Þa þ 12

443

ðbÞ;

where ðbÞ ¼ ½bðb 5Þ þ 5a2 þ 2½bðb 1Þ 1a þ b. For ðbÞ, we have 0 ðbÞ ¼ 1 þ að4b þ 2ba 5a 2Þ > ð3a þ 1Þð1 aÞ > 0 for b > 1, hence, ðbÞ > ð1Þ ¼ a2 2a þ 1 > 0. Therefore we have l0 ðbÞ > 0 for b > 1. Since lim lðbÞ ¼ 0, lðbÞ < 0 for b > 1. b!1 Thus, db=db < 0, which implies that b is decreasing in b. (ii) From (20), we have 1 ðb 1Þb1 1 lim b ¼ þ > ; a!0 2bb 2 2 and 1 ðb 1Þ lim b ¼ þ ¼ 1: 2 2ðb 1Þ

a!1

(iii) From (20), we have b¼

Pe;R 1 ðb 1Þb1 ½ðb 2Þa þ 1 þ ¼ ; Pe;M þ Pe;R 2 2ðb aÞb

then 1 ð1 aÞ ¼ 1; lim b ¼ þ b!1 2 2ð1 aÞ and lim b ¼ 12 þ lim eb log½ba lim b1

b!1

b!1

b!1

n

ðb2Þaþ1 2ðbaÞ

o

ð1aÞ

¼ 12 þ e

a

2

> 12 .

□

Proposition 3 illustrates the behavior of the retailer’s profit share b. It is showed that the retailer can always extract more than 50% of the channel profit even if he does not incur any portion of the channel cost. From (20), the retailer’s and the manufacturer’s profit shares are not affected by demand uncertainty. Following the measurement of the retailer’s dominance defined in (1), at the isoprice-elastic demand case, the retailer’s dominance is as follows: drm ¼

2ðb 1Þb1 ½ðb 2Þa þ 1 ðb aÞb þ ðb 1Þb1 ½ðb 2Þa þ 1

:

(21)

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S. Li et al.

The following Proposition 4 describes the impacts of supply chain parameters b and a on the retailer’s dominance. It indicates that the retailer is always dominant over the manufacturer in the decentralized supply chain, and is the same as the meanings of Proposition 3, i.e., the retailer can always extract more than 50% of channel profit. Proposition 4. With the iso-price-elastic demand model, the retailer’s dominance d rm (i) is increasing in a for any given b > 1, and decreasing in b for any given a. 2 (ii) approaches 1þðb1Þ 1b b > 0 as a ! 0 and 1 as a ! 1. b a

(iii) approaches e 2ae + aea > 0 as b ! 1 and 1 as b ! 1. Proof. (i) From (21), we have ddrm 2ðb 1Þbþ2 ðb aÞb1 ½ðb 2Þa þ b ; ¼ 2 da fðb 1Þðb aÞb þ ðb 1Þb ½ðb 2Þa þ 1g and ddrm 2ðb 1Þbþ1 ðb aÞb ½aðb 2Þ þ 1lðbÞ ; ¼ 2 db fðb 1Þðb aÞb þ ðb 1Þb ½ðb 2Þa þ 1g where lðbÞ ¼

að1 aÞðb 1Þ b1 : þ log ðb aÞ½aðb 2Þ þ 1 ba

Since b > 1 and 0 < a < 1, ðb 2Þa þ b ¼ ðb 1Þa þ ðb aÞ > 0, then ddrm =da > 0. From the proof of Proposition 3, lðbÞ < 0 for 8b > 1, then dd rm =db < 0: (ii) From (21), we have lim d rm ¼

a!0

2ðb 1Þb1 bb

b1

þ ðb 1Þ

¼

2 1 þ ðb 1Þ1b bb

;

and lim d rm ¼ 1:

a!1

(iii) From (21), we have lim drm ¼ 1,

e

b!1 2aea + aea .

and

lim d rm ¼ lim

b!1

b!1 1þ

2 ðbaÞb ðb1Þb1 ½ðb2Þaþ1

¼

2 1þ lim

ðbaÞb

lim

ðb1Þ

b!1 ðb1Þb b!1 ðb2Þaþ1

¼ □

Supply Chain Coordination Under Consignment Contract with Revenue Sharing

445

Proposition 4 implies that the retailer dominance d rm is always greater than zero, that is to say that the retailer is dominant over the manufacturer in the decentralized supply chain under the consignment contract with revenue sharing. From (21), the retailer dominance is not affected by demand uncertainty, and is related to supply chain parameters b and a in the iso-price-elastic demand case.

4.2

The Linear Demand Case

In the linear demand case, the stocking factor of inventory is defined as z ¼ q y(p), then the problem of choosing p, q and r is equivalent to choosing p, z and r. The manufacturer’s and the retailer’s expected profit functions are Pe;M ðp; zÞ ¼ ð1 aÞcq þ ð1 rÞpE½minfq; Dg ¼ ð1 rÞp½a bp þ z LðzÞ ð1 aÞc½a bp þ z; Pe;R ðp; zÞ ¼ acq þ rpE½minfq; Dg ¼ rp½a bp þ z LðzÞ ac½a bp þ z:

(22)

(23)

Substituting (22) and (23) into the cooperative game model (4), and solving the optimal problem of cooperative game model (4), we have the following conclusion. Theorem 3. For any fixed z 2 ½A; B, if F is IFR, the cooperative game model (4) has a unique game equilibrium solution ðpe ; ze ; re Þ, where ze is uniquely determined by Fðze Þ ¼

a þ ze Lðze Þ bc ; a þ ze Lðze Þ þ bc

(24)

pe ðze Þ ¼

a þ ze Lðze Þ þ bc ; 2b

(25)

and Þ ð1 aÞð2r 1Þ Fðz Þ 2a 1 r a z z þ Lðz Þ Lðz Þ 1 Fðz e d e d e d d d : þ re ¼ þ Þ bc 2 Fðze Þ 2ð1 rd Þ 4 1 rd Fðz d (26) Proof. By solving the first order conditions of cooperative game model (4) in the linear demand case, we can have the feasible solutions, which are determined by the following equations:

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Fðze Þ ¼

a þ ze Lðze Þ bc ; a þ ze Lðze Þ þ bc

(27)

pe ðze Þ ¼

a þ ze Lðze Þ þ bc ; 2b

(28)

and e Þ ð1 aÞð2r 1Þ Fðz Þ 2a 1 r a z ze þ Lðz Þ Lðze Þ 1 Fðz d d d d d re ¼ þ : Þ þ 4 2 Fðze Þ 2ð1 rd Þ 1 rd bc Fðz d (29) According to Theorem 1, if 2h2 ðzÞ þ dhðzÞ=dz > 0, there is a unique optimal e Lðze Þbc solution ze 2 ðA; BÞ satisfying Fðze Þ ¼ aþz aþze Lðze Þþbc . Because zd and rd are unique, (27)–(29) determine a unique feasible solution ðpe ; ze ; re Þ if dhðzÞ=dz > 0. It is known that, if the Hessian matrix of cooperative game model is a negative definite matrix at ðpe ; ze ; re Þ, ðpe ; ze ; re Þ is the optimal solution to the cooperative game model (4). The Hessian matrix is a negative definite matrix at ðpe ; ze ; re Þ if the following two conditions are satisfied (see, Gottfried and Weisman 1973): ½1 Fðze Þ2 2bpe f ðze Þ < 0;

(30)

Pe;M Pd;M ¼ Pe;R Pd;R > 0:

(31)

Let Rðpe ; ze Þ ¼ 2bpe f ðze Þ ½1 Fðze Þ2 , (28) gives 1 Fðze Þ : Rðpe ; ze Þ ¼ f ðze Þ a þ ze Lðze Þ þ bc hðze Þ

(32)

From the proof of Theorem 1, we have GðzÞ ¼ ½a þ z LðzÞ½1 FðzÞ bc ½1 þ FðzÞ is a unimodal function if 2h2 ðzÞ þ dhðzÞ=dz > 0. dGðzÞ Sincen GðAÞ ¼ a þ A bc > 0 and GðBÞ ¼ 2bc < 0, o dz < 0, i.e., f ðze Þ a þ ze Lðze Þ þ bc

1Fðze Þ hðze Þ

z¼ze

< 0.

Therefore, Rðpe ; ze Þ > 0, and (30) holds if 2h2 ðzÞ þ dhðzÞ=dz > 0. From the first order optimality conditions, it can be derived that Pe;M Pd;M ¼ Pe;R Pd;R ¼

ðPe;M þ Pe;R Þ ðPd;M þ Pd;R Þ : 2

From Table 1, we have 1a ½a þ zd Lðzd Þ½1 Fðzd Þ gbc½1 þ Fðzd Þ ¼ 0, where g ¼ 1r > 1. d

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447

Let Yðzd Þ ¼ ½a þ zd Lðzd Þ½1 Fðzd Þ gbc½1 þ Fðzd Þ; then dYðzd Þ 1 Fðzd Þ ; ¼ f ðz Þ a þ z Lðz Þ þ gbc d d d dzd hðzd Þ and 0 d2 Yðzd Þ f ðzd Þ dYðzd Þ f ðzd Þ dhðzd Þ 2 ½1 Fðz : ¼ þ Þ 2h ðz Þ þ d d f ðzd Þ dzd h2 ðzd Þ dzd dzd 2 If 2h2 ðzÞ þ dhðzÞ dz > 0, then d2 Yðzd Þ f ðz Þ ¼ h2 ðzd Þ ½1 Fðzd Þ½2h2 ðzd Þ þ dhðzd Þ=dzd < 0, dz 2 d

dYðzd Þ=dzd ¼0

which

d

implies that Yðzd Þ is a unimodal function. 1a and g ¼ 1r Recall that a < rd < 1 ð1aÞbc , so we have YðAÞ ¼ a þ A aþA d

gbc > 0. Since Yðzd Þ is continuous on ½A; B, and YðBÞ ¼ 2gbc < 0, thus zd is the unique solution to Yðzd Þ ¼ 0 and @Yðzd Þ=@zd zd if g > 1. It can be directly observed that Pc , Pd;M þ Pd;R and Pe;M þ Pe;R have the same function form. According to Theorem 1, Pc achieves its maximum if and only if ðzc ; pc Þ ¼ ðzc ; pc Þ. It can also be observed from their expressions that ðze ; pe Þ ¼ ðzc ; pc Þ, thus ðze ; pe Þ is the unique optimal solution to Pe;M þ Pe;R . Because ze > zd , so we have Pe;M þ Pe;R > Pd;M þ Pd;R , and (31) holds. Therefore, the optimal solution ðpe ; ze ; re Þ determined by (24)–(26) is the Nash equilibrium solution to the cooperative game model (4) if dhðzÞ=dz > 0. □ In Theorem 3, zd and rd are the optimal stocking factor of inventory and the optimal revenue share of the decentralized supply chain in the linear demand case respectively, see Table 1. From the proof of Theorem 3, the unique equilibrium solution ðpe ; ze ; re Þ always meets the acceptable decision condition Z if the demand distribution F is IFR. This also reveals that both the manufacturer and the retailer are better off at cooperation than at non-cooperation in the linear demand case, i.e., Pe;M > Pd;M and Pe;R > Pd;R , which is the same as that in the iso-price-elastic demand case. Since the expressions of pe and ze are the same as those of pc and zc respectively, we have the following properties about pe and ze according to Proposition 1: if F is IFR, then ze is decreasing in b , and is also decreasing in c; pe is decreasing in b, and may be increasing, constant or decreasing in c. The following Proposition 5 characterizes the characters of the optimal decision in the cooperative situation, its relationship to the optimal decision in the noncooperative situation and to the optimal decision of the centralized supply chain.

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Proposition 5. In the decentralized supply chain with the linear demand model, (i) ze and pe do not depend on re or a, but depends on b and c; re depends on b, c and a. (ii) pe ¼ pc . (iii) ze ¼ zc > zd . Proof. (i)–(ii) They can be easily verified from the expressions of ðpd ; zd ; rd Þ, ðpe ; ze ; re Þ and ðpc ; zc Þ. (iii) The uniqueness of ze and zc gives ze ¼ zc . According to the proof of Theorem 3, we have ze > zd . Therefore ze ¼ zc > zd . □ It follows from Parts (ii) and (iii) of Proposition 5 that ðpe ; ze Þ ¼ ðpc ; zc Þ, which means that the decentralized supply chain with the linear demand model is perfectly coordinated under the consignment contract with revenue sharing ðpe ; ze ; re Þ. Since it is difficult to provide the closed forms of the manufacturer’s and the retailer’s expected profits, we cannot describe the impacts of parameters b, c and demand uncertainty s on supply chain decisions, b and drm analytically. We perform numerical studies to investigate the impacts.

5 Numerical Analysis To better illustrate the ideas in this chapter, we develop a numerical sample for the linear demand case. In the numerical studies, we assume that e ~ N(10,5), a ¼ 100, c ¼ 1.0. Given a ¼ 0.25, the optimal solutions can be obtained by applying Theorem 3 at a given b. The manufacturer’s and the retailer’s expected profits can then be computed numerically as well as the retailer’s profit share and dominance. When b varies from 1 to 80, the computational results are reported in Table 2. As shown in Propositions 3 and 4, we let a ¼ 0 and a ¼ 0.999 to analyze the behaviors of the retailer’s profit share b and dominance drm as a ! 0 and a ! 1, respectively. The computational results are reported in the following Tables 3 and 4. Results in Tables 2–4 show that, in the linear demand case, the revenue share r, the retail price p and the production quantity q is decreasing in b for any given a. Furthermore, the retailer’s profit share b and dominance drm are also decreasing in b for any given a, these observations are similar to Propositions 3 and 4 in the isoprice-elastic demand case. From Propositions 3 and 4, the retailer can always extract more than 50% of the channel profit in the iso-price-elastic demand case; however, in the linear demand case, the manufacturer can earn more than the retailers. For example, as shown in Table 3, when b > 36.07 (it corresponds to the italic and bold face), the retailer’s profit share is less than 0.5 and its dominance is negative. Table 4 shows that the retailer’s profit share and dominance approach 1 as a ! 1, which is similar to the

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449

Table 2 The optimal solutions in linear demand case, e ~ N(10,5), a ¼ 100, c ¼ 1.0, a ¼ 0.25 b

zd

1 13.518 5 11.247 10 10.057 15 9.255 20 8.616 25 8.064 30 7.566 40 6.656 50 5.792 80 2.804

rd pd 0.950 62.325 0.863 13.621 0.790 7.198 0.731 4.991 0.681 3.863 0.637 3.175 0.597 2.710 0.527 2.119 0.466 1.757 0.321 1.195

ze 20.491 16.804 14.835 13.520 12.490 11.618 10.849 9.491 8.262 4.402

re pe qe Pe;M Pe;R b drm 0.827 55.738 64.753 479.028 2,507.213 0.840 0.809 0.705 11.522 59.194 134.528 412.508 0.754 0.674 0.627 5.997 54.865 70.754 174.030 0.711 0.593 0.572 4.155 51.195 45.651 99.750 0.686 0.542 0.530 3.233 47.830 31.959 64.644 0.669 0.506 0.496 2.680 44.618 23.333 44.656 0.657 0.477 0.466 2.312 41.503 17.441 32.002 0.647 0.455 0.417 1.850 35.492 10.054 17.353 0.633 0.421 0.377 1.572 29.638 5.800 9.572 0.623 0.394 0.290 1.151 12.295 0.681 0.963 0.586 0.293

Table 3 The optimal solutions in linear demand case, e ~ N(10,5), a ¼ 100, c ¼ 1.0, a ¼ 0 b

zd

rd

pd

ze

re

pe

Pe;M

qe

Pe;R

b

drm

1.00 13.173 0.940 63.061 20.491 0.740 55.738 64.753 729.604 2,256.637 0.756 0.677 5.00 10.867 0.833 13.857 16.804 0.582 11.522 59.192 194.331 352.705 0.645 0.449 10.00 9.664 0.741 7.337 14.835 0.482 5.997 54.867 100.420 144.365 0.590 0.304 15.00 8.858 0.667 5.090 13.520 0.413 4.155 51.200 64.203 81.198 0.558 0.209 20.00 8.217 0.603 3.940 12.490 0.360 3.233 47.822 44.675 51.928 0.538 0.140 25.00 7.667 0.545 3.237 11.618 0.315 2.680 44.610 32.470 35.519 0.522 0.086 30.00 7.172 0.493 2.760 10.849 0.278 2.312 41.503 24.183 25.260 0.511 0.043 36.07 6.616 0.434 2.354 10.003 0.238 2.001 37.830 17.205 17.205 0.500 0.000 40.00 6.272 0.399 2.154 9.491 0.215 1.850 35.492 13.863 13.544 0.494 0.024 50.00 5.421 0.316 1.781 8.262 0.165 1.572 29.638 7.965 7.407 0.482 0.075 80.00 2.501 0.107 1.200 4.402 0.051 1.151 12.295 0.933 0.712 0.433 0.309

Table 4 The optimal solutions in linear demand case, e ~ N(10,5), a ¼ 100, c ¼ 1.0, a ¼ 0.999 b

zd

rd

pd

ze

re

pe

qe

Pe;M

Pe;R

b

d rm

15.00 20.00 25.00 30.00 40.00 50.00 80.00

13.459 12.449 11.590 10.827 9.478 8.253 4.398

0.999 0.999 0.999 0.999 0.999 0.999 0.999

4.163 3.238 2.684 2.314 1.851 1.573 1.151

13.520 12.490 11.618 10.849 9.491 8.262 4.402

0.999 0.999 0.999 0.999 0.999 0.999 0.999

4.155 3.233 2.680 2.312 1.850 1.572 1.151

51.200 47.822 44.610 41.503 35.492 29.638 12.295

0.048 0.052 0.044 0.036 0.022 0.013 0.001

145.353 96.551 67.945 49.407 27.385 15.359 1.644

1.00 1.00 1.00 1.00 1.00 1.00 1.00

1.00 1.00 1.00 1.00 1.00 1.00 1.00

results in Propositions 3 and 4. One issue should be illuminated in Table 4, when b is small (e.g., b 10), the manufacturer’s profit is negative at cooperation, and then the manufacturer and the retailer will not bargain to cooperate. For more detailed illustrations of impacts of price sensitivity b on the retailer’s profit share and dominance, we give another example that e ~ N(10,5), a ¼ 100, c ¼ 1.0 and a ¼ 0.5. The optimal solutions of the supply chain and the expected profits of the manufacturer and the retailer are then computed numerically as well as

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Table 5 The optimal solutions in linear demand case, e ~ N(10,5), a ¼ 100, c ¼ 1.0, a ¼ 0.5 b

zd

10.00 11.161 20.00 8.804 30.00 7.060 40.00 5.527 50.00 4.046 60.00 2.506

rd 0.838 0.760 0.702 0.656 0.617 0.583

pd 6.994 3.731 2.611 2.039 1.690 1.454

ze 17.809 14.036 11.381 9.167 7.140 5.137

re 0.760 0.690 0.643 0.608 0.580 0.557

pe 6.079 3.258 2.318 1.847 1.563 1.373

qe 57.015 48.878 41.852 35.293 28.972 22.761

Pe;M Pe;R 45.662 206.125 21.219 77.024 11.777 37.986 6.879 20.388 4.012 11.052 2.240 5.720

b 0.819 0.784 0.763 0.748 0.734 0.719

drm 0.778 0.725 0.690 0.663 0.637 0.608

Table 6 The optimal solutions in linear demand case, e ~ N(10,5), a ¼ 100, c ¼ 1.0, b ¼ 10.0 a

zd

0.01 0.05 0.10 0.15 0.20 0.25 0.50 0.75 0.99

9.678 9.735 9.810 9.888 9.970 10.057 10.586 11.424 14.112

rd 0.743 0.751 0.760 0.770 0.780 0.790 0.843 0.905 0.992

pd 7.332 7.312 7.285 7.258 7.229 7.198 7.019 6.758 6.118

ze 14.835 14.835 14.835 14.835 14.835 14.835 14.835 14.835 14.835

re 0.488 0.511 0.541 0.570 0.598 0.627 0.763 0.890 0.998

pe 5.997 5.997 5.997 5.997 5.997 5.997 5.997 5.997 5.997

qe 54.8650 54.8650 54.8650 54.8650 54.8650 54.8650 54.8650 54.8650 54.8650

Pe;M 99.192 94.315 88.293 82.357 76.510 70.754 43.491 19.222 0.168

Pe;R 145.592 150.469 156.491 162.427 168.275 174.030 201.293 225.562 244.616

b 0.595 0.615 0.639 0.664 0.687 0.711 0.822 0.921 0.999

drm 0.319 0.373 0.436 0.493 0.545 0.593 0.784 0.915 0.999

the retailer’s profit share and dominance. The computational results are reported in Table 5. Table 5 shows that the impacts of price sensitivity b on the manufacturer’s and the retailer’s expected profits, the retailer’s profit share and dominance are the same to those shown in Table 2. For analyzing the impacts of parameter a on the supply chain decisions, the retailer’s profit share and the retailer’s dominance, we next set b ¼ 10.0. When a varies from 0.01 to 0.99, the computational results are listed in Table 6. From Table 6, the revenue share r is increasing in a, however, for any given b > 0, the retailer’s cost share a has no impact on the production quantity q and the retail price p, i.e., the production quantity q and the retail price p are constant in a. As shown in Fig. 1, the retailer’s profit share b and dominance drm are increasing in a for any given b > 0. These observations are similar to Propositions 3 and 4 in the iso-price- elastic demand case. It can also be found that the retailer’s dominance is more sensitive in the price-elasticity index and the retailer’s cost share. To analyze the effects of demand uncertainty on the supply chain decisions and performance, we let b ¼ 10.0, a ¼ 0.25 and s varies from 5 to 50. Applying Theorem 3, we obtain the optimal solutions, the expected profits of supply chain members, the retailer’s profit share and dominance for any given demand uncertainty level s. Table 7 shows the computational results when the demand uncertainty varies.

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1.20 1.00 0.80 0.60 0.40 0.20 0.00

0.01

0.05

0.10

0.15

0.20

Retailer's profit share

0.25

0.50

0.75

0.99

Retailer's dominance

Fig. 1 The impacts of cost share on retailer’s profit share and dominance

Table 7 The optimal solutions in linear demand case, e ~ N(10,s), a ¼ 100, c ¼ 1.0, b ¼ 10.0, a ¼ 0.25 s

zd

5 10 15 20 25 30 35 40 45 50

10.057 10.563 11.478 12.749 14.335 16.201 18.317 20.658 23.203 25.934

rd 0.790 0.780 0.772 0.765 0.758 0.752 0.747 0.743 0.739 0.736

pd 7.198 7.146 7.139 7.153 7.187 7.238 7.305 7.386 7.480 7.586

ze 14.835 19.845 25.102 30.581 36.272 42.166 48.258 54.541 61.008 67.654

re 0.627 0.610 0.596 0.583 0.571 0.561 0.551 0.542 0.534 0.527

pe 5.997 6.156 6.369 6.591 6.819 7.052 7.290 7.533 7.782 8.036

qe 54.8650 58.2850 61.4120 64.6710 68.0820 71.6460 75.3580 79.2110 83.1880 87.2940

Pe;M 70.754 80.113 92.161 105.197 119.070 133.795 149.410 165.954 183.463 201.972

Pe;R 174.030 179.027 188.322 198.603 209.582 221.281 233.749 247.031 261.162 276.177

b 0.711 0.691 0.671 0.654 0.638 0.623 0.610 0.598 0.587 0.578

drm 0.593 0.553 0.511 0.470 0.432 0.395 0.361 0.328 0.298 0.269

Results in Table 7 show that, in the linear demand case, the production quantity and the retail price will increase in the demand uncertainty, and the retailer’s and the manufacturer’s expected profits is also increasing. These mean that the more demand uncertainty will lead to high expected profits for the manufacturer and the retailer because of the high risk in the market. As shown in Fig. 2, the retailer’s profit share b and dominance drm are decreasing in s for any given a > 0 and b > 0, which indicate that higher demand uncertainty will cut the retailer’s profit share and dominance in the decentralized supply chain. It is also can be found that the retailer’s dominance is more sensitive than the retailer’s profit share in the demand uncertainty. Comparing the optimal decisions at cooperation with those at non-cooperation in the above numerical analyses, it is found that the supply chain channel will supply more product quantity with lower retailer price in the market at cooperation. This means that the supply chain members can earn more profit by their cooperation and the consumers also obtain benefits from the supply chain cooperation.

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5

10

15

20

25

Retailer's profit share

30

35

40

45

50

Retailer's dominance

Fig. 2 The impacts of demand uncertainty on retailer’s profit share and dominance

6 Conclusion In this chapter, we investigate the coordination of a decentralized supply chain consisting of an upstream manufacturer and a downstream retailer, in which a single-period product is produced and sold in the retail market. The bargaining process between the supply chain members is governed by a consignment contract with revenue sharing. We present a cooperative model utilizing Nash bargaining model with two demand cases: the iso-price-elastic demand and the linear demand. In the proposed model, the retailer is the leader and the manufacturer is the follower in the non-cooperative game, and the two players are willing to cooperate if their profits at cooperation are no less than those at non-cooperation. They bargain and try to make an equilibrium agreement on the retail price, production quantity and revenue share. The research shows that the cooperative game between the manufacturer and the retailer has a unique equilibrium solution under a very mild restriction on the demand distribution. Under the equilibrium contract, the decentralized supply chain can be perfectly coordinated and both the manufacturer and the retailer are better off. It is found that the retailer’s profit share and dominance is increasing in retailer’s cost share for any given price elasticity, decreasing in price elasticity for any given retailer’s cost share. The retailer can always extract more than 50% of the channel profit even if he does not incur any portion of the channel cost in the multiplicative demand case; however, the manufacturer can earn more than the retailer by their cooperation in the linear demand case. Acknowledgements The research is supported by National Natural Science Foundation of China (No.71001024; No.70801014), Program for New Century Excellent Talents in University of China (NCET-09-0292) and PhD Programs Foundation of Ministry of Education of China (No.20100092120042).

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References Barlow RE, Proschan F (1975) Statistical theory of reliability and life testing probability models. Holt, Rinehart, and Winston, New York Bernstein F, Federgruen A (2003) Pricing and replenishment strategies in a distribution system with competing retailers. Oper Res 51(3):409–426 Bolen WH (1978) Contemporary retailing. Prentice-Hall, Englewood Cliffs, NJ Bose S, Pekny JF (2000) A model predictive framework for planning and scheduling problems: a case study of consumer goods supply chain. Comput Chem Eng 24:329–335 Cachon GP (2001) Stock wars: inventory competition in a two echelon supply chain. Oper Res 49(5):658–674 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 (2004) The allocation of inventory risk in supply chain: push, pull, and advancepurchase discount contracts. Manage Sci 50(2):222–238 Cachon GP, Lariviere MA (2001) Contracting to assure supply: how to share demand forecasts in a supply chain. Manage Sci 47(5):629–646 Canie¨ls MCJ, Gelderman CJ (2007) Power and interdependence in buyer supplier relationships: a purchasing portfolio approach. Ind Mark Manage 36:219–229 Chen KB, Xiao TJ (2009) Demand disruption and coordination of the supply chain with a dominant retailer. Eur J Oper Res 197(1):225–234 Choi KS, Dai JG, Song JS (2004) On measuring supplier performance under vendor-managedinventory programs in capacitated supply chains. Manuf Serv Oper Manage 6(1):53–72 Dana JD, Spier KE (2001) Revenue sharing and vertical control in the video rental industry. J Ind Econ 49(3):223–245 Dong Y, Xu KF (2002) A supply chain model of vendor managed inventory. Transp Res E 38(2):75–95 Dukes AJ, Gal-Or E, Srinivasan K (2006) Channel bargaining with retailer asymmetry. J Mark Res 43:84–97 Gerchak Y, Wang Y (2004) Revenue-sharing vs. wholesale-price contracts in assembly systems with random demand. Prod Oper Manage 13(1):23–33 Giannoccaro I, Pontrandolfo P (2004) Supply chain coordination by revenue sharing contracts. Int J Prod Econ 89(2):131–139 Goto JH, Lewis ME, Puterman ML (2004) Coffee, tea, or . . .?: a Markov decision process model for airline meal provisioning. Transp Sci 38(1):107–118 Gottfried BS, Weisman J (1973) Introduction to optimization theory. Prentice-Hall, Englewood Cliffs, NJ G€ um€us¸ M, Jewkes EM, Bookbinder JH (2008) Impact of consignment inventory and vendor managed inventory for a two-party supply chain. Int J Prod Econ 113(2):502–517 Hua ZS, Li SJ (2008) Impacts of demand uncertainty on retailer’s dominance and manufacturerretailer supply chain cooperation. Omega Int J Manage Sci 36(5):697–714 Hua ZS, Li SJ, Liang L (2006) Impact of demand uncertainty on supply chain cooperation of single-period products. Int J Prod Econ 100(2):268–284 Khouja M, Robbins SS (2005) Optimal pricing and quantity of products with two offerings. Eur J Oper Res 163(3):530–544 Kumar N (1996) The power of trust in manufacturer-retailer relationships. Harv Bus Rev 74(6):92–109 Lariviere MA, Porteus EL (2001) Selling to the newsvendor: an analysis of price-only contract. Manuf Serv Oper Manage 3(4):293–305 Lau AHL, Lau HS (2003) Effects of a demand-curves shape on the optimal solutions of a multiechelon inventory/pricing model. Eur J Oper Res 147(3):530–548

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Part IV

Technological Advancements and Applications in Supply Chain Coordination

.

DEAL: A Heuristic Approach for Collaborative Planning in Detailed Scheduling J. Benedikt Scheckenbach

Abstract Aroused by ongoing globalization, the division of labor steadily increases and future competition will likely take place between whole supply chains and not only between single companies. This compels suppliers and manufacturers to better coordinate their production plans in order to save production and holding costs. Despite this necessity, the willingness to share sensitive production data such as cost factors or resource availability has remained very limited. Usually the companies consider these data as vital to their business. However, today’s production planning models and solvers used in industry are of monolithic type and require full visibility of data in order to compute a solution. Hence, they cannot be used to tackle supply-chain-wide planning problems involving different companies. In recent years, “Collaborative Planning” as a joint decision making process under information asymmetry has received increased attention. The majority of present research in this field breaks with the monolithic approach but assumes that planning problems can be solved to optimality. On the contrary, industry struggles in operational business with large detailed scheduling problems that are only solvable by heuristics – violating this fundamental assumption. However, being computed only shortly before execution, badly aligned detailed schedules most obviously demand a coordinated solution. We propose a decentralized evolutionary algorithm (DEAL) for coordinating large-sized detailed schedules that does not demand the exchange of sensitive data but only transmits delivery dates and ordinal rankings. Experimental results prove that DEAL computes in the same time solutions of similar quality as monolithic heuristics are able to.

This work originated from a joint research project between the University of Hamburg, Institute for Logistics and Transport and the SAP AG Walldorf, Germany. Today, the author works as Supply Chain Consultant at Bayer Technology Services GmbH, Germany. J.B. Scheckenbach (*) Cranachstr. 16, 50733 Koeln, Germany e-mail: benedikt.scheckenbach@gmail.com 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_18, # Springer-Verlag Berlin Heidelberg 2011

457

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Keywords Collaborative planning • Detailed scheduling • Evolutionary algorithm • SAP APO

1 Introduction In the last decades, advanced planning systems (APS) have been put forward by industry and academia to solve supply chain (SC) planning problems. To cope with the complexity of the planning problems, APS are typically organized in a hierarchical manner. That is, on a mid-term level so-called master planning centrally calculates rough cut plans, generating enterprise-wide material quantity targets for subsequent production planning and detailed scheduling, which are then applied for each production site separately. Production planning is concerned with translating material quantity targets into concrete production activities, whereas detailed scheduling is about computing a good sequence and a feasible resource assignment for the activities. As detailed schedules per plant have to respect superordinated rough cut plans on enterprise level, APS inherently employ a hierarchical form of coordination. The problem with the above approach is that it is not applicable to companies that are interdependent on delivered items but legally separate organizations. A superordinated decision-making process fails because of the missing central entity and the reluctance of SC members to disclose sensitive data, such as production and holding cost or resource availability. As a remedy, successive planning strategies are employed today, a typical example is upstream planning. In upstream planning, the more powerful downstream entity calculates its production plan first, generating demand for subsequent upstream entities. Most of today’s “collaboration” systems, such as vendor-managed inventory, are actually implementations of successive planning strategies. From a mathematical perspective the interorganizational optimization problem is split into separate interdependent subproblems. The subproblems are then solved sequentially, whereas already-solved subproblems define the constraints for yet unsolved subproblems. For example, suppose a manufacturer setting up his production plan according to current demand and forecasts. By doing this, the manufacturer creates the demand for his suppliers that have to plan their production accordingly. However, successive planning pays no respect to the suppliers’ actual production capacities. Hence, successive planning strategies lead to suboptimal results, which often will result in wasted production capacity, larger safety stocks, decreased service levels or increased production costs. In recent years, academia has put forward advanced coordination mechanisms that try to attain the results of hierarchical coordination (or centralized planning) without requiring the parties to exchange sensitive data. Subsuming the different approaches, the term Collaborative Planning (CP) has emerged as a new research area. From a practical perspective, CP mechanisms have to cope with three difficulties: First, they should support complex operational planning problems. Second, the exchange of sensitive data must be avoided. Third, the mechanisms are required to be incentive compatible to prevent opportunistically acting partners

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from supplying systematically biased input that changes overall results to their advantage. It is very difficult to consider all three requirements to the fullest. This, and the still-prevailing paradigm of successive planning have hampered the introduction of CP mechanisms in practice. This work presents a decentralized evolutionary algorithm (DEAL) for aligning detailed schedules of several suppliers delivering to one manufacturer. In such a setting, production processes of the manufacturer cannot start until the suppliers have provided the necessary items. Being computed only shortly before execution, badly aligned detailed schedules most obviously demand a coordinated solution. Restricting to the limited scope of detailed scheduling allows to fulfill the above requirements to a degree acceptable for practical implementations. This work is organized as follows. Section 2 provides a brief summary of stateof-the-art CP approaches. In Sect. 3, the term “detailed scheduling” is further clarified by presenting the optimization problem underlying our further discussion. Section 4 introduces evolutionary algorithms. Section 5 presents the proprietary SAP metaheuristic to solve detailed scheduling problems – the so-called Production Planning and Detailed Scheduling (PP/DS) Optimizer. Building on this background, DEAL is presented in Section 6. Finally, we discuss experimental key results in Sect. 7 followed by concluding remarks in Sect. 8.

2 Literature Review on Collaborative Planning According to Stadtler (2007) Collaborative Planning (CP) can be defined as: . . . a joint decision making process for aligning plans of individual SC members with the aim of achieving coordination in light of information asymmetry.

Information asymmetry is a term stemming from game theory and describes the state where no SC member possesses all the information or preferences of other SC members. There exist several definitions for coordination, depending on the exact field of research. Roughly, literature can be divided into analytical and operational models. Analytical models employ very restrictive assumptions that allow a rigorous mathematical treatment. Here, coordination is often defined as optimal solution and Nash equilibrium. In contrast, the complexity of operational models might not even allow to numerically compute the optimal solution in reasonable time. Here, many authors speak of coordination, if the initial situation could be at least improved. The next two subsections provide a brief overview of the two approaches.

2.1

Analytical Models

Regarding analytical models, a large body of literature is concerned with multiechelon systems. Such a system includes several stages (e.g., representing supplier and manufacturer sites) to be coordinated.

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Static multi-echelon systems rely on restrictive assumptions, such as static demand in a single-item, single-production-stage scenario and aim at finding the optimal trade-off between the holding and fixed-order costs. The difficulty lies in the assumption that different costs accrue for each echelon. For example, large orders might be beneficial to suppliers, because of potential savings in order processing costs. However, a manufacturer might prefer more frequent, smaller orders due to high inventory cost (cf. Bhatnagar and Chandra 1993). If the manufacturer is the more powerful party, the supplier’s problem is to persuade the manufacturer to change his order policy by making concessions such as price discounts or compensation payments. Reviews of contributions assuming complete information are provided by Goyal and Gupta (1989) and Thomas and Salhi (1998). Papers that explicitly deal with incomplete information include Sucky (2006), Fransoo et al. (2001), and Corbett and de Groote (2000). Stochastic multi-echelon systems assume a stochastic demand distribution, generally with negligible fixed-order costs; and are concerned with retrieving the optimal safety stock for a given service level. For a single stage facing stochastic demand, the optimal safety stock is a quantile of the demand distribution. Research dates back to 1960, with the paper of Clark and Scarf (1960) who derive an exact algorithm for a serial multi-echelon system. Diks et al. (1996) and van Houtum et al. (1996) give an overview about relevant literature since then. Also contract theory focuses on the analytical investigation of the relationship between a supplier and a buyer of a good. Most of the literature in contract theory is about determining optimal contract parameters given the functional form of a contract, cf. Cachon (2003). As a main result, multi-echelon systems and contract theory provide the insight that lower system-wide costs can be achieved if some SC members provide compensation payments to create incentives for other members to accept a coordinated solution. Auction theory analyzes price-settling mechanisms under pure market conditions. There exist two possibilities for applying an auction mechanism to SC planning. First, the SC partners can be determined by an auction. A typical example is the choice of third-party logistics providers. However, it is questionable if market conditions really exist within an SC: Here, the choice for partners is usually a strategic one, binding a manufacturer to a supplier for a longer time period, cf. Stadtler (2007). Second, auction mechanisms can be employed for choosing the best interorganizational plan out of a set of alternatives. The typical proceeding is that the auctioneer grasps total visibility of resources and sells resource capacity to competing bidding agents, representing, for example, different company divisions. A survey can be found in Wellman and Walsh (2001). Most research focuses on incentive compatibility. In an incentive compatible mechanism, players fare best when they truthfully reveal any private information requested, i.e. truth-telling is a dominant strategy (cf. Myerson 1979). Incentive compatibility can be achieved by introducing compensation payments demanding each player to pay the opportunity cost that his presence introduces to all the other players. Auction mechanisms only focus on the problem of selecting the optimal

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production schedule but does not address the construction of candidate schedules under information asymmetry. A proposal on order quantities or delivery times can also be interpreted as a good offered from one SC member to the other. While each player knows his internal value of the good, he is assumed to have only limited information (in the form of a probability function) of his antagonist’s valuation. For such cases of information asymmetry, publications related to Bargaining theory analyze the efficiency of equilibrium strategies. Chatterjee and Samuelson (1983) provide such an analysis for a bilateral monopoly. Within a nonrecurring sealed-bid double auction, a buyer and a seller of an indivisible good simultaneously reveal their offers. If trading takes place (the buyer’s offer is higher than the seller’s claim) a settlement is computed according to an a priori known mechanism (e.g., by splitting the difference between both offers equally). Though the seller’s and buyer’s actual values of the good are assumed to be private knowledge, a probability distribution of likely values is supposed to be common knowledge. Being rational players, buyer and seller implement “best response strategies,” giving a price offer dependent on the own value and the counterpart’s probability distribution. These interlinked strategies lead to a Nash equilibrium. Related contributions have been put forward by Harsanyi and Selten (1972), Myerson and Satterthwaite (1983), Samuelson (1984), and Radoport and Fuller (1995). Bargaining theory provides interesting analytical insights into strategies leading to Nash equilibria and the related efficiency of the employed price-settlement mechanisms, i.e., the percentage of mutual beneficial agreements that can be actually attained by bargaining under the employed mechanism. However, most approaches assume the parties’ probability distributions of the value of the good to be public knowledge. It is questionable if this assumption can be justified for practical purposes. As auction theory, bargaining theory is not concerned how the good is created. In general, analytical models provide mathematical insight about optimal properties of coordination mechanisms, such as incentive compatibility. However, restrictive assumptions complicate a direct application to practical planning problems of an operational type.

2.2

Operational Models

In contrast to analytical models, operational models explicitly focus on generating production plans under information asymmetry. Using a metaheuristic, Fink (2004, 2006) investigates a supplier-manufacturer scenario where the sequence of delivery influences the quality of the related production plans. A central algorithm (assumed to be common knowledge) repeatedly proposes randomized delivery sequences to be accepted or rejected. The last jointly accepted delivery sequence is the final outcome of the coordination process. To sustain a fair outcome, both SC members are required to accept a certain percentage of the proposals, forcing them to also accept worse proposals.

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The internal acceptance probabilities are calculated according to a cooling scheme taken from Simulated Annealing. Though proposing purely randomized delivery sequences by a third party can be considered as a very fair approach, it might be regarded as inefficient in a real-world environment, where evaluating each proposal takes a considerable amount of time due to the size of the problems. An important aspect of CP mechanisms is security. Security can be defined by means of a trusted mediator (cf. Li and Atallah 2006). Assume an idealized mechanism, where each SC member supplies sensitive information to the trusted mediator, the mediator solves the global problem and reports back the individual solutions without disclosing private information to other SC members. A mechanism is said to be secure if any adversary interacting in the real mechanism can do no more harm than in the ideal scenario. The idea of security can be illustrated by a simple example. Assume three parties A, B, and C and each party holding a stock of items, e.g., A has three items, B has five items and C has two items in stock. Suppose that the parties aim at computing the sum of items without revealing how much items each single party has in stock. A secure sum algorithm can be set up as follows. Party A adds a random number, say 7, to its stock and transmits the sum of 10 to B. B adds its five items and transmits the sum of 15 to C. In turn, C adds 2 and transmits the sum of 17 to A. Eventually, A subtracts the initial random number and the correct sum of stock, i.e. ten items, has been computed. However, no party knows the exact number of items of any other party. Similar, though more complex, secure multiparty algorithms are available for other algebraic operations and different number of parties. These algorithms can be regarded as “one-way functions,” functions that are easy to evaluate but hard to invert (cf. Yao 1982). In simple terms, each protocol takes shares of the input and produces shares of the output (cf. Kerschbaum and Deitos 2008). Each SC member can encrypt his input with a private key, such that it cannot be decrypted by other parties without the private key, whereas the function can still be evaluated despite the encrypted values. Li and Atallah (2006) present a decentralized secure Simplex method for two parties. Atallah et al. (2003) present secure allocation protocols that prevent a disclosure of bidders’ sensitive information during auctions. Being a sophisticated encryption method, secure multiparty computation itself is not incentive compatible and does not prevent users from supplying systematically biased input. Operational planning problems are often formulated as linear problems (LP) or mixed-integer linear problems (MILP) and the solution is computed by an LP/ MILP-solver. Lagrangian relaxation (LR) is a common decomposition technique (cf. Conejo et al. 2006; Williams 1999) for problems having a sparse coefficient matrix, where most of the non-zero coefficients can be ordered into a block-angular structure. Consider the linear programming problem min

x1 ;x2 ;...;xn

n X j¼1

cj xj

(1)

DEAL: A Heuristic Approach for Collaborative Planning in Detailed Scheduling n X

s.t:

dij xj b fi

8i ¼ 1; . . . ; q

463

(2)

j¼1 n X

aij xj ¼ bi

8i ¼ 1; . . . ; m

(3)

j¼1

0 b xj

8j ¼ 1; . . . ; n;

(4)

where constraints (2) have a decomposable structure in n blocks. In the particular case of n ¼ 2, the problem can be written as 0 1 x½1 T T @ A; c½1 j c½2 min x½1 ;x½2 x½2 where the superindices in brackets refer to partitions, subject to 0

D½1 B B B B @ A½1

1 j 0 ½1 1 b C C x @ A b j D½2 C C ½2 A x ¼ j A½2

1 f ½1 B C C B B f ½2 C: C B @ A b 0

The problem can be decomposed by “dualizing” the coupling constraints 3. That is, adding them to the objective function. The problem 1–4 can be rewritten as " max

l1 ;l2 ;...;lm

s.t:

n X

min

x1 ;x2 ;...;xn

dij xj b fi

n X j¼1

cj xj

m X i¼1

li bi

n X

!# aij xj

j¼1

8i ¼ 1; . . . ; q

j¼1

0 b xj 0 b li

8j ¼ 1; . . . ; n; 8i ¼ 1; . . . m

where l1, l2, . . ., lm denote the Lagrangian multipliers. The idea of LR is to iteratively solve the inner minimization problem while updating the multipliers until the procedure has converged to a satisfying solution. Since the inner problem has a true block-angular constraint structure, it can be partitioned accordingly and solved separately. Subgradient procedures are frequently used to iteratively increase multiplier values in proportion to their constraint violation in the primal problem.

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Many SCM problems have a block-angular constraint matrix consisting of the intra-company constraints of each SC member, connected by inter-company material-flow-balance constraints. Subgradient methods only require realized material flows to update the multipliers, not the information of domain-internal production constraints. Hence, LR inherently supports scenarios with information asymmetry and is used by many authors to solve inter-company planning problems information asymmetry, cf. Kutanoglu and Wu (1999), Kutanoglu and Wu (2006), Ertogal and Wu (2000), Nie et al. (2006) and Walther et al. (2008). Multiplier values are dual information. Another possibility is to exchange primal information, target right-hand side (rhs) values for decision variables included in the coupling constraints, cf. Jung et al. (2005). Dudek and Stadtler (2005) decomposes the problem by fixing the material flow from a supplier and a manufacturer to certain values by setting the related rhs values. At the beginning, upstream planning is used to compute the initial “supply pattern.” Then the SC members alternatingly propose supply quantities and associated cost increases trying to compute reasonable proposals by approximating its antagonist’s cost changes within the local objective function. This is realized by penalizing the deviation from the currently confirmed supply quantities. The penalty coefficients are calculated on the basis of the antagonist’s cost reports. Thus, only effective proposals – that is, supply patterns with the highest trade-off of decrease of internal cost and increase of approximated “external” cost – are computed. After coordination, the worse-off player needs to be compensated by the player experiencing an improvement in its local objective function. Above approaches share the drawback that the adherence to imposed multipliers, rhs values or penalties cannot be monitored. An SC member can simply increase its local objective value by relaxing these bounds. Also the introduction of compensation payments does not necessarily ensure incentive compatibility. In a setting where the set of alternatives is already known to all players (as assumed for most analytical models), incentive compatibility in the sense of choosing the best interorganizational solution from a set of alternatives might be ensured by compensation payments. However, the situation is different if the reported values are also used for constructing new solutions. During construction, the future outcome of the search process is not determined yet. Thus, players are tempted to exaggerate their reported costs for receiving a higher compensation payment in order reap a good piece of the savings pie (as long they have the possibility for doing so). If an SC member reports exaggerated costs, its own objective gets a larger influence. At the end, those solutions will get explored that are, ceteris paribus, close to the locally optimal solution of the exaggerating domain obscuring the true global optimum with maximum saving. A possible solution to this problem is to decouple compensation payments from the coordination process. Albrecht (2010) suggests to compute a lumpsum (based on historical information) a priori to jointly constructing the solution. Under this premise the search process fosters truth telling of costs of intermediate solutions, since cheating obscures the true global optimum leading to an outcome where all partners are worse-off. For operational models of a certain structure, Albrecht

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(2010) proves that his mechanism is able to attain the global optimum within a finite number of proposal exchanges. Summarizing the above literature, coordination mechanism must support complex planning problems of operational type under information asymmetry, must not lead to a disclosure of sensitive data and must be incentive compatible. The difficulty lies in equally considering all the three aspects: For example, multi-echelon systems and contract theory support incentive compatibility but require very restrictive assumptions and simplified models. Secure multiparty computation is able to deal with complex models but is not incentive compatible. With the exception of Albrecht (2010), this critique also holds for coordination schemes relating to mathematical decomposition techniques. Moreover, although critical information is not explicitly exchanged, operational mechanisms relying on cost reporting provide opportunistically acting players the possibility to infer sensitive data by systematically probing other SC members. Thus, even if the mechanisms do not directly require the exchange of sensitive data, they are insecure in protecting these data. Practical requirements introduce further difficulties. If we assume planning problems to be NP-hard, the solution with maximum global saving is not guaranteed to be found in a reasonable amount of time – violating a fundamental assumption of many incentive compatible mechanisms. Moreover, in most approaches, the value of each generated alternative is measured as the difference to an initial solution. However, SC members are usually not of equal power in practice. A powerful party might be tempted to create bad initial settings for the weaker parties if there is the possibility to gain future compensation payments from the weaker parties’ “cost decrease” through coordination. For example, a (powerful) manufacturer could initially order items very early, resulting in huge overtime costs for the supplier and additional storage costs for the manufacturer, whereas storage costs are assumed to be less than overtime costs. Now, the coordination mechanism will search for a solution that avoids these costs. In the example, it is likely that such a solution will be found (since the costs have been generated by purpose) and that the coordinated solution will come with a higher cost decrease for the supplier than for the manufacturer. Hence, if savings are split “fairly”, the supplier will have to compensate the manufacturer. Thus, the manufacturer used his power to generate an additional margin at the expense of his supplier. Concluding, even if a mechanism was incentive compatible given a set of assumptions, its application in practice might suffer from a violation of some of these assumptions. Furthermore, real-world optimization usually requires considerable fine-tuning of parameters and data belonging to a company’s specific optimization problem. It is likely that companies would not accept a coordination mechanism that required a major change of the parameters or even abandoned their long-time tested and approved planning system. Moreover, companies often have specific planning problems that cannot be tackled by out-of-the-box solutions. In practice, the landscape of optimization tools is hence very heterogeneous and several software companies compete with different optimization methods with proprietary intrafirm developments. Finding a real-world setting with homogeneous optimization tools is unlikely. Hence, the coordination mechanism should rather treat the underlying

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solver as a black box. As will be discussed later, most models for production planning share commonalities in their structure. Our intention is to construct a coordination mechanism that dynamically changes the input data relating to common model structures but not the solver itself. Different solvers can then be supported by adapting the interface for changing the data.

3 The Resource-Constrained Project-Scheduling Problem The Resource-Constrained Project-Scheduling Problem (RCPSP) belongs to the class of NP-hard problems (cf. Blazewicz et al. 1983; Neumann et al. 2001) and represents a generalizations of different scheduling problems, such as job-shop or flow-shop problems. In our study, the RCPSP forms the underlying scheduling problem of each SC member. We present its basic problem definition below in order to provide the extension to collaborative planning in Sect. 6. We introduce the following definitions. The planning interval is defined as the 2 N0 consisting of several periods with a given discrete time interval ½0; . . . ; T period length (e.g., seconds), in which all activities start and end. An activity j refers to a single production activity (e.g., the assembly of parts). An activity has a start and a duration dj 0, defining the finish date fdj ¼ sdj + dj. date sdj 2 ½0; . . . ; T The set of all activities is referred to as J. Moreover, there exists a set R of renewable resources. Each resource r has a constant capacity of ar units that is renewed every period t ∈ T. When executed, an activity j places a constant resource usage ujr on resource r throughout its duration. The time for executing an activity is bounded by its release date rdj. The activities have to be executed in a predefined partial order, the so-called activity-precedence relation. For convenience, we introduce the sets Pj ¼ {i ∈ J | i is the immediate predecessor of j} and Sj ¼ {i ∈ J | i is the immediate successor of j}. An instance of the RCPSP is then given by • • • • •

A planning interval ½0; . . . ; T, A set of activities J, A set of renewable resources R, A precedence relation given by Pj or Sj for each activity j, respectively, The precedence and capacity constraints sdj r rdj sdj r fdi fdj ¼ sdj þ dj X j2Jjsdj b t b fdj

ujr b ar

8j 2 J

(5)

8j 2 J; i 2 Pj

(6)

8j 2 J

(7)

r2R 8t 2 ½0; . . . ; T;

• An objective function that shall be minimized or maximized.

(8)

DEAL: A Heuristic Approach for Collaborative Planning in Detailed Scheduling

467

4 Evolutionary Algorithms Because of the extensive run times of mixed integer approaches and their complex implementation, many researchers have turned to metaheuristics. The term metaheuristic usually describes a generic optimization principle that is widely applicable to many different problem domains. Very often, the basic idea behind a metaheuristic is inspired by nature. Evolutionary algorithms (EAs) are stochastic iterative optimization heuristics inspired by natural evolution. Starting with a set of candidate solutions (population), in each iteration (generation), promising solutions are selected as potential parents (mating selection), and new solutions (individuals) are constructed by mixing information from the parents (crossover) and slightly modifying them (mutation). The resulting offspring are then inserted into the population, replacing some old or less fit solutions (environmental selection). By continually selecting good solutions for reproduction and then creating new solutions based on the knowledge represented in the selected individuals, the solutions “evolve” and become better and better adapted to the problem to be solved, just like in nature, where the individuals become better and better adapted to their environment through the means of evolution. In line with Deb (2001), an EA can be outlined as follows, t 0 initialize population: P(t) evaluate P(t) repeat select mating pool: M(t) s(P(t)) c(M(t)) construct offspring: M0 (t) evaluate P(t) \ M0 (t) update: P(t + 1) u(P(t) \ M0 (t)) t t+1 until terminated, with t denoting the generation counter, P(t) the population at generation t, and s, c and u representing the different genetic operators for selection, construction and update. For a more detailed introduction to EAs, the reader is referred to Eiben and Smith (2003).

5 The SAP Production Planning and Detailed Scheduling Optimizer SAP offers the PP/DS Optimizer as proprietary tool to solve RCPSPs of extended complexity, including multiple modes, setup times, precedence relations with maximum time constrains, varying capacity profiles and different objectives. Here, the

468

J.B. Scheckenbach

above problem definition shall suffice. As objective we consider lateness minimization. In practice, soft-constrained due dates per activity ddj are often used to model customer preferences. In a simple way the objective can be put mathematically as Minimize

X

max fdj ddj ; 0 :

j2J

Due to the combinatorial complexity of the RCPSP, approaches for computing exact solutions (cf. Manne 1960; Blazewicz et al. 1996; Herroelen et al. 1998; Brucker and Knust 2000; Klein 2000; Neumann et al. 2001; Damay et al. 2007) are only applicable to small-sized problems of minor practical relevance. In line with the research of Hartmann and Kolisch (2000); Hartmann (2002); Alcaraz and Maroto (2001); Kolisch and Hartmann (2006), the PP/DS Optimizer employs an EA based on an activity sequence encoding and a serial schedule generation schemes (SSGS). An SGSS consists of |J| stages in each of which an activity is scheduled at the earliest precedence- and resource-feasible time. SSGS can be used to construct active schedules. Informally spoken, an active schedule is a schedule where no activity can be started earlier without delaying the start of another activity. It has been proved that for regular performance measures, such as lateness minimization, the optimal schedule lies within the set of all active schedules (cf. Sprecher et al. 1995; Neumann et al. 2001). Employing different SGSS as decoding functions, the PP/DS Optimizer uses several mutation and crossover operators to find the best sequence of activities. In other words, the search is not carried out in the space of all possible schedules (spanned by all start dates sdj not violating precedence and capacity constraints), but in the space of all sequences; using SGSS to evaluate the fitness of candidate sequences. Being a metaheuristic, the PP/DS Optimizer does not guarantee to find the optimal schedule. Thinking about compensation payments, it is hard to relate lateness to real monetary terms (cf. Stadtler 2005, p. 585). Especially on an operational level, machine-, workforce- and material-related costs are usually considered as sunk costs from an accountancy perspective. Using lateness costs, we primarily aim at seeking schedules that make optimal use of the available resources. As a detailed discussion of the PP/DS Optimizer would be beyond the scope of this paper, the interested reader is referred to Engelmann (1998) and Scheckenbach (2010). Earliness as a non-regular performance measure precludes the use of SSGS; the optimal schedule no longer lies within the set of all active schedules. However, from a practical point of view, the minimization of earliness is important to reduce holding costs. The strategy is then to modify the schedule constructed by SSGS in a subsequent right-alignment step. We define a right-aligned schedule as a schedule where no activity can be finished later without advancing some other activity, or violating the constraints, or increasing the objective function. Right-alignment considers activities from “right” to “left” (relating to their positions in a Gantt chart) and shifts each activity to the latest possible date. Also SAP provides a rightalignment algorithm.

DEAL: A Heuristic Approach for Collaborative Planning in Detailed Scheduling

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6 DEAL: A Decentral Evolutionary Algorithm This section presents a decentral evolutionary algorithm (DEAL) for aligning detailed schedules of n suppliers and one manufacturer. The algorithm works decentrally in the sense that a top-level evolutionary algorithm iteratively changes local problem instances that are solved locally and separately by the PP/DS Optimizer of each SC member. By doing this, each SC members evolves a population of candidate schedules. The top-level metaheuristic controls the composition of local schedules to global feasible schedules and the evolution of local populations, without demanding the exchange of sensitive data. We start by formulating the interorganizational problem in Subsection 6.1. Subsections 6.2 and 6.3 are about constructing candidate solutions. A solution relates to a composition of local schedules, feasible from an SC-wide perspective. In Subsection 6.4 we address the evaluation of solutions, the selection of the mating pool and the update of the population. Subsection 6.5 presents means to further speed up the search process.

6.1

The Interorganizational Problem

We assume each SC member having its unique set of resources. We further assume the activities between the different SC members to be connected by precedence constraints. That is, some activities of the manufacturer can only be started after the suppliers’ preceding activities have been finished, neglecting transportation times. Moreover, we assume that there is no cyclic dependency between the SC members. Thats is, neither suppliers nor manufacturer deliver to other suppliers taking part in coordination. Nevertheless, suppliers and manufacturer might delivery to external sinks and might be supplied by external sources, not participating in coordination. We extend our previous RCPSP formulation by defining • ddj as static, non-varying due date of activity j relating to external sinks, • rdj as static, non-varying release date of activity j relating to external sources, • e as index of an SC member (agreeing on the notation that the manufacturer is e ¼ 0 and suppliers are e > 0) and E as the set of all SC members, • Je J as the set of activities j of SC member e, Je, • Re as the set of resources r of e, • Pej as the set of predecessors within the problem of SC member e, • Sej as set of successors within the problem of SC member e, activities directly depending • JU0 as the set of manufacturer’s upstream-related on the suppliers’ delivery, i.e. JU0 ¼ j 2 J 0 j9i 2 J e ^ i 2 Pej with e 2 Enf0g , activities directly influencing • JDe as the set of supplier e’s downstream-related the manufacturer’s problem, i.e. JDe ¼ j 2 J e j9i 2 J 0 ^ i 2 S0j .

470

J.B. Scheckenbach

We assume the set of all jobs to be distributed between the SC members e, i.e. 8e 2 E : J e J; [ J e ¼ J; e2E

8e; f 2 E; e 6¼ f : J e \ J f ¼ ;: The same is true for resources. The intra-company RCPSPs of supplier and manufacturer interlink as follows. Finish dates of suppliers and static, nonvarying, release dates rdj determine the manufacturer’s release dates; expressed by including the following equations in the manufacturer’s RCPSP formulation. ( ) rdj ¼ max

max

k2Pej ;e2Enf0g

ðfdk Þ; rdj

8j 2 JU0

(9)

A supplier’s due dates of downstream-relating activities are defined by ( ddj ¼ min

min ðpdk Þ; ddj k2S0j

) 8j 2 JDe ; e 2 Enf0g;

(10)

where pdk denote proposed dates, send by the manufacturer to the supplier during coordination.

6.2

Initialization

During initialization, the manufacturer proposes dates that “myopically” optimize his intradomain problem. To do this, all release dates rdj are first set to their static, non-varying values rdj . Based on this relaxed problem, the manufacturer applies the PP/DS Optimizer for a predefined amount of time, resulting in a solution consisting of start dates sdj and finish dates fdj, j ∈ J0. The proposed dates pdj are then derived from the manufacturer’s related right-aligned schedule. Each supplier receives those proposed dates relating to his scheduling problem. The suppliers translate the proposed dates into downstream-related due dates using Equations 10, apply the PP/DS Optimizer to solve their own scheduling problems and report the realized finish dates back to the manufacturer. Applying (9), the manufacturer in turn converts these finish dates into upstream-related release dates and recalculates his schedule once again using the PP/DS Optimizer. The outcome is a feasible, though suboptimal, SC-wide schedule. After the above procedure, remaining candidates are computed according to the principles discussed in the next section, until the initial population has reached its size. Note that the strategy “propose everything as early as possible” does not provide an initial advantage to the manufacturer, since the suppliers’ capacity is limited

DEAL: A Heuristic Approach for Collaborative Planning in Detailed Scheduling

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(the RCPSP does not provide an option for overtime). Rather than absolute values, relative trade-offs between the proposed dates communicate the manufacturer’s preferences.

6.3

Construction

The construction of a new candidate solution is done analogously to initialization with the difference that the manufacturer tries to compute promising proposed dates based on the current population. To sustain an evolutionary process, the manufacturer should only change the most conflicting dates, otherwise construction would be similar to initialization. Different operators have been studied on the basis of this general idea. In the following, we will highlight a few examples, a detailed discussion can be found in Scheckenbach (2010). Proposed dates can be calculated by propagating the manufacturer’s due dates to upstream-related activities by means of backward passes. A backward pass propagates the upper bounds from the end of the horizon backward starting with pddn ¼ ddn : for j ¼ jJ 0 j; . . . ; 1; pddj ¼ min ddj ; min pddh dh jh 2 Sj where the manufacturers activities are assumed to be numbered according to a topological sorting (i.e. i < j, if i ∈ Pj, and pddj denotes the propagated due date. Proposed dates are then calculated as pdj ¼ pddj dj ; 8j 2 JU0 . Dates calculated by backward passes communicate the manufacturer’s minimal requirements (neglecting capacity constraints) to the suppliers. For most coordination problems, this strategy is too simple, since bottlenecks arise because of limited capacity and suboptimal activity sequences. An alternative is to generate an intermediate problem instance i by partially relaxing release dates in the manufacturer’s problem. By applying the PP/DS Optimizer to the intermediate instance, the manufacturer searches for better sequences of activities. The proposed dates are then derived from the start dates of the related right-aligned schedule. One possibility to calculate the set of release dates to relax is to take the history of coordination into account. With respect to an existing solution p, we define ðpÞ ðpÞ ðpÞ ðpÞ the amount of correction of activity j 2 JU0 as Dj ¼ rdj pdj , whereas rdj denotes the release date that results from the suppliers counterproposals ðpÞ (cf. Equation 9) to the proposed date pdj during constructing of solution p. The ðpÞ amount of correction Dj measures the suppliers’ ability to fulfill the manuðpÞ facturer’s proposals. Note that Dj can also become negative if a supplier exceeds the manufacturer’s requirements. Intuitively, a further relaxation of release dates of activities with a large positive amount of correction is not promising, as at least one of the suppliers is apparently not able to fulfill the resulting proposed dates. Instead ðpÞ we restrict the critical set to those activities with a low Dj . For the first a percent of

472

J.B. Scheckenbach ðpÞ

ðiÞ

activities with low Dj we set rdj ¼ rdj and for the remaining 1 a percent ðpÞ ðiÞ ðpÞ with high Dj we set rdj ¼ sdj . Lateness to ultimate customers is then reduced by applying the PP/DS Optimizer to the relaxed problem instance. The proposed dates are derived from the start dates of the subsequently right-aligned schedule, whereas right-alignment is not allowed to increase lateness, as discussed in Sect. 5. Another possibility is to shift weakly connected components. Imagine an undirected graph, where activities define the nodes and precedence relations the edges. Usually, this graph can be decomposed into several disjunct weakly connected components. Relaxing release dates of all activities belonging to the same component might have a larger effect on the lateness criterion than doing so for arbitrary selected activities. For each component C J, the maximum difference between finish date and propagated due date is computed as ðpÞ ðpÞ mdC ¼ max fdj pddj : j2C

We might aim at relaxing the release dates of the “most-delayed” activities contained in the first ba nC c components with largest maximum difference, where nC is the total number of components and a a further tuning parameter (the symbol b:c denotes the floor function). Crossover uses the information of several existing candidates to derive new proposed dates. For example, linear crossover can be used to compute proposed dates as ðpÞ

ðqÞ

rdj þ rdj ; 8j 2 JU0 ; pdj ¼ 2

(11)

where q and p denote two existing solutions from the mating pool. Instead of the former release dates, also proposed dates might be combined. Concluding, constructing a solution always takes the preferences of all SC members into account. First, the manufacturer proposes dates to the suppliers representing his most demanded corrections to an existing schedule. Second, the suppliers incorporate the manufacturer’s preferences into their objective function by adapting their downstream-related due dates, cf. Equation 10. However, the suppliers are not forced to comply to the manufacturer’s proposal and the SC-wide solution is based on their counterproposal. In particular, suppliers might not agree to delay any external customer. This goal can be expressed by giving the related delay a larger weight in the lateness objective before the start of coordination.

6.4

Evaluation, Selection and Update

Evaluating a candidate set of solutions is critical for advancing in the right search direction. In our approach, evaluating refers to the task of ranking the different SC-wide schedules in order to remove worst ones from the population and for

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selecting the mating pool. Following the above discussion this ranking scheme must support incentive compatibility, at least from a practical perspective. If compensation payments are not taken into account, we refer to a coordination mechanism as practically incentive compatible if it does not put any SC member worse than in the initial, uncoordinated setting. However, such a deterioration is not necessarily measured by the objective function. Costs of a supplier can be decomposed into coordination-related and domain-related costs. Coordination-related costs are penalty costs resulting from the violation of due dates proposed by the manufacturer. Since these costs mainly anticipate the manufacturer’s decision problem, they cannot be claimed by the supplier in an accountancy sense.1 Hence, in addition to the objective function of the local optimization method, a supplier might have another evaluation function representing his preferences regarding the manufacturer’s proposals. A simple, practical evaluation function distinguishes between acceptable and unacceptable solutions. Unacceptable solutions result from a setting of due dates that leave a supplier’s local optimization method too little freedom to produce acceptable results. We define ue ðpÞ ¼

0; if the proposal p is acceptable, z 2 N; if the proposal p is not acceptable:

Integer z denotes a ranking of unacceptable solutions. The smaller z, the better the related solution from a supplier’s perspective. For each solution, the suppliers submit their ranking to the manufacturer. After having computed his ranking in a similar manner, the manufacturer can combine the single ratings by non-dominated sorting. Solution p is said to dominate q, if 8e 2 E : ue ðpÞ b ue ðqÞ and 9d 2 E : ue ðpÞ < ue ðqÞ: Non-dominated sorting clusters the population into subsets dominating each other. This allows to construct a global ranking g, i.e. individuals that are not dominated by other individuals get a g ¼ 1, individuals with g ¼ 2 are only dominated by individuals with g ¼ 1 and so on. Having this information, the manufacturer decides which solutions to select in the mating pool and which to remove from the population. Although he is free in his single decisions, the ultimate solution to be implemented is required to be accepted by all suppliers, i.e. 8e 2 Enf0g : ue ðpÞ ¼ 0. In each generation, m offspring are created out of a population of l individuals. An offspring is created by either applying a mutation or crossover operator. An individual to be mutated is chosen by tournament selection, cf. Deb (2001). That is, two individuals are randomly drawn from the parent generation, and the one with better ranking wins the 1

Nevertheless, coordination-related costs are in direct relation to domain-related costs. For example, setting a due date earlier might result in a solution with higher lateness of other orders of external clients.

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J.B. Scheckenbach

tournament. The two individuals required for crossover are determined analogously by two tournaments. After constructing the offspring, the manufacturer sorts all individuals according to their global ranking. The manufacturer removes the worst m individuals from the population (environmental selection) and transmits this information to the suppliers who update their local populations accordingly. The above ranking has three advantages. First, the transmission of ordinal rankings disguises objective values considered as sensitive by the suppliers. The manufacturer has only limited possibilities for probing the suppliers to infer sensitive data. Although he knows that a solution is regarded by a supplier as better or worse than another solution, he does not know how much better or worse it is. Second, the dimensionality is reduced. The search process is not driven in dimensions were solutions are already acceptable for some suppliers (having a ranking of zero) but a multiobjective Goal Programming approach, as discussed in Deb (2001), is implemented instead. Third, and most important, suppliers have only limited possibilities for cheating. No suppliers can influence the trade-off between the local objectives by exaggerating his values, as such a trade-off does not exist in non-dominated sorting. Of course, suppliers can label actually acceptable solutions as unacceptable – but that is something they already can do today by reporting that present delivery dates cannot be maintained. In general, we assume that suppliers try to comply to delivery dates to the extent possible, as their status of being a reliable partner depends on it. Thus, we are tempted to label DEAL as a practically incentive compatible mechanism (however, not in a mathematical sense).

6.5

Means to Speed Up the Search Process

This section discusses means to further speed up the search process. First, the above coordination mechanism is characterized by periodic idle times. Suppliers wait for new manufacturer’s proposals to counter. Similarly, the manufacturer waits for counterproposal before the construction of a solution is continued. Today, computational power is a relatively cheap expense. As the single solutions are independent of each other, the coordination can be parallelized, allowing the manufacturer to compute several proposals in parallel, without waiting for counterproposals. Note that in such an environment the term generation becomes elusive – the focus is instead on a high workload in the systems. In contrast to classic EAs, individuals have different states. Intuitively, selection, construction and update operators can only be applied to the set of “complete” individuals. Second, a “warm reboot” was established. In a standard PP/DS run, different heuristics compute the initial activity sequences (e.g., the activities are sorted according to their propagated due dates). As DEAL holds a population of schedules, information of previous local PP/DS runs are available, however. Hence, we added the sequence of each schedule in the population to the set of initial standard sequences. Depending on the chosen due or release dates of the new offspring

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this information might not prove useful and is sorted out during environmental selection of the PP/DS run. However, if the problem instances vary only slightly, using existing activity sequences avoids a computation “from scratch” and gives good results already at the beginning of the PP/DS run. Third, a self-adaption of operators for generating proposals can be set up. We regard the decision of selecting an operator as a dynamically changing optimization problem. For each operator o, we define a value to. The probability for choosing operator o out of the set of all operators O is computed as t Po : to o2O

After an operator has been applied, all values are decreased by a factor d ∈ (0;1), to d, 8o ∈ O. If an operator has been applied successfully (the child c is i.e. to better globally ranked than the parent p, its value increases by the relative ranking improvement: to

to þ

gðpÞ gðcÞ : gðpÞ

The value to can be interpreted as pheromone value in an ant-colonization optimization approach, cf. Merkle et al. (2002). Fourth, redundant computations are avoided. After deriving proposed dates for offspring c, the following redundancy checks apply: If another individual p in the manufacturer’s population exists, that is accepted by all suppliers (ne ðpÞ ¼ 0; 8e 2 ðpÞ ðcÞ Enf0g) with rdj b pdj ; 8j 2 JU0 , the manufacturer discards solution c immediately and does not transmit the proposed dates to the suppliers. After an individual has passed the first check, a second check is carried out by each supplier. For supplier e, an individual c is redundant, if there exists a local candidate schedule ðpÞ ðcÞ p with ne(p) ¼ 0 and ddj b ddj ; 8j 2 JDe . In such a case, the supplier skips the local PP/DS run and counterproposes again the release dates of individual p. Each supplier stores the information how manufacturer proposals relate to local schedules in a table. Obviously, this table has to be looked up for computing the correct ranks and needs to be adapted if the manufacturer transmits which individuals are to be deleted.

7 Experimental Key Results The description of DEAL and the SAP PP/DS Optimizer in the above sections were very limited due to the scope of this contribution. With this background a detailed discussion of experimental results would be a futile endeavor. Hence, this section

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aims at a high-level presentation of key results only. More details are provided in Scheckenbach (2010). Two test data generators were used. The first one models the common procedure in practice. Given periodic demand patterns, available Bill-of-Materials, related resource capacity requirements and fixed lot sizes, a material requirements planning (MRP) algorithm calculates number and length of required production activities, precedence constraints and resource utilization. Usually the MRP run generates several lots to satisfy total demand of a finished good. As solely one Bill-ofMaterial per finished good is available, the generated activities show a repetitive structure akin to flow shop problems. The second test data generator is based on the work of Kolisch et al. (1992, 1995). Adding activity by activity, an algorithm randomly constructs an activity network and also randomly chooses the length, the resource requirements and the due dates of each activity. Restricting these decisions to be within certain bounds results in a test instance with predefined complexity measures. Having created a test instance by using one of the above generators, it is divided into the interconnected subinstances belonging to the different SC members. The division is carried out on the basis of a predefined assignment of resources to SC members. By respecting this assignment already during the generation of a test instance we assure that no cyclic dependency results between the different SC members (as assumed in Sect. 6.1). Independent of the chosen test data generator, the above procedure results in one fractioned test instance for the coordination mechanism and one integral benchmark instance to be “centrally” solved by a single PP/DS Optimizer. Several runs for different test instances with up to 10,000 activities have been carried out systematically to increase statistical confidence. Although the coordination mechanism has to cope with information asymmetry, neither a deterioration in solution quality nor run time could be observed with regard to the “central” solution. Several factors contributed to this result. First, proposed dates imply certain tradeoffs in the local optimization problem. Rather than the dates’ absolute values, these tradeoffs steer the local optimization methods. Hence, a crude alignment of delivery dates by DEAL seems to be sufficient – the exact setting of delivery dates is done subsequently by the local optimization methods themselves. Second, DEAL can be regarded as a decomposition method working problems that exhibit specific properties. For example, there are no cyclic dependencies between suppliers and manufacturer. It can be argued that DEAL makes use of these problem structures but not the generic PP/DS Optimizer. Third, the “warm reboot” property carries over information between several solutions. Avoiding an initialization from scratch allows to shorten runtimes of the locally applied PP/DS Optimizer without deteriorating the solution quality. In turn, shorter local runtimes allow more proposals to be exchanged. Due to the combinatorial complexity of the RCPSP, a frequent exchange of proposals is advantageous. Moreover it was observed that the warm reboot gets more effective as the process converges to a good setting of proposed or release dates, as previous activity sequences better fit to an offspring problem instance. Thus, the warm reboot

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supports the seamless shift from the strategy “try as much proposed dates as possible” at the beginning of coordination to the strategy “generate the best schedules for a fixed setting of proposed dates” towards the end of coordination. Fourth, even for larger problem instances a good coordinated solution was found in reasonable time. Although complexity increases, larger problem instances usually allow more freedom of rescheduling, i.e., a crude alignment of proposed dates suffices. Moreover, larger problem instances apparently come with a greater probability that large parts of previous activity sequences can be used for the warm reboot. Fifth, it could be observed that larger population sizes lead to better results. This is a common property of EAs: Larger populations have a better chance of overcoming local optima, at the cost of a slower convergence, however. By computing and evaluating proposals in parallel, DEAL allows larger population sizes without prolonging runtime. To further clarify the test methodology, Fig. 1 exemplary shows the results for a medium-sized test instance, also including multiple modes and setup times. Two suppliers deliver to one manufacturer, but also have other customers that should not be harmed by coordination.2 The convergence of the centrally applied PP/DS Optimizer is shown by the solid graph.3 The graph shows the average delay of 700000

central, monolithic solution sequential coordination parallel coordination

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Fig. 1 Exemplary results of a test instance with more than 2,700 activities in total, two suppliers and one manufacturer. Shown is the lateness to the manufacturer’s ultimate customers of central and coordination solutions, average of 10 runs each

2 Cf. Scheckenbach (2010), par. 8.2.2.1 and 8.2.2.4 for a detailed description of the underlying business problem. 3 The initial 500 s are required for problem generation and initialization.

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deliveries from the manufacturer to ultimate customers of ten different runs and of each run’s best-found solution at a specific point in time. It can be seen that the fitness of the initial population is gradually improved each generation, although the speed of improvement declines. The dashed graphs show the result of sequential and parallel coordination, respectively. An additional idle time can be observed required for the construction of the initial coordinated solution. For computing and evaluating a proposal, each partner runs his PP/DS Optimizer for 50 s. It should be highlighted that the graphs do not show the convergence of a local PP/DS run within these 50 s, but only the results of coordination, which explains the rather step-wise convergence. In order to compute acceptable solutions both suppliers prioritize the lateness to external customers in their local objective (cf. Sect. 6.3). Additionally the double-ranking scheme of Sect. 6.4 was implemented. Hence, improving acceptability can lead to a temporary increase in the lateness to the manufacturer’s customers (only the latter figure is drawn). It is worth mentioning, that all approaches led to acceptable solutions in the end of all runs, although the initial situation was not always accepted by the suppliers. Finally and most surprisingly, it can be seen that the parallel coordination (also employing a larger population size) does not only show better performance than the sequential coordination, but was also able to beat the central solution. It seems that the DEAL framework as a top-level metaheuristic was able to introduce new information that steered the overall search process in the right manner. Adjusting the due dates of the suppliers’ RCPSP instances by an evolutionary process and repeatedly solving these sub problems finally led to the good results. One might argue that the above results do not prove the effectiveness of DEAL but only the ineffectiveness of the PP/DS Optimizer. This is not true. First of all, it has to be mentioned that, for NP-hard planning problems, an optimal solution can usually not be computed in reasonable time. Moreover, one has to keep in mind that the PP/DS Optimizer was constructed to tackle RCPSPs of general type. For evaluating DEAL, we only consider a subset of all possible scheduling problems: problems, that exhibit regions that are only interdependent in a noncyclic manner. DEAL can be regarded as a specialized heuristic working on such problem structures. Hence, by decomposing the central scheduling problem, more detailed knowledge over the subproblems becomes available. According to the No Free Lunch Theorem of computer science, we have to expect that the results of a specialized heuristic are better than those of a general one.

8 Concluding Remarks When designing a coordination mechanism, three fundamental requirements have to be considered: Complex planning problems have to be supported, sensitive data must not be disclosed and incentive compatibility has to be guaranteed. To ensure the latter, most present approaches rely on compensation payments, payments made in an agreement by one or more parties to other parties to induce them to join the

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agreement. Though this idea seems to be logical at first sight, compensation payments are not without problems. As argued, such payments may – if not properly designed within the mechanism – provide possibilities for cheating and probing the other partner to infer sensitive information instead of preventing them from doing so. The presented heuristic restricts to generating ultimate solutions that are acceptable to all SC members without compensation payments. Naturally, this decision prunes the search space, possibly making the global optimum unattainable. However, it is questionable if this is considered as a drawback from a practical point of view. Usually, compensation payments are hardly deducible from control costs of objective functions and for many practical problems the optimal solution is usually not attained anyway. Practical acceptability adds further requirements: The mechanism should handle multiple domains, different optimization engines and different optimization models. It should be scalable to large problems and able to handle degraded, suboptimal solutions. Moreover, due to the limited time before escalation, short-term scheduling most urgently demands coordination. We believe EAs to be the logical choice under such conditions. They are not restricted to a particular type of problem requiring only a black-box fitness computation and are, as a population-based approach, easy to parallelize. Essentially, the idea is to adjust the data of underlying planning problems by an evolutionary process, using local optimization engines for decoding. As only the local data, but not the related optimization models are changed, different optimization engines can be connected to our framework. With regard to detailed scheduling, most practically used heuristics build up on release and due dates. If the possibility of a warm reboot is given, many proposals can be exchanged as well. For evaluating different proposals, we disguised sensitive data and limited possibilities for cheating by transmitting only ordinal rankings (reflecting a proposal’s quality) between the partners. Concluding, we claim that DEAL fulfills the three requirements complexity, security and incentive compatibility – at least from a practical perspective.

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Kolisch R, Sprecher A, Drexl A (1995) Characterization and generation of a general class of resource-constrained project scheduling problems. Manage Sci 41:1693–1703 Kutanoglu E, Wu SD (1999) On combinatorial auction and lagrangean relaxation for distributed resource scheduling. IIE Trans 31:813–826 Kutanoglu E, Wu SD (2006) Incentive compatible, collaborative production scheduling with simple communication among distributed agents. Int J Prod Res 44(3):421–446 Li J, Atallah M (2006) Secure and private collaborative linear programming. In: International conference on collaborative computing: networking, applications and worksharing Manne AS (1960) On the job-shop scheduling problem. Oper Res 8(2):219–223 Merkle D, Middendorf M, Schmeck H (2002) Ant colony optimization for resource constrained project scheduling. IEEE Trans Evol Comput 6(4):333–346 Myerson RB (1979) Incentive compatibility. Econometrica 47(1):61–73 Myerson RB, Satterthwaite MA (1983) Efficient mechanisms for bilateral trading. J Econ Theory 29:265–281 Neumann K, Schwindt C, Zimmermann J (2001) Project scheduling with time windows and scarce resources. Springer, Heidelberg Nie L, Xu X, Zhan D (2006) Collaborative planning in supply chains by lagrangian relaxation. In: Proceedings of the First International multi-symposiums on computer and computational science Radoport A, Fuller MA (1995) Bidding strategies in a bilateral monopoly with two-sided incomplete information. J Math Psychol 39:179–196 Samuelson W (1984) Bargaining under asymmetric information. Econometrica 52(4):995–1005 Scheckenbach B (2010) Collaborative planning in detailed scheduling. Ph.D thesis, University of Hamburg Sprecher A, Kolisch R, Drexl A (1995) Semi-active, active and non-delay schedules for the resource-constrained project-scheduling problem. Eur J Oper Res 80:94–102 Stadtler H (2005) Supply chain management and advanced planning – basics, overview and challenges. Eur J Oper Res 163:575–588 Stadtler H (2007) A framework for collaborative planning and state-of-the-art. OR Spectrum 31:5–30 Sucky E (2006) A bargaining model with asymmetric information for a single supplier-single buyer problem. Eur J Oper Res 171:516–533 Thomas PR, Salhi S (1998) A tabu search algorithm for the resource constrained project scheduling problem. J Heuristics 4:123–139 van Houtum GJ, Inderfurth K, Zijm WHM (1996) Materials coordination in stochastic multiechelon systems. Eur J Oper Res 95:1–23 Walther G, Schmid E, Spengler TS (2008) Negotiation based coordination in product recovery. J Prod Econ 111(2):334–350 Wellman MP, Walsh WE (2001) Auction protocols for decentralized scheduling. Game Econ Behav 35:271–303 Williams HP (1999) Model building in mathematical programming, 4th edn. Wiley, New York Yao A (1982) Protocols for secure computation. In: Proceedings of the 23rd Annual IEEE Symposium on Foundations of Computer Science

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Inventory Record Inaccuracy, RFID Technology Adoption and Supply Chain Coordination H. Sebastian Heese

Abstract Most retailers suffer from substantial discrepancies between inventory quantities recorded in the system and stocks truly available to customers. Promising full inventory transparency, RFID technology has often been suggested as a remedy to this problem. We consider inventory record inaccuracy in a supply chain model, where a Stackelberg manufacturer sets the wholesale price and a retailer determines how much to stock for sale to customers. We first analyze the impact of inventory record inaccuracy on optimal stocking decisions and profits. Contrasting optimal decisions in a decentralized supply chain with those in an integrated supply chain, we find that inventory record inaccuracy exacerbates the inefficiencies resulting from double marginalization in decentralized supply chains. Assuming that RFID technology can eliminate the problem of inventory record inaccuracy, we determine the cost thresholds at which RFID adoption becomes profitable. We show that a decentralized supply chain benefits more from RFID technology, such that RFID adoption improves supply chain coordination. Keywords Inventory • Record inaccuracy • RFID • Supply chain coordination

1 Introduction The discrepancy between inventory records and the amount of product effectively available for sale to customers presents a key problem in retail operations. Based on extensive empirical studies, Raman et al. (2001) report they found 65% of inventory This chapter is based on the article “Inventory record inaccuracy, double marginalization and RFID adoption” by H.S. Heese, published 2007 in Production and Operations Management (volume 16, issue 5, pages 542–553). H.S. Heese Kelley School of Business, Indiana University, 1309 East Tenth Street, Bloomington, IN 47405, USA e-mail: hheese@indiana.edu 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_19, # Springer-Verlag Berlin Heidelberg 2011

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records to be erroneous, i.e., the recorded inventory quantity did not match the quantity present at the store. Physical inventory deviated from inventory records by 35% on average, and about 16% of the product physically present at the store was not readily available to customers due to SKU misplacement. Transaction or scanning errors present yet another source of inventory record inaccuracies – all too often cashiers at the checkout use the number key to aggregate products with different flavors but the same price. Raman et al. (2001) conclude that “most retailers cannot, with any degree of precision, identify the number of units of a given item available at a store.” Since even the most sophisticated management system is handicapped if working with flawed data, the problem of lacking inventory record accuracy is often referred to as the missing link in retail execution and it has been estimated that the resulting lost sales and inventory costs reduce profits by more than 10% (Raman et al. 2001; Alexander et al. 2002). In this chapter we do not differentiate between the different sources of discrepancies between inventory records (system inventory or planned inventory) and inventory truly available for sale to customers (shelf inventory or available inventory). Since from the retailer’s perspective, the common underlying problem is the uncertainty of the inventory record, we will also use the term inventory uncertainty to refer to this problem. Among other developments in information technology, Radio-Frequency Identification (RFID) technology is being discussed as a powerful means to solve the problem of inventory uncertainty, among a plethora of other problems in supply chain execution. Enabling virtually full inventory transparency (both location and quantity) at any time, RFID technology, employed at the item-level, has indeed the potential to vastly improve retail execution (cf. McFarlane and Sheffi 2003). In recent years, several major retail chains have strongly promoted – or even mandated – RFID adoption by their suppliers, mostly at the pallet level. While RFID technology is suggested to enable substantial efficiency gains at the different stages of a supply chain, the associated costs are by no means negligible. Besides fixed costs related to the purchase and implementation of the necessary infrastructure, at this point especially the substantial cost of RFID tags seems to prohibit widespread use at the item level. Even if tag-costs decreased substantially, it is unlikely that item-level RFID adoption would be financially profitable at every retailer and for all products – it would likely start with more expensive items (cf. Want 2004). An optimal adoption decision needs to balance the value of full inventory transparency with the costs of RFID. In making this trade-off, it is important to distinguish the different incentives (benefits and costs) at the various stages of the supply chain. For example, adoption may be difficult if the retailer reaps most of the benefits, while the manufacturer bears most of the costs. More generally asked, how does the adoption decision of a decentralized supply chain differ from that of an integrated supply chain? We use the classic supply chain model of a single manufacturer, who as Stackelberg leader determines the wholesale price at which a product is sold to a retailer, who in turn determines how much to stock for sale to consumers (at an exogenous retail price). In making this stocking decision, the retailer faces

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uncertainty in customer demand as well as in inventory records. The latter source of uncertainty can be eliminated through RFID employment. We analyze the impact of inventory record inaccuracies on supply chain performance, and we investigate how double marginalization in a decentralized supply chain affects the different stages’ incentives to adopt RFID. While the possible implications of RFID technology on the supply chain are manifold, we focus on the improvements that result from having accurate inventory information. (With RFID all system inventory is available for sale to customers.) Poor execution at the retail level has been identified as the main driver of out-of-stock situations, which are a key problem in retailing (Andersen Consulting 1996; Gruen et al. 2002). While the benefit of such accuracy to the retailer is immediate, it is less clear, how RFID adoption affects the manufacturer, and the supply chain as a whole. Specifically, we address the following research questions: 1. How does inventory uncertainty affect stocking decisions compared to the standard case with accurate inventory records? 2. How does inventory uncertainty affect profits in a decentralized supply chain as compared to an integrated supply chain? 3. When do the manufacturer and the retailer individually benefit from RFID technology? 4. How does the RFID adoption decision in a decentralized supply chain differ from that of an integrated supply chain? We review the related literature in Sect. 2, and we introduce our base model in Sect. 3. Section 4 contains our model analysis and main results; we discuss the impact of inventory record inaccuracy on supply chain decisions and performance in an integrated and a decentralized supply chain (Sects. 4.1 and 4.2, respectively), before we contrast RFID adoption decisions for these two cases in Sect. 4.3. In Sect. 5, we conclude with a discussion of our results, limitations of our model, and suggestions for future research. All proofs are relegated to Appendix 1.

2 Literature Review The problem of inaccurate inventory records and misplaced SKUs at the retail level is widely discussed (e.g., Alexander et al. 2002; Raman et al. 2001; DeHoratius and Raman 2008). While most classic inventory models are based on the assumption of accurate inventory information, a subset of that research area considers the problem of inventory uncertainty under the term of yield uncertainty. An extensive review of this research stream is provided in Yano and Lee (1995). There is some very recent work that investigates optimal inventory management explicitly under inaccurate inventory records. Camdereli and Swaminathan (2010) consider the case where a fixed and known proportion of the retailer’s order quantity becomes unavailable for sale due to misplacement at the retailer. All misplaced products are recovered and salvaged

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at the end of the period. They analyze the impact of such proportional loss on the retailer’s optimal order quantity. They compare the performance of a decentralized supply chain with that of a vertically-integrated supply chain and investigate how coordination can be achieved by means of buy-back and revenue-sharing contracts. DeHoratius et al. (2008) propose the maintenance of a probabilistic inventory record instead of the commonly used point estimate to account for the presence of inventory record inaccuracy. Sources of such inaccuracies are modeled through an additional random variable called invisible demand. They suggest a simple Bayesian procedure to periodically update the inventory record. In K€ok and Shang (2007), the phenomenon of inventory inaccuracy is represented through random errors that change the physical inventory level at the end of each period. These errors keep accumulating, until the inventory record is updated by means of a costly inspection. They propose a near-optimal joint inventory inspection and replenishment heuristic and find that order quantities should increase as the number of periods since the last inspection increases, to accommodate for the added uncertainty. The inspection frequency should be higher for items with high value and larger error variance relative to demand variance. While explicitly addressing the impact of inventory record inaccuracy on inventory management, none of the above articles quantifies the value of inventory accuracy or the problem of RFID adoption. ¨ zer (2007) argue that most of the existing Dutta et al. (2007) and Lee and O estimates of the value of RFID are at best educated guesses, often lacking any comprehensible basis. Since most of these estimates are overly optimistic, the ¨ zer (2007) suggest that results of implementation are often frustrating. Lee and O this discrepancy between wild value propositions and sobering results has created a credibility gap and they call for the academic community to provide solid models to enable more realistic estimates of the RFID value. Atali et al. (2009) consider a single stage inventory control problem, where inventory records are inaccurate due to three additional demand streams: shrinkage, misplacement, and transaction errors. By prioritizing the different demand streams, they derive upper and lower bounds on the optimal solution as well as a simple heuristic. They compare the costs under inventory record inaccuracy with those under perfect transparency to obtain what they call the value of inventory visibility, which represents an upper bound on the benefits of RFID technology. Sahin (2004) considers a single-stage inventory system with inventory record inaccuracy. Considering different possible sources of such inaccuracies as well as different mathematical forms to represent these relations, Sahin investigates the consequences of inaccurate inventory records and thereby provides a quantifiable value for Auto ID technology. Rekik et al. (2008, 2009) study product loss through misplacement or theft. They demonstrate the importance of explicitly considering this problem in the stocking decision and derive threshold cost values, at which RFID adoption would become cost-effective. Karaer and Lee (2007) study the value inventory visibility in a manufacturer’s reverse channel. Their findings suggest that RFID technology might enable

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substantial benefits if the product return flow is highly volatile, if the duration of the reverse channel process is long, and if a significant portion of the returned products needs to be reworked. While these four articles explore the problem of inventory control under inaccurate inventory information and explicitly consider the role of RFID technology in mitigating the problem of inventory inaccuracy, their focus lies on the value of inventory visibility at a single stage or firm. In contrast, our work specifically focuses on the consequences of inaccurate inventory records in the presence of double marginalization in a decentralized supply chain, contrasting RFID adoption incentives of the individual supply chain stages with that of the integrated firm. Kang and Gershwin (2005) use simulation to study the consequences of inventory record inaccuracies in a setting where the inventory management process is automated. They find that even small undetected losses can lead to severe disruptions and stock-outs, especially in lean systems. They suggest several approaches to mitigate this problem, including the adoption of Auto-ID technology. Even though Auto-ID technology in their model enables full inventory transparency, their results suggest that simple heuristics that compensate for the stock loss can achieve near-optimal performance – at much lower cost. Fleisch and Tellkamp (2005) simulate the consequences of inventory record discrepancies in a three-stage supply chain. They find that an elimination of such inaccuracies, which could for example be achieved by RFID technology, can substantially reduce supply chain cost and stock-outs. While the simulation models of Kang and Gershwin (2005) and Fleisch and Tellkamp (2005) present a valuable first step in analyzing the value of inventory visibility in a supply chain, they assume a vertically integrated supply chain and hence cannot derive any insights with respect to the value of RFID adoption in a decentralized supply chain. Using simulation to estimate of the value of RFID technology in a four-stage supply chain under different degrees of collaboration, Sari (2010) finds that RFID adoption to be most beneficial in environments with longer lead times and limited demand uncertainty. Rekik et al. (2005) consider a supply chain with one manufacturer and one retailer both in the presence of inventory inaccuracies and under full inventory transparency due to RFID implementation. Comparing these two settings, they derive the value of RFID technology and they determine tag cost threshold values for profitable adoption. Similar to Camdereli and Swaminathan (2010), Rekik et al. (2005) also investigate the inefficiencies due to double marginalization in a decentralized supply chain and derive coordinating buy-back contracts. Gaukler et al. (2007) investigate the impact of RFID technology on a vertically-integrated supply chain vis-a`-vis a decentralized supply chain with one manufacturer and one retailer. They capture possible discrepancies between shelf inventory (available to satisfy customer demand) and backroom inventory through a parameter y that represents the efficiency of the retailer’s replenishment process. More specifically, they define y as the conditional probability that, given ample backroom inventory, a customer will find the product available on the retail shelf. If true demand is normally distributed, so is the effective demand that can be satisfied from the shelf.

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Assuming that neither the manufacturer nor the retailer include y in their decision making and that losing sales is the only penalty for empty shelves, the problem can be formulated as a standard newsvendor problem. Gaukler et al. (2007) analyze the benefits of full replenishment efficiency ðy ¼ 1Þ in a vertically-integrated supply chain and derive some insights into the threshold cost, at which RFID adoption would become profitable. While Rekik et al. (2005) and Gaukler et al. (2007) address problems that are similar to the one investigated in this article, their works differ in several important aspects. Most importantly, their research assumes that the impact of inventory record inaccuracy on the order yield is always negative, i.e., the retailer’s inventory available for sale is always less than ordered. However, it is reasonable to believe that inventory misplacements might eventually be corrected, implying available inventory that is larger than ordered in a specific period. Similarly, scanning errors will have a negative effect on the inventory records for some products and a positive for others (e.g., if scanning a vanilla yogurt as a strawberry yogurt). In such settings, RFID technology promises two potential benefits. On the one hand, it could help reduce the problem of shrinkage. On the other hand, RFID improves inventory transparency, reducing the uncertainty associated with inaccurate inventory records. We believe that these are two different value propositions, each of them important enough to merit attention. However, under the assumption that the source of inventory record inaccuracy always reduces order yield, the impact of inventory record inaccuracy can only be analyzed in the presence of such product losses, which are likely to have a confounding effect in estimating the value of RFID technology in achieving inventory transparency. The problem of uncertainty in inventory records cannot even be considered in a model where such inaccuracies are deterministic and known (as in Rekik et al. 2005). In our model, inventory record inaccuracies can be both positive or negative. Specifically, we capture this discrepancy between system inventory and available inventory by assuming that the ratio of the two follows a random variable. While this setup allows us to model shrinkage by assuming a mean below unity, we show that even in the absence of such losses, the mere uncertainty with respect to available inventory has a substantial impact on the supply chain. Gaukler et al. (2007) further assume that the retailer acted as if she was not aware of the inventory inaccuracy problem. This approach greatly simplifies the mathematical analysis, reducing it to a standard newsvendor problem. In our model, the retailer is conscious of the uncertainty with respect to inventory and she adjusts her orders accordingly, such that her execution problems also affect the manufacturer. As a consequence, in our setting both supply chain stages have an incentive to consider the costly adoption of RFID technology. Finally, the focus of Gaukler et al. (2007) is on the impact of RFID adoption on supply chain performance, rather than the adoption decision, while the question of how the adoption decision in a decentralized supply chain differs from that of an integrated supply chain is of central concern in our work.

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3 The Model In this section, we introduce our base model with its underlying assumptions. While we present the model for the integrated supply chain (which is analyzed in Sect. 4.1), the adjustments for the decentralized scenario (Sect. 4.2) are straightforward. A table with an overview of the notation is provided in Appendix 2. Generally, our model is based on the familiar newsvendor model. Given a starting inventory xn in period n, the decision maker needs to determine the optimal stocking quantity Qn in order to maximize expected (discounted) infinite-horizon profits. Demands in the different periods Dn are independent random variables with the identical support on ½0; D, density fD ðÞ and mean mD . A product is produced at cost c and offered for sale at a retail price r > c. We assume the firm has no market power and hence is a price-taker (i.e., the retail price r is exogenous). We do not explicitly consider penalty costs to capture the negative implications of shortages. Including such costs in our model would be straightforward and would not affect any of our insights. The key difference between our model and the standard newsvendor model with accurate inventory records is that the quantity available for sale to customers (shelf inventory) might differ from the planned stocking quantity. We assume that supply is reliable, so that such differences occur due to execution inefficiencies at the retailer, e.g., shrinkage, scanning errors, or SKU misplacement (cp. Sect. 1). Whereas some products might be lost forever due to theft, others might be physically present at the retailer (stored or lost in the backroom or on the wrong shelf), but not readily available for sale. However, in some periods these units can be made available to customers – since record inaccuracies are certainly not planned, such recoveries might come as a surprise and hence might actually increase available inventory beyond the planned stocking level. As a consequence, besides demand uncertainty, the decision maker faces uncertainty with respect to the validity of the inventory records. For example, let Q denote the number of units on the inventory record (and supposedly in the store) and let QA denote the number of units actually available for sales to the customers (on the right shelf). If inventory is misplaced, QA < Q. If those units are then found and placed at the right spot, it might also occur that QA > Q. On average, the available inventory should not be larger than the planned inventory, but it can be smaller, if there is shrinkage. Noting the obvious similarity to the problem of inventory management under yield uncertainty, we build on the model of Inderfurth (2004) and model the discrepancy between planned and available inventory in any given period through a stochastic multiplier Yn on ½0; Y with density fY ðÞ and mean mY 1. The random variable Y represents the ratio of available to recorded (planned) inventory, so on average there is no shrinkage if mY ¼ 1, while there is some loss if mY < 1. To achieve mathematical tractability in an environment complicated by uncertainty in inventory records, we assume that the ending inventory in each period is no larger than the desired stocking quantity of the next period (i.e., xn Qn ).

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Following the seminal work of Veinott (1965), this assumption has frequently been used in the inventory literature to achieve tractability in multi-period settings with non-perishable inventory. This assumption is always satisfied in traditional inventory models with accurate inventory records, if cost parameters are stationary and demand is stochastically non-decreasing, since optimal stocking levels are non-decreasing over time and leftover inventory can never exceed the intended stocking quantity. However, this assumption is non-trivial in our setting, where stochastic order yield might lead to leftover inventory in excess of the intended stocking quantity. With the assumption that left-over inventory in any given period is less than the optimal stocking quantity of the following period, consecutive periods can be decoupled by assuming that there is a salvage value s on end-of-period leftover inventory, which is equal to the (discounted) unit cost of the following period minus a potential per-unit per-period holding cost. Hence, under this assumption, consecutive periods are independent and can be analyzed separately, so that the analysis of a multi-period inventory model with inventory holding cost can be reduced to a single-period analysis with the appropriate salvage value and zero starting inventory, without any loss of generality. While we base our following analysis on a single-period model, we ask the reader to keep in mind that our results and insights apply to any period of a multi-period horizon, and thus also for the corresponding multi-period problem. Let pðQÞ denote the supply chain’s expected profit if ordering Q.1 The finite To obtain a continuously differentiable support of fD ðÞ implies a discontinuity at D. representation of the problem, we distinguish two cases depending on whether the Y. 2 order quantity Q is smaller (case A) or larger (case B) than D= D Case A : Q Y

(1)

0YQ 1 ð pA ðQÞ ¼ cQ þ @ ðrD þ sðYQ DÞÞfD ðDÞdDAfY ðYÞdY ðY

ðY

0

B þ @ 0

D Case B : Q > Y 1

0

ðD

0

1

C rYQfD ðDÞdDAfY ðYÞdY

YQ

(2)

Analyzing a single-period model with zero starting inventory, we will use the terms order and stocking quantity interchangeably. 2 In the following, the cases where inequality conditions are satisfied with equality are assigned arbitrarily, without loss of generality.

Inventory Record Inaccuracy, RFID Technology Adoption

0YQ D=Q ð ð @

pB ðQÞ ¼ cQ þ 0

ðY þ D=Q D=Q ð

þ 0

491

1 ðrD þ sðYQ DÞÞfD ðDÞdDAfY ðYÞdY

0

0 D 1 ð @ ðrD þ sðYQ DÞÞfD ðDÞdDAfY ðYÞdY 0 B @

0

ðD

1 C rYQfD ðDÞdDAfY ðYÞdY

YQ

It is easy to confirm that the expected profit functions are strictly concave and to derive sufficient first-order conditions. However, analytical closed form expressions can only be obtained for special types of demand and yield distributions, for example the uniform and the exponential distribution (Inderfurth 2004). To obtain analytically tractable and crisp results, we assume that both yield and demand are uniformly distributed. Numerical studies based on normally distributed demand and yield largely confirmed our analytical results. With these assumptions, (1) and (2) can be simplified to the following expressions. Case A : Q

mD : mY

pA ðQÞ ¼ ðrmY cÞQ

ðr sÞmY 2 2 Q 3mD

Case B : Q >

mD : mY

pB ðQÞ ¼ ðr sÞmD ðc smY ÞQ

ðr sÞmD 2 1 Q 3mY

(3)

(4)

In the following section, we analyze how inventory record uncertainty affects stocking decisions and supply chain performance, focusing specifically on the impact of double marginalization in a decentralized supply chain. Assuming that the adoption of RFID technology implies additional per-unit costs, but leads to full inventory transparency (the problem then becomes a standard newsvendor problem), we derive closed form solutions for the RFID cost thresholds that make adoption profitable.

4 Analysis In this section, we investigate the consequences of inventory uncertainty and we contrast RFID adoption incentives of an integrated supply chain with those of a decentralized supply chain. We first determine the optimal decisions and supply chain profits for the integrated (Sect. 4.1) and for the decentralized supply chain

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(Sect. 4.2). In Sect. 4.3, we derive and contrast the RFID cost thresholds for the two scenarios, below which adoption would be profitable. In the following, superscripts are used to indicate the concerned supply chain stage (M: manufacturer, R: retailer, SC: supply chain), and subscripts (1–4) are used to distinguish between the following four cases (1) an integrated supply chain with inaccurate inventory records, (2) an integrated supply chain with RFID, (3) a decentralized supply chain with inaccurate inventory records, and (4) a decentralized supply chain with RFID.

4.1

The Effect of Inventory Record Inaccuracy in an Integrated Supply Chain

We first derive the optimal order decisions and resulting profits for an integrated supply chain that experiences inventory record inaccuracies. This supply chain faces average procurement/production cost of c=mY and, for notational conveY nience, we use the parameter a ¼ rc=m rs to denote its critical fractile. Proposition 1. The optimal order quantity for the integrated supply chain under 3a otherwise. D, if a 23 , and Q1B ¼ p1ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ D, inventory inaccuracy is Q1A ¼ 4m 2mY

Y

¼ Correspondingly, expected profits are qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ðrsÞðc=mY sÞ pSC , otherwise. 1B ¼ ðr sÞmD 2mD 3 pSC 1A

3ðrc=mY Þ2 4ðrsÞ

3ð1aÞ

mD , if a 23 , and

Reflecting the earlier distinction of two possible regimes [cp. cases A and B in (1) and (2)], we see that the rule for determining the optimal order quantity under inventory uncertainty differs depending on the specific value of the critical fractile ðaÞ.3 While the optimal order rule for small service levels (case A) has a similar structure as in the standard newsvendor problem, it is very different for larger critical fractiles (case B). Shrinkage (if mY < 1) generally has two conflicting effects on the optimal order quantity. On the one hand, one could expect shrinkage to increase the optimal order quantity to compensate for the expected loss in product. On the other hand, shrinkage increases the expected per unit production costs, with a decreasing effect on the target service level and hence the optimal order quantity. The net outcome of these two effects can be both positive or negative, depending on the specific values of the different cost parameters (see Lemma 1 in Appendix 1). For sufficiently high salvage value ðs > r=4Þ, both order quantities strictly increase in mY . If the salvage value is lower, Q1A is increasing-decreasing in mY , and Q1B is either decreasingincreasing (if s > c=2) or even strictly decreasing in mY ðs < c=2Þ. 3 While the specific value of two-thirds is characteristic of the uniform distribution, a threshold based stocking policy is likely for all demand distributions with finite support (see earlier discussion of the two cases). We observed similar threshold based stocking policies in numerical experiments for normally distributed demand and order yield.

Inventory Record Inaccuracy, RFID Technology Adoption 140

493

Q1B

120 100 80 60

Q1A

a = 2/3

40 20 0 20%

µY

100%

Fig. 1 Optimal order quantity of the integrated supply chain

Figure 1 illustrates the optimal order decisions and the contrast between the different stocking rules Q1A and Q1B for D ¼ 100, c ¼ 10, s ¼ 8, and r ¼ 50. Shrinkage lowers the critical fractile under inventory uncertainty, and increasing mY from 20 to 100% raises a from 0 to approximately 95%. Figure 1 demonstrates how the integrated supply chain’s optimal order quantity is raised for higher critical fractiles. Compared to the optimal order quantity for lower critical fractiles, the optimal order quantity for higher critical fractiles is always inflated to buffer against the comparatively high underage costs that might occur in the event of extremely low order yield (Q1B > Q1A , if a > 2=3). Lemma 1 in Appendix 1 proves that the validity of this observation is not limited to this specific numeric example. In this case, the threshold value a ¼ 2=3 corresponds to an average ratio between available inventory and system inventory of about mY ¼ 46%. It is interesting to analyze the optimal order quantity for the case without shrinkage (i.e., mY ¼ 1) and to compare it to the optimal order quantity Q^ for the corresponding newsvendor problem without inventory uncertainty (i.e., Y ¼ 1). It is well-known that Q^ ¼ aD and without shrinkage, deviations from this order quantity can only be due to the inherent inventory uncertainty. The results of Lemma 1 (see Appendix 1) show that without shrinkage ðmY ¼ 1Þ there is a threshold value 2=3 < a < 1, such that the optimal order quantity under inventory uncertainty is larger than the optimal order to the corresponding problem without inventory uncertainty, if and only if a > a, i.e., if the critical fractile is sufficiently large.4 For the specific example presented in Fig. 1, it can easily be verified that the optimal order quantity under inventory uncertainty is always higher than the optimal order quantity for the standard newsvendor problem without inventory uncertainty (Q^ ¼ aD increases from 0 to 95). Employing RFID technology has two effects on supply chain performance. On the one hand, this technology enables better (we assume perfect) inventory 4 This threshold value solves 12a2 ð1 aÞ ¼ 1, so a 89:6%. This threshold result has been mentioned in Inderfurth (2005).

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H.S. Heese

transparency, such that all inventory is available for sale.5 It can be easily confirmed that such accuracy increases expected supply chain profits. On the other hand, there are costs associated with RFID technology. In practice, the cost of item-level RFID is largely driven by the cost of the tags. Correspondingly we assume that employment of RFID increases the per-unit cost by a fixed amount t.6 Assuming full inventory transparency under RFID, the supply chain in this scenario (subscript 2) faces the standard newsvendor problem (cf. Porteus 2002) and expected profits are ðQ

ðD

pRFID ðQÞ ¼ ðc þ tÞQ þ ðrD þ sðQ DÞÞfD ðDÞdD þ rQfD ðDÞdD 0

(5)

Q

Proposition 2 provides the optimal order quantity and expected profits for the integrated supply chain under RFID employment. Proposition 2. The optimal order of the integrated supply chain with RFID 2 yields maximum expected profit pSC ¼ ðrctÞ mD . D employment Q2 ¼ rct 2 rs rs Since inventory transparency increases expected profits, there must be a threshold value for the RFID costs, below which adoption is profitable. Before analyzing these cost thresholds (in Sect. 4.3), in the following section (Sect. 4.2) we derive the optimal decisions for a decentralized supply chain.

4.2

The Effect of Inventory Record Inaccuracy in a Decentralized Supply Chain

As in the previous section, we first study the optimal decisions and resulting supply chain performance for the case of inaccurate inventory records, and then analyze the setting with RFID technology. The previous section studied the decisions in an integrated supply chain; here we assume that the Stackelberg manufacturer sets a wholesale price, anticipating the retailer’s corresponding optimal stocking decision. While the manufacturer is not directly affected by the inventory record inaccuracy, he is so indirectly through the retailer’s adjustments in her order decision. In a setting were the retailer does not consider the inventory uncertainty in making her order decisions (as in Gaukler et al. 2007), the only reason for her to adjust her orders under RFID would be a change in the per-item cost. However, if the retailer 5

An implication of this assumption is that RFID adoption eliminates shrinkage. While this effect can be an important driver of RFID adoption, much of the following discussion focuses on the potential benefit of RFID technology in reducing inventory uncertainty, assuming there is no shrinkage ðmY ¼ 1Þ. However, unless noted otherwise, all results are also valid for the case with shrinkage ðmY < 1Þ. 6 To avoid trivial cases we assume t < r c.

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495

is aware of the inventory uncertainty problem and adjusts the order quantity accordingly, the impact of inventory uncertainty (or RFID adoption) on the manufacturer benefits is not immediately clear. Proposition 3 describes the equilibrium wholesale price and order decisions as well as resulting expected profits for the decentralized supply chain with inventory uncertainty. Proposition 3. In a decentralized supply chain with inventory inaccuracies, the Y and the retailer orders manufacturer charges a wholesale price w3 ¼ cþrm 2 2 3a D. Manufacturer, retailer and supply chain profits are pM ¼ 3mD ðrc=mY Þ , Q3 ¼ pR3

¼

8mY 3mD ðrc=mY Þ2 16ðrsÞ

3

and

pSC 3

¼

9mD ðrc=mY Þ2 16ðrsÞ

8ðrsÞ

, respectively.

Interestingly, we find that in the decentralized supply chain with endogenous wholesale pricing, the optimal order decision no longer follows the two different rules encountered in the previous section, where we distinguished between regimes A and B depending on the critical fractile a. While for high critical fractiles the retailer would still order according to a different function (similar to Q1B in Proposition 1), with the double marginalization (Spengler 1950) under endogenous wholesale pricing she simply never faces such high critical fractiles. Even if the supply chain as a whole had a higher critical fractile (a > 2=3), the manufacturer’s mark-up above cost always reduces the retailer’s critical fractile sufficiently to induce an order following the rule for regime A. It can easily be seen that, for a 2=3, double marginalization in the decentralized supply chain reduces the order quantity to 50% of the quantity that maximizes integrated supply chain profits (Q1A ). We know from Proposition 1 that the integrated supply chain follows a different stocking rule (Q1B ) for higher critical fractiles and it can be shown that Q1B > Q1A for a > 2=3 (see Lemma 1 in Appendix 1). Hence, under inventory uncertainty, the integrated supply chain inflates the order quantity for high critical fractiles (compared to case A). For a > 2=3, the decentralized supply chain keeps following the same rule (case A) and hence orders even less than half of what the integrated supply chain orders. As a result, we find that for products with high critical fractiles inventory inaccuracies exacerbate the problem of double marginalization in a decentralized supply chain. We now explore the benefits of RFID employment on the decentralized supply chain with double marginalization. Similar to Proposition 2, Proposition 4 provides the equilibrium decisions and resulting performance measures after RFID adoption for the decentralized supply chain. Proposition 4. In a decentralized supply chain with RFID technology, the manufacturer charges a wholesale price w4 ¼ rþcþt and the retailer orders Q4 ¼ 2 2 1 rct D. Manufacturer, retailer and supply chain profits are pM ¼ ðrctÞ mD , 2

pR4

rs Þ2 ¼ ðrct 4ðrsÞ

4

mD and

pSC 4

¼

3ðrctÞ2 4ðrsÞ

2ðrsÞ

mD , respectively.

Recall that the costs of RFID technology are borne by the manufacturer, but that he can, with endogenous wholesale pricing, pass some of these costs on to the

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H.S. Heese

retailer. Comparing the wholesale prices before and after RFID adoption, we see YÞ , so the manufacturer’s surcharge on the wholesale price that w4 w3 ¼ tþrð1m 2 increases with the severity of the retailer’s pre-RFID shrinkage problem (it decreases in mY ). If there is no shrinkage at the retailer (i.e., if mY ¼ 1), the costs of RFID technology are split evenly. Consistent with the earlier observation, we find that the decentralized supply chain again orders exactly half of what the integrated supply chain would order. However, since there is only one stocking rule for the case with inventory transparency, here the impact of double marginalization on the optimal order quantity is consistent for all values of a. The fact that RFID adoption restores this proportionality of order quantities for high critical fractiles hints at the coordinating effect of RFID adoption. In the following section we derive and analyze the adoption thresholds for the integrated and the decentralized supply chain, and we show that the inventory transparency following RFID adoption might indeed improve supply chain coordination.

4.3

Double Marginalization and RFID Adoption

Proposition 5 provides the RFID cost threshold below which the integrated supply chain would find RFID adoption profitable. Proposition 5. An integrated supply chain would adopt RFID technology pﬃﬃ for t < tA ¼ ðr cÞ 23 ðr c=mY Þ if a 23 , and for t < tB ¼ ðr cÞ rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ qﬃﬃﬃﬃﬃﬃﬃ ðr sÞ 1 2 1a 3 , otherwise. As expected, we see that the benefit of RFID adoption differs depending on the balance of underage and overage costs, captured by the supply chain’s critical fractile a. It can be easily verified that both cost thresholds are positive, confirming the value of inventory transparency. The following Proposition 6 provides the analogue to Proposition 5 for the decentralized supply chain. Proposition 6. In a decentralized supply chain where the manufacturer set the wholesale price as a Stackelberg leader, both the retailer and the manufacturer are better off with RFID technology if and only if t < tA . Interestingly, we find that with endogenous wholesale pricing, the retailer’s and the manufacturer’s RFID adoption decisions are perfectly aligned, i.e., the retailer is better off under RFID adoption if and only if the manufacturer is better off. As a consequence, the (leading) manufacturer’s adoption decision is also optimal in terms of total supply chain profits. Surprisingly, Proposition 6 also shows that, if the critical fractile is relatively low ða 2=3Þ, the cost threshold of the decentralized supply chain is identical to that of the integrated supply chain. Consequently, the double marginalization in the

Inventory Record Inaccuracy, RFID Technology Adoption

497

decentralized supply chain in this case does not distort the evaluation of RFID. However, adoption decisions are different for higher critical fractiles ða > 2=3Þ, as described in the following Proposition 7, which provides one of the key insights of this chapter. Proposition 7. For a > 2=3, the cost threshold for profitable RFID adoption is strictly larger in a decentralized supply chain than in an integrated supply chain ðtA > tB Þ. No matter if the RFID adoption is undertaken by the manufacturer or mandated by the retailer, as long as the manufacturer is able to adjust wholesale prices as Stackelberg leader, for products with a sufficiently high critical fractile, the decentralized supply chain has a larger incentive to adopt RFID than the integrated supply chain, as it extracts a higher relative benefit from RFID technology. As a consequence, the performance gap due to double marginalization is decreased and RFID adoption improves supply chain performance. Figure 2 illustrates the difference between the two cost thresholds and contrasts the RFID adoption decisions of an integrated and a decentralized supply chain. For this graph, we use the same parameter values as in Fig. 1 (D ¼ 100, c ¼ 10, s ¼ 8, and r ¼ 50). Shrinkage in our model increases the expected procurement cost and thus has a significant impact on the supply chain’s critical fractile. For this specific example, the critical fractile a increases from 0 to approximately 95% as the average ratio of available-to-planned inventory mY increases from 20 to 100%. For settings with substantial shrinkage and relatively smaller critical fractiles under inventory uncertainty (a 2=3 for mY < 46%), we see that the incentives of the integrated and the decentralized supply chain are perfectly aligned ðtA ¼ tB Þ. The two threshold values are the same, even if the product would have a high critical fractile in the absence of shrinkage. However, for higher critical fractiles under inventory record inaccuracy, the decentralized supply chain extracts a higher value from RFID and finds RFID 40

8

35

tA

7 30

a=2/3

6

25 20

5 tA = tB

4

tA > tB

15

3 80%

10 5 0 20%

µY

Fig. 2 Cost thresholds for profitable RFID adoption

tB

100%

µY

100%

498

H.S. Heese

adoption profitable at a higher per-unit cost compared to the cost threshold of the integrated supply chain ðtA > tB Þ. In settings with very high shrinkage, the value of RFID as a tool to prevent such losses is substantial (for this specific numeric example we have tA ¼ tB ¼ 40 for mY ¼ 20%). In order to better illustrate the value of RFID in reducing the uncertainty in inventory records, in the small frame of Fig. 2, we contrast the threshold values for higher values of mY (from 80 to 100%). We see that the relative difference between the two thresholds might be substantial for such settings with minor shrinkage problems (higher mY ) – arguably the case in most real life instances. In the absence of shrinkage ðmY ¼ 1Þ, the integrated supply chain in this example would invest up to tB ¼ 3:67 per unit into RFID technology, whereas the decentralized supply chain would find RFID adoption profitable up to a significantly higher cost threshold of tA ¼ 5:36. Why does the decentralized supply chain gain proportionally the same from RFID adoption for relatively low critical fractiles ða 2=3Þ, but more if the critical fractile is higher ða > 2=3Þ? As discussed before, if the critical fractile is relatively low, both the decentralized and the integrated supply chain increase their orders by the same fraction ðQ3 =Q1A ¼ Q4 =Q2 ¼ 1=2Þ, and both supply chains make the same RFID adoption decision (Proposition 6). For higher critical fractiles however, the optimal order quantity for the integrated supply chain is proportionally larger under inventory inaccuracies (Q1B > Q1A , see Lemma 1 in Appendix 1). While both the integrated and the decentralized supply chain make the same proportional adjustments to their optimal order quantities when employing RFID in settings with relatively low optimal service levels, we see that with inaccurate inventory information for higher critical fractiles, the integrated supply chain inflates the optimal order quantity (cp. Fig. 1). Since the order quantities are proportionally the same for both supply chain structures with inventory transparency, the change after RFID adoption is proportionally larger for the decentralized supply chain, counteracting the problem of double marginalization. Note that improved supply chain coordination in our model is a result of individually rational RFID adoption decisions, and coordination was not an explicit objective at the outset. Clearly the performance of the decentralized supply chain could be further improved, for example by means of buy-back contracts. However, the focus of our research lies on RFID adoption decisions under inventory record inaccuracy. The general problem of supply chain coordination has been addressed by a wide body of research (see Cachon (2003) for a review).

5 Conclusion The discrepancy between inventory records and the amount of product effectively available for sale to customers presents a key problem in retail operations. Promising full transparency, RFID technology is often proposed as a remedy to this problem. However, RFID technology is not free of cost, so an optimal

Inventory Record Inaccuracy, RFID Technology Adoption

499

adoption decision needs to quantify the achievable benefits and balance them against these costs. We consider a simple supply chain model, where a Stackelberg manufacturer sets the wholesale price and a retailer determines how much to stock for sale to customers. Besides demand uncertainty, the retailer faces uncertainty with respect to her inventory records, i.e., she is uncertain by how much available stock differs from the planned stocking quantity. We first analyze how inventory record uncertainty affects optimal stocking decisions in an integrated supply chain. We find that in the presence of such inaccuracies, the optimal stocking rule can follow two different types, depending on the balance between underage and overage costs. For relatively low critical fractiles, the optimal stocking rule resembles the solution to the standard newsvendor problem without inventory inaccuracies. However, for higher critical fractiles, we find that the optimal stocking quantity for the integrated supply chain is increased to reduce the risk of costly stock-outs in the event that available inventory turns out to be much lower than system inventory. We then determine the optimal wholesale price and stocking decisions in the decentralized supply chain. We show that with endogenous wholesale pricing by a Stackelberg manufacturer, the manufacturer’s and the retailer’s RFID adoption decisions are perfectly aligned, i.e., the manufacturer is better off with RFID technology if and only if the retailer is better off. As a consequence, the manufacturer’s optimal RFID adoption decision is also optimal in terms of total supply chain profits. Interestingly, we find that with endogenous wholesale pricing, the critical fractile to the retailer is never sufficiently high to warrant an stocking quantity increase as encountered for the integrated supply chain. In the absence of inventory record uncertainty, both supply chains resort to stocking rules of the same structure and our findings are consistent with the classic result that the stocking quantity of the decentralized supply chain with double marginalization is half of that of the integrated supply chain. We show that this proportionality of stocking quantities is maintained under inventory record uncertainty, as long as the (integrated) supply chain’s critical fractile is sufficiently low. However, for higher critical fractiles the (integrated) supply chain’s optimal stocking quantity is increased. Since the decentralized supply chain with double marginalization fails to adjust stocking quantities sufficiently to maintain proportionality, we find that for relatively high critical fractiles, the double marginalization in a decentralized supply chain may exacerbate the negative consequences of inventory record inaccuracy. Analyzing the cost thresholds for profitable RFID adoption, interestingly we find that for low critical fractiles the incentives for RFID adoption in the decentralized supply chain are perfectly aligned with those in the integrated supply chain, such that both supply chains would find RFID technology attractive at the same cost. However, for products with higher critical fractiles, the relative benefit of RFID is larger for the decentralized supply chain and consequently adoption would become profitable at a strictly higher cost. The rationale behind this surprising finding lies in the decentralized supply chain’s failure to sufficiently increase stocking quantities

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H.S. Heese

under inventory record inaccuracy. Since RFID adoption eliminates the need for such stocking quantity inflation and restores the proportionality of stocking quantities of the two supply chains, it mitigates the consequences of double marginalization and hence improves coordination of the decentralized supply chain.

Appendix 1: Additional Results and Proofs Lemma 1. (a) Q1A is increasing in mY for 0 < a 2=3, if s > r=4, and increasing-decreasing otherwise. (b) Q1B is increasing in mY (for 2=3 < a < 1), if s > r=4, decreasing if s < c=2, and decreasing-increasing otherwise. ^ and Q1B > Q^ , a > a, where 2=3 < a < 1. (c) If mY ¼ 1, then Q1A < Q, (d) If a > 2=3, then Q1B > Q1A . Proof of Lemma 1. 2c 2 3c 1A Part (a): @Q @mY > 0 , mY < r ; a 3 , mY rþ2s . Q1A is increasing in mY for 3c 2c r 0 < a 2=3, if rþ2s < r , s > 4 , and increasing-decreasing otherwise. c 2 3c 1B Part (b): @Q @mY > 0 , mY > 2s ; a > 3 , mY > rþ2s . Q1B is increasing in mY on 3c 2=3 < a < 1, if 2sc < rþ2s , s > 4r , decreasing if 2sc > 1 , s < 2c , and decreasingincreasing otherwise. 1 ﬃ < a , 12a2 ð1 aÞ > 1. Part (c) Q1A < Q^ by inspection; Q1B < Q^ , pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 3ð1aÞ The left side of this inequality is decreasing in a(for a > 2=3). At a ¼ 2=3, it is equal to 16=9 > 1, and it is equal to zero at a ¼ 1. Hence there exists a 2 23 ; 1 , where the inequality is satisfied with equality. 3mD a 1 9a2 D ﬃ> Part (d) Q1B >Q1A , pmﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2mY , 3ð1aÞ > 4 (since both sides of the mY 3ð1aÞ □ inequality are positive) ,14ð3aþ1Þ ð23aÞ2 >0,a6¼23. Proof of Proposition 1. The optimal order quantities are derived in Inderfurth (2004). The resulting expected profits can be obtained by substituting these quantities into (3) and (4). □ Proof of Proposition 2. The optimal order quantity Q2 is the well-known critical fractile solution to the classic newsvendor model. The expected profits follow from □ substituting Q2 into the expected cost function (5) and simplification. Proof of Proposition 3. The retailer’s optimal order quantities are as given in 3 =mY Proposition 1, but with a critical fractile of a3 ¼ rwrs instead of a. The M manufacturer’s profit equals p3 ¼ ðw3 cÞQ3 . Two cases need to be distinguished 3 ¼ ðr þ 2sÞ m3Y to denote the depending on the resulting value of a3 . Use w

Inventory Record Inaccuracy, RFID Technology Adoption

501

3 (i.e., a3 2=3), a3 ¼ 2=3. For w3 w @ 2 pM @pM 3mD rw3A =mY 3mD 3A ðw3A Þ 3A ðw3A Þ ¼ ðw3A cÞQ3A ¼ ðw3A cÞ 2m < 0; @w ¼ ; @w3A 2 ¼ ðrsÞm rs 3A Y Y cþrmY 3 , a3 jw3A < 2=3 , 3ðc mY sÞ þ ðr sÞmY > 0. w3A > w For 0 ! w3A ¼ 2 ; qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ @pM ðw Þ mD 3B M rs 3B 3 (i.e., a3 > 2=3), p3B ¼ ðw3B cÞQ3B ¼ ðw3B cÞ m 3ðw3B =m sÞ; @w w3 < w ¼ 3B Y Y wholesale

price

at

which

pM 3A

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

mD cþw3B 2smY 1 3ð1a3 Þ 2w3B 2smY mY M pM 3A a3 ¼2=3 ¼ p3B a3 ¼2=3

3 (both cases are equivalent at w 3 ). Since > 0 ! w3B ¼ w 3 , the optimal wholesale price is w3 ¼ w3A . and w3A > w

Substituting this optimal wholesale price gives the expressions in the Proposition.□ Proof of Proposition 4. The retailers optimal order quantity Q4 is the standard critical fractile solution to the newsvendor problem in (5) – the critical fractile 4 The manufacturer’s profit equals pM is rw 4 ¼ ðw4 ðc þ tÞÞQ4 ¼ rs . rw @ 2 pM @pM 4mD rþcþt 4 ðw4 Þ 4 ðw4 Þ 4 ðw4 ðc þ tÞÞ2mD rs ; @w4 ¼ 0 ! w4 ¼ 2 . @w4 2 ¼ ðpþrsÞ < 0; Substitution of the optimal wholesale price and simplification gives the expressions in the Proposition. □ 2

2

ðrctÞ 3ðrc=mY Þ SC Proof of Proposition 5. If a 23 : pSC 2 > p1A , rs mD > 4ðrsÞ mD p ﬃﬃ ﬃ 2 2 , 4ðr c tÞ > 3ðr c=mY Þ , 2ðr c tÞ > 3ðr c=mY Þ (both sides are pﬃﬃ Þ2 SC positive) , t < ðr cÞ 23 ðr c=mY Þ; If a > 23 : pSC , ðrct 2 > p1B rs mD > qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Y sÞ ðr sÞmD 2mD ðrsÞðc=m 3 qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ qﬃﬃﬃﬃﬃﬃ qﬃﬃﬃﬃﬃﬃ rct2 12=3 1a 1a , rs > 1 2 3 1 2 3 > 1 2 ¼ 1=3 > 0 3

, rct rs >

rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ qﬃﬃﬃﬃﬃﬃﬃ 12

1a 3

, t < ðr cÞ ðr sÞ

rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ qﬃﬃﬃﬃﬃﬃﬃ 12

1a 3 .

□

M Proof of Proposition 6. The inequalities pR4 > pR3 and pM 4 > p3 (hence by definition SC SC also p4 > p3 ) can both easily be transformed to 4ðr c tÞ2 > 3ðr c=mY Þ2 , which is the same inequality as in the proof of Proposition 5 for case A. □ pﬃﬃ Proof of Proposition 7. tA > tB , ðr cÞ 23 ðr c=mY Þ > ðr cÞ ðr sÞ rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ qﬃﬃﬃﬃﬃﬃﬃ qﬃﬃﬃﬃﬃﬃﬃ pﬃﬃ qﬃﬃﬃﬃﬃﬃ 3 1a 1a a , 1 2 > 3 a2 (both sides of the > 1 2 3 , 1 2 1a 2 3 q3ﬃﬃﬃﬃﬃﬃ 4 inequality are positive for a > 2=3) , 1 34 a2 > 2 1a 3 (both sides of the inequal 3 3 2 9 4 4 9 ð2 þ aÞ a 23 > ity are positive for a > 2=3) , 1 2 a þ 16 a > 3 43 a , 16

0 , a > 23 :

□

502

H.S. Heese

Appendix 2: Overview of Notation Symbol c r s t Y U½0; Y mY D U½0; D mD w Q p

a

Description Unit cost Unit revenue Unit salvage value RFID tag cost (per unit) Yield (random variable) Mean yield Demand (random variable) Mean demand Unit wholesale price (decision variable) Order quantity (decision variable) Expected profit Superscript (M ¼ manufacturer; R ¼ retailer; SC ¼ supply chain) Critical fractile for the supply chain

References Alexander K, Birkhofer G, Gramling K, Kleinberger H, Leng S, Moogimane D, Woods M (2002) Focus on retail: applying Auto-ID to improve product availability at the retail shelf. White paper, Auto-ID Center, MIT, Cambridge Anderson Consulting (1996) Where to look for incremental sales gains: the retail problem of outof-stock merchandise. The Coca-Cola Retailing Research Council, Atlanta ¨ zer O ¨ (2009) If the inventory manager knew: value of visibility and RFID under Atali A, Lee HL, O imperfect inventory information. Working paper, Stanford University, Stanford, CA Cachon GP (2003) Supply chain coordination with contracts. In: Graves S, de Kok T (eds) Handbooks in OR/MS: supply chain management: design, coordination and operation. North-Holland, Amsterdam, pp 229–239 Camdereli AZ, Swaminathan JM (2010) Misplaced inventory and radio-frequency identification (RFID) technology: information and coordination. Prod Oper Manage 19(1):1–18 DeHoratius N, Raman A (2008) Inventory record inaccuracy: an empirical analysis. Manage Sci 54(4):627–641 DeHoratius N, Mersereau A, Schrage L (2008) Retail inventory management when records are inaccurate. Manuf Serv Oper Manage 10(2):257–277 Dutta A, Lee HL, Whang S (2007) RFID and operations management: technology, value and incentives. Prod Oper Manage 16(5):646–655 Fleisch E, Tellkamp C (2005) Inventory inaccuracy and supply chain performance: a simulation study of a retail supply chain. Int J Prod Econ 95(3):373–385 Gaukler GM, Seifert RW, Hausman WH (2007) Item-level RFID in the retail supply chain. Prod Oper Manage 16(1):65–76 Gruen TW, Corsten DS, Bharadwaj S (2002) Retail out-of-stocks: a worldwide examination of extent, causes, and consumer responses. The Grocery Manufacturers of America, Washington, DC Inderfurth K (2004) Analytical solution for a single-period production-inventory problem with uniformly distributed yield and demand. Cent Eur J Oper Res 12:117–127

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Inderfurth K (2005) Incorporating demand and yield uncertainty in advanced MRP systems. In: Lasch R, Janker CG (eds) Logistik management – innovative logistikkozepte. Deutscher Universitaets-Verlag, Wiesbaden Kang Y, Gershwin SB (2005) Information inaccuracy and inventory systems: stock loss and stockout. IIE Trans 37(9):843–859 ¨ , Lee HL (2007) Managing the reverse channel with RFID-enabled negative demand Karaer O information. Prod Oper Manage 16(5):625–645 K€ ok AG, Shang KH (2007) Inspection and replenishment policies for systems with inventory record inaccuracy. Manuf Serv Oper Manage 9(2):185–205 ¨ zer O ¨ (2007) Unlocking the value of RFID. Prod Oper Manage 16(1):40–64 Lee HL, O McFarlane D, Sheffi Y (2003) The impact of automatic identification on supply chain operations. Int J Logistics Manage 14(1):1–17 Porteus EL (2002) Foundations of stochastic inventory theory. Stanford University Press, Stanford, CA Raman A, DeHoratius N, Ton Z (2001) Execution: the missing link in retail operations. Calif Manage Rev 43(3):136–152 Rekik Y, Jemai Z, Sahin E, Dallery Y (2005) Involving the performance of retail stores subject to execution errors: coordination versus Auto-ID technology. Technical report, LGI – Ecole Centrale Paris, Paris Rekik Y, Sahin E, Dallery Y (2008) Analysis of the impact of the RFID technology on reducing product misplacement errors at retail stores. Int J Prod Econ 112(1):264–278 Rekik Y, Sahin E, Dallery Y (2009) Inventory inaccuracy in retail stores due to theft: an analysis of the benefits of RFID. Int J Prod Econ 118(1):189–198 Sahin E (2004) A qualitative and quantitative analysis of the impact of the auto ID technology on the performance of supply chains. PhD thesis, LGI – Ecole Centrale Paris, Paris Sari K (2010) Exploring the impacts of radio frequency identification (RFID) technology on supply chain performance. Eur J Oper Res 207(1):174–183 Spengler JJ (1950) Vertical integration and antitrust policy. J Polit Econ 58(4):347–352 Veinott AF (1965) Optimal policy for a multi-product, dynamic, nonstationary inventory problem. Manage Sci 12(3):206–222 Want R (2004) RFID: a key to automating everything. Sci Am 290(1):56–65 Yano CA, Lee HL (1995) Lot sizing with random yields: a review. Oper Res 43(2):311–334

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Possibilistic Mixed Integer Linear Programming Approach for Production Allocation and Distribution Supply Chain Network Problem in the Consumer Goods Industry Bilge Bilgen

Abstract In the consumer goods industry there is an ongoing trend towards an increased product variety and shorter replenishment cycle times. Hence, manufacturers seek a better coordination of production and distribution activities. Our study is motivated by the production-distribution problem encountered by a soft-drink company operating in consumer goods industry. The problem is to determine the optimal allocation of products and routing decisions for a multiechelon supply chain to minimize the total supply chain cost comprising of production, setup, inventory and distribution costs. A mixed integer linear programming (MILP) model is proposed to describe the optimization problem. However, a real supply chain operates in a highly dynamic and uncertain environment. The ambiguity of cost parameters is considered in the objective function of the model. The proposed approach uses the strategy of minimizing the most possible cost, maximizing the possibility of obtaining lower cost, and minimizing the risk of obtaining higher cost. Zimmermann’s fuzzy multi objective programming method is then applied for achieving an overall satisfactory compromise solution. The applicability of the proposed model is illustrated through a case study in consumer goods industry.

1 Introduction A supply chain system is comprised of all organizations that are involved in transforming raw materials to a final product. Supply chain management has received considerable attention from academicians and practitioners during the last several decades. In today’s world fast economic changes and the increasing pressure of market competition lead firms to focus on integrated supply chains. The coordination and integration of the production (supply), inventory, and distribution B. Bilgen Department of Industrial Engineering, Dokuz Eylul University, 35160 Izmir, Turkey e-mail: bilge.bilgen@deu.edu.tr 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_20, # Springer-Verlag Berlin Heidelberg 2011

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(demand) operations is widely perceived to be a route to obtain a competitive advantage. Arshinder et al. (2008) present a survey and classification of studies on supply chain coordination found in the literature. In the consumer goods industry the focus in production planning and scheduling is shifting from the management of plant-specific operations to a holistic view of the entire supply chain comprising value adding functions like purchasing, manufacturing and distribution. A closer coordination of production and distribution activities is required in order to avoid excessive inventories at the manufacturers’ warehouses. While traditionally minimizing production costs has been considered as the major objective, attention has shifted towards faster replenishment and improved logistical performance. Thus, finished product inventories are merely regarded as buffers between the manufacturing and the distribution stage of the supply chain. As a result, distribution costs have to be included in the overall objective function (Bilgen and Guenther 2010). The dynamic and complex nature of supply chain imposes a high degree of uncertainty in supply chain planning decisions and significantly influences the overall performance and effectiveness of the configuration and coordination of supply chain network (Klibi et al. 2010). The significance of accounting for uncertainty has prompted the researchers to address uncertain parameters in supply chain planning. Several authors have analyzed the sources of uncertainty present in a supply chain. In a recent review Peidro et al. (2009a) have presented an analysis of the literature on supply chain planning under uncertainty conditions by adopting quantitative approaches. In their review, some of the strengths and weaknesses of the approaches currently used have been pointed. Interested reader could refer to excellent reviews of Klibi et al. (2010) and Peidro et al. (2009a) regarding the review of supply chain network design problems under uncertainty, and review of quantitative models for supply chain planning under uncertainty, respectively. In this chapter, the optimal design and operation of a multi-product, multiperiod, multi-echelon, supply chain consisting of multiple manufacturers, multiple production lines and multiple distribution centers is considered. The problem is to assign products to the production lines, and to determine the routes to be traveled to coordinate the production and transportation routing operations so that the customer demand, capacity constraints, production, and inventory constraints are all satisfied, while the resulting cost (i.e. the sum of the production, inventory, setup, and transportation costs) over a given planning horizon is minimized. The problem is formulated as a MILP model. Because of price fluctuations in a dynamic market, assigning crisp values for parameters is no longer appropriate for dealing ambiguous decision problems. Possibility distribution offers an effectual alternative for proceeding with inherent ambiguous phenomena in determining cost parameters. Therefore, in this study a possibilistic MILP model is developed. The purpose of this study is twofold: first to develop more relatively sophisticated MILP model able simultaneously to form production and distribution network in the consumer goods industry, secondly to demonstrate the usefulness and significance of the fuzzy programming through a possibilistic programming approach.

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The remainder of this chapter is organized as follows. The relevant literature is reviewed in Sect. 2. In Sect. 3 the key characteristics of the problem are outlined, and the model is described in detail. The proposed possibilistic mixed integer linear programming is elaborated in Sect. 4. The results of a numerical investigation which show the practical applicability of the developed possibilistic optimization model is presented in Sect. 5. Finally, concluding remarks and future directions for further studies are stated in Sect. 6.

2 Literature Review The efficient coordination of production and distribution systems becomes a challenging problem as companies move towards higher collaborative and competitive environments. In the academic literature, integrated production and distribution planning problem has been the subject of many studies during the last decade. For general in-depth description of the field of production and distribution planning in supply chain management, the reader is referred to Ereng€uc¸ et al. (1999), Sarmiento and Nagi (1999), Bilgen and Ozkarahan (2004), Stadler (2005), Chen and Vairaktarakis (2005) Arshinder et al. (2008), Peidro et al. (2010a), Chen (2010). Research effort on the supply chain coordination issues have been overwhelming and many fruitful research results have obtained. e.g. Chandra and Fisher (1994), Fumero and Vercellis (1999), Dhaenens-Flipo and Finke (2001), de Matta and Miller (2004), Park (2005), Lei et al. (2006), Cordeau et al. (2006), Eksioglu et al. (2007), Stadtler and Kilger (2008), Rizk et al. (2008), Elhedhli and Gzara (2008), Tsiakis and Papageorgiou (2008), and Bilgen and Guenther (2010). The models stated above assume the parameters that influence the design decisions to be deterministic. However, much of decision making in real world takes place in an environment where the objectives, constraints, or parameters are not known precisely. Several approaches take into account sources of uncertainty are arising in the area of supply chain management. These approaches can be roughly classified into analytic approaches, and simulation-based approaches (Guillen et al. 2005). On the other hand another classification is done by Peidro et al. (2009a) as analytic models, models based on artificial intelligence (AI), simulation based models and hybrid models. A number of researchers have proposed stochastic supply chain management models that are closer to real situations. Most research has modeled the supply chain uncertainty (e.g., uncertain demand) by probability distribution that is usually predicted from historical data. However, whenever statistical data is unreliable or even unavailable, stochastic models may not be the best choice. Fuzzy set theory may provide an alternative approach for dealing with the supply chain uncertainty (Lai and Hwang 1992a). Fuzzy set theory was proposed by Zadeh and has been found extensive applications in various fields such as operations research, management science, control theory and artificial intelligence. Fuzzy mathematical programming (FMP) is one of the most popular decision making approaches based on fuzzy set theory. Fuzzy sets theory has been

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implemented in mathematical programming since 1970 when Bellman and Zadeh (1970) introduced the basic concepts of fuzzy goals, fuzzy constraints, and fuzzy decisions. Many of the developments in the area of FMP are based on this seminal paper. A detailed discussion of the FMP procedures can be found in (Lai and Hwang 1992a; Zimmermann 1996). In the context of FMP, two research directions that are being pursued are flexible programming, and possibilistic programming. Flexibility is modeled by fuzzy sets and may reflect the fact that constraints and goals are linguistically formulated. Some applications of flexible programming can be found in Tsai et al. (1997), Miller et al. (1997), and Mula et al. (2006). On the other hand there is uncertainty, corresponding to an objective variability in the model parameters (randomness), or a lack of knowledge of the parameter values (epistemic uncertainty). Epistemic uncertainty is concerned with ill-known parameters modeled by fuzzy intervals in the setting of possibility theory (Mula et al. 2007). Different applications of possibilistic programming approach can also be found in the literature. Hsu and Wang (2001) develop a possibilistic linear programming model to determine appropriate strategies regarding the safety stock levels for assembly materials, regulating dealers’ forecast demands, and number of key machines. Wang and Liang (2005) present a novel interactive possibilistic programming approach for solving the multi-product aggregate planning problem with imprecise forecast demand, related operating costs, and capacity. In a more recent work, Sakalli et al. (2010) develop a possibilistic aggregate production planning (APP) model for blending problem in a brass factory. In the proposed model, the Lai and Hwang’s fuzzy ranking concept is relaxed by using either-or constraints. The important works concerning application of possibilistic programming in SCM are presented by Wang and Shu (2005, 2007). Other applications of possibilistic programming in production and distribution planning problems can be found in Chen and Chang (2006), Mula et al. (2007), Liang (2008), Torabi and Hassani (2008a), Liang and Cheng (2009), Mula et al. € (2010), Kabak and Ulengin (2011). In recent years there is a significant growth in the study of production and distribution models using the FMP approach. In a pioneering study, Sakawa et al. (2001) have presented fuzzy programming for the production and transportation planning in the light of obscure estimation of parameters. Chen and Chang (2006) simultaneously handle multi-product, multi-echelon, and multi-period supply chain model with fuzzy parameters. And they propose a solution procedure that is able to calculate the fuzzy objective value of the fuzzy supply chain model. Subsequent research on FMP include by Aliev et al. (2007). They develop a fuzzy integrated multi-product, multi-period production and distribution planning model within supply chain. The model is formulated in terms of fuzzy programming and solution is provided by Genetic Algorithm (GA). Torabi and Hassani (2008a) develop a novel multi-objective possibilistic MILP model for a supply chain master planning problem consisting of multiple suppliers, one manufacturer and multiple

Possibilistic Mixed Integer Linear Programming Approach for Production Allocation

509

distribution centers. Their model integrates the procurement, production and distribution plans considering various conflicting objectives simultaneously as well as the imprecise nature of some impractical parameters such as market demands, cost/time coefficients, and capacity levels. In another study they develop a more comprehensive multi-objective supply chain master planning framework for a multi-echelon supply chain by extending their previous work. They propose a FGP model which is solved through a novel auxiliary crisp formulation (Torabi and Hassani 2008b). Liang (2008) presents an interactive fuzzy multi-objective linear programming (FMOLP) model for solving integrated production and transportation problem with multiple fuzzy goals in fuzzy environments. Liang and Cheng (2009) proposes a FMOLP model to simultaneously minimize total costs and total delivery time with reference to inventory levels, available machine capacity and labor levels at each source, as well as market demand and available warehouse space at each destination, and the constraint on total budget. More recently, Peidro et al. (2009b, 2010b) adopt two well-known fuzzy programming approaches to solve the tactical supply chain planning problem arises in the automobile industry. The fuzzy model integrally handles all the epidemistic uncertainty sources identified in tactical supply chain planning problem. The proposed model in their papers integrates procurement, production, and distribution planning activities into a multi-echelon, multi-product, multi-level and multi-period supply chain network. Bilgen (2010) apply flexible programming approach on the integrated production and distribution planning problem within consumer goods industry. The effects of different fuzzy operators on the model under different demand granularity levels are investigated. Jolai et al. (2010) consider a multi-objective, multi-product, and multi-period collaborative production and distribution planning model using Fuzzy Goal Programming (FGP) approach. A simple genetic algorithm, a particle swarm optimization (PSO) and hybrid GA are developed to solve the problem. Kabak and € Ulengin (2011) propose a possibilistic linear programming model to make strategic resource planning decisions in the SC context. In its current form, the proposed model is expected to provide an important guide to SC managers in preparing their strategic plans, taking into account the fuzziness of long-term plans. Table 1 summarizes the relevant studies. In this chapter, we present a novel model consisting of multiple production lines, multiple plants, and multiple distribution centers considering imprecise nature of the integrated production and distribution planning problem. Our study also extends this literature in terms of the model scope. It enhances the above mentioned studies by considering detailed distribution routing, minor and major setup time and costs, and assignments of products to production lines. Its main advantage lies in its ability to simultaneously coordinate production allocation and transportation operations of the entire planning horizon. The main contribution of this chapter is the integration of production allocation and distribution supply chain network problem through a possibilistic programming approach, accompanied by experiments on real data in consumer goods industry.

S S

M M M M

S

S M S

S

S

Chen and Chang (2006) Aliev et al. (2007)

Torabi and Hassani (2008a) Torabi and Hassani (2008b) Liang (2008) Liang and Cheng (2009)

Peidro et al. (2009b)

Mula et al. (2010) Jolai et al. (2010) Bilgen (2010)

€ Kabak and Ulengin (2011)

Proposed research

M

M

M M M

M

M M S S

M M

S

M

M

M M M

M

M M S M

M M

M

Number of products

M

S

M M M

M

M M S M

M M

S

Number of periods

T

S/T

T T T

T

T T S T

T T

T

Decision level

þ

þ

þ þ

þ

þ

þ

þ þ þ

þ þ

þ

þ

þ

Objective Constraints function

Source of uncertainty

S strategic, T tactical, O operational, S single, M multi, V vagueness, A ambiguity, H hypothetic

S

Sakawa et al. (2001)

Objective (S vs. M)

Number of echelons

Problem characteristics

Table 1 Literature on integrated production and distribution planning models using FMP

A A

þ þ

A V V

þ þ

A

þ

A A A A V A

þ þ

þ þ

V

Ambiguity vs. Parameters vagueness

Zimmermann’s approach Zadeh’s extentions principle Genetic algorithm Possibilistic programming FGP FMOLP FMOLP Chanas and Verdegay method Possibilistic programming FGP, GA, PSO Flexible programming Possibilistic programming Possibilistic programming

Solution approach

þ

þ

þ H þ

þ

H H þ þ

H þ

þ

Industrial application

510 B. Bilgen

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3 Problem Description and Model Formulation Our study is motivated by the production-distribution problem encountered by a soft-drink company, which has to decide routinely the best way of delivering a set of orders to its customers over a multi-day planning horizon. Often it is not economical to install production equipment at each plant for the entire portfolio of products due to the high investment costs. Therefore, dedicated lines for the production of a specific range of products are established at each plant. Each individual plant has two production lines. No plant can produce the whole product range but just a part of the product range. Each production line at each individual plant can be viewed as a single stage process capable of producing several product groups. Setup costs are incurred at each plant whenever a production line changes production to a different product group. Each plant has an attached distribution center (DC) which serves as a buffer for the local production, and storage for products, which cannot be produced at the corresponding plant. Each DC has to be able to deliver the whole range of products. Products which cannot be produced in the corresponding plant must be delivered to a DC from another plant or another DC. Delivery takes place by means of homogenous vehicles with limited capacity. The distribution of goods from the plants to the DCs occurs in one of two ways: on a straight-and-back basis, i.e. there is only one DC on a given delivery vehicle’s path, which is the case if the customer requirements constitute a full truck load (FTL), or in a route involving multiple DCs on the route have individual requirements less than a truckload (LTL). For LTL multiple customers are served in a single route. In this case, transportation costs only depend on the transportation distance, not on the specific load. Usually, optimization models for production and distribution planning are based on the simplifying assumption that transportation costs are linear with respect to quantity and distance. This assumption, however, is not rectified in real transportation systems since most often different modes of transportation can be used. Particularly, in the consumer goods industry road transport is the most preferred mode in the distribution of finished goods due to its high flexibility (Guenther and Seiler 2009). The problem is to assign products to the production lines, and to determine the routes to be traveled to coordinate the production and transportation routing operations so that the customer demand, capacity constraints, production, and inventory constraints are all satisfied, while the resulting cost (i.e. the sum of the production, inventory, setup, and transportation costs) over a given planning horizon is minimized.

3.1

Problem Assumptions and Notation

The following considerations further define and delimit the problem: • The supply network consists of several plants which deliver the final products to various distribution centers.

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• Each plant comprises several not necessarily identical production lines. Each line produces a given range of products. Multiple assignments of products to the lines are allowed. • For each product inventory balances at distribution centers are updated on a daily basis according to the production output from the various lines at the plants, the inbound and outbound transportation quantities, and the given external demand. • All vehicles used in LTL transportation are assumed to be identical. • No specific handling capacities and costs at distribution centers are considered. • Transportation activities are carried out within a single day. Nevertheless, lead times for long-distance transportation can be modeled simply by offsetting the time index of the respective decision variables. • Each vehicle can only travel according to pre-defined route with fixed operational cost. It is assumed that a vehicle can pick up products from a plant in a travel. Each vehicle is allowed to discharge cargo at most two distribution centers.

3.2

Model Formulation

The MILP model for the considered problem proposed by Bilgen (2010) is adopted as the basis of this work. It reflects major issues arising in the consumer goods industry, e.g. major and minor setups, daily demand assignments, use of different transportation modes etc. Nevertheless, its main features are also relevant for other types of industries. The sets, indices, parameters and variables used to formulate the problem mathematically are described below.

3.2.1

Notation

Sets I J Ji L P Lp IL W T Ji Rp Rw

Set of products (i ¼ 1,2,. . .,I) Set of product groups (j ¼ 1,2,. . .,J) Set of products that belong to group j Set of production lines (l ¼ 1,2. . .,L) Set of plants (p ¼ 1,2,. . .,P) Set of production lines at plant p Set of products that can be processed on line l Set of distribution centers (DC) (w ¼ 1,2,. . .,W) Set of time periods Set of product groups including product p Set of routes that begin with plant p Set of routes including distribution center w

Possibilistic Mixed Integer Linear Programming Approach for Production Allocation

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Parameters PCapl ECapl diwt ail ai V tsil TSjl ~ l EC C~prod il C~SMin il C~SMaj

Capacity of production line l Extra capacity of production line l External demand of product i at DC w in period t Time consumed to produce product i on line l Factor for converting quantities of product i into unit loads, e.g. pallets Loading capacity of a vehicle Minor setup time of product i on line l Major setup time of product group j on line l Extra cost for using production line l Processing cost of product i on production line l

C~Inven iw C~LTL

Inventory holding cost of product i at DC w in time period t Transportation cost per vehicle on route r

Minor setup cost of product i on production line l Major setup cost of product group j on production line l

jl

r

Decision variables xilt elt yirwt qipwt Iiwt nrt zilt ujlt

Production volume for product i produced on production line l in time period t Extra capacity used on production line l in time period t Quantity of product i delivered to warehouse w via route r in time period t Quantity of product i shipped from plant p to DC w in time period t Inventory level of product i at DC w at the end of time period t The number of vehicle used on route r in time period t. (identical vehicles are used) 1, if product i is setup on line l in period t 0, otherwise 1, if product group j is setup on line l in period t 0, otherwise

3.2.2

Objective Function

Minimize XXX i2I l2L t2T

XXX i

w

t

od xilt þ C~Pr il

XXX

zilt þ C~SMin il

i2I l2L t2T

C~Inven iw Iiwt þ

XX

XXX j2J l2L t2T

ujlt þ C~SMaj jl

XX r

t

C~LTL r nrt

ðCapÞ EC~l elt

l2L t2T

(1)

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The first term in objective function defines the production costs, the second and the third terms represent the minor and major setup costs for products, and product groups, respectively. Finally last three terms represent transportation ~ l , C~prod , cost of the system, and the inventory costs, and extra capacity costs. EC il SMin ~SMaj ~Inven ~LTL ~ Cil , Cjl , Ciw , Cr are imprecise coefficients.

3.2.3

Model Constraints

subject to X

ail xilt þ

X

i

tsil zilt þ

X

i

TSjl ujlt PCapl þ elt

8l; t

(2)

j

Constraints (2) are the time capacity constraints. Capacity is the upper bound on the total time that can be consumed to produce products. It specifies that the time used for processing (manufacturing) on a line cannot exceed the capacity of that line in time period t. ail xilt ðPCapl þ ECapl Þzilt

8i; l; t

(3)

Constraints (3) enforce the production quantity i on line l to zero, if no corresponding setup operation is performed (i.e. zilt ¼ 0). elt ECapl

8l; t

(4)

Constraints (4) guarantee that the extra capacity used for production line l and time period t cannot exceed maximum available extra capacity for that production line. X zilt Mujlt 8j; l; t (5) i2Ji

Constraints (5) ensure that product i belong to product group j can only be set, if the line is set up for the product group j. XX X X yirwt þ qipwt ¼ xilt 8p; i; t (6) r2Rp

w

w

l2Lp

Total output quantities achieved from producing product i at the production lines in plant p must be equal to the shipping quantities on the routes starting from plant p and including DC w plus the shipping quantities to the DCs which are supplied from plant p. That is, it ensures the availability of the product i at plant p in time period t. XX yirwt ai Vnrt 8r; t (7) i

w2Wr

Possibilistic Mixed Integer Linear Programming Approach for Production Allocation

515

Constraints (7) guarantee that, the total quantity of products (converted into unit loads) to be transported to the various DCs w 2 Wr included in route r determines the number of vehicles nrt required for that route in time period t. Note that variable nrt is defined as an integer number of identical vehicles each having a transportation capacity of Vunit loads. X X Iiwt1 þ yirwt þ qipwt diwt Iiwt ¼ 0 8i; w; t with Iiw0 ¼ given r2RðwÞ

p

(8) The daily demands of products must be satisfied. Constraints (8) ensure the inventory flow balance at DCs, and require each DC to have enough supply (from either inventory and/or the quantity arrived in that period) to meet the demand. That is, the inventory of product i at distribution center w at the end of time period t is determined by the ending inventory of the previous time period, the quantities received, and the external demand to be satisfied on the respective time period. xilt ; yirwt ; qipwt ; 8i; j; l; p; w; r; t

I iwt 0;

nrt 0;

and integer,

zilt ;

ujlt 2 f0; 1g

(9)

Finally, constraints (9) are integrality constraints of integer and binary variables, and non-negativity constraints of continuous variables. In real-life situation for a production and distribution planning problem, many input information related to the process are not known with certainty. Fuzzy and/or imprecise natures of the problem cannot be described adequately by the conventional approach. Fortunately, possibility distribution offers an effectual alternative for proceeding with inherent ambiguous phonemia in determining environmental coefficients and related parameters. In our model, cost coefficients are all imprecise. Therefore a possibilistic programming model is developed to determine the optimum production allocation and distribution decisions. Based on Lai and Hwang’s (1992b) approach, a possibilistic linear programming model is transformed into crisp multiobjective programming models. Finally, Zimmermann’s fuzzy programming method (1978) is applied to obtain composite single objective. In this study, production, setup, transportation, and inventory costs are represented by triangular possibility distributions. The parameters of a triangular possibility distribution are given as the optimistic, the most possible, and the pessimistic values, which were estimated by decision maker.

4 Solution Methodology 4.1

Converting Imprecise Cost Coefficients to Crisp Numbers

There are many studies to tackle with the imprecise cost coefficients in the objective function in the literature. The first method for getting a compromise solution was

516

B. Bilgen

proposed by Tanaka et al. (1984). They adapt a weighted average as a substitute for the fuzzy objective with a crisp compromise objective. An a-Pareto optimal solution proposed by Sakawa and Yano (1989) restricts the fuzzy coefficients to a-level sets. A similar concept is the b-possibility efficient solution of Luhandjula (1987). He obtained a single objective semi-infinite linear programming problem, and provided a cutting plane method to solve this semi-infinite problem. Rommelganger et al. (1989) proposed a multiple objective linear programming by using r a-level sets and establishing membership functions of the upper and lower bound for each a-level set to solve linear programming problem with triangular fuzzy costs. In contrast to the approach of Rommelganger et al. (1989) has restricted their solution method to one a-level. They used decomposition theory, and considered a convex set with extreme points defined by the lower and upper bound of the n a-level sets of the fuzzy coefficients. On the other hand, Lai and Hwang (1992b) converted the fuzzy objective with a triangular possibility distribution into three crisp objectives. Following the Lai and Hwang’s (1992b) approach, the approach developed here minimizes cm, maximizes (cm co), and minimizes (cp cm), rather than simultaneously minimizing cm, cp, co. Figure 1 represents a triangular possibility distribution of imprecise numbers. Geometrically, this imprecise objective is fully defined by three corner points (cm,1), (cp, 0) ,(co,0) in Fig. 1. Using Lai and Hwang’s approach (1992b), we substitute minimizing cm, maximizing (cm co), and minimizing (cp cm). That is the approach used in this work involves minimizing the most possible value of the imprecise costs, cm, maximizing the possibility of lower costs (cm co), and minimizing the risk of obtaining higher cost (cp cm). The three replaced objective functions can be minimized by pushing the three prominent points towards left. In this way, our problem can be transformed into a multi-objective linear programming as follows:

π~ Ci 1

Fig. 1 The triangular possibility distribution for cost coefficients

~ Ci(o)

~ (m) Ci

~ (p) Ci

Possibilistic Mixed Integer Linear Programming Approach for Production Allocation

XXX

z1 ¼

Min

XX

mðPr odÞ C~il xilt þ

i2I l2L t2T

þ

XXX

þ

t

r

z2 ¼

Max

XXX

mðSMinÞ C~il zilt þ

C~rmðLTLÞ nrt

þ

moðPr odÞ xilt þ C~il

þ

t

r

Min z3 ¼

XXX

w

XX r

t

moðCapÞ

elt

XXX

moðSMajÞ C~jl ujlt

j2J l2L t2T

w

i pmðPr odÞ C~il xilt þ

moðInvenÞ Iiwt C~iw

t

XX

EC~l

pmðCapÞ

elt

l2L t2T pmðSMinÞ C~il zilt þ

i2I l2L t2T

þ

EC~l

XXX

C~rmoðLTLÞ nrt þ

XXX

mðInvenÞ Iiwt C~iw

t

XX

moðSMinÞ C~il zilt þ

i2I l2L t2T

þ

mðSMajÞ C~jl ujlt

l2L t2T

i2I l2L t2T

XX

elt

j2J l2L t2T

i

XXX

XXX

XXX

i2I l2L t2T

þ

mðCapÞ

l2L t2T

i2I l2L t2T

XX

EC~l

517

C~rpmðLTLÞ nrt þ

XXX

pmðSMajÞ C~jl ujlt

(10)

j2J l2L t2T

XXX i

w

pmðInvenÞ Iiwt C~iw

t

To solve, many multi-objective linear programming techniques can be used, such as utility theory, goal programming, and so on. Lai and Hwang (1992b) suggested using Zimmermann’s (1978) fuzzy programming method to convert the auxiliary MOLP model into an equivalent single goal LP problem. Initially, the positive ideal solutions (PIS) and negative ideal solutions (NIS) of the three objective functions should be obtained to construct the linear membership functions of the objectives (Lai and Hwang 1992b). For each objective function, the corresponding linear membership function is computed as:

mz1 ¼

8 > < 1NIS z

if

9 z1 =

8 > : 1 : NIS ; 0 if z1 >z1 0 9 8 if z3 3 ;> = < 1NIS z3 z3 NIS ; : 3 3 0 if z3 >zNIS 3

if if

9 z2 >zPIS 2 ;> =

PIS zNIS 2

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