Planning Production and Inventories in the Extended Enterprise: A State-of-the-Art Handbook, Volume 2

International Series in Operations Research & Management Science Volume 152 Series Editor: Frederick S. Hillier Stanf...

Author: Karl G Kempf | Pnar Keskinocak | Reha Uzsoy

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International Series in Operations Research & Management Science

Volume 152

Series Editor: Frederick S. Hillier Stanford University, CA, USA

Special Editorial Consultant: Camille C. Price Stephen F. Austin, State University, TX, USA

For further volumes: http://www.springer.com/series/6161

Karl G. Kempf Pınar Keskinocak Reha Uzsoy Editors

Planning Production and Inventories in the Extended Enterprise A State of the Art Handbook, Volume 2

ABC

Editors Karl G. Kempf Decision Technologies Group Intel Corporation 5000 W. Chandler Blvd., Chandler, Arizona, USA 85226 [email protected] Pınar Keskinocak Georgia Institute of Technology School of Industrial and Systems Engineering Atlanta, Georgia 30332-0205, USA [email protected]

Reha Uzsoy Edward P. Fitts Department of Industrial and Systems Engineering 300 Daniels Hall Campus Box 7906 North Carolina State University Raleigh, NC 27695-7906, USA [email protected]

ISSN 0884-8289 ISBN 978-1-4419-8190-5 e-ISBN 978-1-4419-8191-2 DOI 10.1007/978-1-4419-8191-2 Springer New York Dordrecht Heidelberg London Library of Congress Control Number: 2011922912 c Springer Science+Business Media, LLC 2011 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

Acknowledgements

The process by which these volumes came to fruition has been a long one, and we are grateful for the support of many different people for their support, advice and contributions. First among these are the contributing authors – without them, there would be no book, and we are deeply grateful to them for bearing with our repeated requests for materials and revisions while providing the high-quality contributions worthy of these volumes. The many reviewers who gave of their time and effort to help improve the chapters are also gratefully acknowledged. Thanks are also due to Professor F. Hillier, the Editor of this Series, for supporting the project, and to Gary Folven, under whose guidance the project took shape, and who sustained us through the first four years until his well-deserved retirement. His many contributions to operations research over a long and distinguished publishing career will stand for a long time. Neil Levine and Matthew Amboy of Springer saw the volumes through production, nudging the editorial team when necessary and exhibiting resourcefulness and patience above and beyond the call of duty. Thanks also to Carrie Brooks of North Carolina State University for preparing the lists of authors for indexing at very short notice. An undertaking of this magnitude and duration would simply not be possible without a collegial, mutually supporting editorial team. It has been a privilege to work together on this project; we have all learnt from each other and from the authors who have contributed, and are grateful to have had this opportunity to work with some of the best in our field. Finally, we would like to thank our families, who have had to put up with late nights and grumpy mornings over the duration of the project, and without whose support and understanding our lives would be much poorer in many ways.

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Contents

1

Production Planning Under Uncertainty with Workload-Dependent Lead Times: Lagrangean Bounds and Heuristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . Gregory Dobson and Uday S. Karmarkar

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Production Planning and Scheduling: Interaction and Coordination .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 15 Yiwei Cai, Erhan Kutanoglu, and John Hasenbein

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The Effects of Production Planning on the Dynamic Behavior of a Simple Supply Chain: An Experimental Study . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 43 Seza Orcun and Reha Uzsoy

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Supply and Demand Synchronization in Assemble-to-Order Supply Chains . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 81 Markus Ettl, Karthik Sourirajan, Pu Huang, Thomas R. Ervolina, and Grace Y. Lin

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Quantitative Risk Assessment in Supply Chains: A Case Study Based on Engineering Risk Analysis Concepts . . .. . . . . . . . . . . . . . . . .105 Léa A. Deleris and Feryal Erhun

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A Practical Multi-Echelon Inventory Model with Semiconductor Manufacturing Application . . . . . . . .. . . . . . . . . . . . . . . . .133 Kaan Katircioglu and Guillermo Gallego

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A Mechanism Design Approach for Decentralized Supply Chain Formation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .153 Dinesh Garg, Y. Narahari, Earnest Foster, Devadatta Kulkarni, and Jeffrey D. Tew

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Contents

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Procurement Network Formation: A Cooperative Game Approach . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .185 T.S. Chandrashekar and Y. Narahari

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Designing Flexible Supply Chain Contracts with Options .. . . . . . . . . . . . . .207 Feng Cheng, Markus Ettl, Grace Y. Lin, Maike Tonner, and David D. Yao

10 Build-to-Order Meets Global Sourcing: Planning Challenge for the Auto Industry . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .231 Melda Ormeci Matoglu and John Vande Vate 11 Practical Modeling in Automotive Production . . . . . . . . . . .. . . . . . . . . . . . . . . . .249 Jonathan H. Owen, Robert R. Inman, and Dennis E. Blumenfeld 12 Why Is It So Hard to Build and Validate Discrete Event Simulation Models of Manufacturing Facilities? . . . . . . . .. . . . . . . . . . . . . . . . .271 Seth A. Fischbein and Edward Yellig 13 A Practical Approach to Diagnosing and Tuning a Statistical Forecasting System .. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .289 Ying Tat Leung and Kumar Bhaskaran 14 The Ongoing Challenge: Creating an Enterprise-Wide Detailed Supply Chain Plan for Semiconductor and Package Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .313 Kenneth Fordyce, Chi-Tai Wang, Chih-Hui Chang, Alfred Degbotse, Brian Denton, Peter Lyon, R. John Milne, Robert Orzell, Robert Rice, and Jim Waite 15 Production Planning in the Plastics Industry . . . . . . . . . . . .. . . . . . . . . . . . . . . . .389 Rajesh Tyagi and Srinivas Bollapragada 16 Model Predictive Control in Semiconductor Supply Chain Operations ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .403 Karl Kempf, Kirk Smith, Jay Schwartz, and Martin Braun 17 Models and Methods for Production Scheduling in the Pharmaceutical Industry . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .429 Dario Pacciarelli, Carlo Meloni, and Marco Pranzo 18 Developing a Computerized Scheduling System for the Steelmaking–Continuous Casting Process . . . . . . .. . . . . . . . . . . . . . . . .461 Hubert Missbauer, Wolfgang Hauber, and Werner Stadler

Contents

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19 A Multi-Model Approach for Production Planning and Scheduling in an Industrial Environment . . . . . . . . . . .. . . . . . . . . . . . . . . . .489 Abdelhakim Artiba, Valerie Dhaevers, David Duvivier, and Salah E. Elmaghraby 20 Fuzzy Logic-Based Production Scheduling and Rescheduling in the Presence of Uncertainty .. . . . . . .. . . . . . . . . . . . . . . . .531 Sanja Petrovic, Dobrila Petrovic, and Edmund Burke 21 The Summing-Up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .563 Karl Kempf, Pınar Keskinocak, and Reha Uzsoy Author Index. . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .571 Subject Index . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .583

Contributors

Abdelhakim Artiba Professeur des Universités, Vice-Président Recherche et Valorisation, Université de Valenciennes et du Hainaut Cambrésis, UVHC – LAMIH, F-59313 Valenciennes Cedex 9, France, [email protected] Kumar Bhaskaran IBM Thomas J. Watson Research Center, P.O. Box 218, Yorktown Heights, NY 10598, USA, [email protected] Dennis E. Blumenfeld Department of Industrial and Operations Engineering, University of Michigan, 1205 Beal Avenue, Ann Arbor, Michigan 48109, USA, [email protected] Srinivas Bollapragada General Electric Global Research Center, Niskayuna, NY 12309, USA, [email protected] Martin Braun Customer Planning and Logistics Group, Intel Corporation, 5000 W. Chandler Blvd., Chandler, AZ 85226, USA, [email protected] Edmund Burke Automated Scheduling, Optimisation and Planning Research Group, School of Computer Science and IT, University of Nottingham, Nottingham, UK, [email protected] Yiwei Cai Graduate Program in Operations Research and Industrial Engineering, Department of Mechanical Engineering, The University of Texas at Austin, Austin, TX 78712, USA, yw [email protected] T.S. Chandrashekar Department of Computer Science and Automation, Indian Institute of Science, Bangalore, India, [email protected] Chih-Hui Chang IBM Corporation Strategic Systems Department, 1000 River Road, Essex Junction, VT 05452, USA, [email protected] Feng Cheng Federal Aviation Administration, 800 Independence Ave, S.W., Washington, DC 20591, USA, [email protected] Alfred Degbotse IBM Corporation Strategic Systems Department, 1000 River Road, Essex Junction, VT 05452, USA, [email protected]

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Contributors

Léa A. Deleris IBM Research – Smarter Cities Technology Center, Damastown Industrial Estate – Mulhuddart, Dublin 15 – Ireland, [email protected] Brian Denton Edward P. Fitts Department of Industrial and Systems Engineering, North Carolina State University, Raleigh, NC 27695-7906, USA, [email protected] Valerie Dhaevers Louvain School of Management, Catholic University of Mons, B-7000 Mons, Belgium, [email protected] Gregory Dobson Simon School, University of Rochester, Rochester, NY 14627, USA, [email protected] David Duvivier LISIC-ULCO Université du Littoral Côte d’Opale, Laboratoire d’Informatique Signal et Image de la Côte d’Opale, Maison de la Recherche Blaise Pascal, 50, rue Ferdinand Buisson – BP 719, 62228 CALAIS Cedex, France, [email protected] Salah E. Elmaghraby University Professor Emeritus, North Carolina State University, Department of Industrial and Systems Engineering and The Graduate Program in Operations Research, Raleigh NC, 27695-7906, USA, [email protected] Feryal Erhun Department of Management Science and Engineering, Stanford University, Stanford, CA 94305, USA, [email protected] Thomas R. Ervolina IBM Thomas J. Watson Research Center, P.O. Box 218, Yorktown Heights, NY 10598, USA, [email protected] Markus Ettl IBM Thomas J. Watson Research Center, P.O. Box 218, Yorktown Heights, NY 10598, USA, [email protected] Seth A. Fischbein Intel Corporation, Mailstop CH3–113, 5000 W. Chandler Blvd., Chandler, AZ 85248, USA, [email protected], [email protected] Kenneth Fordyce IBM Corporation Strategic Systems Department, 227 Evergreen Lane, Hurley, NY 12443, USA, [email protected] Earnest Foster General Motors Research and Development, Warren, MI, USA, [email protected] Guillermo Gallego Department of Industrial Engineering and Operations Research, Columbia University in the City of New York, New York, NY, USA, [email protected] Dinesh Garg Department of Computer Science and Automation, Indian Institute of Science, Bangalore, India, [email protected] John Hasenbein Graduate Program in Operations Research and Industrial Engineering, Department of Mechanical Engineering, The University of Texas at Austin, Austin, TX 78712, USA, [email protected] Wolfgang Hauber Solentia Software & Consulting GmbH, Denkstrasse 30, 4030 Linz, Austria, [email protected]

Contributors

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Pu Huang IBM Thomas J. Watson Research Center, P.O. Box 218, Yorktown Heights, NY 10598, USA, [email protected] Robert R. Inman General Motors Company, Global R&D, 30500 Mound Road, Warren, Michigan 48090, USA, [email protected] Uday S. Karmarkar UCLA Anderson School of Management, 110 Westwood Plaza, PO Box 951481, Los Angeles, CA 90095-1481, USA, [email protected] Kaan Katircioglu IBM T. J. Watson Research Center, Yorktown Heights, NY 10598, USA, [email protected] Karl Kempf Decision Technologies Group, Intel Corporation, 5000 W. Chandler Blvd., Chandler, Arizona, USA 85226 [email protected] Pınar Keskinocak Georgia Institute of Technology, School of Industrial and Systems Engineering, Atlanta, Georgia 30332-0205, USA, [email protected] Devadatta Kulkarni General Motors Research and Development, Warren, MI, USA, [email protected] Erhan Kutanoglu Graduate Program in Operations Research and Industrial Engineering, Department of Mechanical Engineering, The University of Texas at Austin, Austin, TX 78712, USA, [email protected] Ying Tat Leung IBM Almaden Research Center, 650 Harry Road, San Jose, CA 95120, USA, [email protected] Grace Y. Lin VP, Advanced Research Center, Institute for Information Technology, 1F., No.133, Sec. 4, Minsheng E. Rd., Taipei, Taiwan, 105, [email protected] Peter Lyon IBM Corporation Strategic Systems Department, 1000 River Road, Essex Junction, VT 05452, USA, [email protected] Melda Ormeci Matoglu Faculty of Economics and Administrative Sciences, Ozyegin University, Istanbul, Turkey, [email protected] Carlo Meloni Dipartimento di Elettrotecnica ed Elettronica, Politecnico di Bari, via E. Orabona, 4 – 70125 Bari, Italy, [email protected] R. John Milne Neil’64 and Karen Bonke Assistant Professor in Engineering Management, Clarkson University, School of Business, 107 B. H. Snell Hall, P.O. Box 5790, Potsdam, NY 13699-5790, [email protected] Hubert Missbauer Department of Information Systems, Production and Logistics Management, University of Innsbruck, Universitätsstrasse 15, 6020 Innsbruck, Austria, [email protected] Y. Narahari Department of Computer Science and Automation, Indian Institute of Science, Bangalore, India, [email protected]

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Contributors

Seza Orcun Laboratory for Extended Enterprises at Purdue, e-Enterprise Center at Discovery Park, Purdue University, West Lafayette, IN 47907, USA, [email protected] Robert Orzell IBM Corporation Strategic Systems Department, 1000 River Road, Essex Junction, VT 05452, USA, [email protected] Jonathan H. Owen General Motors Company, Global R&D, 30500 Mound Road, Warren, Michigan 48090, USA, [email protected] Dario Pacciarelli Dipartimento di Informatica e Automazione, Università “Roma Tre”, via della Vasca Navale, 79-00146 Roma, Italy, [email protected] Dobrila Petrovic Control Theory and Applications Centre, Faculty of Engineering and Computing, Coventry University, Coventry, UK, [email protected] Sanja Petrovic Automated Scheduling, Optimisation and Planning Research Group, School of Computer Science and IT, University of Nottingham, United Kingdom [email protected] Marco Pranzo Dipartimento di Ingegneria dell’Informazione, Università di Siena, via Roma, 56 – 53100 Siena, Italy, [email protected] Robert Rice IBM Corporation Strategic Systems Department, 227 Evergreen Lane, Hurley, NY 12443, USA, [email protected] Jay Schwartz Decision Technologies Group, Intel Corporation, 5000 W. Chandler Blvd., Chandler, AZ 85226, USA, [email protected] Kirk Smith Customer Planning and Logistics Group, Intel Corporation, 5000 W. Chandler Blvd., Chandler, AZ 85226, USA, [email protected] Karthik Sourirajan IBM Thomas J. Watson Research Center, P.O. Box 218, Yorktown Heights, NY 10598, USA, [email protected] Werner Stadler Siemens VAI, Turmstr. 44, 4031 Linz, Austria, [email protected] Jeffrey D. Tew General Motors Research and Development, Warren, MI, USA, [email protected] Maike Tonner Euler Hermes Risk Management GmbH & Co. KG, 22763 Hamburg, Germany, [email protected] Rajesh Tyagi General Electric Global Research, Niskayuna, NY 12309, USA, [email protected] Reha Uzsoy Edward P. Fitts Department of Industrial and Systems Engineering, 300 Daniels Hall, Campus Box 7906, North Carolina State University, Raleigh, NC 27695-7906, USA, [email protected] John Vande Vate School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, Georgia, USA, [email protected]

Contributors

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Jim Waite IBM Corporation Strategic Systems Department, 1000 River Road, Essex Junction, VT 05452, USA Chi-Tai Wang School of Management, National Central University, 300, Jhongda Road, Jhongli City, Taoyuan County 32001, Taiwan, [email protected] David D. Yao IEOR Department, Columbia University, New York, NY 10027, USA, [email protected] Edward Yellig Intel Corporation, Mailstop CH3-113, 5000 W. Chandler Blvd., Chandler, AZ 85248, USA, [email protected]

Chapter 1

Production Planning Under Uncertainty with Workload-Dependent Lead Times: Lagrangean Bounds and Heuristics Gregory Dobson and Uday S. Karmarkar

1.1 Introduction Advances in modeling production system behavior over the past three decades have made it apparent that work-in-process (WIP) inventories, created by the dynamics of manufacturing systems, are an important characteristic of these systems. Furthermore, the recognition of the importance of lead times and the variability of lead times for system performance has lent additional emphasis to this behavior. Consequently, consideration of WIP and lead times has been a key aspect addressed in the analysis and development of various types of scheduling and job release systems, of which Kanban and workload-based methods are examples. It is well known that capacity loading is the key determinant of lead time and WIP (although there are many other ancillary factors, such as batching and system variability). The loading effect makes itself felt as the explosive .1=.1 u// factor in queuing and lot-sizing models, where u represents the level of resource utilization or loading. It is especially striking then that there are very few investigations of capacity planning methods or models that incorporate or account for the effects of loading on WIP and lead time. It is also notable that very few planning methods are devised to interface with job release systems that account for these phenomena. In this paper, we continue the investigation of an aggregate production planning model that explicitly considers the effect of capacity loading on WIP and lead times. To begin with, consider the role of such a production planning or capacity planning system, in a production control approach that already addresses these factors at the release or scheduling level. In other words, what does the planning level add? There are in fact three key issues that are not handled at the detailed level. One is the management of seasonal variations in demand through inventories, which are accumulated and drawn down. The second is the management and planning of safety stocks. The third is short-term capacity planning (e.g., through overtime and/or subcontracting) that affects available capacity.

G. Dobson () Simon School, University of Rochester, Rochester, NY, 14627, USA e-mail: [email protected]

K.G. Kempf et al. (eds.), Planning Production and Inventories in the Extended Enterprise, International Series in Operations Research & Management Science 152, c Springer Science+Business Media, LLC 2011 DOI 10.1007/978-1-4419-8191-2 1,

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G. Dobson and U.S. Karmarkar

Seasonal planning is often thought of as a matter of finished goods inventory management. The reality is that in a dynamic production system with substantial lead times, there are strong interactions between finished goods inventories and WIP. At any rate, it is clear that there are very few release system models that correctly incorporate seasonal demand and inventory planning against seasonal variation and uncertainty. Indeed, one could argue that this is not the proper role of release models. The second issue of safety stocks is driven by the uncertainty in demand over the horizon of and at the resolution level of planning models. It is important not to confuse this uncertainty, essentially due to ignorance about future demand levels (forecast uncertainty) with uncertainty or variability at the level of job release or operation timing. The latter is the province of release models and dispatching or scheduling methods. However, planned safety stocks of finished goods are another matter since they depend on seasonal patterns, forecast uncertainty over longer horizons, and production plans at the aggregate level. It is especially important to note that the interaction between lead times and safety stocks is significant and cannot be ignored. This lead time depends on capacity loading and hence on production plans over the planning horizon. In this paper, we develop a multiperiod planning model that is able to incorporate the effect of capacity loading on WIP and lead times, as well as demand uncertainty. Consequently, it also implicitly handles the interaction between lead time and demand uncertainty, in determining safety stocks. The key construct in this model is the so-called “clearing function” that models the relationship between capacity, workload, and output (Fig. 1.1). This function effectively forces recognition of the WIP and lead-time consequences of high capacity loading. It replaces the capacity constraints of conventional planning models. The planning model tracks WIP as well as finished goods inventories through balance equations in each period (see Fig. 1.2 for a representation of the mass balance equations). While traditional models assume that production output is the primary decision variable, our model instead recognizes that output actually depends on job release as well as capacity loading. The central decision variables in the model are not production output levels, but release quantities in each period and these then in concert with the clearing model, determine the production output endogenously. Concurrently, the model determines

Output Rate

Fig. 1.1 A saturating clearing function giving output rate as a function of load

Load

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Production Planning Under Uncertainty with Workload-Dependent Lead Times

Fig. 1.2 Material balance relationships for the discrete period capacity planning model with one product class and one production facility. The variables represent Release Quantity .R/, WIP .W /, Production output .X/, Finished Inventory .I / and Demand .D/; the subscripts represent time periods

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Rt

Wt-1

Wt

Xt It-1

It

Dt

the level of capacity loading that must be maintained in each period, and balances the costs of WIP (and implicitly lead time), finished goods inventories, and backorders. From a hierarchical perspective, the model thus produces a release plan, and a complementary capacity loading or workload plan. Depending on the release approach being used at a detailed level, these plans can be used to provide guidelines either for a release policy in each period or a workload level to be maintained. In short, the planning model is consistent with a workload-based control scheme, with the additional feature that it captures the seasonal variation necessary to correctly match seasonal inventory accumulation against seasonal demand variations. We do not model short-term capacity decisions (e.g., overtime or subcontracting) in this paper, but they can be included without significant change in model complexity. However, our main purpose here is to explore the capture of workload and lead time considerations at the aggregate production planning level. It is worth noting that the only stochastic element in this model is demand. The clearing function is a deterministic relationship. While the clearing concept has some relationship to queuing models, it abstracts away from the details of the queuing and congestion processes. It is also reasonable to interpret the clearing function model as representing expected or average values, but this is not a formal part of this model. In the next section, we briefly survey the existing literature in the topics that we have touched on in the discussion above. We formulate the workload and release planning problem with stochastic demands. Small versions of this problem (e.g., a single product) can be solved optimally. However, that is not feasible with larger versions, because of the size of the state space that results. We develop Lagrangean methods for lower bounds, which also provide computational heuristics.

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Decomposition of the problem by Lagrangean methods seems difficult at first sight, because of the stochastic nature of inter-period state transitions. However, we get around this issue by using a “restricted Lagrangean” approach that gives valid lower bounds. We then examine a set of problems for which optimal solutions are computed by dynamic programming. This is a complex nonlinear stochastic multiperiod multistate model, and as noted above, optimal solutions are only possible for small problems. We also compute lower bounds for these problems that are quite tight for the cases investigated. These bounds can be computed for large problems as well, and provide solutions that are feasible in the first period of the problem. Hence, they can be used as heuristic methods by implementing the first period solution, and then recomputing the solution for the next period on a rolling horizon basis. This approach has been shown to be effective in other stochastic multiperiod problems. Here, we conduct some studies on this heuristic approach, which show it to be a viable direction here as well.

1.2 Literature Survey There has been a substantial amount of work dealing with lead time and WIP issues at the work release level. Bertrand (1985), Karmarkar (1987b, 1993), Karmarkar et al. (1985), and Zipkin (1986) have illustrated the importance of these considerations in batching and release decisions. Research on release mechanisms that use a fixed WIP or inventory level to control WIP and lead times has been presented under several different labels including the traditional base stock models (Kimball 1988), Kanban system models (Deleersnyder et al. 1989; Karmarkar and Kekre 1989), and workload, or load-based control (Bertrand 1981, 1983; Bechte 1982; Denardo and Tang 1992; Spearman and Zazanis 1992; Land and Gaalman 1996; Perona and Portioli 1998). Zijm and Buitenhoek (1996) have studied the estimation of lead times for capacity planning. In general, these models do not extend in an obvious way to the dynamic seasonal demand, varying production mix, and multiple capacity constraint scenario of capacity planning. Early formulations of aggregate capacity planning decisions employed linear programming (Bowman 1956; Hanssmann and Hess 1960) and quadratic cost models (Holt et al. 1960) to represent the problem. Linear programming models, especially, have also been widely used in practice for planning in production and logistics management. Extensions of the LP model included its use in hierarchical control schemes, sometimes through column generation techniques (e.g., Manne 1958; Lasdon and Terjung 1971). Surveys of early aggregate planning models are presented by Hax (1978) and Nam and Logendran (1992). Of course, as noted earlier, these models do not capture the effect of loading on WIP and leadtimes, and cannot interface effectively with workload based release methods. There have been some attempts to capture the issue of WIP and leadtimes within a planning context. Graves (1986) addressed the problem by assuming that lead times are planned and maintained at a certain level, independent of loading. This

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approach is technically tractable and can handle uncertainties, but fails to capture the effect of loading on lead times, since the latter are assumed to be constant. Karmarkar (1989) used “clearing functions” to model the relationship between loading and lead times. This model extends the traditional material balance constraints to include WIP material balance, and replaces linear constraints with functions that capture congestion effects, WIP and (implicitly) lead times. More current work on such models includes Homem-de-Mello et al. (1999), Missbauer (2002), Riaño (2002), Pahl et al. (2005), Asmundsson et al. (2006). Uncertainty is critical in certain kinds of planning problems. The fashion goods planning problem has been tackled in papers by Hausman and Sides (1974), and Bitran et al. (1986). Chance constrained linear programming (Charnes and Cooper 1959) captures the effects of stochastic parameters through constraints rather than through cost implications. A typical constraint might set limits on the expected value, or the fractile of a distribution, or a probability. The use of such models in planning has been explored by Bitran and Yanasse (1984), who develop a technique for deterministic approximation. Subsequent research on planning under uncertainty includes Lin (1986), Karmarkar and Lin (1988), Lasserre and Mercé (1990), Gferer and Zäpfel (1995), Riaño (2002), and Sahinidis (2004). An approach that appears to be quite effective in dealing with multiperiod stochastic problems is the “restricted Lagrangean” method used by Karmarkar (1987a) to address multilocation multiperiod inventory problems. It has been applied to planning problems with uncertainties in demand (Lin 1986) as well as in yields or technology coefficients (Karmarkar and Lin 1988). The method has also been applied to mixed nonlinear integer stochastic problems with variable results (Karmarkar and Yoo 1994; Karmarkar and Rajaram 2002). From the viewpoint of numerical calculations, this is the methodology that we will subsequently employ.

1.3 Problem Formulation We formulate the problem as a stochastic multiperiod dynamic program, with the state variables being the WIP level and the finished inventory (or backorder level). The objective is taken to be the minimization of expected costs over the horizon, as is typical in planning models. An extension to consider risk preferences would be nontrivial, and would then also need to include revenues. The formulation assumes a single product and a single production stage. However, the extension of the formulation to multiple products and stages is feasible. Of course, any increase in the state space will make the formulation significantly harder computationally, to the point of intractability with respect to optimal solutions. The reason to limit the present discussion to a simpler context is to permit numerical investigation of the optimal solution in comparison with lower bounds and heuristics. The latter can be applied to much larger problems but then even the valuation of the heuristics becomes difficult, let alone finding optimal solutions.

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Define: Rt Wt f .Wt ; Rt / Xt It Dt Yt Zt ct wt kt bt ht Lt .Yt / t t

Releases in period t WIP at end of period t The clearing function Production in period t Inventory at end of period t (Random) demand in period t Target inventory level in t, including current production Target production quantity in t Variable cost of releases Holding cost for WIP Variable cost of production Cost of backorder Holding cost for inventory Expected holding and backorder costs given Yt Multiplier for inventory balance equations Multiplier for production balance equations

The variables R; W; X , and Z should be nonnegative throughout the following. If the function Lt .Yt / is convex, then these problems below are, of course, easier to solve. For our computational work, we use the standard form, namely, ht E.Yt Dt /C Cbt E.Dt Yt /C . An example of the clearing function would be f .Wt ; Rt / D M.Wt C Rt /=.˛ C Wt C Rt /, where M is the maximum theoretical throughput of the process in a given period and ˛ affects the shape of the clearing function. We also use this form in our numerical tests. For other examples of clearing functions, see Asmundsson et al. (2006). The stochastic multiperiod planning problem can now be formulated as follows. Period T: vT .WT 1 ; IT 1 / D Min.cT RT CkT XT CwT WT /CLt .YT /salvage value if any s:t: Xt f .WT 1 ; RT /

(1.1T)

YT D XT C IT 1

(1.2T)

WT 1 C RT D XT C WT :

(1.3T)

Period t: vt .Wt 1 ; It 1 / D Min.ct Rt C kt Xt C wt Wt / C Lt .Yt / C E.vt C1 .Wt ; Yt Dt // s:t: Xt f .Wt 1 ; Rt /

(1.1t)

Yt D Xt C It 1

(1.2t)

Wt 1 C Rt D Xt C Wt :

(1.3t)

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7

1.4 Lagrangean Decompositions The approach used here is the same as that used in Karmarkar (1987a), Karmarkar and Lin (1988), Karmarkar and Yoo (1994), Karmarkar and Rajaram (2002). For more details, see those papers. Two decompositions are presented. Both result in the separation of the stochastic inventory problem from the capacitated (deterministic) planning problem. The first decouples the problem into a multiperiod deterministic clearing function-based planning model, and a multiperiod stochastic dynamic newsboy problem. The second decomposition results in a similar multiperiod deterministic model as one subproblem. However, the stochastic part is broken up into a sequence of single period stochastic newsboy problems. It is likely that the first decomposition is superior in the sense of tighter lower bounds, since the stochastic transitions between periods are still recognized. Decomposition 1: In period t, change .1:2t/ to Yt D Zt C It 1 : Add constraint: Xt D Zt and relax the latter constraint, using multipliers t . The resulting problem for period T can be stated as: uT .WT 1 ; T / D Min cT RT C .kT T /XT C wT WT .salvage value for WT / s:t: .1T; 3T / uT .IT 1 ; T / D Min T ZT C LT .YT / .expected salvage value for IT / s:t: YT D ZT C IT 1 UT .IT 1 ; WT 1 ; T / D uT .WT 1 ; T / C uT .IT 1 ; T /: Let t D .t ; : : :; T /. In each period t apply the same decomposition, replacing vt C1 by Ut C1 , so that the lower bound character is maintained. ut .Wt 1 ; ƒt / D Min ct Rt C .kt t /Xt C wt Wt C ut C1 .Wt ; ƒt C1 / s:t: .1t; 3t/ D Min

T X

Œci Ri C .ki i /Xi C wi Wi

i Dt

.salvage value for WT / s:t: .1i; 3i /; for i D t; : : : ; T ut .It 1 ; ƒt / D Min t Zt C Lt .Yt / C E.ut C1 .Yt Dt ; ƒt C1 // s:t: Yt D Zt C It 1 Ut .It 1 ; Wt 1 ; ƒt / D ut .Wt 1; ƒt / C ut .It 1 ; ƒt /:

8


Decomposition 2: Relax .1:2t/ using multipliers . The period T problem becomes: g.WT 1 ; T / D Min cT RT C .kT T /XT C wT WT .salvage value forWT / s:t: .1T; 3T / g T .IT 1 ; T / D Min T YT C LT .YT / .expected salvage value for IT / GT .IT 1 ; WT 1 ; T / D gT .WT 1 ; T / C g T .IT 1 ; T /: As in the previous example, defining Mt D .t ; : : :; T /, and replacing v by G, the problem in period t is written as gt .Wt 1 ; Mt / D Min ct Rt C .kt t /Xt C wt Wt C gt C1 .Wt ; Mt C1 / s:t: .1t; 3t/ D Min

T X

Œci Ri C .ki i /Xi C wi Wi .salvage value for WT /

i Dt

s:t: .1i; 3i /; for i D t; ; T g t .It 1 ; Mt / D Min t Yt C Lt .Yt / t C1 Yt and Gt .It 1 ; Wt 1 ; Mt / D gt .Wt 1 ; Mt / C g t .It 1 ; Mt / C

T 1 X

i C1 E.Di /:

i Dt

The main purpose of the Lagrangean relaxations is, of course, to provide lower bounds and an approach for the construction of heuristic solutions. Sometimes such relaxations can also have some intuitive interpretations, or conceptual value. Here, we do not feel that there is a strong intuitive meaning to the decomposition. Rather the motivation is structural. What makes this problem computationally very difficult is the combination of the stochastic factors, multiple time periods, multidimensional state space, and nonlinear constraints. The decompositions separate the stochastic factors from the nonlinear constraints, leaving tractable subproblems. Of course, the multipliers communicate marginal cost information across the subproblems, as always.

1.5 Solution of the Subproblems The deterministic subproblem for both decomposition examples is of the form Min

T X

Œct Rt C k t Xt C wt Wt

t D1

s:t: .1t; 3t/; for t D 1; ; T;

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9

where the coefficients k t represent the expression kt – the Lagrangean multiplier. Note that the coefficients k t will be negative for some periods; if not, the optimal solution is to simply set all variables to zero. This problem can be solved by a standard nonlinear programming algorithm. It can also be solved by linear programming methods by linearizing the nonlinear constraints (1.1). The resulting LP then appears like a network problem with side constraints. Another alternative is to dualize the nonlinear constraints, placing them in the objective function. The problem is then of a convex cost network form. To linearize the constraints in (1.1), let pt D @f=@Wt 1 ; qt D @f=@Rt , and rt D f , all evaluated at given values of Wt 1 ; Rt . Then a tangent approximation to the constraint at this given point can be written as Xt pt Wt 1 qt Rt rt : Several such approximating linear constraints can be generated initially. Subsequently, given a current solution to the problem, a new constraint is generated at the value of W and R in the current solution. Nonbinding constraints could be dropped at each iteration. A realistic problem might be six to ten periods, with two to three stages of production. In the simplest case, we might have five to six constraints per stage-period and three variables per stage-period; i.e., with six periods and two stages, approximately 70 constraints and 40 variables. Of course, iterations are required, as prices (multipliers) are changed (subgradient optimization or other method). In the case of the first decomposition, the stochastic subproblem can be solved by dynamic programming. In the second decomposition, when the loss function L has newsvendor form, the solution of the decoupled newsvendor problem is well known, and is essentially a one-line expression.

1.6 Numerical Tests for Method Validation The decomposition approaches suggested above, to give lower bounds and heuristics, have been tried elsewhere on multiperiod stochastic planning problems, generalized from LP planning models. The computational experience in those cases was very favorable (Lin 1986, Karmarkar and Lin 1988). There is therefore some evidence suggesting that the approach would work. As an initial validation of the model described here, some preliminary numerical tests were carried on problem instances small enough to permit optimal solutions by dynamic programming. Thus, the performance of lower bounds could be compared against the optimum. Note that computing optimum solutions for problems larger than the single-item problems dealt with here is generally infeasible, because of the size of the state space. A computer program has been written, which implements the original dynamic programming formulation of the planning model from the first section of this paper.

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The computer implementation of the algorithm is different from the model only to the extent that the following parameters are used to limit the size of the DP arrays: Imax D The maximum allowable inventory (actual and order-up-to). Wmax D The maximum allowable level of work-in-progress inventory. Dmax D The maximum possible value of demand in any time period. Solving the DP requires determining the optimal release quantity in each time period for each possible .W; I / combination. Since releases are limited by both Imax and Wmax , the computational complexity of the algorithm is O.T Wmax Imax ŒminfImax ; Wmax g/. Assuming that Imax and Wmax are approximately equal, the complexity is linear in time and cubic in inventory size. In the second section, two Lagrangean decompositions of the original DP formulation were derived. The first decomposition was implemented in PASCAL, and several test problems have been run in order to ascertain the tightness of the lower bound provided by the decomposed problem. Judging from the results of various test problems, as shown in the table below, the lower bound is typically within 3% of the optimal solution, for the data set considered. In each period, demand was assumed to be discrete and uniformly distributed on [0,1,. . . ,Dmax ]. The parameter Pmax represents the maximum possible production in each period. The parameter ’ is used in the clearing function, where larger values of ’ correspond to slower production response, or equivalently, longer lead times. Table 1.1 below compares the results obtained for lower bounds and optimal solutions. The “percent error” gives the difference between the bound and the optimum as a percentage of the optimal value. The complexity of the algorithm, which implements the decomposition, is Imax /. Assuming Wmax D Imax D n, the decomposition has reduced O.T Wmax the complexity of the original problem from O.T n3/ to O.T n2/ . An additional advantage of the decomposition technique is reduced storage requirements. The storage requirements of the original algorithm are O.T Wmax Imax /, whereas the storage requirements of the decomposition technique are O.T maxfImax; Wmax g/.

Table 1.1 Results of computational tests: optimal solutions and lower bounds Time Optimal Lower periods Wmax Imax Dmax Pmax ˛ solution bound 5 25 25 40 20 4 1173.5 1153.7 5 25 25 30 20 4 864.6 852.0 5 20 20 30 20 4 866.1 852.8 10 20 20 30 15 3 1732.7 1685.2 10 20 20 25 15 3 1378.8 1345.7 10 15 15 20 10 3 1262.9 1249.0 20 15 15 20 10 3 2607.9 2596.6 20 15 15 15 10 3 1875.3 1842.0 20 10 10 10 10 3 1140.5 1133.1

% error 1.69 1.46 1.54 2.74 2.40 1.10 0.43 1.78 0.65

Note: Each of the problems represented in the table was solved using identical procedures for optimizing over the Lagrange multipliers

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In the more general case of multiple products, the gains are much greater, accruing primarily in the reduction of the state space for dynamic programming. In the latter case, the state space would be of the order of (Wmax Imax /m when there are m products. Decomposition would reduce this to .maxfWmax ; Imax g/m , or (Wmax /m depending on the approach used. Furthermore, since the state variables .W; I / would be associated with the multiperiod deterministic problem rather than with the stochastic newsboy problems, they can be treated as a vector in Rm rather than as a discrete state space. It is also possible to devise decompositions that would essentially decouple the problem into single period nonlinear subproblems that would eliminate the state space issue at the expense of possibly weaker bounding performance. Since the optimal solution is difficult to obtain, heuristic methods will prove necessary in practice. Based on the past experience (e.g., Karmarkar 1987a), we expected that the information from the relaxation would be useful in providing a heuristic algorithm. To this end, we defined a heuristic based on the release quantity for each inventory level, for each WIP level, and for each period, Rt .W; I /. The heuristic consists of using the first period decision from the Lagrangean relaxation in each period. Computing the effectiveness of the heuristic requires evaluation of the expected solution from the heuristic, at the start of each period, conditional on the state. The expected value of the heuristic then must be computed by working backward from the end period. The computation can be summarized as follows. qt .Wt 1 ; It 1 / D ct Rt .Wt 1 ; It 1 / C kt Xt C wt Wt X CLt .Yt / C P fDt D d gqt C1 .Wt ; .Yt d /C / d

where Xt D f .Wt 1 ; Rt .Wt 1 ; It 1 // Wt D Wt 1 Xt C Rt Yt D Xt C It 1 : To use the heuristic in practice, one would store the table of values Rt .W; I / and at each period, observe the level of WIP, W , and inventory, I , after demand in the previous period occurred. The heuristic would then release Rt .W; I / new work in period t. The value of the heuristic would obviously depend on the demand realization observed but the calculation above computes the expected value of using the heuristic, qt .W; I /, from period t to period T , starting with WIP, W , and inventory, I . To test this heuristic, we ran a series of experiments with different demand patterns and different costs for holding and backorder. The purpose of the test problems was to conduct an initial evaluation of the performance of the heuristic, and to see whether there were situations in which the heuristics worked better or worse. In particular, we considered different patterns of demand. For each problem, we computed

12 Table 1.2 Computational results of the heuristic compared to the lower bound of the relaxation

G. Dobson and U.S. Karmarkar Periods 4 4 4 4 8 8 8 8

Demand pattern Level Step (low demand) Step (high demand) Staggered Level Step (low demand) Step (high demand) Staggered

E.q/=ZLB 1.01 1.06 1.03 1.02 1.00 1.11 1.01 1.06

the heuristic value q D q0 (0,0) and the lower bound provided by the relaxation, ZLB , and calculated the ratio q=ZLB . Table 1.2 above provides a summary of these results. The three demand patterns we tested were “level,” “step,” and “staggered.” Let M denote the maximum theoretical throughput of the process in a given period. Level demand was random demand uniformly distributed between approximately 0.6 M and 1.6 M in each period. Step demand was deterministic demand equal to either 2M or .T 2/M in the last period. Staggered demand was level demand in the odd periods and no demand in the even periods. The results for four periods represent an average of six problems, where the holding cost was either 0.5 or 4 and the backorder cost was 25, 50, or 150. The results for eight-period problems are for a single example each where the holding cost was four and backorder cost was 50. The results of this test suggest that the approach is quite effective, since the gaps between the value of using the heuristic and the lower bounds are typically very small.

1.7 Summary This paper has posed and tackled a multiperiod stochastic, dynamic, convex programming problem, in which the constraints include “clearing functions” to model capacity constraints. These constraints provide a way of capturing the effect of capacity loading on lead times and WIP. The formulation is of interest because it addresses some issues in planning which are difficult to capture in traditional models. In particular, this formulation captures the increase in lead times for release, which result from loading capacity. It also captures the effect of lead times on safety stocks. Lower bounds were developed using a restricted Lagrangean technique. As in some previous studies (Karmarkar 1987a; Karmarkar and Lin 1988, Karmarkar and Yoo 1994, Karmarkar and Rajaram 2002), they appear to be effective for the cases studied. A preliminary test of using the lower bounding results as a heuristic solution, suggests that this could be a promising direction. Our next efforts will address the further development of algorithms and solution techniques for large-scale problems. Many other refinements of the model are also possible. For example, the state

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space could be enlarged to distinguish work already in process, from newly introduced loads. The model could be enlarged to multiple stages, much like a queuing network (this would also identify the “age” of work-in-process, in terms of physical progress). Acknowledgements The authors appreciate the assistance provided by Mike Pangburn in conducting numerical experiments with the algorithms.

References Asmundsson J, Rardin RL, Uzsoy R (2006) “Tractable nonlinear production planning models for semiconductor wafer fabrication facilities,” IEEE Trans Semicond Manuf 19(1):1–17 Bechte W (1982) “Controlling manufacturing lead time and work-in-process inventory by means of load-oriented order release”. 25th Annual International Conference. Proceedings, APICS, APICS Fall Church VA, pp 67–72 Bertrand J (1981) “The effect of workload control on order flow-times”. Proc. IXth IFORS Conference on Operations Research. North Holland, Amsterdam Bertrand JWM (1983) “The use of workload information to control job lateness in controlled and uncontrolled release production systems”. J Oper Manag 3:67–78 Bertrand JWM (1985) “Multiproduct optimal batch sizes with in-process inventories and multiwork centres”. IIE Trans 17:157–163 Bitran GR, Haas EA, Matsuo H (1986) “Production planning of style goods with high setup costs and forecast revisions.” Oper Res 34(2):226–236 Bitran GR, Yanasse HH (1984) “Deterministic approximations to stochastic production problems.” Oper Res 32(5):999–1018 Bowman EH (1956) “Production scheduling by the transportation method of linear programming.” Oper Res 4(1):100–103 Charnes A, Cooper WW (1959) “Chance constrained programming.” Manag Sci 5:73–79 Deleersnyder JL, Hodgson TJ, Muller H, O’Grady PJ (1989) “Kanban controlled pull systems: an analytic approach”. Manag Sci 35:1079–1091 Denardo EV, Tang CS (1992) “Linear control of a markov production system”. Oper Res 40: 259–278 Gferer H, Zäpfel G (1995) “Hierarchical model for production planning in case of uncertain demand.” Eur J Oper Res 86(1):142–161 Graves SC (1986) “A tactical planning model of a job shop”. Oper Res 34:522–533 Hausman WH, Sides RGS (1974) “Mail-order demands for style goods: Theory and data analysis,” Manag Sci 20:191–202 Hanssmann F, Hess SW (1960) “A linear programming approach to production and employment scheduling,” Manag Tech 1 Hax AC (1978). “Aggregate production planning”. In: Moder J, Elmaghraby SE (eds.) Handbook of operation research, Chap. 5, vol. 2. Van Nostrand, Reinhold Homem-de-Mello T, Shapiro A, Spearman ML (1999) “Finding optimal material release times using simulation-based optimization.” Manag Sci 45(1):86–102 Holt CC, Modigliani F, Muth JF, Simon HA (1960) Planning production, inventories and work force. Prentice-Hall, Englewood Cliffs, NJ Karmarkar US (1987a) “The multilocation multi-period inventory problem: bounds and approximations,” Manag Sci 33:86–94 Karmarkar US (1987b) “Lot sizes, lead times and in-process inventories”. Manag Sci 33:409–418 Karmarkar US (1989) “Capacity loading and release planning with work-in-progress (WIP) and leadtimes”. J Manuf Oper Manag 2:105–123

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Karmarkar US (1993) “Manufacturing lead times, order release and capacity loading”. In: Graves S, Rinnooy Kan AHG, Zipkin P (eds.) Handbook in operations research and management science, Vol. 4. Logistics of Production and Inventory. North Holland Karmarkar US, Kekre S (1989) “Batching policy in Kanban systems”. J Manuf Syst 8:317–328 Karmarkar US, Kekre S, Kekre S, Freeman S (1985) “Lotsizing and lead time performance in a manufacturing cell”. Interfaces 15:1–9 Karmarkar US, Lin S (1988) “Production planning with uncertain yields and demands”. Simon School of Management Working Paper, University of Rochester Karmarkar US, Rajaram K (2002) “Product cycling with uncertain yields: analysis and application to the process industry” Oper Res 50:680–691 Karmarkar US, Yoo J (1994) “Stochastic dynamic product cycling problem”. Eur J Oper Res 73:360–373 Kimball GE (1988) “General principles of inventory control”. J. Manuf Oper Manag 1:119–130 Land M, Gaalman G (1996) “Workload control concepts in job shops – a critical assessment”. Int J Prod Econ 46–47:535–548 Lasdon LS, Terjung RC (1971) “An efficient algorithm for multi-item scheduling”. Oper Res 19(4):946–969 Lasserre JB, Mercé C (1990) Robust hierarchical production planning under uncertainty. Ann Oper Res 26:73–87 Lin SC (1986) “Production with uncertainty”. Ph.D. Dissertation, William E. Simon Graduate School of Business Administration, University of Rochester, unpublished Manne AS (1958) “Programming of economic lot sizes.” Manag Sci 4:115–135 Missbauer H (2002) “Aggregate order release planning for time-varying demand”. Int J Prod Res 40(3):699–718 Nam S, Logendran R (1992) “Aggregate production planning – a survey of models and methodologies”. Eur J Oper Res 61:255–272 Pahl J, Voß S, Woodruff DL (2005) “Production planning with load dependent lead times”. 4OR Q J Oper Res 3(4):257–302 Perona M, Portioli A (1998) “The impact of parameters setting in load oriented manufacturing control”. Int J Prod Econ 55:133–142 Riaño G (2002) “Transient behavior of stochastic networks: application to production planning with load-dependent lead times.” Ph.D. Dissertation, Georgia Institute of Technology Sahinidis NV (2004) “Optimization under uncertainty: state-of-the-art and opportunities”. Comput Chem Eng 28:971–983 Spearman ML, Zazanis MA (1992) “Push and pull production systems: issues and comparisons”. Oper Res 40:521–532 Zijm WHM, Buitenhoek R (1996) “Capacity planning and leadtime management”. Int J Prod Econ 46–47:165–179 Zipkin PH (1986) “Models for design and control of stochastic, multi-item batch production systems”. Oper Res 34:91–104

Chapter 2

Production Planning and Scheduling: Interaction and Coordination Yiwei Cai, Erhan Kutanoglu, and John Hasenbein

2.1 Introduction In many organizations, production planning is part of a hierarchical planning, capacity/resource allocation, scheduling and control framework. The production plan considers resource capacities, time periods, supply and demand over a reasonably long planning horizon at a high level. Its decision then forms the input to the more detailed, shorter-term functions such as scheduling and control at the lower level, which usually have more accurate estimates of supply, demand, and capacity levels. Hence, interaction between production planning and production scheduling/control is inevitable, not only because the scheduling/control decisions are constrained by the planning decisions, but also because disruptions occurring in the execution/control stage (usually after schedule generation) may affect the optimality and/or feasibility of both the plan and the schedule. If the overall performance of the production system is to be improved, disruptions must be managed effectively, with careful consideration of both planning and scheduling decisions. This chapter focuses on the interaction between production planning and scheduling, emphasizing the coordination of decisions, with special emphasis on making robust decisions at both levels in the face of unexpected disruptions. We provide examples and realistic scenarios from semiconductor manufacturing. To capture the interaction between production planning and scheduling, we suggest an intermediate model between the two levels. One can view this as a lowerlevel planning model or a higher-level scheduling model, but ultimately it provides a middle ground between the two levels of the decision-making framework. In many systems, the longer-term, aggregated production plan is used to facilitate scheduling. This is usually achieved by creating specific work orders or jobs of different product types that collectively resemble the output required by the production plan and generating a release schedule for the jobs. For example, semiconductor manufacturers E. Kutanoglu () Graduate Program in Operations Research and Industrial Engineering, Department of Mechanical Engineering, The University of Texas at Austin, Austin, TX 78712, USA e-mail: [email protected]


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rely on detailed simulation-based models to fine tune the release schedule (also referred to as “wafer starts”), which ultimately determines the product mix in the wafer fab. The scheduling function typically dispatches the jobs according to their perceived or assigned priorities to align the processing sequence of the jobs with the production plan while using the latest information on job and machine availabilities. Dispatching inherently utilizes local information (typically one-job, one-machine at a time) to make a decision. It is very hard, if not impossible, to calculate the effects of an individual dispatching decision on the long-term system performance, or even on the performance of the upstream and downstream machines on the shop floor. The idea of an intermediate model between planning and scheduling is to provide additional useful information to the scheduling system, whether it is dispatching based or otherwise. With the interaction and coordination between production planning and scheduling being the main theme of this chapter, we first review the literature, focusing on a selected set of production planning and scheduling-based papers. In Sect. 2.3, we present two versions of our approach that attempt to fill the gap between the classical planning and dispatching-based scheduling models using an intermediate decision model. Section 2.4 describes our computational study with a simplified reentrant system that represents a small wafer fabrication facility. Section 2.4 also discusses the implementation issues that must be addressed to properly compare the proposed approach with the conventional planning-to-scheduling approach, using this minifab model. In Sect. 2.5, we present the experimental results focusing on insights that can be obtained from our preliminary experiments. We conclude with a summary and future research directions in Sect. 2.6.

2.2 Literature Review We first review the production planning models that are representative of the existing models in the literature. This review is not exhaustive by any means, and our main focus is on the interaction between planning and scheduling, not necessarily on the other aspects of the models, such as batching or setups, that may be potentially critical. We also try to give examples and applications from the literature on semiconductor manufacturing to make the discussion more concrete, but the methods discussed in this section also pertain to more discrete parts manufacturing systems. Due to increasing competition and the rapid development of technology, manufacturing managers, especially those in semiconductor industry focus strongly on cycle time, which is defined as the time between the release of an order to the shop floor and its completion time of that order. Long cycle times imply a high work-in-process (WIP) level, and thus high inventory costs. Therefore, we choose cycle time as the main performance measure with which we evaluate the effectiveness of the coordination between planning and scheduling. This choice further limits our literature review to studies that emphasize cycle time as a potential issue to be addressed between planning and scheduling.

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2.2.1 Production Planning Models An extensive literature on production planning has been developed over almost five decades. In this section, we focus on only a few of these optimization models. Interested readers are referred to the chapter by Missbauer and Uzsoy (2010) in the first volume of this handbook which reviews the basic formulations that are most commonly used in academic research and industrial practice. A capacitated Material Requirement Planning (MRP)-based model is proposed by Horiguchi et al. (2001). The goal is to calculate a planned release date for each order during each of its visits to a bottleneck station, and to estimate when the order will be completed. The authors aggregate the times available across machines over discrete time periods (time buckets) that are used to incorporate capacity factors. The model explicitly considers capacity only for specified near-bottleneck stations, and assumes that all other stations have infinite capacity, which is different from the conventional MRP approach. They perform two experiments. One examines the effect of the predictability of the capacity model. In their paper, predictability is defined as the deviation of the realized completion time in the simulation model from the predicted completion time in the planning model. The results show that finite capacity planning gives better predictability than dispatching rules such as Critical Ratio (Rose 2002). The lot with the lowest value has the highest priority. The second experiment tests the effects of using a “safety capacity” in planning, that is, the reduction of the planned capacity of a given station by some amount to keep processing capacity in reserve to deal with unexpected events such as machine breakdowns. Their results show that increasing the safety capacity reduces tardiness and improves predictability, without adversely affecting other performance measures. There are a wide variety of linear programming-based planning models for production planning. Hackman and Leachman (1989) propose a general production planning framework based on a linear programming model. They take into consideration specific components such as processing and transfer time in order to provide an accurate representation of the production process. However, the time delays in the model do not capture the load-dependent nature of the lead times. Thus, the aspects of production captured in the model are limited. In addition, the LP formulation accommodates noninteger values for cycle times as well as planning time buckets of unequal length. Expanding the model in Hackman and Leachman (1989), Hung and Leachman (1996) incorporate time-dependent parameters representing partial cycle times from job release up to each operation into the LP planning model. Furthermore, they provide a framework that iteratively updates the plan through an LP model that develops a plan for a given set of lead times and a simulation model that evaluates the system performance for a given production plan. They estimate the cycle times from the simulation results and show that they can achieve better results by iterating between the LP and the simulation model. It is well known that the relationship between cycle time and machine utilization is nonlinear. Therefore, the iterative LP-simulation process provides a good way to approximate such a nonlinear relationship. The process stops when satisfactory agreement in cycle times is achieved. Hung and Leachman’s experiments with deterministic and random

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machine breakdowns indicate that the difference between the LP and simulation cycle times can be reduced to 5% or less within a few iterations. However, other researchers have not been able to easily replicate such results, and our own research indicates that similar results are difficult to achieve. In particular, the convergence behavior appears to be unpredictable, and is certainly not well understood. Some planning models that try to capture the relationship between cycle time and utilization without resorting to simulation make use of so-called clearing functions (Graves 1986). Clearing functions express the expected throughput of a machine in a planning period as a function of the expected WIP inventory at the machine over the period. (Our focus on load-dependent cycle times is mainly due to the observation that the load as a function of releases determined as part of the plan affects cycle times that result in scheduling. There are recent studies that try to capture the dependency between the load level and/or utilization and cycle times. In that sense, the underlying approaches can also be viewed as “hybrid” models attempting to link planning and scheduling.) Missbauer (2002) considers clearing functions for an M/G/1 system. He uses a piece-wise linear approximation for the clearing function to model the effective capacity for bottleneck stations, and considers fixed, load-independent time delays between bottleneck stages to represent the delays at nonbottleneck machines. His planning model determines the release plan and uses a short-term order release policy to select specific orders for release into the job shop. The product mix is not considered in their clearing function, i.e., the clearing function only depends on the total planned production quantity, which means the total output of a station can be allocated arbitrarily to different products. Asmundsson et al. (2006) propose a clearing function-based planning model, which explicitly considers the product mix. They approximate the clearing function using an empirical approach, together with two sets of constraints enforcing flow conservation for WIP and finished goods inventory (FGI). There is no need for explicit cycle time parameters in their model. Due to the product mix, different products may have different capacity needs (capacity allocation), and a particular difficulty is estimating the throughput as a function of the product mix currently represented in the WIP. To overcome this, they assume that all products see the same average cycle time, which allows them to use a convex combination of the capacity allocation parameters to approximate the WIP levels of different products, which in turn leads to approximated clearing functions. Exploiting the concavity of clearing functions, they use outer linearization to approximate the functions, which results in an LP model. The objective in their model is to minimize the total production cost, the WIP cost, the FGI holding cost, and the raw material cost. The approximation of the clearing function is also done by simulation with several randomly generated realizations of the demand profile, which are evaluated using the release schedules obtained from the fixed cycle time production planning model of Hackman and Leachman (1989). They perform extensive experiments to evaluate the benefit of the clearing function-based model. Different dispatching rules are used to compare planned throughput and actual throughput. Based on these experiments, one of their conclusions is that:

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“If planning is done properly, the role of a detailed schedule can be viewed as rescheduling the jobs to adhere to the original production plan that has been distorted by equipment failures and other unpredictable occurrences. Although it is unlikely that the scheduler can restore the original plan in every instance, its ability to do so is highly dependent on the planning algorithm’s ability to represent the shop floor dynamics correctly.”

Such a conclusion shows that we need to consider the coordination between planning and scheduling to achieve better performance, which further motivates our study. Pahl et al. (2005) give an extensive survey of planning models, which consider load-dependent cycle times. In addition to the use of clearing functions, there are other approaches. Interested readers are referred to Sect. 3 and the corresponding references in Pahl et al. (2005) for more details. Model predictive control, or MPC (Qin and Badgwell 2003), is a method of process control that has been used extensively in processing industries (Kleindorfer et al. 1975). MPC encompasses a group of algorithms that optimize the predicted future values of the plant output by computing a sequence of future control increments. This optimization model is implemented through a rolling-horizon approach at each sampling time. MPC attempts to model the dependence between the sequence of predicted values of the system output, and the sequence of future control increments. With knowledge of the system model, disturbance measurements, and historical information of the process, the MPC model calculates a sequence of future control increments that must satisfy appropriate constraints. Vargas-Villami and Rivera (2000) propose a two-layer production control method based on MPC. Extending this work, Vargas-Villami et al. (2003) propose a three-layer version. The first layer, called the adaptive layer, is used to develop a parameter estimation approach. The second layer, the optimizer, solves an MILP model by branch and bound to generate a good-quality production plan. The third layer (direct control) uses dispatching to control the detailed discrete-event reentrant manufacturing line in a simulation model. The computational results show that the method is less sensitive to initial conditions than “industrial-like” policies examined by Tsakalis et al. (2003). Furthermore, the three-layer approach with the adaptive parameter estimation model achieves reduced variation at high production loads as compared to the two-layer approach. They did not perform any cycle time comparisons with existing methods, but point out that an MPC-based model could be a promising tool for planning. Jaikumar (1974) proposes a methodology, which decomposes the planning and scheduling problem into two subproblems. The first problem is a long range planning problem, which maximizes the profit subject to resource constraints. The Lagrange multiplers obtained in the first problem are used in the objective function of the second short range scheduling model. They propose a heuristic algorithm to reduce the second model to a sequential allocation of production facilities to products.

2.2.2 Scheduling Models Apart from planning models, there are models and studies that focus on scheduling only. Leachman et al. (2002) summarize their effort, dubbed Short Cycle Time

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and Low Inventory in Manufacturing (SLIM), to improve scheduling at Samsung Electronics. They try to control the production line by managing the WIP level at particular bottleneck stations. A dispatching-based method is used to achieve target WIP levels at the bottleneck stations. An important task is to determine appropriate target WIP levels. Based on an overall target cycle time for each product, the total buffer time is calculated as the difference between the target cycle time and theoretical (or raw) cycle time. The total buffer time is then proportionally allocated to each bottleneck step. The resulting buffer time allocation is in turn used to estimate the target WIP levels using Little’s Law. By using different strategies according to the characteristics of different production stages (bottleneck, batching, nonbottleneck steps, etc.) and prioritizing the jobs in different stages to meet target WIP levels (and hence bottleneck utilization) as closely as possible, they show that one can reduce average cycle times. In contrast to the fab-wide approach in Leachman et al. (2002), other authors focus on bottleneck steps only, with explicit controls the WIP level. Lee and Kim (2002) try to implement WIP control at bottleneck steps to balance a production line in semiconductor manufacturing. Assuming a given target throughput rate, they calculate the buffer time and the associated target WIP level. The work focuses on short-term scheduling for the steppers, which are usually the bottleneck machines in most wafer fabrication facilities. One of the two proposed MILP models minimizes the total weighted deviation from the target WIP level, and the other maximizes the total wafer production for all steppers in the hope that this will lead to high utilization of bottleneck machines. Kim et al. (2003) use a similar idea to determine a single-shift schedule for the steppers for a given WIP status. Using an MILP formulation, they try to maintain the WIP levels close to the “desired” levels so that the flow of material through the factory is balanced. The objective is to meet the predetermined WIP targets. Three proposed heuristics to solve the underlying MILP model can find schedules within 5% of the optimum values in a reasonable amount of time. Queueing theory is useful in scheduling in several ways. One way is to analyze the stability (that is, boundedness of average WIP) of scheduling policies. Another way is to model manufacturing systems with multiclass queueing networks to develop scheduling policies. Generally speaking, queueing theory is based on long-term steady-state analysis and may not be optimal in a finite period. However, the following two papers implement queueing theory using fluid models that focus on transient analysis and are a reasonable approximation for short periods of time. Dai and Weiss (2002) develop a fluid-relaxation-based heuristic to minimize makespan in job shops. In a fluid model, discrete jobs are replaced with flow of a continuous fluid and machines are replaced with valves that affect the flow rate of the of fluid. The proposed online (dispatching-based) heuristic uses safety stocks for WIP and tries to keep the bottleneck machine busy at almost all times, with the idea that the nonbottleneck machines are paced accordingly. The heuristic is constructed in three steps: (1) reduce the job shop problem to a reentrant line scheduling problem, which has the same lower bound; (2) define an infeasible backlog schedule that keeps the bottleneck machine busy (here the schedule is infeasible because a

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machine is allowed to start work on a job step even if the previous step of the job on a different machine has not been completed); (3) introduce safety stocks to make the backlog schedule feasible. Similarly, Bertsimas et al. (2003) use a fluid model to solve a job shop scheduling problem with the objective of minimizing the holding cost. The proposed algorithm uses the optimal fluid solution as a guide. Comparison with several other commonly used heuristics shows that the proposed algorithm outperforms the other heuristic methods. This section provided a brief review of planning and scheduling models with a focus on the interaction between the two models. We note that most of the planning models in the literature do not consider WIP allocation across stations, while many scheduling models consider WIP level explicitly. Therefore, we propose an approach in Sect. 2.3 to explicitly consider WIP allocation in an intermediate model called high-level scheduling, and try to control WIP allocation in a way which facilitates the coordination between planning and scheduling.

2.3 Coordination of Planning and Scheduling 2.3.1 Overall Approach In most manufacturing companies, the planning and scheduling functions belong to two different departments, as planning is viewed as a tactical activity and scheduling as more operational. Sometimes it is necessary to separate planning and scheduling because it is almost impossible to obtain a comprehensive system-wide solution that encompasses both planning and scheduling concerns. Such a model would have to provide detailed decisions for each machine in each period. In general, it is impossible to solve such a model since it is inherently too complex with too many constraints and variables. Hierarchical decomposition into planning and scheduling provides an easy way not only to obtain reasonable solutions to both subproblems, but also to generate decisions aligned with the current organizational structure between planning and scheduling functions. However, the conventional hierarchical separation of these two functions may cause several problems. One drawback is that a solution that is good at the planning level might not be easy to implement as a detailed schedule; the plan may not be even feasible when the scheduling issues are explicitly considered. One reason behind this is that the dynamics of the production system are modeled at an aggregate level, and detailed execution may be infeasible even if aggregate constraints are satisfied. Another issue is that objectives are usually different between planning and scheduling. The planning function focuses more on how to meet the demand, and reduce inventory and backorders, while scheduling emphasizes more operational measures such as minimizing cycle times and maximizing bottleneck utilization. Ideally, if the hierarchical decomposition were done properly, the objectives of the two levels would be aligned. However, due to the complexity of the overall problem,

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the computational effort involved in solving the two levels to optimality, and the differences in preferences between the organizational areas representing the two levels, this is not the case in the real world. In this chapter, we test an idea that seeks to overcome the drawbacks of such separation of planning and scheduling. The idea is to introduce an “intermediate” module between planning and scheduling, which overall modifies the conventional hierarchical approach. The goal is to solve the discrepancy in objectives between production planning and detailed scheduling. In the following, we define “normal planning” as a planning model that does not consider WIP levels explicitly, and a “high level scheduling” model as one that does. In this approach, we have a normal planning model (denoted by “P”) and a high level scheduling model (denoted by “H”) that both feed into detailed scheduling (denoted by “D”) (See Fig. 2.1). To provide a framework for our discussion, we represent each “stage” in the process with an associated model: the planning model, the high-level scheduling model, and the detailed scheduling model. In fact, in the following discussion, the first two models are linear programming problems and the last one is a simulation model of the system that represents the actual implementation of the plan and schedule. First, we explain the proposed version of the P–H–D approach: The planning model tries to meet the demand while minimizing inventory and backorder costs. The output of the planning model specifies how much of each product should be produced by the end of each period. Then, we use that output as a modified demand profile, which is input to the high level scheduling model, which explicitly tries to minimize the average WIP level, and thus the average cycle time. The high level scheduling model determines the release policy and processing targets for each product in each period, which form the input to detailed scheduling. Figure 2.1 compares the proposed P–H–D approach with the P–D approach.

Fig. 2.1 Three approaches for planning and scheduling

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Consideration of the three potential levels in the overall process gives rise to another structure, in which the planning step is skipped: the original demand data is fed directly to the high-level scheduling model whose output is input to the detailed scheduling model. Intuitively, using a high level schedule without a planning model consideration may achieve lower cycle times, but may result in higher costs since it pays more direct attention to WIP levels than to costs. We now discuss our mathematical formulations for the planning and high level scheduling models.

2.3.2 Planning Model For the planning model, we use a fixed cycle time version of a well-known linear programming model found in Hung and Leachman (1996). This model assumes that the next planning horizon is divided into equal length time periods (or time buckets, e.g., representing individual shifts). The planning horizon is assumed to be long enough to capture varying demand levels (especially across product types) over time. We assume that we have accurate forecasts for demand levels for each product type, say, in every week, i.e., every 14 shifts. (To be consistent with the length of the time bucket used in the model, the demand profile used in the experiments has nonzero demand at the end of every 14 shifts). We finally assume that the production process is divided into production stages or steps, each of which represents a unique operation to be performed on a particular machine group (station). We first introduce the notation that supports the model. 2.3.2.1 Sets I : set of products, indexed by i T : set of time periods (say, shifts), indexed by t K: set of processing steps D f1; :::; g, indexed by k, where is the number of steps, assumed to be the same for all products M : set of stations, indexed by m

2.3.2.2 Input Parameters pi;m;k : processing time of product i on station m at step k (say, in minutes) cm : available running time of station m in one shift (cm D 12 h for all m in our model) di;t : demand for product i in shift t (in number of jobs), assumed to be nonzero every 14 shifts with the availability of weekly forecasts and 2 shifts per day fi;k : average partial cycle time for product i to finish step k (estimated from simulation results or historical data), i.e., the average difference between the time when the job is released and the time it finishes step k(in shifts) u: length of one shift (12 h)

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˙

qi;k : smallest integer greater than fi;k , i.e. qi;k D fi;k bi : unit backorder cost for product i , set to a large number to discourage backorders hi : cost for holding one unit FGI of product i for one shift ıi;k : coefficients used in constraints (2.4) for product i at step k (we explain this term later in detail)

2.3.2.3 Decision Variables (All Variables Are Nonnegative) Di;t;k : amount of product i that depart from step k in shift t (in number of jobs) Ri;t : amount of product i released in shift t (in number of jobs) Bi;t : backorder for product i in shift t (in number of jobs) Ii;t : inventory of product i at the end of shift t at the end of production line (in number of jobs).

2.3.2.4 Model Formulation min

XX

.et Ri;t C hi Ii;t C bi Bi;t /

i 2I t 2T

subject to XX

pi;m;k Di;t;k cm 8t 2 T; m 2 M

(2.1)

i 2I k2K

Di;t; Ii;t C Ii;t 1 Bi;t 1 C Bi;t D di;t 8i 2 I; 1 < t < jT j 1 (2.2) Di;t; Bi;t 1 C Bi;t D di;t 8i 2 I; t jT j 1

(2.3)

ıi;k Ri;t qi;k C .1 ıi;k /Ri;t qi;k C1 D Di;t;k 8i 2 I; k 2 K:

(2.4)

In this model, the objective is to minimize the total costs of releases, inventory, and backorders, across all products and shifts. The discounted raw material release costs are used to release raw material into the factory as late as possible so as to indirectly manage the WIP in the factory. Constraints (2.1) limit the capacity of each station with the given amount of time. Constraints (2.2) and (2.3) ensure that the end product demand in each period is either met by finished product inventory or backlogged. Constraints (2.4) capture the dynamic properties of the cycle time. From simulation results or historical data, we can estimate fi;k , which is the average partial cycle time for product i to finish step k (from the beginning of first step of the product until and including step k). Thus, we consider the production process as a fluid model, and estimate the relation between the released quantity and the departing quantity by backtracking the production flow along the time horizon. Figure 2.2 shows the details of the relationship between Ri;t and Di;t;k .

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Fig. 2.2 Partial cycle times, and the relationship between Ri;t and Di;t;k

All products of type i finished at step k in shift t, denoted by Di;t;k should have been released fi;k time units ago. Thus, by backtracking we can determine the time when these products are released. In Hung and Leachman (1996), a formula is provided to address all the cases where the cycle time is either longer or shorter than the length of the time bucket. Since we use a constant length for each time bucket, and in our mini-fab model (see Sect. 2.4.1) the partial cycle times are less than the length of the time bucket (shift), Di;t;k is composed of two parts. Thus, the relation between the products released, Ri;t , and Di;t;k is as follows: Di;t;k D ıi;k Ri; t qi;k C .1 ıi;k /Ri; t qi;k C1 8i 2 I; 1 t jT j 1; k 2 K; where the portions of releases to be completed by step k are estimated by ıi;k and (1 ıi;k ). Here, we use ıi;k ’s proportional to time: ıi;k D .fi;k mod u/=u, 8i 2 I and 8k 2 K.

2.3.3 High Level Scheduling Model As mentioned before, the purpose of the high level scheduling model is to find a balance between the cycle times and the inventory/backorder costs. Thus, its objective function and constraints should take both factors into consideration. Although it is hard to model cycle times directly, we know from Little’s Law that they are proportional to WIP levels for a fixed throughput. Thus, we consider the WIP level instead and try to represent minimization of cycle times by minimizing the WIP levels at the end of each shift.

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In this model, the relationship between releases and departures is captured in the same manner as in the planning model. The main difference is that we explicitly capture WIP levels and their distribution across stations and stages of production in detail. In this model, we manage to keep a certain level of WIP at bottleneck station to prevent its starvation, which may lead to reductions in the throughput rate. These levels are formulated using a WIP control constraint that allows the WIP to be within a tolerance around the target value. To enable the model to build up WIP to meet the target level, the LP model enforces the WIP control constraints only after a certain number of periods elapsed. With this in mind, we introduce additional notation.

2.3.3.1 Sets KB : set of steps in the bottleneck station

2.3.3.2 Input Parameters C ; : upper and lower tolerance of WIP control (currently set at C D 1:1 and D 0:9) vi : target cycle time for product i (in minutes) i : total raw processing time for product i (in minutes) yi : total buffer time for product i , i.e. the difference between the vi and i (in minutes) zi;k : target WIP level of product i at station k (in number of jobs) i;k : the average cycle time for product i to travel to the kth bottleneck step from its previous bottleneck step (in minutes) i : average demand rate for product i (in jobs per minute) ts : the last shift in which the WIP control constaints are not enforced; the high level scheduling model is assumed to be in steady state after shift ts

2.3.3.3 Decision Variables Wi;t;k : WIP level of product i at step k at the end of shift t (in number of jobs).

2.3.3.4 Model Formulation The calculation of the target WIP level is adapted from the method proposed by Leachman et al. (2002), which uses Little’s Law (as will be explained in detail later). They use target WIP levels as criteria for dispatching. Here, we incorporate the target WIP levels into the high level scheduling model. The target WIP calculation allocates the buffer time to bottleneck steps in proportion to the partial cycle time between two consecutive bottleneck steps. Cycle times are estimated

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from a simulation model of the mini-fab in our experiments, but practice they can be obtained from historical data. One can calibrate the estimated cycle times through multiple simulation runs, but here we obtain cycle times in a single run. We first compute the buffer time as the difference between the target cycle time and the theoretical (or raw) cycle time (which consists of only processing times) for each product: yi D vi i 8i 2 I: To allocate this overall buffer time (slack) into bottleneck steps, we compute the time between two consecutive bottleneck steps of each product: i;k D fi;k fi;k 0 8i 2 I; k; k 0 2 KB; where k and k 0 are two consecutive bottleneck steps. Finally, we set the target throughputs equal to the average demand rates, allocate the overall buffer time and convert the allocations to target WIP levels as follows: 8i 2 I; k 2 KB : zi;k D i yi Xi;k i;k 0 k 0 2KB

The final mathematical model is as follows: min

XX X i 2I t 2T

! Wi;t;k C hi Ii;t C bi Bi;t

k2K

subject to XX

pi;m;k Di;t;k cm ; 8t 2 T; m 2 M

(2.5)

t 2T k2K

Di;t; Ii;t C Ii;t 1 Bi;t 1 C Bi;t D di;t ; 8i 2 I; 1 < t < jT j 1 (2.6) Di;t; Bi;t 1 C Bi;t D di;t ; 8i 2 I; t jT j 1

(2.7)

Ri;t Di;t;1 C Ii;t;1 D Wi;t;1; 8i 2 I; t 1

(2.8)

Di;t;k1 Di;t;k C Ii;t;k D Wi;t;k ; 8i 2 I; t > 1; k > 1 X X Wi;t;k C zi;k ; 8t 2 T; k 2 KB

(2.9)

i 2I

X

(2.10)

i 2I

Wi;t;k

X

i 2I

zi;k ; 8t 2 t; k 2 KB

(2.11)

i 2I

ıi;k Ri; t qi;k C .1 ıi;k /Ri; t qi;k C1 Di;j;k ; 8i 2 I; 1 t ts k 2 K ıi;k Ri; t qi;k C .1 ıi;k /Ri; t qi;k C1

(2.12) D Di;j;k ; 8i 2 I; ts < t jT j 1 k 2 K: (2.13)

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The main difference between this model and the previous planning-based one is that we now have WIP control constraints for the bottleneck steps. Here, we first calculate the target WIP level for each step in the bottleneck station, and then use the target WIP level in the LP model. Leachman et al. (2002) use a dispatching rule to execute WIP control for individual buffers (product and step), while in our LP model we control the total WIP level across all products at the same step in the bottleneck station. The reason is that we think the purpose of setting target WIP levels is to keep feeding the bottleneck station, so we only need to track the total WIP level across all products at the same step in the bottleneck station instead of controlling WIP levels for individual buffers.

2.4 Experimental Study To test the idea of high level scheduling, we perform several experiments on a threestation six-step hypothetical production system, which represents a small wafer fabrication facility (mini-fab). Below, we first describe the mini-fab system, and then present the experimental results. These experiments focus on (1) evaluating the overall merit of the P–H–D approach, (2) testing the impact of different cost settings, and (3) testing the impact of machine breakdowns at nonbottleneck stations on the cycle times and cost. For testing, we run the mathematical models that represent the planning and/or high-level scheduling problems with the same inputs, and then evaluate the system performance (cycle time, WIP levels, costs) with a simulation model that imitates the implementation of decisions made by the mathematical models. For this, the simulation model relies on a dispatching-based methodology that tries to follow the release and production targets set by the outputs of the mathematical models. Thus, the dispatching logic in the simulation model acts as a detailed scheduling system.

2.4.1 Mini-Fab Model The three-station six-step mini-fab model is depicted in Fig. 2.3. There are two different products, each of which must complete six operational steps. Each step is to be performed by a machine at one of the stations. The process flows (routings), which are the same for both products, are also shown in Fig. 2.3. Table 2.1 shows the basic settings of the raw processing times for each product at each step (in minutes). The last column shows the total raw processing time (RPT) for both products. We consider a time horizon of 150 days, each day with 2 shifts of 12 h, leading to a 300-shift long horizon. All the processing times are assumed to be deterministic. In the experiments with machine breakdowns, the first station’s machines may fail. Times between failures follow an exponential distribution with a mean of 42 h. The repair times follow an exponential distribution with a mean of 45 min. The base

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Fig. 2.3 Six-step three-station mini-fab model Table 2.1 Raw processing time for the mini-fab model Step 1 Step 2 Step 3 Step 4 Step 5 Product 1 47.5 30 75 40 52.5 Product 2 38 24 60 32 42 Table 2.2 Traffic intensity (expected utilization) for all stations

Step 6 30 24

Total RPT 275 220

Traffic intensity Station 1 Station 2 Station 3 No machine breakdowns 0.78 0.74 0.94 With machine breakdowns 0.80 0.74 0.94

demand rates are 55 jobs per week for product 1 and 44 jobs per week for product 2. The backorder cost is $50 per unit per shift, and the inventory cost is $1 per unit per shift. We vary these values in the next section to evaluate impacts from different factors. Table 2.2 gives the traffic intensities (or utilizations) with and without machine breakdowns for all stations. We observe that station 3 is an overall bottleneck, and even when breakdowns are considered, the bottleneck station does not change.

2.4.2 Simulation Settings To evaluate the performances of the different approaches in combining planning and scheduling, we use a simulation model that imitates the execution of the planning/scheduling decisions using a dispatching-based methodology. Here, the simulation model represents the actual system, where the planning/scheduling decisions are executed at the detailed scheduling level, during which the actual costs and performances are incurred and measured. The simulation model first creates production jobs according to the release schedule obtained in the mathematical model being used. It then dispatches the jobs at the machines with some level of adherence to the planning and scheduling decisions. Although there are several ways to carry out dispatching for a given plan/schedule, we apply two different dispatching rules: First-in-first-out (FIFO) and “target following.” FIFO dispatches the job that arrives at the station earliest, when a machine at the station becomes free, and is used primarily as a benchmark. The target following rule tries to follow the planning or high level scheduling production targets (captured by the optimal values of

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departure variables in both models) as closely as possible. Since the solution to the high-level scheduling LP model provides the processing targets for each product at each step by the end of each shift, the target following rule gives the highest priority to the buffer (product/step combination), which is the most behind (or least ahead) of its cumulative processing target. The simulation run length is determined by the completion of all jobs that are released during the time horizon, 300 shifts. To obtain stable statistics on cost and other performance measures such as cycle times, the first and last 100 jobs are discarded. For the experiments with machine breakdowns (i.e. the scenarios in which the first station may break down), we run 50 replications with stochastic machine failure times and repair times to obtain the statistics.

2.4.3 Implementation There are several issues to be addressed regarding the simulation experiments, especially in terms of how the planning and scheduling decisions from the mathematical models are converted to inputs for simulation testing. One issue is how to convert the release policy to a usable input for simulation. The LP model solution produces a noninteger numbers of jobs to be released during each shift, which should be converted to integer values before they are used in simulation. We first round the noninteger release values in the LP solution to integer values. However, due to the run length of 300 shifts, the difference (denoted by ) between the sum of the rounded values and the sum of the originally fractional releases can sometimes be significant. As a result, if we use the rounded value for the release policy in the simulation, we may incur unnecessary backorders or extra inventory. To overcome this problem, we first calculate , and then try to evenly distribute this difference over the 300 shifts. For example, suppose D 30 (i.e., the rounded values lead to an accumulated release of 30 more jobs over 300 shifts than the fractional values do). Then we would ideally release one additional job every 10 shifts for an even distribution over 300 shifts. We follow this ideal distribution whenever we can, for example when there is already a scheduled release in a shift and the even distribution causes an additional job release in that shift. However, for shifts with no scheduled releases, this may cause a release of one job by itself. Hence, if the even distribution of suggests releasing a job by itself in a shift with no originally scheduled release, we release the extra job in the next shift with a nonzero release in the original LP solution. This way, we can minimize the discrepancy between the rounded release values and the original ones, and follow the original release schedule as closely as possible. A similar issue must be resolved while coordinating the planning and the high level schedule. However, we do not address the noninteger issue here, because the output result from the planning model is not a “real” demand profile, but rather a guideline that will allow the high level schedule to meet the actual demand. Thus, the fractional solution from the planning model should be adequate for this purpose.

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Therefore, when the planning model provides its decisions, i.e. how many jobs need to be produced at the end of each shift, to the high level schedule as the demand profile, it retains the original fractional values. Finally, we discuss the criteria used to evaluate the simulation results. A common performance measure in practice is cycle time, which measures how much time a product spends in the system. Average cycle time is a typical aggregated measure of the cycle time performance for each product category. In simulation, we collect the cycle times for finished jobs of each product and divide the average cycle time for each product by its raw processing time to obtain the so-called X-factors. A productbased X-factor in a way measures how many multiples of the raw processing time a product spends in the system, on average, before completion. For the one product experiments, it is obvious that a lower X-factor indicates a better performance. When we compare different approaches or parameters in the two-product experiments, it is possible that one product’s X-factor is smaller and the other’s is larger across different approaches and models. To facilitate a reasonable overall comparison, we compute the weighted average of the individual X-factors, with the weights being set equal to the product demand rates, thus producing a demand-weighted X-factor as a common measure across both products.

2.5 Experimental Results Before we present the experimental results, we describe our terminology in this part of the chapter. In the charts that present the simulation results in the following discussion, we use the notation in Fig. 2.1. “P–D” means the traditional planning approach feeding detailed scheduling simulation; “P–H–D” means planning followed by high-level scheduling which feeds detailed scheduling; “H–D” means high-level scheduling feeding detailed scheduling without prior planning. In detailed scheduling, “F” means using FIFO as a dispatching rule, and “T” means using the target following dispatching rule. Aggregate performance measures obtained from simulation are shown as a function of the two arguments – the approach that feeds detailed scheduling (whether it is P–D, P–H–D, or H–D), and the dispatching rule used in detailed scheduling (whether it is F or T). The two major performance measures are the weighted average X-factors and total costs. “X” shows the product 1 X-factor in the one-product setting, and the demand-weighted average X-factor in the two product setting. “C” denotes the total inventory and backorder costs. To understand how the system performance changes, we vary certain parameters and observe their effects on the relative performances of the tested planning/scheduling approaches. In particular, we vary the unit inventory costs and demand variation across time periods, and examine different scenarios for machine breakdowns. The mini-fab setting explained in Sect. 2.4 is our base case, and in all the following experiments, we only change one parameter at a time and keep other parameters unchanged.

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2.5.1 One Product Results First, we test the approaches with one-product experiments. In these experiments, we only have demand for product 1. In order to keep the traffic intensity (or utilization) consistent with the two-product settings, we change the demand profile in the base case setting for product 1 to 90 jobs every 14 shifts, which results in a traffic intensity of 93%. In these experiments, we keep the backorder cost fixed at $50 per unit per shift and modify the inventory cost per unit per shift according to Table 2.3. Figure 2.4 shows the impact of varying the unit inventory cost on X-factor and total costs. We see that the proposed P–H–D approach and the H–D approach always produce lower X-factors than the planning model alone (with either dispatching rule). We also find that the changes in inventory cost affect the overall X-factors level very little. The total costs generally increase as the unit inventory costs increase, regardless of the approach, with small differences across alternative approaches. To see whether there is any interaction between the approaches and the components of the total costs (inventory and backorder), we plot their levels separately in Fig. 2.5. We see that as the unit inventory cost increases, we keep less inventory and incur more backorders for a given approach, as expected. Since the P–H–D and H–D approaches try to control the WIP levels, they have a little more inventory and much fewer backorders than the traditional P–D approach. We also observe that an increase in the unit inventory cost (and a decrease in the relative cost of backorders) causes significantly sharper increases in the conventional P–D approach, regardless of the dispatching rule used for detailed scheduling. The experiments using the FIFO dispatching rule with the P–H–D and H–D approaches also show that FIFO is in general worse than the target following dispatching rule. We only report the performance of P–D and FIFO as a benchmark to make the graphs more legible. Table 2.3 Change in inventory cost per unit per shift Experiment index 1 2 3 4 Unit inventory cost for product 1 1 2 4 6

Fig. 2.4 Impact of varying unit inventory costs on X-factor and total costs for single product experiments

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Fig. 2.5 Impact on inventory and backorder quantity of inventory cost change for single product

Table 2.4 Demand distributions for the single product experiments Experiment index 1 2 3 4 Demand (Uniform) 90 (88, 93) (85, 95) (83, 98)

Fig. 2.6 Impact on X-factor and cost of demand variance

2.5.1.1 Demand Variation In actual manufacturing systems, demand may vary over time. This is especially true in the semiconductor industry, where the technology evolves so fast that it affects the demand levels for different categories of products differently. In the experiments in this section, we modify the variability of the demand. Hence, for each week, we generate a random demand value drawn from a discrete uniform distribution. We change variability of demand over time by extending the underlying range of the distribution as shown in Table 2.4. Again, as seen in Fig. 2.6, the X-factors obtained by P–H–D and H–D are much lower than those obtained by P–D, regardless of the dispatching rule used with P–D. The H–D approach achieves the lowest total costs among the approaches tested. From these results, we see that increase in demand variation does not cause a significant increase in the X-factor levels for a given planning/scheduling approach, while the increase in total costs can be high, especially for P–D and P–H–D.

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Fig. 2.7 Impact of demand variation on inventory and backorder quantities

Figure 2.7 shows the average inventory and backorder level. We find that the backorder level of the P–D approach increases much more sharply than in the other approaches as the demand variation increases, regardless of the dispatching rule used in conjunction with it. Since the P–H–D approach obtains the implied demand profile from planning and does not directly consider the original demand profile, it cannot perform well in controlling backorders as the demand variation increases. Although the P–H–D approach has higher backorder levels than the H–D approach, it has lower backorders than the P–D approach in three out of four experiments, and a comparable backorder level for the other one. However, the demand variation almost has no impact on the H–D approach; the inventory and backorder levels do not climb significantly while the demand variation increases. The reason could be that the H–D approach takes the original demand (and its variation) into account directly and, to some extent, anticipates the variations in demand. Thus, it performs better than the alternatives in the case where the demand variation is a dominating factor.

2.5.2 Two Product Results We now present the results of the two-product experiments. Since the two products compete for capacity in this setting, the analysis is not straightforward. However, we still find some useful insights here.

2.5.2.1 Unit Inventory Cost In this set of experiments, we evaluate the impact of varying unit inventory cost, which takes a value of $1, $2, $4, or $6 per job per shift. When we change the inventory cost for one product, we fix the inventory cost for the other to the base level, $1 per job per shift. The changes are displayed in Table 2.5. Figures 2.8 and 2.9 show the simulation results of these experiments varying unit inventory cost for product 1 and for that of product 2, respectively.

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4 6 1

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5 1 2

6 1 4

7 1 6

Fig. 2.8 Impact of unit inventory cost of product 1

Fig. 2.9 Impact of unit inventory cost of product 2

As before, increasing unit inventory costs results in increasing total costs and relatively stable X-factor values for the tested approach. Comparing the different planning/scheduling approaches, the P–D approach with FIFO has the largest weighted average X-factor, and the P–D approach with target following the second largest. In general, the P–D approach with either dispatching rule has lower total costs, and the approaches with high level scheduling (P–H–D and H–D) have larger total costs. For all the experiments in which we modified the unit inventory cost for product 1, the weighted average X-factors are similar for the P–H–D and H–D approach. Overall, P–H–D and H–D are the best in terms of the X-factor and they are comparable to P–D with FIFO or target following dispatching. 2.5.2.2 Demand Variation In the previous 2-product experiments, we have demand levels constant over time, at 55 units and 44 units per 14 shifts for products 1 and 2, respectively. To make it

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more realistic, we change shift demands over time, setting the means to the levels in the constant demand case. The distributions used to create individual shift demand levels are discrete uniform as in the one product experiment. We change the demand variation by extending the range of the uniform distribution, taking range values of 0 (no variation, fixed demand scenario), 5, 10, or 15. For example, when we set the variation range to 5, the demand distribution is discrete uniform between 53 and 58 (both inclusive, represented by Uniform(53, 58)) for product 1, and Uniform (42, 47) for product 2. These levels are described in Table 2.6. Figures 2.10 and 2.11 show the impact of demand variation of product 1 and 2, respectively. Again, we have lower weighted X-factors in the P–H–D approach and the H–D approach. For the planning model, the cost increases when the demand variation increases for the first product. As in the one product case, it seems that the high level scheduling approach is robust with respect to the demand variation. Table 2.6 Demand profiles with different distributions Experiment 1 2 3 Product 1 demand (Uniform) 55 (53, 58) (50, 60) Product 2 demand (Uniform) 44 44 44

Fig. 2.10 Impact from demand variation of product 1

Fig. 2.11 Impact from demand variation of product 2

4 (45, 61) 44

5 55 (41, 47)

6 55 (38, 50)

7 55 (34, 53)

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2.5.2.3 Machine Breakdowns In the base experimental setting, described in Sect. 2.4.1, the time between machine failures and the time to repair follow exponential distributions. The mean time between failures (MTBF) at station 1 is 42 h, and the mean time to repair (MTTR) is 45 min, which produces an availability of 98.2% (long-term percentage of “up” time out of total time, i.e., MTBF/(MTBF+MTTR)). In this section, we evaluate the impact of varying levels of machine breakdowns. However, to keep the evaluation simple, we keep the traffic intensity of station 1 less than that of station 3, so that the bottleneck station does not change when we vary the downtime parameters. As before, when we modify one of the factors we keep the others at the base levels. We first vary the mean time between failures at station 1, and keep the mean repair time at 45 min. The MTBF changes according to Table 2.7. Figure 2.12 shows the impact of different mean times between failures at station 1 on the weighted average X-factors for the tested planning/scheduling approaches. In these experiments, the P–H–D approach produces the lowest weighted X-factor, and its cost is comparable to the cost of the planning model. Although the H–D approach has a slightly higher weighted X-factor than P–H–D (still lower than P–D with FIFO or target following), it has lower total costs than the other three approaches when breakdowns are more frequent. As the mean time between failures increases (i.e., as the availability increases), the differences among the total cost of the various approaches decrease. Also, the weighted X-factor decreases as the disruptions are less frequent for all four approaches. Again, the FIFO dispatching rule with the planning model gives the worst performance. As part of the experiments in this section, we also vary the MTTR at station 1 according to the values in Table 2.8. By keeping the MTBF at 42 h, we generate availability levels comparable to those in the MTTR experiments. Figure 2.13 Table 2.7 Mean times between failures at station 1 MTBF (h) 7.5 9 12.5 Station 1 availability 90% 91.6% 94% Station 1 traffic intensity 0.87 0.85 0.83

Fig. 2.12 Effect of mean time between failures at station 1

25 97% 0.81

42 98.2% 0.8

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150 94% 0.83

75 97% 0.81

45 98.2% 0.8

Fig. 2.13 Impact to weighted average X-factor and cost from mean repair time at station 1

shows the effect of different mean repair times on X-factor and total costs. In these experiments, the approaches with high-level scheduling (P–H–D and H–D, both with target following dispatching) produce lower weighted X-factors than P–D with FIFO and target following. Again, P–H–D and H–D have similar weighted X-factors. In total costs, these approaches yield lower values than the planning-onlybased approaches (P–D), especially when the repair times are long and availability is low. Comparing this with the previous set of experiments, we find that the impact from the breakdown duration (MTTR experiments) is more pronounced than the impact from the frequency of breakdowns (MTBF experiments).

2.5.3 Gantt Chart Analysis To analyze why P–H–D or H–D works better than P–D, we examine the Gantt charts representing the processing of jobs in some of the above experiments. As mentioned before, we choose the same random seed for all three approaches so that they see the same realization of machine breakdowns. Figures 2.14 and 2.15 are two typical time periods in one replication of the same setting, which corresponds to the second column in Table 2.7. As we can see from Fig. 2.14, in the P–D approach, during the 1.05–1.08 (104 ) min time interval, there are no new releases. Therefore, when the machine breaks down at station 1 around 1:12 104 min, there is not enough WIP at the bottleneck station 3, and the station becomes idle at around 1:13 104 min. However, for the P–H–D and H–D approach, the same disruption does not affect the bottleneck station at all. In Fig. 2.15, although the machine break down in station 1 affects the bottleneck station in all three approaches, we can see clearly that the idle time of the bottleneck station in the P–H–D and H–D approaches is significantly smaller than the one

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Fig. 2.14 Gantt chart for three approaches between 0.9 and 1.3 (104 ) min

in the P–D approach. If we look at the Gantt chart along the whole time horizon, we find that situations such as those seen in Figs. 2.14 and 2.15 are very common. In the P–H–D and H–D approaches, jobs are “pushed” to the bottleneck station, thus some jobs can avoid the machine-break-down situation without being held at station 1. On the contrary, the total number of jobs released is almost the same for all three approaches since the demand profiles are the same. Therefore, the weighted average cycle times in the P–H–D and H–D approach are smaller than the cycle time achieved by the P–D approach.

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Fig. 2.15 Gantt chart for three approaches between 2 and 2.3 (104 )min

2.6 Summary Production planning is a critical process for every manufacturing company since it directly affects the performance of detailed scheduling, which ultimately determines the overall performance of the manufacturing system. Despite this interdependency, in many manufacturing companies planning and detailed scheduling activities are separated, with a limited coordination between them. Usually, planning decisions

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obtained from a long-term model are fed into a scheduling model as restrictions, jobs to be released, and due dates. Detailed scheduling is typically handled through dispatching. In this paper, we suggest incorporating an intermediate stage into the usual planning-scheduling hierarchy to seek coordination between planning and detailed scheduling. This approach consists of the usual planning model and a high level scheduling model, both of which feed dispatching-based detailed scheduling. The high level scheduling model explicitly controls the WIP over time at each stage in the system, thus providing a more specific guide to detailed scheduling. Our numerical results indicate that the proposed approach results in shorter cycle times (realized as a lower weighted X-factor) than the conventional two-stage approach of feeding planning results into a detailed scheduling algorithm. In most cases, the use of the high level scheduling model, either as an intermediate step between planning and detailed scheduling or as an initial step before detailed scheduling, results in lower inventory and backorder costs. This approach without a major planning step turns out to be especially suitable for situations with high demand variability. All these results indicate that if we consider more scheduling details in the planning level and/or at an intermediate level before detailed scheduling, we would have better performance on the shop-floor in terms of both cycle times and system costs including inventory and backorders. In actual manufacturing systems, all planning and scheduling systems are implemented in a rolling horizon fashion. This is also true for the three-level planning=high-level scheduling=detailed scheduling approach. For example, the planning model would generate a plan for one quarter every month, and produce the first several months’ demand profile, release schedule, and production targets, for the upcoming weeks. This information would then be released to the high level scheduling model. Then the high level scheduling model would generate a more granular (say by day or shift) release policy and processing targets, for each major processing step pertaining to the upcoming week or month. The detailed scheduling step would try to implement the high-level scheduling decisions made for the next few days on the shop floor. High level scheduling and planning steps can be rerun with some regular frequency (say every week and month, respectively). During the execution of detailed scheduling, the current system status, such as WIP level, machine availability, may provide feedback that would initiate more frequent runs of the high level scheduling and planning models. We believe the rolling horizon approach would improve the performance of the proposed approach that incorporates a high-level scheduling model. Simulation of such a rolling horizon approach requires significant effort and we leave this as a topic for future research.

References Asmundsson J, Rardin R, Uzsoy R (2006) Tractable nonlinear production planning models for semiconductor wafer fabrication facilities. IEEE Trans Semicond Manuf 19:95–111 Bertsimas D, Gamarnik D, Sethuraman J (2003) From fluid relaxation to practical algorithms for job shop scheduling: the holding cost objective. Oper Res 51(5):798–813

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Dai J, Weiss G (2002) A fluid heuristic for minimizing makespan in job shops. Oper Res 50(4):692–707 Graves S (1986) A tactical planning model for a job shop. Oper Res 34:552–533 Hackman S, Leachman R (1989) A general framework for modeling production. Manag Sci 35:478–495 Horiguchi K, Raghavan N, Uzsoy R, Venkateswaran S (2001) Finite-capacity production planning algorithms for a semiconductor wafer fabrication facility. Int J Prod Res 39:825–842 Hung YF and Leachman RC (1996) A production planning methodology for semiconductor manufacturing based on iterative simulation and linear programming calculations. IEEE Trans Semicond Manuf 9(2):257–269 Jaikumar R (1974) An operational optimization procedure for production scheduling. Comput Oper Res 1:191–200 Kim S, Yea S, Kim B (2003) Shift scheduling for steppers in the semiconductor wafer fabrication process. IIE Trans 34:167–177 Kleindorfer PR, Kriebel CH, Thompson GL, Kleindorfer GB (1975) Discrete optimal control of production plains. Manag Sci 22 Leachman R, Kang J, Lin V (2002) SLIM: Short cycle time and low inventory in manufacturing at samsung electronics. Interfaces 32(1):61–77 Lee Y, Kim T (2002) Manufacturing cycle time reduction using balance control in the semiconductor fabrication line. Prod Plann Contr 13:529–540 Missbauer H (2002) Aggregate order release planning for time-varying demand. Int J Prod Res 40(3):699–718 Pahl J, Voˇ S, Woodruff D (2005) Production planning with load dependent lead times. Q J Oper Res 3:257–302 Qin SJ and Badgwell TA (2003) A survey of industrial model predictive control technology. Contr Eng Practice 11:733–764 Rose O (2002) Some issues of the critical ratio dispatch rule in semiconductor manufacturing. Proceedings of the 2002 Winter Simulation Conference, December 2002 Tsakalis K, Godoy JF, Rodriguez A (2003) Hierarchical modeling and control for re-entrant semiconductor fabrication lines: A mini-fab benchmark. pp. 578–587 Vargas-Villami F, Rivera D (2000) Multilayer optimization and scheduling using model predictive control: application to reentrant semiconductor manufacturing lines. Comput Chem Eng 24:2009–2021 Vargas-Villami F, Rivera D, Kempf K (2003) A hierarchical approach to production control of reentrant semiconductor manufacturing lines. IEEE Trans Contr Syst Technol 11(4):578–587

Chapter 3

The Effects of Production Planning on the Dynamic Behavior of a Simple Supply Chain: An Experimental Study Seza Orcun and Reha Uzsoy

3.1 Introduction Sophisticated supply chain planning systems, also known as Advanced Planning and Scheduling (APS) systems, have become commonplace in industry, and constitute a multibillion dollar software industry (Musselman and Uzsoy 2001; de Kok and Fransoo 2003; Stadtler and Kilger 2004). Many of these models rely to some degree on mathematical programming formulations of multistage productioninventory systems, which have been discussed extensively by (Saad 1982; Voss and Woodruff 2003; Johnson and Montgomery 1974; Hax and Candea 1984) and in this volume by Missbauer and Uzsoy. However, there has been little study in the literature of the effects of these production planning models on the dynamic behavior of supply chains. The dynamic behavior of supply chains over time has been studied in the system dynamics literature for several decades (Sterman 2000; Forrester 1962), leading to a growing understanding of the effects of information and material delays on the behavior of these systems, such as the bullwhip effect (Chen et al. 1998; Chen et al. 2000; Dejonckheere et al. 2003; Dejonckheere et al. 2004). However, the production planning procedures used in these models are generally feedback control procedures, with little ability to predict future states of the system and behave in a reactive manner. It is also quite difficult to interface optimization-based production planning models to standard system dynamics software. Hence, there is very little work of which we are aware that examines the effect of optimization-based planning procedures on the dynamic behavior of the supply chain in a systematic manner. In this chapter, we use an agent-based supply chain simulation environment to address this question experimentally using a simplified supply chain based on an industrial example. The specific question we focus on is that of how the use of different production planning models affects the dynamic behavior of a

S. Orcun () Laboratory for Extended Enterprises at Purdue, e-Enterprise Center at Discovery Park, Purdue University, West Lafayette, IN 47907, USA e-mail: [email protected]


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production-inventory system. The system dynamics literature has focused on simple feedback control policies based on reorder point inventory models, which gives little insight into the possible advantages and disadvantages of the more complex optimization-based procedures. As the use of such systems becomes more prevalent in industry, we believe it is important to develop a fundamental understanding of their effects on the performance of both the firm using them, and the other firms interacting with this firm. The work in this chapter is an exploratory first step in this direction. We approach the question incrementally by first implementing the simple feedback control policies studied by the system dynamics community in both our simulation environment and a commercially available system dynamics package (VenSim), and calibrating the two models to ensure exact correspondence between their results. This exercise is interesting in itself as it highlights a number of difficulties that may befall the unwary system dynamics modeller. An important characteristic of the policies modeled in the system dynamics literature is how they represent the dynamics of capacitated production systems. Most system dynamics models of which we are aware assume a fixed mean lead time that is independent of resource utilization, which contradicts well-known results from queueing indicating that production lead times increase nonlinearly with increasing utilization (Hopp and Spearman 2001). Using both our own simulation environment and VenSim, we show that the conventional feedback control policies assuming fixed lead times will give quite different dynamic behavior if the production facility is subject to congestion, which we represent using nonlinear clearing functions (Karmarkar 1989; Asmundsson et al. 2006; Asmundsson et al. 2009). These results suggest that the dynamic behavior of a supply chain is affected by the degree of agreement between the behavior assumed by the production planning/control model and the actual behavior of the system, which we shall refer to as the internal fidelity of the model or policy. A second aspect of the models we examine is their ability to predict future conditions and incorporate this information into their decisions. In the production planning context, the most important external variable to be predicted is usually the product demand. While the feedback control policies use predicted demand in their decisions, accurate prediction of future demand is widely held to be crucial to the success of optimization-based models. We shall refer to the degree to which the model has access to accurate predictions of external demand as its predictive fidelity. Our results suggest that even with very limited predictive fidelity, an optimizationbased model with high internal fidelity, i.e., that is capable of representing the internal operations of the production system accurately, yields significantly better performance in the face of changing demand patterns than a feedback policy with the same internal fidelity. In comparing the behavior of the different models, we emphasize the dynamic behavior of production and inventories over time, since these variables are, in our experience, those monitored by production planners and managers and hence those to which they react on a daily basis. The cost comparisons of the different models which we also present highlight that the aggregation of production, inventory

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holding, and shortage costs across the planning horizon may mask significant differences in dynamic behavior between models, which in practice would very likely lead to quite different responses both from internal management and from the firm’s external customers and suppliers. To gain insight and focus on the specific questions addressed, we use a simplified version of a semiconductor supply chain consisting of two manufacturing echelons and an inventory echelon. While considerably simpler than the real-world supply chain upon which it is based, this system includes many of the complexities encountered in the real thing: long cycle times at the first manufacturing stage, much shorter delays at the second, and co-production of two different products from a common base product, which results in mismatches between supply and demand of a given product. We follow conventional system dynamics approaches of subjecting the system to deterministic pulse changes in external demand and observing their effect on dynamic behavior under the different production planning models studied. In the following section, we review previous related work on simulation of supply chains. Section 3.3 presents an overview of the simulation environment used in the experiments, and Sect. 3.4 the different production planning models and policies compared in the experiments. Section 3.5 describes the supply chain system used as a testbed throughout the study, and presents the results of the initial simulations used to validate the implementation in our environment against the proven commercial tool, VenSim. In Sect. 3.6, we compare the results of the optimization-based models to the feedback policy under deterministic demand conditions. We conclude the paper with a summary of our conclusions, and a discussion of future research directions.

3.2 Previous Related Work There is an extensive literature spanning several different disciplines addressing various aspects of supply chain management and design. A comprehensive review of this literature is clearly beyond the scope of this chapter. Instead, we shall highlight only the basic research streams, focusing on the main modeling aspects and their limitations and citing only a few representative references. Enterprise Resource Planning (ERP) systems (Ptak and Schragenheim 1999; Davenport 2000) have proved enormously useful because they permit different functional areas of an enterprise to make decisions and generate business transactions on the basis of commonly understood, often highly detailed data sets. However, ERP systems on their own are unlikely to provide insight to management on how to develop innovative approaches to supply chain configuration and management. Lead times, prices, and the like are taken as fixed, and little consideration is given to the varying goals pursued by different players in the supply chain or the distinct constraints confronting each. While a number of such systems allow limited exchange of data among different firms, their general emphasis is heavily on making data from different parts of the firm uniformly available within the firm, rather than on

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developing decision models based on this data. Furthermore, the need to support the transaction level of enterprise operation makes the granularity of ERP representations far too detailed for easy scalability across enterprises of different sizes and for quick prototyping of supply chain innovations. Over the past 10–15 years, APS systems (Musselman and Uzsoy 2001; Stadtler and Kilger 2004; Ovacik 2011) have introduced a new layer of decision support above the transaction level of ERP that tries to account for capacities and other elements in operations planning. The emphasis here is on transaction-based systems for demand management, order acceptance, and finite-capacity scheduling. Like ERP, however, such decision support systems rarely consider more than a single firm, and they are far too detailed for timely modeling of the effects of major changes in the structure of extended enterprises. The presence of multiple companies with different data structures and operating procedures, together with the vast amounts of data captured by today’s ERP and other information systems, renders the development of models of any kind for large supply chains extremely challenging. To facilitate the development of models of supply chains as well as protocols for interactions between members of supply chains, and to provide a unified frame of discourse on supply chain issues, there have been a number of efforts in both academia and industry to develop unified data and decision architectures for supply chains. An early example of such work is the GRAI approach (Maloubier et al. 1984; Akif 1991), which focuses on the decision systems for production management. Current examples include the Supply Chain Operations Reference (SCOR) Model developed by the Supply Chain Council (http://www.supply-chain.org), and the Collaborative Forecasting, Planning and Replenishment (CFPR) effort developed by the Volunteer Interindustry Commerce Solutions Association (VICS) (http://www.vics.org/). These data models serve as natural points of departure for developing models of various aspects of supply chains. Buckley and An (2005) discuss the use of the SCOR model in formulating a supply chain simulation system. Much ongoing academic research in supply chain management focuses on extending existing inventory models to incorporate situations encountered in industry, such as information sharing and vendor-managed inventory, as well as microeconomic aspects, such as principal–agent theory and customer behavior models. Excellent overviews of this research can be found in (Tayur et al. 1998; Simchi-Levi et al. 2004; Thomas and Griffin 1996; Goetschalckx et al. 2002; Min and Zhou 2002), and (Huang et al. 2003), among others. A number of authors have developed large, time-indexed mathematical programming formulations that consider all nodes of the supply chain explicitly and model the flow of material through the resulting network over time. Examples of such models are given by (Shapiro 1993; Vidal and Goetschalckx 2001) and (Arntzen et al. 1995) among others. Particularly notable is the Asset Management Tool (AMT) developed by researchers at IBM (Ettl et al. 2000), which integrates an optimization model with queuing analyses and simulation to evaluate and optimize supply chain configurations.

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The high degree of uncertainty and inherent complexity in supply chains would seem to suggest that they can be analyzed using the discrete-event simulation techniques (e.g., Law and Kelton 1991) that have been used so successfully for analyzing many individual elements of these supply chains, such as manufacturing or inventory systems. However, the effort to develop and maintain a simulation model that considers all entities at the same level of detail is prohibitive when considering large supply chains. A number of authors have addressed the latter issue by constructing models using aggregate representations of specific nodes in the supply chains to reduce both model complexity and computation time. An example is Duarte et al. (2007), who propose a compact representation of manufacturing nodes that consist of three basic distributions: the number of units produced in a period, the delay between these parts entering and leaving the system, and the fraction of these parts that are not scrapped. The authors calibrate the parameters of their model using a detailed discrete event simulation model of a production facility, and demonstrate that the proposed compact model is capable of obtaining cycle time estimates that are close to those from the detailed simulation. However, the obvious drawback here is the need for the detailed simulation model to calibrate the aggregate one. A number of authors, for example (Petrovic 2001; Persson and Olhager 2002) present simulation models of specific supply chains, but the results of these are hard to generalize, and the supply chains simulated are generally quite small. An extended body of work at IBM (Chen et al. 1999; Huang et al. 2004; Buckley and An 2005) presents a simulation modeling environment for supply chains that integrates system dynamics and discrete-event simulation models to analyze the effects of different decisions to remedy deviations from plans noted in a sense-and-respond environment. (Kleijnen 2004) presents a review of supply chain simulation techniques with emphasis on model validation. An alternative solution to the combined problems of long execution times and model complexity is the use of a distributed simulation approach, where different participants in the supply chain integrate their individual models over a messaging network that synchronizes simulation clocks across models and ensures transfer of the necessary data between models. Terzi and Cavalieri (2004) provide an extensive review of simulation models in supply chains, focusing heavily on distributed simulation techniques. Lendermann et al. (2001) give an interesting discussion of such a framework that addresses the integration of APS procedures (Stadtler and Kilger (2004) with the simulation, as well as issues of maintaining data confidentiality between partners interacting in the supply chain. In summary, attempts to apply conventional Monte Carlo discrete-event simulation techniques to large supply chains encounter a number of significant problems: the high data requirements, which lead to difficulty and delay in model building and validation; high run times required to obtain statistically significant results for such large systems; and very limited optimization capability due to the high computational burden of the resulting large simulation models. A number of efforts have been made to address these issues by utilizing more aggregate models of supply chain entities that allow the complex interactions of multiple entities at a high level of detail to be aggregated into a few probability distributions (e.g., Duarte, Fowler

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et al. 2007) by distributed simulation architectures, or by using continuous flow models on a fixed, discrete time grid as in system dynamics (e.g., Lee et al. 2002; Schieritz and Grossler 2002). However, the types of decisions we wish to evaluate in this chapter require explicit optimization of entity goals, which is difficult in most simulation environments. The simulation user may have to do a great deal of hard-coding to represent protocols of interaction. The systems dynamics work pioneered by Forrester (1962) and since applied frequently to supply chain problems (Coyle 1977; Sterman 2000; Angerhofer and Angelides 2000; Burns and Sivazlian 1979; Towill 1996; Towill 1982; Towill and Del Vecchio 1994) is based on a dynamic control theory framework, which limits the complexity of the decision logic that can be used in representing the components of the supply chain. A major advantage of system dynamics models is their computational efficiency relative to discrete-event models, which is due to their use of continuous approximations for system state variables. Another advantage of these models is that under the right circumstances they can be approached with the well-developed tools of classical control theory, as a number of authors (Towill and Del Vecchio 1994; Dejonckheere et al. 2002; Dejonckheere, Disney et al. 2003; Dejonckheere, Disney et al. 2004) have done to examine the well-known bullwhip effect (Chen, Drezner et al. 2000; Lee et al. 1997). Several authors have exploited this aspect of system dynamics models by integrating them into agent-based models (Schieritz and Grossler 2002), or discrete-event simulation models (Lee, Cho et al. 2002). The manner in which production planning methods are modeled in the system dynamics literature bears some comment. The majority of the work in this domain applies relatively simple gap-based feedback approaches, where the system observes the deviation from a target inventory level caused by finished goods being shipped to the customer, and then computes the desired production releases based on the estimated average demand and the deviation, smoothed over a number of time periods. The most common approach to representing production facilities is that described in Chap. 17 of (Sterman 2000), which is very similar to the Inventory and Order Based Production Control System (IOBPCS) described by (Towill 1982; Towill and Del Vecchio 1994). However, optimization-based production planning algorithms are becoming more prevalent in industry, and it is thus of interest to examine the effect of these more complex planning techniques on the behavior of the entire supply chain, especially when a number of firms with different planning models interact. While Kleindorfer et al. (1975) and Kempf (2004) present interesting integrations of optimization-based production planning models with control theoretic approaches, it is quite difficult to integrate optimization models for production planning with system dynamics models for a variety of reasons. The first is simply that most system dynamics software is not designed with such interfacing in mind, so the interfacing requires a great deal of both modeling and software knowledge on the part of the user. A second, more subtle reason is that the time grids used by the two approaches are inherently different. Optimization models of production planning tend to be used for relatively long planning periods, typically representing

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weeks or months of real time. System dynamics models by their nature use a much finer integration interval for the numerical solution of the differential equations used to represent the operation of the system. This requires that the control signals, particularly those related to work releases, generated by the optimization model over larger planning intervals must be disaggregated in a manner that is meaningful relative to the continuous flow model adopted by the system dynamics approach. This can be accomplished in a number of different ways, each of which has its own implications in terms of the performance of the production system being modeled, adding a significant layer of experimental complexity. It is, of course, possible to use system dynamics models with longer time periods, but this may cause complications with commercial software, as we shall discuss below. Given the decentralized nature of supply chains, where multiple entities (e.g., firms or departments of firms) interact with each other to complete transactions in the supply chain by exchanging information, goods and money, agent-based approaches are a natural approach to modeling supply chains. Excellent introductions to agent-based systems are given by (Wooldridge 2001) and (Weiss 2001), among others. While there is no uniformly accepted definition of agents, we shall adopt that of (Wooldridge 2001), which defines an agent to be “. . . a computer system that is situated in some environment, and that is capable of autonomous action in this environment to meet its design objectives”. Agent-based modeling and simulation tools have been used in a wide variety of disciplines (Macal and North 2005), but we shall focus on their use in the supply chain context. Agent-based simulation models of supply chains represent the many entities operating in a supply chain with agents of this type, allow them to execute their specific actions in response to stimuli from the environment and observe the resulting behavior of the individual entities, as well as the emergent behavior of the larger supply chain consisting of these entities. An interesting discussion of supply chains as complex adaptive systems is given by Choi et al. (2001). In the design of any agent-based system, there are a number of fundamental design decisions to be made, and agent-based models of supply chains are no exception. Van Dyke Parunak (2001) gives an excellent discussion of these issues, which can be summarized as follows: (a) (b) (c) (d) (e)

What parts of the supply chain being modeled are represented as an agent? How does each agent model the environment? How are agents structured internally? How do agents communicate, with what protocols and sharing what information? How do agents coordinate their actions?

In most agent-based approaches to supply chain modeling, a hierarchical agent structure is used, where the various functionalities of an entity operating in the supply chain are represented as agents. Examples of these are agents responsible for production scheduling, production planning, managing procurement from suppliers using a bidding process, and transportation planning. A number of agents of this type representing a firm may be grouped under the control of an agent representing the firm or division to which these agents report, and which has some strategic planning functionality at a coarser time granularity than the operational

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decisions taken by the functional agents. Agent-based approaches of this type have been proposed by (Fox et al. 2000; Van Dyke Parunak et al. 1999; Sadeh et al. 2001) and (Swaminathan et al. 1998), among others, and used to evaluate different aspects of supply chain design and operation, such as lateral coordination efforts based on coordinated scheduling (Sadeh, Hildum et al. 2001) propagation of demand variability up the supply chain (Van Dyke Parunak, Savit et al. 1999), and distribution network design (Swaminathan, Smith et al. 1998). The work we have found that is closest to ours in spirit is the system developed by Biswas and Narahari (2004), who develop an object-oriented model that is very similar to ours. Different nodes of the supply chain are represented as objects, from which one can generate a variety of model formulations such as linear programs, inventory models, or simulation models. A strength of this approach is that, like our approach, it can be used at different levels of time granularity, for both strategic and tactical decisions. The authors illustrate the operation of their system with a case study involving the design and operation of a supply chain for liquid petroleum gas, which begins with decisions on location of bottling plants and proceeds down to decisions on detailed inventory optimization at individual dealers. In summary, there appears to be a strong need for modeling tools that combine the computational tractability of aggregate approaches with the ability to accurately represent the nonlinear dynamics of capacitated, congestion prone systems to provide an accurate, high-level picture of how the supply chain will behave over time. The development of such tools can sometimes require the development of novel models of capacitated components of supply chains, such as factories and distribution centers, which are computationally tractable but still represent the nonlinear behaviors of such systems accurately. The clearing function models used in this study represent such a class of models that are of considerable interest in their own right, and are discussed at length in (Missbauer and Uzsoy 2011). The SCOPE environment described in the next section is an attempt to develop such a system.

3.3 The SCOPE Environment SCOPE views a supply chain as a directed graph whose nodes represent facilities such as manufacturing plants and distribution centers, while arcs represent both information and material flows. Information flows are divided into a backward information flow (backward pass) and forward information flow (commit pass) in addition to forward material flow (realization pass). The purpose of this distinction is to approximately represent the iterative negotiations that occur between elements of a supply chain. During the backward pass, information flows from the market back up the supply chain toward the raw material suppliers, with each entity developing its production plans and placing orders with its suppliers. Once the information from the market has propagated upstream to the origin of the supply chain, the forward, or commit, pass is initiated. During the commit pass information flows from the raw material suppliers back toward the consumer, with each entity

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adapting its original plans to the amounts of inputs its suppliers are willing to provide. The realization pass updates the state of the system as it evolves in time, which may also capture realizations of process uncertainty within each node. This allows us to model many different situations where the realization of a system variable differs from that predicted by the models used to control the behavior of the system, such as the presence of random variables or the failure of planning models to accurately represent the behavior of the system. The inclusion of the realization layer allows us to address the question of how the internal fidelity of a planning model, the degree to which the model accurately represents the actual operation of the production system under study, affects the performance of the plans generated by the model. This separation of forward and backward information flows provides significant flexibility in modeling information flows and modes of interaction between elements of the supply chain. During both forward and backward passes, time lags of different durations as well as different levels of information aggregation can be used to model different modes of interaction. The Commit Pass, in particular, on the one hand, allows the user to embed a wide variety of logic to determine how elements of the supply chain allocate their outputs among different customers, and how the effects of decisions by individual elements of the supply chain affect the rest of the system. The Backward Pass, on the other hand, provides considerable scope for modeling different degrees and forms of information sharing between elements of the supply chain. For example, the CFPR approach advocated in the retail sector and studied extensively by several researchers (e.g., Aviv 2001) can be implemented as a module invoked during the backward pass of a SCOPE run. A SCOPE node is composed of five agents as shown in Fig. 3.1. The Demand Estimation Agent gathers demand information and predicts future demands. The Production Planning Agent allocates the capacity at the node among the products to create master plans using the demand forecasts collected by the Demand Estimation Agent in the backward pass, and allocates the raw material received from the suppliers to orders in the realization pass. The Raw Material Planning Agent calculates the raw material needs based on the master plan generated by the Production Planning Agent. The Procurement Agent is responsible for distributing

Fig. 3.1 Agents within a node in SCOPE

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the raw material requirements estimated by the Raw Material Planning Agent to suppliers, in other words, determining what and how much to order from which supplier. Finally, the Order Fulfillment Agent decides how to allocate the realized production to its customers, represented by downstream nodes. It should be noted that each agent can have different roles during each pass. Based on its role, each agent may use totally different algorithms and models, or the same model populated with different data, in its decision-making process. For example, the production planning agent may use a linear programming (LP) model assuming unlimited raw material availability to develop an initial plan and place orders with suppliers during the backward pass. However, if suppliers are unable to meet the desired order quantities, it may then execute the same LP model with additional constraints on raw material availability during the forward pass. The advantage of this architecture is that we can incorporate specific decision models into different nodes, as opposed to the relatively simple decision rules observed in most system dynamics work, and observe their effects on system behavior. This capability is particularly valuable in today’s environment, where sophisticated planning systems represent both an opportunity and a very significant investment for companies. Such tools are also being widely used by many companies that have installed APS systems to exploit their ERP implementations. A prototype SCOPE environment with the described characteristics has been implemented using Microsoft Excel. XML is used to describe the supply chain topology in terms of both material and information flows, the roles of each SCOPE agent and their data requirements and the mapping of the agents to the decision modules that will be executed in each pass depending on the role of the agent. Information and material flow between the nodes of the supply chain are handled through property visibility in a manner analogous to message boards. The decision to implement the prototype in Excel was based on the need for a rapid prototyping environment for the software that is easily accessible and allows us to perform functions such as data extraction and graphical presentation of output with minimum effort. This is clearly a suboptimal decision from the point of view of computation time, and a run of the prototype software currently takes of the order of 20 min on a Pentium IV laptop. However, the purpose of this implementation is to develop proof of concept, and once the system is sufficiently tested in this form we plan to migrate to a more effective implementation using an industry strength database such as DB2 or Oracle. A model of a specific supply chain is currently implemented using a series of flat files describing the system under consideration. In a previous paper (Orcun et al. 2007), we have illustrated the use of our simulation environment in both centralized and decentralized supply chains.

3.4 The Supply Chain Testbed and Capacity Models In this chapter, we use a simple supply chain derived from that of a major semiconductor manufacturer. The process begins with the fabrication of the integrated circuit on silicon wafers in the Fab/Sort (F/S) facility. There is a yield rate of 0.85

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on this process, at the end of which the wafers are tested (probed) to assess their performance characteristics. As a result of this test, some fraction ’1 are determined to be potential high-speed devices, which may be discovered to be medium-speed devices at the end of final testing. The remaining fraction ’2 D 1 ’1 are classified as medium-speed devices, which may be downgraded to low-speed devices at the end of final testing. The wafers then enter the assembly and test (A/T) facility according to their grade at sort, where they are packaged and tested. The final disposition of each circuit is made at the end of the A/T process, where the circuits pass into a finished goods inventory from which they are shipped to the customer. The cycle time at the F/S facility is given to be 90 shifts on average, with the facility operating two shifts a day seven days a week. Since all other cycle times and shipping delays in the system are far shorter than the F/S cycle time, we ignore them in our experiments as they are unlikely to qualitatively affect the results. For the exploratory research in this chapter, we will follow the system dynamics literature in treating all system variables as deterministic quantities, to gain insight into the behavior of the system under different conditions. An overview of the basic semiconductor manufacturing process is given in Uzsoy et al. (1992). Two aspects of this system are worth highlighting. The first of these is the presence of co-production at both the Fab-Sort (F/S) and Assembly/Test echelons. In both cases, if we release more products into the F/S line to obtain more of a given product, we are likely to obtain more of the associated co-product as well: in an effort to obtain more high-speed devices from the F/S line, we will inevitably obtain more medium-speed devices as well, regardless of whether there is demand for the additional medium-speed devices or not. The second is that the vast majority of the system’s lead time is at the Fab/Sort stage, which is not the stage closest to the customer demand. This introduces a substantial delay between market demand, or even inventory levels after Assembly/Test, being observed and material launched into the system as a result of those observations becoming available to meet customer demand. In the course of our experiments, we use three different models to describe the behavior of the F/S line. The first of these, drawn directly from the system dynamics literature, represents the factory as a third-order delay with a fixed mean cycle time, which is defined as an exogenous parameter. The second uses conventional LP models with fixed lead times (Johnson and Montgomery 1974; Hackman and Leachman 1989). The third class of models is based on the concept of clearing functions, whose purpose is to capture the nonlinear relationship between workload and lead times in capacitated production systems governed by queueing behavior (Asmundsson, Rardin et al. 2006; Karmarkar 1989; Missbauer and Uzsoy 2011). The use of clearing functions will be discussed in detail in Sect. 3.6 of this chapter. We use these three different models of the internal operation of a capacitated production node to examine the effects of internal model fidelity on the dynamic behavior of the supply chain. The basic question here is that of whether failure by the planning model to represent accurately the realized dynamic behavior of the production system will affect the behavior of the system controlled based on its outputs. This situation arises when the planning model uses a different representation

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of system behavior than that governing the realized behavior. Hence, for example, a planning model using fixed lead times when the realization is governed by a continuous clearing function represents a case of imperfect internal fidelity – the model does not correctly capture the actual realized behavior of the production system. We shall explore this effect in our experiments presented in the following sections. To represent the entire F/S facility using a single clearing function, we follow an approach similar in spirit to the Practical Worst Case analysis of Hopp and Spearman (2001). We assume that the average cycle time of 90 shifts corresponds to a bottleneck utilization level of 0.90. Since the F/S process requires many manufacturing steps, the shortest possible time in which material that is introduced can emerge as finished product is given by the raw process time (RPT), which represents the average time for a lot to pass through the empty facility as defined by Hopp and Spearman (2001). We assume that the average cycle time CT will follow a relationship of the form CT D RPT C ˛.WIP=Cd /;

(3.1)

where RPT is the raw processing time, ˛ the desired average utilization, WIP the work-in-progress inventory level, and Cd the desired average capacity that is utilized. The desired capacity in our case is thus given by 0.90CT , where CT is the maximum theoretical capacity of the facility. Then, by Little’s Law, the average throughput rate of the facility per period is given by TH D WIP=CT:

(3.2)

Substituting (3.1) in (3.2), we obtain TH D WIP.Cd =˛/=Œ.Cd =˛/RPT C WIP;

(3.3)

where CT D Cd =˛ and the term .Cd =˛/RPT represents the time required to clear fully utilized plant. Equation (3.3) generates the behavior shown in Fig. 3.2 when 1600 1400 1200 1000 Th 800 600 400 200 0 0

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Fig. 3.2 Clearing function representing F/S facility in experiment

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Table 3.1 Nomenclature for models in experiments Demand visibility IOBPCS Feedback SF Simple forecasting PF Perfect forecasting PGDV Perfect global demand visibility Planning representation DELAY PW CF2

Third-order delay Piecewise linear clearing function Two-segment clearing function

Realization representation DELAY PW CF

Third-order delay Piecewise linear clearing function Continuous clearing function

Cd ; ˛ and RPT are set to 1,400 units, 0.90 and 9=14 D 0:6428 weeks, respectively. The calculation of RPT is based on an assumption of 14 shifts per week. Under these parameter values, the desired capacity utilization is achieved when WIP is 9,000 units, leading to an average cycle time of 90=14 D 6:43 weeks at the desired utilization level of 0.90. To facilitate the implementation of the conventional LP planning models by having cycle time equal to an integer multiple of the basic planning period, we rounded this down to 6 weeks. In our experiments exploring the effects of internal fidelity, we use two forms of clearing function: a continuous function derived as above, representing the actual operation of the system, and a piecewise linear approximation to the continuous function, which we use to obtain the LP models to optimize the resulting systems. Details of the models used are given in Sect. 3.6, where we discuss the results of our experiments with models using optimization-based production planning. To refer to the models used in the experiments in a concise manner, we use a fourfield notation representing demand visibility, planning representation, realization representation, and replanning policy. The possible values of the first three fields are summarized in Table 3.1. The fourth field that denotes replanning policy will be set to R if replanning is present, and will be left blank if a single planning model is solved for the entire horizon and its results implemented without modification. Detailed descriptions of the different fields used will be given as they are introduced into the models in the subsequent sections. We now discuss the implementation of the feedback control policy used widely in the system dynamics literature that we will use to calibrate the SCOPE models and establish a baseline for the performance of the optimization-based planning models.

3.5 System Dynamics Models The system dynamics models used for calibration were implemented in the VenSim environment, available from Ventana Systems (www.vensim.com). The production

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policy implemented was the Inventory and Order-Based Production Control System (IOBPCS) described in Towill (1982) and Sterman (2000), which the former suggests “. . . is representative of much industrial practice associated with manual production control systems” (Coyle 1977), an observation borne out by our own industrial experience. This makes it particularly interesting to compare the performance of this approach with that of the optimization-based procedures, we implement in the SCOPE environment in the following section. In our implementation of the IOBPCS model, we assume that no inventory is held between the F/S and A/T facilities, and that work release into the F/S facility is governed by the status of the finished goods inventory of tested circuits between the A/T facilities and the external demand. This does not match the manner in which most semiconductor supply chains are managed; common industrial practice is to maintain an inventory of sorted die between the Fab/Sort and Assembly/Test facilities. However, in the models studied in this chapter, this will only introduce an additional delay and control point between the customer demand and the releases into the Fab/Sort facility, so we will not consider it here. The system dynamics model in which we implement IOBPCS for the testbed supply chain is shown in Fig. 3.3. To implement the IOBPCS procedure, we need to first specify target inventory levels for each of the post-A/T facility inventories (the “A/T Hi & Med Speed Inventory” and “A/T Low & Med Power Inventory” stocks in Fig. 3.3). The target inventory levels are set to the mean demand over replenishment lead time, which is known with certainty and equal to the cycle time at the F/S facility. We assume in all the models that the F/S cycle time is equal to 6 weeks, and ignore shipping times between facilities. This is done to maintain the system cycle time equal to an integer number of the planning periods used in the optimization models. Recall that demand is deterministic in this model, so no additional safety stock is incorporated into the inventory targets. Given the inventory target, we first determine the desired inventory adjustment as AT1 InvAdj D .AT1 Hi & Med Speed Inventory AT1 H=M Target Inv/=.AT1 H=M InvAdjTime/;

(3.4)

where AT1 H/M InvAdjTime is the number of time periods over which the difference between current inventory level and the target will be eliminated. The desired production rate at F/S for wafers leading to H/M speed products is then given by AT1 HM Desired Production Rate D AT1 H=M Inv Adj C AT1H/M Mean Demand: (3.5) The AT1 ML Desired Production Rate for wafers that will lead to Medium/Low Power devices is calculated in the same manner using the demand for these devices, denoted by AT1 ML Mean Demand in Fig. 3.3. The two desired production rates, which reflect the production required to eliminate the deviation of the current inventory levels from the inventory targets, are then combined to yield the Desired Net Production Rate for the F/S facility. Since both types of products (Hi & Med


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Fig. 3.3 System dynamics (VenSim) model for Testbed supply chain

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Speed and Med & Lo Power) result from the same wafer, the amount of wafer starts required to produce Q units of Hi & Med Speed product is given by AT1 HM Desired Production Rate ; (Hi & Med Speed Split)(F/S 1 Success Rate)

(3.6)

where the first term in the denominator denotes the fraction of F/S output that will classify as Hi & Med Speed product, and the second term denotes the F/S line yield. Calculating the required starts for the Med & Lo Power products in the same fashion, the Desired Net Production Rate is given by max

AT1 HM Desired Production Rate ; (Hi & Med Speed Split)(F/S 1 Success Rate) AT1 ML Desired Production Rate : (Med & Low Power Split)(F/S 1 Success Rate)

(3.7)

However, to determine the start rate, this quantity must be combined with another adjustment aimed at maintaining the WIP level. In this model, the WIP level at the F/S facility is modeled as a stock with inflow equal to F/S1 Start Rate and outflow equal to F/S1 Production Rate, which is represented by a third-order delay of the inflow (Sterman 2000). The cycle time of the F/S facility, F/S1 Cycle Time, is specified as an exogeneous parameter, whose value is given as 6 weeks. The desired WIP level for the F/S facility, FS1 Desired WIP Level, is then set to a level that will sustain the current production rate based on Little’s Law, as (Desired Net Production Rate)(F/S1 Cycle Time). The adjustment for WIP, denoted by FS1 Adj for WIP in the model, is calculated in a similar manner to that for inventory adjustment described above. The model was hot-started from steady state using the technique described in Chap. 17 of Sterman (2000). The model was then run for 200 periods, where the time period was set to 1 week, and subjected to a demand pulse that raises demand for each product type from 300 to 600 units per week at the start of week 10, and reduces it to 300 units per week again at the end of week 80. The length of the integration interval used created several anomalies in VenSim, which had to be addressed by examining the calculations in the software and modifying them to ensure that flow was conserved. The principal results of this model, denoted by IOBPCS-DELAY-DELAY, are shown in Figs. 3.4 and 3.5. The former figure shows the behavior of the F/S production facility. When the demand pulse is observed at week 10, a significant gap is created between the actual and target inventory levels, resulting in a rapid increase in F/S starts. The production rate and WIP increase more gradually, and all three quantities exhibit a mild oscillation that causes an undershoot followed by an overshoot before stabilizing again in the vicinity of week 50. When the demand returns to its initial level at the end of week 80, the inventory target is reduced, creating significant excess inventory that causes the start rate to be reduced to zero for

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Fig. 3.4 Production performance of IOBPCS-DELAY-DELAY

Fig. 3.5 Inventory performance of IOBPCS-DELAY-DELAY

several weeks. This, in turn, is mirrored in the production rate and WIP level, both of which actually drop to zero. The maximum WIP level achieved is approximately 16,000 units, right after the demand pulse arrives in week 10. The response of the inventory of Hi & Med Speed products is shown in Fig. 3.5. The shipments from the inventory track demand exactly, with no shortages. The inflow from the production facility follows the same pattern as the output of the F/S line shown in Fig. 3.4, while the inventory level shows significant overshoot and undershoot at the beginning and the end of the demand pulse due to the time lag between demand being observed and the starts based on that information affecting system output. The damped oscillation behavior observed at the ends of the demand pulse is typical of those observed for stable systems operating under feedback control.

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S. Orcun and R. Uzsoy F_S 1 Plot 1

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Fig. 3.6 Production performance of SCOPE implementation of IOBPCS-DELAY-DELAY A_T 1 Plot 1 6000

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Fig. 3.7 Inventory performance of SCOPE implementation of IOBPCS-DELAY-DELAY

Figures 3.6 and 3.7 show the results of the IOBPCS-DELAY-DELAY model implemented in the SCOPE environment to ensure that the SCOPE model produces the same results as the VenSim model. Inspection of the figures shows that this is indeed the case.

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F_S 1 Plot 1 14000

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Fig. 3.8 Production performance of IOBPCS-DELAY-CF

Our final experiment with the feedback control policy was implemented in SCOPE, and is labeled IOBPCS-DELAY-CF. In this model, the IOBPCS feedback policy is implemented, but the realized response of the F/S factory follows the continuous clearing function described in the previous section and in Fig. 3.2. The purpose of this experiment is to examine how the behavior of the system under the IOBPCS policy is affected by the manner in which the behavior of the F/S line is modeled. i.e., by the internal fidelity of the IOBPCS model. The results of this experiment are shown in Figs. 3.8 and 3.9. Even a casual examination of Figs. 3.8 and 3.9 reveals a qualitative change in the behavior of the system when congestion in the F/S line is modeled explicitly, giving better internal fidelity. The high peak of the WIP followed by an oscillation in Fig. 3.4 is now replaced by a smooth initial accumulation of WIP that peaks and then declines smoothly until the end of the demand pulse. The releases surge at the start of the demand pulse, after an initial period of low starts, and then decline smoothly, without oscillation, to a stable level. The WIP peak here (13,000) is significantly lower than that of the previous models (16,000). The inventory behavior in Fig. 3.9 is also quite different; inventory at the A/T lines accumulates steadily throughout the demand pulse, with a significant jump at the end of the pulse that is, again, due to the reactive nature of the feedback policy – it takes a full system cycle time of 6 weeks for the reduction in demand to be translated into a reduction in output. These delayed responses to changes in the demand are a necessary consequence of the reactive nature of the feedback control policy – since it has no visibility of future events, it is unable to be proactive in any way, and the long delay imposed on the system by the F/S cycle time results in serious peaks in both WIP and finished goods inventories.

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S. Orcun and R. Uzsoy A_T 1 Plot 1

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Fig. 3.9 Inventory behavior of IOBPCS-DELAY-CF

The accumulation of inventory is interesting to observe, and suggests a mismatch between the control policy and the reality of the system it models – poor internal fidelity in the terms we have introduced. The cause of this is the effect of the clearing function on the system’s ability to react to changes in WIP level. As the WIP level increases, the cycle time also increases nonlinearly, requiring longer time for the system to convert releases into output. In contrast, when production is modeled with a third-order delay with fixed mean duration, the system does not capture the increase in cycle time due to congestion in the production facility. It is also interesting that the third-order delay model does not limit the amount that can be produced in a given time period. Since it appears that the third-order delay is widely used in the system dynamics literature to model capacitated production systems (Sterman 2000), this raises interesting questions as to the accuracy of some system dynamics models when used to model production systems such as semiconductor wafer fabs that operate at high utilization levels.

3.6 Optimization-Based Planning Models Having discussed the system dynamics models using the IOBPCS policy in the previous section, we now examine the behavior of the optimization-based planning models. These models are characterized by several components: the manner in which they model the production facility (approximated clearing function or fixed lead time); the manner in which they forecast future demand (simple

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forecasting, perfect forecasting, perfect global demand visibility); and the use of periodic replanning (used or not). Under simple forecasting, the models assume that the last realized demand will represent demand for the next 7 periods, which was chosen to be one period longer than the average cycle time. Under perfect forecasting, the models have perfect visibility of demand for 7 periods into the future, while under perfect global demand visibility they see all demand values for the remainder of the planning horizon. When periodic replanning is used, the planning model is rerun every period and the results implemented for the next period. Otherwise, a single model is solved for the entire planning horizon of 200 periods and executed without any revision. The optimization models and comparison experiments discussed in this section are summarized in Tables 3.2 and 3.3, respectively. We begin our discussion with a simple model where the F/S facility is represented using a piecewise linear approximation of the actual continuous clearing function as described in Sect. 3.6 above. The model assumes that the current period’s realized demand is an accurate forecast of the demand for the next 7 periods. The planning model is implemented on a rolling horizon basis, with a new plan being generated every period, the first period’s decisions being implemented, and the process repeated at the start of the following period.

Table 3.2 Planning model formulations (the WIP and inventory balances are repeated for each period, t, and the capacity constraints for each period t and line segment i ) Clearing function planning model Fixed lead time planning model Objective function Objective function P P t CW Wt C Cj jt C Cr Rt C CS St C CX Xt t CW Wt C Cj jt C Cr Rt C CS St C CX Xt WIP balance Wt D Wt1 C Rt Xt

WIP balance Wt D Wt1 C RtLT C1 Xt

Inventory balance jt D jt1 C Xt Dt C St

Inventory balance jt D jt1 C Xt Dt C St

Capacity limit Xt ˛i Wt C ˇi

Capacity limit Xt C

Nomenclature CW : Cost of holding unit work in progress inventory C j: Cost of holding unit finished goods inventory Cr : Unit raw material cost Cs : Penalty cost of unit shortage Cx : Unit production cost Wt : Work in progress inventory at time period t It : Finished goods inventory at time period t Rt : Releases at period time period t St : Shortage at period time period t

Xt : Throughput at period time period t Dt : Demand at period time period t C : Total capacity ˇi : y-intercept of piecewise linear segment i of clearing function (capacity) ˛i : slope of piecewise linear segment i of clearing function (capacity)

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Table 3.3 Summary of experiments Demand Planning Experiment visibility model IOBPCS– Feedback (last Third-order DELAY– observed delay DELAY demand)

Realization model Replanning Results Third-order No Figures 3.4–3.7 delay F/S WIP D 1173442 A/T Inv D 1041952

IOBPCS– Feedback (last Third-order DELAY– observed delay CF demand)

Clearing function

No

Figures 3.8–3.9 F/S WIP D 882764 A/T Inv D 1168347

SF–PW– CF–R

Feedback (last Piecewise observed linearized demand) clearing function

Clearing function

Yes


PF–PW– CF–R

Perfect demand visibility for next 7 periods Perfect global demand visibility

Piecewise linearized clearing function

Clearing function

Yes



Clearing function

No


Piecewise linearized clearing function Clearing function

No


No


PGDV– PW–CF

PGDV– PW– PW

Perfect global demand visibility


PGDV– CF2– CF


Two segment clearing function

PGDV– CF2– CF–R


Two segment clearing function

Clearing function

Yes


PGDV– LP–CF


Fixed lead-time Clearing linear function programming

No


PGDV– LP– CF–R


Fixed lead-time Clearing linear function programming

Yes


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F_S 1 Plot 1 9000

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Fig. 3.10 Production performance of SF-PW-CF-R A_T 1 Plot 1 3000

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Fig. 3.11 Inventory behavior of SF-PW-CF-R

Figures 3.10 and 3.11 summarize the behavior of the system under this new planning procedure. The peaks and oscillations of the system dynamics models are replaced by a series of stable production regimes, where WIP is increased twice by large additional releases in periods 13 and 27. The WIP peak is now significantly

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lower than that of IOBPCS-DELAY-CF, which was in turn lower than that of IOBPCS-DELAY-DELAY; it is now less than 8,000 units, as opposed to 12,000 in IOBPCS-DELAY-CF and 16,000 in IOBPCS-DELAY-DELAY. The difference in inventory behavior shown in Fig. 3.11 is striking: the overall level of inventory maintained is significantly lower than that in the feedback models. The model initially accumulates inventory due to the initial WIP level in the system, then draws this down during the initial periods of the demand pulse. The large jump in inventory level at the end of the demand pulse is a direct consequence of the model’s limited forecasting capability; it does not see the reduced demand coming until it is actually realized, and thus overproduces during the last seven periods of the demand pulse. Once the pulse is observed, its forecasts are again accurate, and it reduces the inventory level to zero for the rest of the planning horizon. The conclusion appears to be inescapable: the dynamics of the supply chain are qualitatively different under optimization-based planning model than under the IOBPCS. This is actually not very surprising: the IOBPCS model appears to have been originally postulated to represent manual production control policies (Towill 1982; Coyle 1977), and upon reflection one would expect the behavior of an optimization model to be quite different, especially one that uses predictions of the future upon which to base its decisions. The magnitude of the differences in supply chain behavior suggests that conventional system dynamics models assuming feedback production control logic of the IOBPCS type must be treated with caution when used to model supply chains where significant portions of the planning process make use of optimization models. With the increasing growth of ERP systems and the related APS systems (Musselman and Uzsoy 2001), there appear to be an increasing number of such supply chains in operation. To examine the effect of forecast accuracy on the performance of the preceding model, we perform an additional experiment where the model now has perfect visibility of future demand for the next 7 periods at each planning execution. The results of this experiment are shown in Figs. 3.12 and 3.13. Given the simple demand pattern used, the perfect forecast will differ from the simple one only at the beginning and end of the demand pulse. Hence, the performance of the production system in this experiment is quite similar to that of the previous case. The most notable difference is that the inventory peak at the end of the demand pulse is significantly reduced under perfect forecasts. Pursuing our examination of the effects of forecast accuracy, we run another experiment where the optimization model has perfect global visibility of demand for the entire 200-period planning horizon, whose results are shown in Figs. 3.14 and 3.15. Again, the result is qualitatively similar to those of the previous two models, with the peaks at the beginning and end of the demand pulse now eliminated. However, the WIP levels are significantly lower, and the inventory behavior quite different; inventory is initially built up to a level slightly higher than that of the previous models, depleted to a level of zero at the end of the demand pulse, and then held at a level of zero for the rest of the planning horizon except for the brief jump arising at the end of the demand pulse.

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F_S 1 Plot 1 10000 9000 8000 7000 6000 5000 4000 3000 2000 1000 0 1

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Fig. 3.13 Inventory behavior of PF-PW-CF-R

An issue that is very seldom explored in the production planning context is that of the effect of internal fidelity – the effect of differences between the model used to perform the planning and the actual realization of the behavior induced by

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Fig. 3.15 Inventory behavior of PGDV-PW-CF

those decisions in the production system. The scenario above assumes a built-in mismatch: the continuous clearing function representing the behavior of the F/S facility is approximated by an inner piecewise linearization, rendering the planning

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Fig. 3.16 Production performance of PGDV-PW-PW A_T 1 Plot 1 3500

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Fig. 3.17 Inventory behavior of PGDV-PW-PW

model more conservative in terms of its predictions of output than the actual facility. Figures 3.16 and 3.17 show the behavior of the system when the piecewise linear approximation is a perfect representation of the behavior of the F/S facility. This is clearly not the ideal experiment, which would be to plan and execute using the

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continuous clearing function. However, that would require using a non-LP model for the planning engine, and our objective here is simply to examine what happens under perfect internal fidelity when the planning model’s representation of production system behavior is perfectly accurate. There is little difference between this model and that with perfect forecasts. Hence, we experiment further with simpler representations of the clearing function, consisting of two line segments – a linear segment whose inverse slope is equal to the fixed lead time of 6 periods assumed for the original supply chain with an upper limit of 1,555 (1,400/0.9) units. Once again, the model assumes perfect visibility of demand for the entire planning horizon of 200 periods and a continuous clearing function in the realization step. The behavior of the system, shown in Figs. 3.18 and 3.19, has again changed substantially; instead of WIP increasing to its peak level in two discrete steps, it now increases gradually over the entire period of the demand pulse, with a significant inverse response at the end of the demand pulse. The inventory behavior is totally different; instead of declining and finally settling at a level of zero, the inventory rises after the demand pulse to a level of 1,500 units and remains there until the end of the horizon begins to affect decisions. Although not shown in the figures, the system also misses demand by a small amount throughout the entire duration of the demand pulse. We attribute this behavior to the fact that the planning model assumes that lead time remains unchanged (the first segment of the clearing function) until capacity is fully utilized, when the second segment becomes binding. This overestimates the system’s ability to respond to changes in demand as long as the system remains below capacity.

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Fig. 3.18 Production behavior of PGDV-CF2-CF

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Fig. 3.19 Inventory behavior of PGDV-CF2-CF

To examine the effects of replanning with imperfect internal fidelity, we ran another experiment with perfect forecasting, where in each period t the model plans production for the next 200-t periods using the two-segment clearing function of the previous experiment and perfect visibility of demand for the remainder of the horizon. In this scenario, illustrated in Fig. 3.20, WIP rises more sharply at the start of the demand pulse, and levels off at the peak value of 7,000 units for most of the duration of the pulse. The inverse responses are again present at the end of the pulse, but once the pulse is over both WIP and starts exhibit severely oscillatory behavior. This is due to the mismatch between the planning model and the realization of production. The planning model underestimates the amount of production for low WIP levels, which results in a mismatch between the planned and realized WIP and inventory levels at the end of each period. At each re-planning period, the planning model will reduce its planned releases to account for the unexpected additional inventory generated by this mismatch, causing a shortfall in the next period, which forces an increase in starts at the next replanning. This oscillating behavior is not seen during the demand pulse when the system is at high utilization, since in this production regime the system is producing at its maximum capacity, which is accurately captured by the second, horizontal segment of the clearing function used in the planning model. The inventory behavior shown in Fig. 3.21 is now very similar to that of the models with perfect forecasts and no replanning, except for the oscillations in inventory level after the demand pulse which are a consequence of the oscillating starts and WIP discussed above. These results are themselves interesting, since frequent replanning is often held up as the way to address uncertainties in the planning environment. Uncertainties,

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Fig. 3.21 Inventory behavior of PGDV-CF2-CF-R

in the end, create a mismatch between the reality of execution and the world as represented in the planning model, so our experiment above can be viewed as representing a very simple type of uncertainty. The use of replanning causes quite different types of behavior here depending on the nature of the mismatch – oscillations

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when the planning model underestimates actual production, but no major ill effects when the system is at maximum utilization. This contrasts with the results of Figs. 3.18 and 3.19, where the only difference is the lack of replanning. A production manager might well argue that when forecasts are good enough, the oscillatory behavior caused by replanning may be less desirable than the stable production and starts levels we get without it. Our last analysis of the effects of imperfect internal fidelity uses a conventional LP model that assumes a fixed lead time equal to that assumed in the system dynamics models. The realized production is again driven by the continuous clearing function, and results are shown in Figs. 3.22 and 3.23. The WIP increases smoothly but continuously during the demand pulse, suggesting that the model overestimates the ability of the system to produce output, while in reality the starts that are made are not being completed and add to the congestion in the system. The inventory behavior is characterized by a long interval of stock-outs, seen in Fig. 3.23, while Fig. 3.24 indicates that demand is being missed over this interval, although in small quantities. The replanning version of this model is illustrated in Figs. 3.25 and 3.26. The oscillating behavior observed previously is even more severe here, which is natural since the LP model used in this scenario is an even cruder representation of the production system than the two-segment clearing function used previously. This suggests that the oscillating starts behavior often criticized by managers using optimization-based production planning models may be due in some degree to the model’s poor representation of the production system. While this does not explain the oscillating behavior that other authors such as Zäpfel and Missbauer (1993a)

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S. Orcun and R. Uzsoy A_T 1 Plot 1

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Fig. 3.23 Inventory behavior of PGDV-LP-CF Market 1 Plot 1 700

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Fig. 3.24 Delivery performance of PGDV-LP-CF

have observed in solving models without replanning, it certainly suggests that model mismatch may significantly contribute to oscillating behavior. Figure 3.27 compares the realized costs for the different scenarios compared in this chapter. One would expect that the scenarios with maximum internal and

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Fig. 3.25 Production behavior of PGDV-LP-CF-R A_T 1 Plot 1 4000

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Fig. 3.26 Inventory behavior of PGDV-LP-CF-R

external fidelity, i.e., with an exact match between the planning model and the realization and perfect global demand visibility, to yield the minimum cost. The results in the figure for the PGDV-PW-PW scenarios show that this is indeed the case. The system dynamics models with the feedback control policy incur both higher

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Fig. 3.27 Comparison of realized costs for different scenarios

production costs and higher inventory costs, but meet demand effectively. The LP models incur very high shortage costs due to their poor internal fidelity, i.e., their faulty representation of the behavior of the F/S facility caused by the fixed lead times that ignore congestion. This is in spite of the fact that the LP models achieve the lowest production costs among all the models compared. The PGDV-CF2-CF scenario supports the hypothesis that poor internal fidelity leads to poor cost performance, although its results are not as bad as those for the LP models. Surprisingly, the use of replanning in the PGDV-CF2-CF-R scenario leads to increased shortage and reduced inventory costs, which can be attributed to the oscillatory behavior induced by the frequent replanning combined with poor internal fidelity. The improvement in performance between IOBPCS-DELAY-DELAY and IOBPCS-DELAY-CF is at first sight somewhat surprising; one would have expected the continuous CF model with its heavy congestion effects to perform rather worse than the third-order delay in IOBPCS-DELAY-DELAY. While this observation is valid at the high utilization levels the system experiences during the demand pulse, over the remainder of the planning horizon the cycle time is far lower when the system is at low utilization. This illustrates one of the difficulties in comparing models of this type based on total cost alone: compensating errors may well blur the picture of what is truly going on with the models over time, underscoring the importance of examining dynamic behavior in detail rather than relying directly on cost comparisons over the entire planning horizon.

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3.7 Summary and Conclusions It is important not to read too much into these preliminary experimental results conducted in a very exploratory manner over a very simple supply chain system and a single deterministic demand scenario. However, the results of this study seem to suggest that the dynamics of supply chains operating under optimization-based planning models are qualitatively different from those operating under the relatively simple feedback policies used extensively in the system dynamics literature. While this will not come as a surprise to anyone familiar with this field, we believe this is the first time that these have been illustrated with the differences in dynamic behavior in mind. It is also interesting to observe the effects of periodic replanning on system behavior; it is not always productive of desirable behavior, suggesting that this issue needs considerable further study. We conjecture that replanning may be desirable when the mismatch between the desired and expected states of the world is large, but that sometimes it can produce undesirable oscillating behavior simply due to model mismatch rather than uncertainty in the environment. This type of model mismatch is actually quite similar to uncertainty, in that it creates a situation where there is a difference between the realized state of the world and that predicted by the planning models, prompting the latter to take corrective actions they would not otherwise require. It is also clear that the dynamic behavior of complex supply chains using optimization-based production planning models is an interesting area that does not seem to have been explored extensively in the literature to the best of our knowledge. However, there are a number of large optimization models in use for complex supply chains in a number of industries, notably semiconductor manufacturing (Leachman et al. 1996; Leachman et al. 2002; Fordyce et al. 2011). A number of commercial software packages now also include optimization models (Stadtler and Kilger 2004), so understanding the effects of these models on the dynamics of the supply chains we intend to manage based on the decisions they produce would seem to be an important direction for future research. Topics such as the relationship between replanning frequency, model mismatch and uncertainty in model parameters such as demand estimates need to be explored in more detail, taking advantage of today’s advanced software and computing capabilities. Finally, we believe this study illustrates the usefulness of tools such as the SCOPE environment for rapidly prototyping and studying different supply chain control logics across an entire supply chain, with a focus on dynamic behavior. Acknowledgements We thank Dr. Karl Kempf of Intel Corporation for providing the data for the testbed, and for his thoughtful suggestions throughout this work. The development of the SCOPE environment has been supported by The Laboratory for Extended Enterprises at Purdue (LEEAP), the UPS Foundation, and NSF Grants DMI-0075606 and DMI-0122207.

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Ptak CA, Schragenheim E (1999) ERP: tools, techniques and applications for integrating the supply chain. St. Lucie Press, Boca Raton, FL Saad GH (1982) “An overview of production planning models: structural classification and empirical assessment.” Int J Prod Res 20(1):105–114 Sadeh NM, Hildum DW, et al. (2001) “MASCOT: an agent-based architecture for dynamic supply chain creation and coordination in the internet company.” Prod Plann Contr 13:212–223 Schieritz N, Grossler A (2002) Emergent structures in supply chains: a study integrating agentbased and system dynamics modeling. 36th Hawaii International Conference on System Sciences, IEEE Computer Society Shapiro JF (1993). Mathematical programming models and methods for production planning and scheduling. In: Graves SC, Rinnooy Kan AHG, Zipkin P (eds.) Handbooks in operations research and management science, vol. 4, logistics of production and inventory. Elsevier Science Publishers B.V. Simchi-Levi D, Wu SD, et al. (eds.) (2004) Handbook of quantitative supply chain analysis: modeling in the e-business era. International series in operations research and management science. Kluwer, New York Stadtler H, Kilger C (eds.) (2004) Supply chain management and advanced planning. Springer Sterman JD (2000) Business dynamics: systems thinking and modeling for a complex world. McGraw-Hill, New York Swaminathan JM, Smith SF, et al. (1998) “Modeling supply chain dynamics: a multiagent approach.” Decis Sci 29(3):607–632 Tayur S, Magazine M, et al. (1998) Quantitative models for supply chain management. Kluwer, Amsterdam Terzi S, Cavalieri S (2004) “Simulation in the supply chain context: a survey.” Comput Ind 53:3–16 Thomas DJ, Griffin PM (1996) “Coordinated supply chain management.” Eur J Oper Res 94:1–15 Towill DR (1982) “Dynamic analysis of an inventory and order based production control system.” Int J Prod Res 20(6):671–687 Towill DR (1996) “Industrial dynamics modeling of supply chains.” Logist Inform Manag 9(4):43 Towill DR, Del Vecchio A (1994) “The application of filter theory to the study of supply chain dynamics.” Prod Plann Contr 5:82–96 Uzsoy R, Lee CY, et al. (1992) “A review of production planning and scheduling models in the semiconductor industry part i: system characteristics, performance evaluation and production planning.” IIE Trans Schedul Logist 24(47–61) Van Dyke Parunak H (2001). Industial and practical applications of DAI. In: Weiss G (ed.) Multiagent systems: a modern approach to distributed artificial intelligence. MIT, Cambridge, MA Van Dyke Parunak H, Savit R, et al. (1999) Dasch: dynamic analysis of supply chains. Center for Electronic Commerce, ERIM Inc. Ann Arbor, MI Vidal CJ, Goetschalckx M (2001) “A global supply chain model with transfer pricing and transportation cost allocation.” Eur J Oper Res 129:134–158 Voss S, Woodruff DL (2003) Introduction to computational optimization models for production planning in a supply chain. Springer, Berlin; New York Weiss G (ed.) (2001) Multiagent systems: a modern approach to distributed artificial intelligence. MIT, Cambridge, MA Wooldridge M (2001). Intelligent agents. In: Weiss G (ed.) Multiagent systems: a modern approach to distributed artificial intelligence. MIT, Cambridge, MA, pp. 27–77 Wooldridge M (2001) Multiagent systems. Wiley, Chichester, U.K. Zäpfel G, Missbauer H (1993a) “Production Planning and Control (PPC) systems including loadoriented order release - Problems and research perspectives.” Int J Prod Econ 30:107–122

Chapter 4

Supply and Demand Synchronization in Assemble-to-Order Supply Chains Markus Ettl, Karthik Sourirajan, Pu Huang, Thomas R. Ervolina, and Grace Y. Lin

4.1 Introduction In this chapter, we describe a methodology for effectively synchronizing supply and demand through the integrated use of supply and demand flexibilities. While most prior literature focuses on the concept of Available-To-Promise (ATP) to determine product availability, we propose a new methodology called Available-To-Sell (ATS) that incorporates firm-driven product substitutions into capitalize on up-sell and alternative-sell opportunities in the production planning phase. ATS aims at finding marketable product alternatives that replace demand on supply-constrained products while minimizing expected stock-out costs for unfilled product demand and holding costs for leftover inventory. It enables a firm to maintain a financially viable and profitable product portfolio, taking effective actions to avoid excess component inventory, and articulating marketable product alternatives. We formulate a mathematical programming model to analyze the performance of ATS, and show how to exploit the structural properties of the model to develop an efficient solution procedure utilizing column generation techniques. The model can easily be embedded into a firm’s supply chain operations to improve day-to-day flexibility. Several leading companies are beginning to adopt ATS principles to better deal with the difficulties of supply and demand synchronization in highly uncertain markets. Companies such as IBM, Dell, or Hewlett Packard continuously monitor the supply position of every part. If a part is in short supply due to higher demand, sales teams are requested to check whether demand can be moved to alternative products. For example, if 120 gigabyte hard drives are in short supply, marketing teams could offer products with a 160 gigabyte drive to the technologically savvy customer, or 80 gigabyte models to the price-sensitive customer. Firms that interact with their customers directly through a website or telesales systems can highlight these products on-the-fly based on current component availability and steer customers toward product configurations that they can supply easily and profitably. Existing ATS

M. Ettl () IBM Thomas J. Watson Research Center, P.O. Box 218, Yorktown Heights, NY 10598, USA e-mail: [email protected]


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applications are manual and rely heavily on expert’s knowledge using partial sets of data and are generally invoked at the time of order promising to customers. We present a systematic approach that incorporates ATS principles using mathematical models into the production planning phase. Consider the following motivating example that illustrates the benefits of ATS over a traditional ATP availability management process. A firm sells three types of computer products, denoted M1 ; M2 , and M3 , to its customers in a given planning period. Each product is assembled from components such as processors, hard drives, and optical drives as shown in Table 4.1. Assuming that the demand for each computer is 1,000 units, the supply of components that is required to fill all customer orders is shown in the third column of the table. Suppose that the firm’s hard disk supplier experiences a short-term production constraint on 120 GB hard drives. To compensate for the shortfall, the supplier promises to ship an additional 500 units of 160 GB hard drives as shown in the next column. Given these component supply quantities, a traditional ATP planning model would match the actual supply to the demand and create an ATP allocation as shown at the bottom of Table 4.1. As a result of the supply constraint, the firm would incur 500 unfilled orders for product M2 . To mitigate short-term supply and demand imbalances, our proposed ATS model seeks to identify marketable product substitutions that can replace demand on supply-constrained products while minimizing inventory from leftover components. In the above example, the ATS model creates an alternative product A1 shown in the shaded column of the table that can be substituted as an up-sell product for the constrained product M2 , possibly in

Table 4.1 Example of availability management with ATP and ATS Supply

Products

2.8 GHz Xeon 3.2 GHz Xeon 3.4 GHz Xeon

Requirement 1,000 1,000 1,000

Actual 1,000 1,000 1,000

P1 1 – –

P2 – 1 –

P3 – – 1

A1 – 1 –

Hard drives

80 GB 7200 rpm 120 GB 7200 rpm 160 GB 7200 rpm 250 GB 7200 rpm

1,000 1,000 1,000 –

1,000 500 1,500 –

1 – – –

– 1 – –

– – 1 –

– – 1 –

Software

XP professional XP media center Small business

1,000 1,000 1,000

1,000 1,000 1,000

1 – –

– 1 –

– – 1

– 1 –

Optical drives

48x/24x Combo 48x/32x Combo

1,000 2,000

1,000 2,000

– 1

1 –

– 1

1 –

Demand

1,000

1,000

1,000

–

ATP allocation ATS allocation

1,000 1,000

500 500

1,000 1,000

– 500

Components System processors

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conjunction with a price discount providing an incentive for customers to accept the substitution. The ATS allocation shown in the table eliminates any unfilled orders and excess component inventories, thus increasing sales revenue and lowering the firm’s supply chain operating costs. Several papers have analyzed firm-driven product substitutions from a manufacturing perspective (e.g., Barnhart et al. 1999; Hale et al. 2001; Balakrishnan and Geunes 2000, 2003; Gallego et al. 2006). Balakrishnan et al. (2005) investigate how a firm can improve profitability by shaping demand through dynamic pricing. The product portfolio is static and product alternatives are not considered. In our problem, product alternatives and their allocations are optimized simultaneously based on anticipated supply and demand. In this respect, our model has similarities to the vanilla box problem considered in Swaminathan and Tayur (1998) and the component allocation problem described in Akcay and Xu (2004). Although in a different context, Meacham et al. (1999) analyze a related problem in a reverse logistics application. They develop a fast column generation algorithm determining the optimal set of products that should be disassembled in order to meet a specified demand for recovered components. Netessine et al. (2002) describe an example of firmdriven product substitution in the context of service delivery where customers may be upgraded to a higher level of service at no additional cost to the customer. In their model, a firm invests in service capacity before demand is known and assigns available capacity to customers when demand is realized. The authors find that significant benefits can be gained if service substitutions are considered at the time of the capacity investment decision, rather than only at the time of service delivery. In the production planning area, there is an extensive literature dealing with real-time order promising and ATP. Ball et al. (2004) develop a general modeling framework for order promising and present industrial applications at Dell and Toshiba. Chen et al. (2002) develop a mixed integer programming model for order promising of batch orders. Ervolina and Dietrich (2001) describe a constraintbased planning technique called implosion to create a feasible production plan for promising customer orders. Chen-Ritzo (2006) studies a similar problem in a configure-to-order supply chain with order configuration uncertainty. The closest work in this stream is Ervolina et al. (2006) in which the authors develop a simulation modeling framework for availability management that allows them to compare ATS and ATP in terms of expected profitability and expected order delivery performance. The remainder of this chapter is organized as follows. In Sect. 4.2, we present three different business practices for availability management: a traditional approach (ATP), a sequential approach (ATP/ATS), and an integrated approach (ATS). In Sect. 4.3, we describe a mathematical programming formulation that can be incorporated into these business processes to determine optimal product substitutions and demand allocations. We also develop an efficient solution procedure based on column generation. Our numerical findings are presented in Sect. 4.4. We highlight the advantages of the sequential and integrated approach over ATP through numerical experiments with realistic production data. Section 4.5 concludes this chapter.

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4.2 Business Processes for Availability Management Availability management is the task of matching available resources to customer orders to maximize a firm’s return on assets and ensure a profitable match of supply and demand. After introducing the traditional ATP process, we propose two novel business processes that incorporate ATS concepts into availability management. Both approaches account for the possibility of product substitutions at the time of planning. Figure 4.1 shows the three different processes. The traditional approach to availability management is based on ATP as illustrated in Fig. 4.1a. ATP as defined by The Association for Operations Management (APICS) is “the uncommitted portion of a company’s inventory and planned production, maintained in the master schedule to support customer order promising” (e.g., Vollmann et al. 1997). The aim of ATP is to determine whether an incoming order can be promised for a specified customer request date. Frequently, ATP relies on optimization techniques such as constraint-based planning to match a firm’s available inventory with demand so as to optimize financial or operational objectives, e.g., maximizing order profitability, customer satisfaction or production efficiency. The ATP allocation plan is the essential output of the ATP optimization. As a customer order arrives, the order promising function must check the contents of the order against the ATP allocation to respond to the customer order with a quantity and a delivery date commitment. Because the traditional ATP process has no means of dealing with inventory overages that may arise from supply

Fig. 4.1 Different business processes for availability management

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and demand imbalances, a separate (nonintegrated) business process is often created to manage excess supply, e.g. by exercising buy-back agreements with component suppliers. To more effectively deal with short-term supply and demand imbalances, we propose a two-stage sequential process as depicted in Fig. 4.1b. In the first stage, an ATP allocation plan is generated to ensure that available supply is allocated to the firm’s most critical customer demand. After determining any leftover supply and unfilled customer orders, an ATS module is executed in the second stage to generate an ATS allocation plan consisting of product alternatives that best consume the excess supply. The ATP and ATS allocation plans are subsequently combined and handed over to the order promising and fulfillment process. The ATS module is powered by a mathematical programming model (described in the next section) that seeks to identify product alternatives that can be substituted to satisfy unfilled customer orders while consuming excess inventory. The product substitutions may come in the form of up-sells, alternate-sells, or down-sells. An up-sell is a more richly configured product that is above the customer’s selected price range; an alternative-sell refers to the sale of a similar product that falls within the customer’s selected price range; and a down-sell to a sale of a product that falls below the price range selected by the customer. In the sequential process, the available supply is assigned to standard products before product substitutions are considered. Hence, the resulting allocation plan may not fully capitalize on product substitution opportunities. The integrated approach illustrated in Fig. 4.1c addresses this limitation. In the integrated approach, product substitutions are considered at the time when the available inventory is matched to the customer demand. The outcome is an integrated ATS allocation plan in which the available inventory is allocated partly to standard products and partly to product substitutions that are best suited to restore balance between supply and demand. The integrated process is most effective in an assemble-to-order environment where products are configured from pluggable components, such as hard drives, microprocessors, optical drives. In such an environment, customers are generally more flexible, and the simplified product structure facilitates finding saleable product substitutions. Figure 4.2 is an illustration of an ATS allocation plan. Each circle in the figure represents a product, and the size of a circle indicates its supply allocation (larger circles mean a larger allocation). The product portfolio is grouped into three product segments denoted economy, value, and performance. The components used to assemble these products are grouped into component tiers. The position of a circle on the grid indicates whether a product is assembled from low-end, mid-range, or high-end components. Products in the economy segment are shown in the lower left quadrant because they are predominantly assembled from low-end components. Products in the performance segment are shown in the upper right quadrant since they are assembled from high-end components. The dark-colored circles depict alternative products selected by the ATS optimization model to supplement the portfolio of standard products.

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Fig. 4.2 Illustration of an ATS allocation plan

4.3 Mathematical Optimization Model for ATS Next, we formulate a mathematical programming model that determines an optimal ATS allocation plan, and present an efficient solution algorithm for the model that uses decomposition and column generation. We discuss how the model can be used to evaluate the three availability management processes discussed in the previous section. The main inputs to the model are a demand statement of the manufacturer at the product level (usually based on revenue targets) and a supply commitment with possible upside flexibility of the suppliers at the component level. The manufacturer incurs a liability cost for leftover component inventory, and a backorder penalty for unfilled demand. The liability costs are assessed at the end of the planning period (e.g., a quarter) based on the actual component consumption. Although we present a single-period formulation, it is easy to extend the model to a multiperiod formulation that would capture demand trends or time-dependent supply plans. Other inputs to the model include a portfolio of standard products that are grouped into different product segments, their bills-of-materials, and configuration rules for building product substitutions. The product configuration rules restrict the selection or components that can be used to create a product substitution. The goal of the optimization is to allocate components to products so as to satisfy customer demand at a minimum cost, taking into consideration product substitutions that may satisfy unfilled demand at an additional subsitution cost.

4.3.1 Notation Before we discuss the formulation, let us define the notation required.

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4.3.1.1 Parameters I: K: P: Mp :

Number of components indexed by i D 1; : : :; I . Number of component groups indexed by k D 1; : : :; K. Number of product segments indexed by p D 1; : : :; P . Number of standard products in segment p indexed by m D 1; : : :; Mp for p D 1; : : :; P .

4.3.1.2 Demand and Supply p

Dm : Demand for standard product m in segment p for m D 1; : : :; Mp and p D 1; : : :; P . Simin : Downside supply flexibility for component i for i D 1; : : :; I , i.e., the minimum supply consumption required by the component supplier. Simax : Upside supply flexibility for component i for i D 1; : : :; I , i.e., the maximum quantity of supply that can be obtained from the supplier.

4.3.1.3 Bills-of-Materials and Product Configuration Rules upim : 1 if component i is used to assemble standard product m in segment p, 0 otherwise for i D 1; : : :; I; m D 1; : : :; Mp and p D 1; : : :; P . Bik : 1 if component i belongs to component group k, 0 otherwise for i D 1; : : :; I and k D 1; : : :; K. wpi : 1 if part i can be used to assemble alternative products in segment p; 0 otherwise for i D 1; : : :; I and p D 1; : : :; P .

4.3.1.4 Costs hi : Unit inventory liability cost of component i for i D 1; : : :; I . p bm : Unit backorder penalty cost for standard product m in segment p for m D 1; : : :; Mp and p D 1; : : :; P . c p : Unit product substitution cost for product segment p for p D 1; : : :; P .

4.3.1.5 Decision Variables Np : Number of product alternatives in product segment p indexed by n D 1; : : :; Np for p D 1; : : :; P . p Xm : ATS allocation of standard product offering m in segment p for m D 1; : : :; Mp and p D 1; : : :; P .

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p Ymn : ATS allocation of alternative product n that meets demand for standard product m in product segment p for m D 1; : : :; Mp ; n D 1; : : :; Np and p D 1; : : :; P . Si W Actual consumption of component i for i D 1; : : :; I . vpin : 1 if component i is used to assemble alternative product n in product segment p; 0 otherwise for i D 1; : : :; I; n D 1; : : :; Np and p D 1; : : :; P .

The decision variable Np tracks the number of columns representing alternative products that are generated in the column generation procedure.

4.3.2 Problem Formulation We can now formulate the ATS optimization problem as follows:

Min

p p Z.Xm ; vin ;

p Ymn /

D

0

Mp P X X

p bm

pD1 mD1

C

I X

p p @Dm Xm

Np X

1 pA C Ymn

nD1

0

hi @Si

i D1

P X X

P X

c

p

pD1

Mp

P X X

mD1 nD1

Np

p p uim Xm

pD1 mD1

pD1 nD1

Mp Np X X

p vin

Mp X

p Ymn

1

pA Ymn

mD1

(4.1) Subject to

p Dm

Si Simax ; 8i

(4.2)

Si Simin ; 8i:

(4.3)

p Xm

Np X

p Ymn 0

(4.4)

nD1

Si

Mp P X X

p upim Xm

pD1 mD1

Np P X X

vpin

pD1 nD1 p

Mp X

(4.5)

mD1

p

vin wi ; 8i; p; n I X

p Ymn 0

Bik vpin D 1; 8k; p; n

(4.6) (4.7)

i D1

p 0; 8p; m Xm

(4.8a)

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Si 0; 8i

(4.8b)

p Ymn 0; 8p; m; n

(4.8c)

Np 0 & Integer; 8p

(4.8d)

vpin

2 f0; 1g; 8i; n; p

(4.8e)

The objective function (4.1) minimizes the total manufacturer’s cost, which consists of three components: 1. Backorder penalties. Costs incurred if the ATS allocation for a standard product (including product substitutions) falls short of the demand for this product. 2. Product substitution costs. Costs incurred if a product substitution is used to (partially) fulfill demand for a standard product. 3. Component liability costs. Costs incurred if the actual component consumption is lower than the minimum supply consumption requirement. Although the details of inventory liability settlements between a supplier and a manufacturer differ from contract to contract, most supply contracts track demand forecast updates coupled with a liability window to determine what amount of inventory already built up at a supplier will become a liability for the manufacturer (e.g., Ettl et al. 2006). Constraints (4.2) and (4.3) ensure that the quantity Si of component i ordered from the supplier is bounded by maximum and minimum values as determined by the supply contract. Constraints (4.4) state that the total allocation for a standard product p . m in segment p (including product substitutions) must not exceed the demand Dm Constraints (4.5) ensure that the number of components allocated to standard or alternative products does not exceed the available supply. Constraints (4.6) ensure that the bill-of-material of each product alternative adheres to the selection menu specified for its product segment. Constraints (4.7) require that a product alternative uses exactly one component from each component group (this constraint is motivated by our application of computer manufacturing and can be easily relaxed). Finally, constraints (4.8) are nonnegativity and integrality constraints on the decision variables. It is important to note that even if the number of product alternatives Np in prodp uct segment p were fixed, there is a nonlinear term (vpin Ymn ) in the objective function (4.1). If we enumerate all possible product alternatives, the optimization problem (4.1)–(4.8e) reduces to a linear program (LP). This LP, however, would become very large because the number of possible configurations increases exponentially with the number of components. It would also be possible to linearize the problem using standard techniques (e.g., Barnhart et al. 1998) and convert it into a mixed integer program (MIP). However, given that industry-size problems involve hundreds of components and dozens of product categories (which would result in thousands of binary variables), finding the optimal solution of the MIP is likely to take a prohibitively long time. Because speed of execution is essential for a timely resolution of supply and demand imbalances, we have developed an efficient iterative procedure based on column generation, exploiting similarities between our model and the cutting stock problem studied in Gilmore and Gomory (1961).

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4.3.3 Computational Algorithm The problem described in (4.1)–(4.8e) is decomposed into two subproblems: a master problem that aims at finding the optimal allocation plan for a given set of products, and a set of slave problems in which new product alternatives are determined. The decision variables of the master problem are the product allocation p p p and Si whereas the sets of variables vin and Np are and supply variables Xm ; Ymn assumed to be fixed. The two subproblems are solved iteratively until the optimal solution is reached. The master problem MP is defined as follows:

4.3.3.1 Master Problem (MP)

p p Min Z.Xm ; Ymn /D

Mp P X X pD1 mD1

C

I X

0 p @ p p bm Dm Xm

Np X

1 pA Ymn C

nD1

0

hi @Si

i D1

Mp P X X

p

p uim Xm

pD1 mD1

P X

cp

pD1 Np P X X pD1 nD1

Mp Np X X

p Ymn

mD1 nD1 p

vin

Mp X

1

pA (4.9) Ymn

mD1

subject to constraints (4.2), (4.3), (4.4), (4.5), (4.8a), (4.8b) and (4.8c). p Since vin and Np are fixed parameters in (4.9), the MP reduces to an LP that can be solved very efficiently even for large problem sizes. Next, let i ; i D 1; : : :; I , the shadow prices pertaining to the inventory liability constraints (4.5), p and ˇm ; m D 1: : :; Mp and p D 1; : : :; P denote the shadow prices pertaining to the ATS allocation constraints (4.4) in the optimal solution of MP. Let p p m N p D arg minfbm C ˇm W m 2 Mp g. We can now formulate the slave problem SPp for p D 1; : : :; P as follows:

4.3.3.2 Slave Problem (SPp )

Min Z2p .Nvpi / D

I X i D1

p p p .hi C i /Nvpi bm N p C ˇm Np C c

(4.10)

subject to vN pi wpi ; 8i I X i D1 vN pi

(4.11)

Bik vN pi D 1; 8k

(4.12)

2 f0; 1g; 8i:

(4.13)

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Constraints (4.11), (4.12), and (4.13) are equivalent to constraints (4.6), (4.7), and (4.8e). Our column generation procedure is similar to that of Gilmore and Gomory (1961). In the initial step of the algorithm, the master problem is solved with the set of standard products, and its dual solution is fed into the slave problems for every p. The slave problems subsequently try to improve the solution by finding new product offerings for every product category. Any such new product offerings found are then added back into MP after updating Np for p D 1; : : :; P , and the master problem MP is re-solved. This process iterates until no further improvement is possible. If we were to follow the standard column generation procedure, we would formulate a slave problem for every p and m. This way, we would have just added a p p C ˇm / to the objective function (4.10) for every m. The different constant .bm optimal solution would then be the same for all slave problems for a given p. Thus, finding m N p and solving just one slave problem for every p makes our algorithm more efficient compared to the standard column generation procedure for this problem. The solutions obtained at the end of the column generation procedure are guaranteed to be feasible and optimal (Gilmore and Gomory 1961). The algorithm is outlined below:

4.3.3.3 Algorithm ATS Step 1: Initialize Np WD 0 and set fvpin g D for all p. p . Step 2: Solve the master problem MP and obtain the optimal values of i and ˇm p p Using ˇm and bm , find m N p ; 8p Step 3: Solve the slave problems SPp for every p. Step 4: If the optimal values of Z2p ./ 08p, we have found the optimal solution; STOP and print the results from the current master problem solution. Otherwise, go to Step 5. Step 5: If the optimal value of Z2p ./ < 0 for any p, set Np WD Np C 1 and add vN pi to the set of columns fvpin g and go to Step 1. Although the slave problem SPm is an integer program, it has a special structure that can be exploited to solve it in a greedy fashion: for each product segment p and component group k, consider all components i allowed in the selection p menu wi . Compare their .hi C i / values and choose the one with the minimum .hi C i / value. It is easy to see that repeating this procedure for all component p p groups produces product substitution that minimizes Z2 .Nvi /. The above algorithm solves the allocation problem associated with the integrated ATS process described in Sect. 4.2. The algorithm can be applied with minor modifications to solve the allocation problem pertaining to the traditional and sequential processes. To model the traditional ATP process in which product substitutions are p from the formulation not considered, we simply remove the variables vpin and Ymn p along with all their associated constraints and costs. This yields an LP in which Xm p is the set of decision variables. The ATP allocation plan is determined by the Xm variables, and the manufacturer’s cost is the sum of the inventory liability cost and backorder cost.

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To model the sequential process, we first solve the ATP model to obtain the optimal ATP allocation plan, and then run the ATS model under the following modified demand and supply statement: p p p Dm WD Dm Xm P X P Mp

Simin WD Simin

pD1 mD1

p

p uim Xm and Simax WD Simax

(4.14) P M P Pp pD1 mD1

p

p

uim Xm ;

(4.15)

p values are obtained from solving the ATP allocation problem. The where the Xm allocation plan for the sequential model is obtained by combining the results of the ATP and ATS models. The manufacturer’s cost is the inventory liability cost, backorder cost, and substitution cost of the combined allocation plan.

4.3.4 Limitations of the Model The model formulation has a number of limitations that we discuss next. First, the single-period model does not capture situations in which demand and component supplies are planned over multiple time periods. In practice, suppliers commit to the delivery of components several weeks in advance. Such multiperiod deliveries create correlations of component allocations to product segments that are not captured in a single-period model. Second, we assume that the demand is provided in the form of a point forecast for each product. Given that firms are facing significantly uncertain demand at the time of planning and that point forecasts are invariably wrong, we need to investigate how the model can be extended to accommodate stochastic demand. Third, we assume that all demand in a period is observed before the available supply is allocated to products. In reality, customers place orders continuously and components would need to be allocated to orders on an order-by-order basis. Ervolina et al. 2006 provide some quantitative insights into the dynamics of order execution by simulating different order acceptance policies in conjunction with ATS. Finally, we assume that all pricing decisions are exogenous inputs to the model. It would be interesting to develop an integrated model that determines not only the product allocations but also the ideal pricing policy that would maximize the firm’s profit in view of the customer’s propensity to upgrade to a more profitable product. Such an integrated decision model is beyond the scope of this work, but is a promising area for future research.

4.4 Numerical Study We implemented the solution procedures described in the previous section and applied them to an ATO system for mid-range server computers. In this section, we present and discuss our numerical findings. The numerical study focuses on

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(a) comparing the performance of the sequential and integrated models, in terms of expected backorder and inventory costs, to that of the traditional ATP approach; and (b) evaluating the impact of supply quantity flexibility on supply chain cost under the different availability management approaches.

4.4.1 Comparisons Between ATP and ATS The example scenario for the numerical study is depicted in Fig. 4.3. The data set resembles a real-world problem compiled from actual server supply chain data. The product portfolio consists of three product families that represent low-end, midrange, and high-end server computers. Each product family comprises a number of standard products with bills-ofmaterials as depicted in Table 4.2. Products P1 –P4 are low-end systems, P5 –P7 are mid-range systems, and P8 –P10 are high-end computers. Each configuration is assembled from components of four component groups: system processors, hard drives, software, and optical drives. For example, product P1 is assembled from a 2.8 GHz system processor, an 80 GB hard drive, XP Professional, and a 48x/32x optical drive. Note that all products within the same product segment use the same system processor, i.e., 2.8 GHz for low-end systems, 3.2 GHz for mid-range systems, and 3.4 GHz for high-end configurations. We assume that the inventory liability cost for component i is hi D 5, the backp order cost per unit of unfilled demand is bm D 50, and the product substitution

Fig. 4.3 Assemble-to-order product structure

6,000 4,500 4,500


XP Professional Enterprise Linux Small Business


System processors

Hard drives

Software

Optical drives

1,000

ATP allocation

– 1

1 – –

1 – – –

1 – –

P1

1,500

6,000 9,000

4,000 7,000 4,000

4,000 5,000 4,000 2,000

2,500 10,000 2,500

Actual

Demand

7,500 7,500

4,500 6,000 4,500

4,500 3,000 4,500 3,000

Requirements


Components

1,500

1,500

1 –

1 – –

– –

1

1 – –

P2

–

1,500

– 1

– 1 –

1 – – –

1 – –

P3

Table 4.2 Bills-of-materials, supply requirements, and supply commitments Supply Economy products

–

1,500

– 1

1 – –

– 1 – –

1 – –

P4

1,500

1,500

1 –

– 1

– – 1 –

– 1 –

P5

1,500

1,500

– 1

– 1 –

– – 1 –

– 1 –

P6

Value products

1,500

1,500

1 –

– – 1

– 1 – –

– 1 –

P7

1,000

1,500

1 –

– – 1

– – 1 –

– – 1

P8

–

1,500

1 –

– 1 –

– – – 1

– – 1

P9

1,500

1,500

– 1

– – 1

– – – 1

– – 1

P10

Performance products

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cost is c p D 10. The top-level demand for configuration m in product segment p is p Dm D 1;500 units. To determine the supply requirements for each component, the demand is exploded through the bills-of-materials in a standard MRP calculation. The third and fourth columns of Table 4.2 show the supply requirement and a sample supply commitment pertaining to the top-level demand. The supply commitment indicates a suppliers’ capability to deliver to the manufacturer’s supply requirements. For now, assume that the minimum supply quantity Simin is equal to the maximum supply quantity Simax , i.e., there is no supply flexibility. Comparing the supply requirements with the actual supply indicates a supply constraint on 2.8 GHz and 3.4 GHz processors. To mitigate these constraints, the micro supplier committed a higher than requested supply volume of 10,000 units for the 3.2 GHz processor. Note that the supply commitment matches the supply requirements in each component group, although the individual component mix deviates from the requirements. Given the supply commitment, the traditional ATP approach matches the available supply to the demand and yields the ATP allocation shown in the bottom row of Table 4.2. As a result of the supply constraints, the ATP schedule incurs 3,500 backorders of low-end systems and 2,000 backorders of high-end systems. Next, we apply the sequential and integrated availability management model to p the example scenario. Table 4.3 shows the selection menu (wi ) for the three product segments. To analyze the performance of the sequential model, we first explode the ATP allocation generated above through the bills-of-materials to obtain the actual consumption of each component and determine unallocated supply. Next, we apply the ATS optimization model to create product substitutions. The demand for the ATS model is given by the unfilled demand pertaining to the ATP schedule. The sequential algorithm creates two alternative products, R11 and R31 , with bills-ofmaterials shown in the shaded columns of Table 4.4. The sequential ATS allocation

Table 4.3 Selection menu for the different product segments Components Economy System processors 2.8 GHz Xeon 1 3.2 GHz Xeon 1 3.4 GHz Xeon –

Value – 1 –

Performance – 1 1

Hard drives


1 – – –

– 1 1 –

– – – 1

Software


1 1 –

– 1 1

– 1 1

Optical drives


1 1

1 1

1 1



Software

Optical drives

Demand ATS allocation



Hard drives


1,500 1,000

– 1

1 – –

1 – – –

P1 1 – –

1,500 1,500

1 –

1 – –

1 – – –

P2 1 – –

Table 4.4 ATS allocation generated in the sequential process Economy products

1,500 –

– 1

– 1 –

1 – – –

P3 1 – –

1,500 –

– 1

1 – –

– 1 – –

P4 1 – –

1 – 1,500 1,500

– 1 – 1,500

– 1 –

– – 1 –

1 – – – 1 – –

P5 – 1 –

R11 – 1 –

1,500 1,500

– 1

– 1 –

– – 1 –

P6 – 1 –

Value products

1,500 1,500

1 –

– – 1

– 1 – –

P7 – 1 –

1,500 1,000

1 –

– – 1

– – 1 –

P8 – – 1

1,500 –

1 –

– 1 –

– – – 1

P9 – – 1

1,500 1,500

– 1

– – 1

– – – 1

P10 – – 1


– 500

– 1

– 1 –

– – – 1

P31 – 1 –

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is displayed in the bottom row. As a result of optimization, the backorders for lowend systems are reduced from 3,500 to 2,000, and the backorders for high-end systems are reduced from 2,000 to 1,500. We next analyze the performance of the integrated model. Recall that in the integrated model an ATS allocation plan is generated in a single optimization run based on the initial demand and component supply. The results are displayed in Table 4.5. The integrated algorithm creates four alternative products with bills-of-materials shown in the shaded columns. We can see that the backorders for all three families are significantly reduced: 500 backorders for low-end systems (down from 2,000 in the sequential model), zero backorders for mid-range systems (down from 500), and 1,000 backorders for high-end systems (down from 1,500). Table 4.6 compares the unallocated supply derived from the three different models. The traditional ATP process leaves a substantial portion of the components unallocated, in particular 5,500 units of 3.2 GHz system processors and 5,000 units of 80 GB and 120 GB hard drives. The reason for the large overage is that none of the existing products in Table 4.2 is configured with a 3.2 GHz processor and a low-end hard drive. Both the sequential and the integrated model create alternative products that utilize a 3.2 GHz processor and an 80 GB or 120 GB hard drive in their bills-of-materials. As a result, the excess supply is reduced from 22,000 to 14,000 units in the sequential model and 6,000 units in the integrated model. Under the given inventory liability cost of hi D 5, the sequential and integrated models yield inventory costs of $70,000 and $30,000, respectively. These cost figures are significantly lower than the $110,000 produced by the traditional ATP. The sequential model yields a backorder cost of $175,000, a substitution cost of $20,000 and a total cost of $265,000. The integrated model yields a backorder cost of $75,000, a substitution cost of $45,000 and a total cost of $150,000. Given that the total cost incurred by the ATP model is $385,000, the percentage cost improvement gained in the sequential model is more than 30%, and more than 60% in the integrated model. The individual costs pertaining to the different approaches are summarized in Table 4.7.

4.4.2 Impact of Quantity Flexibility on Supply Chain Costs Next, we investigate the impact of supply flexibility. Supply contracts with supply quantity flexibility contracts are widely used by firms facing significantly uncertain demand. Quantity flexibility is a supplier’s guarantee to deliver up to a certain percentage above the committed supply quantity. The manufacturer can purchase options or other derivative instruments for risk management to protect its supply chain against demand risk (e.g., Cheng et al. 2011). We consider quantity flexibility a key input to our optimization models, allowing additional flexibility in identifying alternative product configurations. As before we assume that the top-level demand for configuration m in prodp D 1;500 units and the minimum supply commit for each uct family p is Dm



Software

Optical drives

Demand ATS allocation



Hard drives


1,500 –

– 1

1 – –

1 – – –

P1 1 – –

1,500 –

1 –

1 – –

1 – – –

P2 1 – –

1,500 1,000

– 1

– 1 –

1 – – –

P3 1 – –

Table 4.5 ATS allocation generated in the integrated process Economy products

1,500 1,500

– 1

1 – –

– 1 – –

P4 1 – –

– 1 – 500

– 1 – 2,500

– 1 –

1 – – –

1 – – – 1 – –

R12 – 1 –

R11 – 1 –

1,500 1,000

1 –

– 1 –

– – 1 –

P5 – 1 –

1,500 1,500

– 1

– 1 –

– – 1 –

P6 – 1 –

Value products

1,500 1,500

1 –

– – 1

– 1 – –

P7 – 1 –

– 500

1 –

– 1 –

– 1 – –

R21 – 1 –

1,500 1,500

1 –

– – 1

– – 1 –

P8 – – 1

1,500 –

1 –

– 1 –

– – – 1

P9 – – 1

1,500 1,000

– 1

– – 1

– – – 1

P10 – – 1


– 1,000

1 –

– 1 –

– – – 1

P31 – 1 –

98 M. Ettl et al.

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Table 4.6 Excess inventory for different availability management processes Excess inventory Components System processors


Supply 2,500 10,000 2,500

ATP – 5,500 –

Sequential – 3,500 –

Integrated – 1,500 –

Hard drives


4,000 5,000 4,000 2,000

1,500 3,500 – 500

– 3,500 – –

– 1,500 – –

Software

XP rofessional Enterprise Linux Small Business

4,000 7,000 4,000

1,500 4,000 –

– 3,500 –

– 1,500 –

Optical drives


6,000 9,000

500 5,000

500 3,000

500 1,000

Table 4.7 Cost comparison between ATP and ATS models ATP Sequential

Integrated

Unit Cost Quantity Backorders $ 50 5,500 Excess inventory $ 5 22,000 Substitutions $ 10 –

Cost Quantity Cost Quantity Cost $275,000 3,500 $175,000 1,500 $75,000 $110,000 14,000 $70,000 6,000 $30,000 – 2,000 $20,000 4,500 $45,000

Total cost

$385,000

$265,000

$150,000

component i; Simin , is as shown in the third column of Table 4.2. To model quantity flexibility, let Simax WD .1 C ˛/Simin denote the maximum supply quantity, where ˛ is a contingency factor that determines the amount of upside supply flexibility. With quantity flexibility, the supplier is committed to provide up to Simax units of component i . The manufacturer assumes inventory liabilities only if the allocated supply is less than Simin . Figure 4.4 compares the optimized supply chain cost pertaining to the ATP schedule and the two ATS-based approaches when the contingency factor takes on values of ˛ D 0; 0:1; 0:2, and 0.3. We observe that the supply chain cost of the ATP model improves from $385,000 for the base scenario (no upside flexibility) to $337,500 for the scenario with 30% flexibility for a 12% overall cost reduction. For the ATS-based approaches, the cost reductions are even more dramatic. In the sequential model, the additional supply flexibility translates into 43% cost savings ($265,000 supply chain cost in the base scenario vs. $148,500 for the scenario with 30% flexibility). In the integrated model, the additional flexibility translates into 45% savings ($150,000 vs. $83,250 supply chain costs). This result is intuitive because higher supply flexibility provides more opportunities for product substitutions without increasing the manufacturer’s inventory liability exposure. These results confirm that even a modest level of upside flexibility can result in significant percentage cost improvements. For example,

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Fig. 4.4 Total supply chain cost as a function of supply quantity flexibility

when ˛ D 0:1 the supply chain cost decreases by 13% in the sequential model and by 24% in the integrated model compared to the base case. Table 4.8 summarizes inventory liability costs, backorder cost, and substitution cost for the different management approaches. Not surprisingly, the backorder costs and inventory liability penalties decrease as the supply flexibility increases. However, the relative impact of the additional supply flexibility on cost is substantially higher in the integrated model than in the sequential model. For example, a 10% upside flexibility yields 40% less backorders for the integrated model but only 17% less backorders in the sequential model compared to the base case. The reason is that the integrated model not only recognizes supply-constrained products, but also is capable of finding an appropriate set of saleable product alternatives that can be offered to customers in place of the constrained products at minimal or no additional cost. Note that the integrated model carries the highest substitution cost, which suggests that the ATS allocation deviates considerably from the original ATP allocation.

4.5 Summary Many firms recognize that supply and demand imbalances differ in magnitude and severity of impacts, but rarely invest in resources to develop a systematic approach for a resolution. Some companies have begun implementing more flexible availability management processes that seek to dynamically adjust featured products to guide marketing and sales teams. However, these are almost always manual processes that rely on expert’s knowledge and partial sets of data. With hundreds or even thousands

No flexibility 10% flexibility 20% flexibility 30% flexibility

Backorders $175,000 $145,000 $115,000 $72,500

Substitution – – – –

Backorders $275,000 $262,500 $250,000 $237,500

Inventory $110,000 $106,250 $102,500 $100,000

Excess

Excess Inventory $70,000 $62,250 $54,500 $43,000

Table 4.8 Detail supply chain cost as a function of supply quantity flexibility ATP ATS (sequential model) Substitution $20,000 $23,500 $27,000 $33,000

Backorders $75,000 $45,000 $30,000 $20,000

Inventory $30,000 $24,250 $21,500 $19,750

ATS (integrated model) Excess Substitution $45,000 $44,500 $44,000 $43,500

4 Supply and Demand Synchronization in Assemble-to-Order Supply Chains 101

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of products in a firm’s product portfolio, it is impossible to manually assemble the data and reach an optimal conclusion. An automated approach based on optimization not only ensures that the resources invested are likely to produce a successful result, but also offers the additional advantage of speed in execution, which is critical for a timely resolution of supply and demand imbalances. In this chapter, we have described an optimization-based approach to availability management and have outlined the business processes required to incorporate availability management into supply chain operations. We formulated the planning problem as a mathematical program and developed an efficient computational procedure to generate optimal solutions. We demonstrated the benefits of the solutions through numerical experiments with realistic production data. The principles and capabilities described in this chapter enable manufacturers to determine a financially viable, profitable, and marketable product portfolio, help resolve supply-demand imbalances, and take effective actions to avoid inventory overages. They can easily be imbedded into supply chain operations to improve day-to-day flexibility. The models featured in this chapter have already contributed to business improvements in real-world supply chains. IBM has implemented the sequential process described in Sect. 4.2 in its complex-configured server product line and has executed an ATS business process since 2002. The process has resulted in a $100M reduction in the first year and more than $20M of inventory reduction in each of the following years, not accounting for additional savings from excess inventory avoidance, which can often be substantial. Acknowledgements The authors thank the two referees for their insightful comments, which have helped improve the presentation of this chapter. The authors also thank Larry Phillips, Rich Bell, Blair Binney, and Dan Peters for sharing their knowledge about demand conditioning and availability management, and Reha Uzsoy for pointing us to the literature on reverse logistics.

References Akcay Y, Xu S (2004) “Joint inventory replenishment and component allocation optimization in an assemble-to-order system.” Manag Sci 50:99–116 Balakrishnan A, Geunes J (2000) “Requirements planning with substitutions: exploiting bill-ofmaterials flexibility in production planning.” Manuf Serv Oper Manag 2(2):166–185 Balakrishnan A, Geunes J (2003) “Production planning with flexible product specifications: an application to specialty steel manufacturing.” Oper Res 51(1):94–112 Balakrishnan A, Xia Y, Zhang B (2005) “Shaping demand to match anticipated supply.” MSOM Conference 2005, Northwestern University. Ball MO, Chen CY, Zhao ZY (2004) “Available to Promise.” In: Simchi-Levi D, Wu, SD, Shen ZJ (eds). Handbook of quantitative supply chain analysis - modeling in the e-business era. Kluwer, pp. 447–480 Barnhart C, Johnson EL, Nemhauser GL, Savelsbergh MWP, Vance PH (1998) “Branch-and-price: Column generation for solving huge integer programs.” Oper Res 46(3):316–329 Cheng F, Ettl M, Lin GY, Tonner M, Yao DD (2011) “Designing flexible supply contracts with options.” In: Kempf KG, Keskinocak P, Uzsoy R (eds.) Planning production and inventories in the extended enterprise: a state of the art handbook, vol 2. Springer, New York, pp. 207–230

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Chen C-Y, Zhao Z, Ball MO (2002) “A model for batch advanced available-to-promise.” Prod Oper Manag 11:424–440 Chen-Ritzo C-H (2006) “Availability management for configure-to-order supply chain systems.” PhD Dissertation, Pennsylvania State University Ervolina T, Dietrich B (2001) “Moving toward dynamic available-to-promise.” In: Gass S, Jones AT (eds.) Supply chain management practice and research: status and future directions pp. 1–19 Ervolina T, Ettl M, Lee Y, Peters D (2006) “Simulating order fulfillment with product substitutions in an assemble-to-order supply chain.” In: Perrone LF et al. (eds.) Proceedings of the 2006 winter simulation conference. pp. 2012–2020 Ettl M, Lu Y, Squillante M (2006) “Liability and serviceability trade-offs in vendor-managed inventory systems,” In: Proceedings of the 2006 M&SOM Conference, Georgia Institute of Technology, Atlanta Gallego G, Katircioglu K, Ramachandran B (2006) “Semiconductor inventory management with multiple grade parts and downgrading.” Prod Plan Contr 17(7):689–700 Gilmore PC, Gomory RE (1961) “A linear programming approach to the cutting-stock problem.” Oper Res 9:849–859 Hale W, Pyke DF, Rudi N (2001) “An assemble-to-order system with component substitution.” Tuck School of Business, Dartmouth Meacham A, Uzsoy R, Venkatadri U (1999) “Optimal disassembly configurations for single and multiple products.” J Manuf Syst 18(5):311–322 Netessine S, Dobson G, Shumsky RA (2002) “Flexible service capacity: optimal investments and the impact of demand correlation.” Oper Res 50(2):375–388 Swaminathan JM, Tayur SR (1998) “Managing broader product lines through delayed differentiation using vanilla boxes.” Manag Sci 44(12):S161–S172 Vollmann TE, Berry WL, Wybark DC (1997) Manufacturing planning and control systems, 4th edn. McGraw-Hill

Chapter 5

Quantitative Risk Assessment in Supply Chains: A Case Study Based on Engineering Risk Analysis Concepts Léa A. Deleris and Feryal Erhun

5.1 Introduction In recent years, numerous events have shown the extent to which companies, and subsequently their supply chains, are vulnerable to uncertain events. We have witnessed many supply chain malfunctions (with substantial consequences) due to supply and demand disruptions: affected companies reported, on average, a 14% increase in inventories, an 11% increase in cost, and a 7% decrease in sales in the year following the disruption (Hendricks and Singhal 2005). Component shortages, labor strikes, natural and manmade disasters, human errors, changes in customer taste, technological failures, malicious activities, and financially distressed and, in extreme cases, bankrupt partners, among many others, can cause disruptions in supply chains: [Component shortages] In March 2000, a fire in a Philips Electronics semi-

conductor plant located in Los Alamos, New Mexico, disrupted cellular-phone chip production for both Nokia and Ericsson. Ericsson delayed the execution of recovery plans and suffered significant financial losses from lost production – estimated at $220 million for the second quarter of 2000 alone (Latour 2001). A parts shortage in April 2005 forced MG Rover to suspend operations at its Longbridge automobile plant, which probably sped the collapse of the company (Rice 2005). [Labor strikes] In 1997, the teamsters’ strike that lasted for 15 days severely affected the United Parcel Service (UPS) Company, which at the time controlled 80% of all the package deliveries in the USA, and crippled the logistics of numerous US manufacturers (Treece 1997). The shutdown of all the West Coast ports due to the dockworkers’ strike that took place in September 2002 idled NUMMI while parts were flown in Pender (2002).

L.A. Deleris () IBM Research – Smarter Cities Technology Center, Damastown Industrial Estate – Mulhuddart, Dublin 15 – Ireland e-mail: [email protected]


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[Natural and manmade disasters] The Taiwan earthquake of September 1999

had a disastrous effect on the semiconductor industry: Dell estimated the cost increase for PC memory at around 25% (Moore 1999). [Changes in customer taste] In 2001, the steep decline of sales due to the dot-com crash and the deteriorating economy caused raw material and workin-process inventories to pile up at contract manufacturer (CM) sites, which eventually led Cisco to disclose a $2.25 billion inventory write-off. Cisco suffered a 6% fall in stock price (Barrett 2001). [Technological failures/malicious activities] A glitch in Nike’s demand planning software in early summer 2000 caused supply shortages for Air Jordans. As a result, Nike announced $100 million in lost sales (Koch 2004). Oftentimes, supply chains are operated through a variety of sophisticated applications (logistics, supplier management, inventory management), which are vulnerable to machine downtime and software issues. A survey that polled 300 randomly selected companies found that almost one third of these companies had major virus incidents in 2003 with an average recovery cost of almost $100,000 per incident (Lemos 2004). [Financially distressed/bankrupt partners] The financial collapse of UPFThompson caused Land Rover to almost suspend the production of its Discovery model, as UPF-Thompson was the sole supplier of chassis for this model. A legal dispute followed to determine the responsibility of Land Rover with regard to the bankruptcy, since the contract between the two firms represented 65% of UPF-Thompson’s business (Macalister 2002). The consequences of many of these adverse events could have been different with a sound risk management plan. For instance, Nokia, unlike Ericsson, successfully reacted to the Philips plant fire. As Nokia was monitoring its inventories closely, the company managed to shift its order to other Philips plants and other suppliers right after the incident and did not suffer a serious supply problem. Consequently, Nokia gained a significant competitive advantage over Ericsson (Latour 2001; Sheffi 2005). Similarly, Dell successfully managed to match supply with demand and in turn increase its market share during the period following the Taiwan earthquake by changing prices, so that the products with the components hit by the earthquake became expensive (Murphy 2005). By and large, the strategic and tactical decisions of a firm affect the extent to which its supply chain is able to mitigate the consequences of adverse events. Beyond reducing costs to the leanest, effective supply chain management should also involve the design of a robust supply network. However, the quest for better and cheaper sometimes leads companies to extend their supply chains haphazardly to gain competitive advantage. Trends such as globalization, lean manufacturing, single sourcing, and offshoring increase the complexities, and interdependencies in supply chains. Such trends have a direct effect on interruptions and have been identified as a source of risk by both practitioners and researchers (Newman 1989; Waters-Fuller 1995; Coleman and Jennings 1998; Aichlmayr 2001). Furthermore, recent literature on risk management suggests that most companies have an intuitive perception of their risk exposure based on informal, qualitative appraisals.

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Quantitative Risk Assessment in Supply Chains

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It appears that those who address risk more systematically often do so a posteriori, using contingency plans and buffer strategies rather than taking actions a priori to reduce the probability of occurrence of an adverse event or the magnitude of its consequences (Lee and Billington 1992; Zsidisin et al. 2000). Therefore, it is important to provide decision makers at all levels of an organization with appropriate information for risk management decision support. This chapter addresses the issue of assessment and characterization of the risk drivers of a supply chain. Using techniques from the engineering risk analysis field, we describe a framework for quantification and management of risk in supply chains. Our approach is novel in supply chain management in the sense that it seeks to account for a diverse set of risks that affect supply chains. The notion of risk is not restricted to specific hazards or to specific interfaces among supply chain parties; it follows a global view of adverse events, which results in a holistic approach. We tailor the chapter mainly to planners who wish to take a systematic perspective toward the assessment and mitigation of supply chain risk that goes beyond financial management measures such as insurance. For that purpose, we rely on a case study to illustrate our framework. Our focus on practical assessment leads us to emphasize numerical estimation, sometimes at the expense of analytical results. Academic researchers will find in our chapter only the first steps toward bridging current supply chain research in risk assessment with the engineering risk analysis literature. Such cross-disciplinary studies are still, to the best of our knowledge, in their development stages and present multiple directions for future research.

5.2 Probabilistic Risk Analysis: A Primer Complex systems, such as supply chains, often fail due to a combination of events (i.e., a class of scenarios1 ) rather than a single point of failure. Probabilistic risk analysis (PRA) studies complex systems by identifying the disruption scenarios that affect the system, and assessing the probability of occurrence (i.e., likelihood) of each possible scenario and its consequences (i.e., impact) on the system’s performance. In this section, we provide a short overview of technical description of the tools used in the traditional PRA process and refer the reader to Kumamoto and Henley (1996) for details. PRA provides a quantitative estimation of the risk of the system. As such, it enables to assess whether a given system performance-risk trade-off is adequate. When combined with an optimization model, PRA also helps a planner set priorities among risk mitigation actions for a given set of resource constraints. Furthermore, the structured approach that underlines PRA can uncover additional insights into the system under scrutiny, such as design flaws, and often leads to the identification of mitigation solutions. 1

In the rest of the chapter, we call “scenario” what in reality is a class of scenarios, meaning that we do not try to capture all possible details of a chain of events. We do this because an excessive level of details of individual scenarios could lead to unmanageable analytical complexities in practice.

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Since the emergence of the field in the late 1950s, PRA has been applied to study the failure of critical systems from diverse fields of engineering: nuclear power plants (Garrick 1984), offshore oil platforms (Paté-Cornell 1993), bridges and dams (Paté-Cornell and Tagaras 1986), chemical plants (Health & Safety Executive Canvey Island Studies 1978, 1981), ships (Drake and Kalelkar 1979), spacecrafts (Paté-Cornell and Fishbeck 1993), and airplanes (Sachon and Paté-Cornell 2000) among others. These systems are characterized as low-likelihood, high-impact (Wreathall and Nemeth 2004). Recently, PRA has also been applied to highlikelihood, high-impact systems, such as health care (Paté-Cornell et al. 1997; also see Wreathall and Nemeth 2004). Our goal is to demonstrate yet another application of this powerful tool; this time to supply chains. The fundamental idea in risk analysis is to have a comprehensive approach toward the system under study. This may often require a decomposition of the system into subsystems and of scenarios into basic events, as expertise and data are often more readily available at the subsystem or single event level. Thus, a key role of the risk analyst is to use probabilistic modeling to aggregate subsystem and events-level information into a system-level analysis, which is the core of PRA. Assume that we have constructed a mutually exclusive, collectively exhaustive set of N scenarios fsi gN i D1 . This means that only one scenario can happen at any time. Let S represent this random scenario; hence, P .S D si / represents the probability that it corresponds to si . For each possible scenario si , we estimate the probability that the value V will be less than v under this scenario for every realization v of the outcome; i.e., we estimate P .V jS D si /. Therefore, for all scenarios, the cumulative distribution function (cdf) of V is: P .V / D

N X

P .S D si / P .V jS D si /:

(5.1)

i D1

The overall goal of PRA is to identify and assess each of the terms of (5.1) in order to quantify the risk of the system; i.e., the distribution on value. Equation (5.1) relies on no assumption beyond having a set of mutually exclusive, collectively exhaustive set of scenarios. Having mutually exclusive scenarios necessitates only a careful description of the scenarios so that two scenarios cannot happen at the same time. Having collectively exhaustive scenarios (i.e., completeness), however, is a challenge, as it is often impossible, in practice, to cover the whole range of failure scenarios. Traditionally, in engineering risk analysis, one seeks to estimate the reliability of the system; i.e., the probability that it will not fail. This is equivalent to defining V as an indicator of the state of the system, being equal to 1 if the system works and 0 if it does not. Equation (5.1) represents a generalization of the concept of reliability and is a straightforward application of the total probability theorem. Once we have the cdf of V , a performance measure of common interest to planners, i.e., the probability of exceeding given loss levels, can then be calculated easily as 1 P .V ). The analytical aggregation of the terms in (5.1) is not always straightforward. Except in special cases, probability distributions do not naturally yield simple

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functional forms when mixed together. Because of the diversity of events that affect them, complex systems rarely fall in one of those special cases. Hence, numerical estimation of (5.1), for instance through Monte Carlo simulation, is a commonly used approach in PRA. Such an approach is readily available through Excel add-ins such as Crystal Ball or @Risk. The applications of PRA come with several limitations. In addition to the completeness, the subjectivity of probability distributions (e.g., lack of data to estimate event likelihood, overreliance on expert judgment, and biased estimation due to an ill-defined system) poses a major concern for decision-makers. There is the possibility of “tunnel vision,” whereby the analysis is not unbiased, but is driven by predetermined conclusions. Furthermore, PRA is rarely applied to systems in a vacuum. Human error (hence, human reliability analysis (HRA)) as well as organizational error should be an integral part of PRA (Rasmussen et al. 1987; Kirwan 1994; Reason 1997). This, however, is challenging and in practice PRA may suffer from such issues as “reduction,” i.e., “simplifying human experience” into PRA (Wreathall and Nemeth 2004). Without denying the validity of these criticisms, risk analysis is a legitimate and rigorous approach for the assessment and management of risks, especially compared to the option of ignoring risk altogether. Apostolakis (2004) remarks that ignoring uncertainties is vain, as “[u]ncertainties exist independently of whether we do a QRA [Quantitative Risk Analysis] or not,” thereby suggesting that risk analysis is a plausible and substantive alternative to relying on intuition alone.

5.3 Literature Review In recent years, risk management in supply chains has attracted significant attention. Tang (2006) provides an extensive summary and classification of the supply chain risk management literature in his review paper, which includes more than 200 citations. The author classifies supply chain risk management literature in four categories: supply management, demand management, product management, and information management. He then discusses robust strategies for mitigating operational and disruption risks in these categories. Commonly advocated mitigation strategies for supply chain risk management include multisourcing (e.g., Chopra and Sodhi 2004; Mart´ınez-de-Albéniz and Simchi-Levi 2005; Tomlin 2006; Yazlali and Erhun 2006), supply contracts (see Cachon (2003) for a review of supply contracts and Heching and King (Chap. 20) and Cheng et al. (Chap. 21) for applications of option contracts to manage demand risk), inventory policies that consider the trade-off between a risk measure and an expected return (Chen and Federgruen 2000), demand shifts across products (time/markets), such as Dell’s successful demand management following the Taiwan earthquake (Murphy 2005), postponement (Lee 1996), managing product transitions (Erhun et al. 2007), and supply chain collaboration (Chap. 8). However, one needs to be careful in interpreting the value of such mitigation strategies. For example, in

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a recent survey, Peck and Juttner (2002) confirm that many companies are indeed reducing their risk exposure by setting up contracts with suppliers. However, except when it provides true performance incentives, this does not generally result in any improvement in the global resilience of the supply chain, but rather transfers the risks to suppliers. Similarly, Davis (1993) advocates the use of inventory for risk reduction, but he recognizes that: “The real difficulty is knowing how much to hold and where to hold it, for the propagation of intrinsic uncertainties is mathematically complicated.” The research field that quantitatively addresses general issues of risk and uncertainty in supply chains is being established. Most of the literature revolves around interorganizational risk sharing, and little is said about intraorganizational actions toward reducing risk; i.e., the likelihood and impact of adverse events. There are a few exceptions. Prater et al. (2001) develop a scoring method and target international supply chains, illustrating their method with five case studies. Kleindorfer and Saad (2005) develop a conceptual framework for risk assessment and mitigation to manage risks arising from disruptions to normal activities in supply chains. This chapter contributes to this stream of literature by developing a framework based on PRA and discussing its application to a high-tech company.

5.4 A Framework for Supply Chain Risk Assessment Our goal in this section is to present a framework (Fig. 5.1), which helps decision makers in assessing and characterizing risk drivers of a supply chain.

Analytical Process

Information

Definition of System And Performance Value

Expert opinion

Tools and Techniques

Risk Identification Risk factors on influence diagram

Risk Quantification

Reassessment

Monitoring

Performance measures

Probability distributions

Risk Management

Fig. 5.1 Steps of the proposed framework

Expert opinion; statistics

Influence diagrams

Expert opinion; statistics

Simulation; probabilistic modeling Simulation; box plots; risk curves; decision analysis

Output

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5.4.1 Step 1: System Definition and Performance Value The first step of our framework constitutes the definition of the system under consideration. It includes not only the geographical and temporal boundaries of the supply chain, but also the description of related parameters, such as products, fixed and variable costs, and revenues. The advances in supply chain visibility and a trend toward increased collaboration in supply chains allow easier interactions between the planners and their partners. This, in turn, permits identification of the interdependencies among operational steps. In parallel, the goal of the supply chain has to be articulated in terms of the value measure, V , which can be multidimensional. For instance, profit can be considered as a proxy for value. Most companies, however, also consider other performance measures such as cost, market share, or leadtime2 . A model of the supply chain is then built in order to relate the operations of the diverse entities in the supply chain to the performance measures. Such a model allows evaluation of the output of the supply chain given the specification of its inputs and the conditions under which it operates (for a given scenario). In other words, the model enables the evaluation of P .V jS D si / of (5.1) for any type of scenario si .

5.4.2 Step 2: Risk Identification As our second step, we identify the uncertain events that disrupt the supply chain under study and their effect on the entities within the supply chain. This step also includes the estimation of probability distributions of basic events that comprise scenarios. Hence, it is designed to support the estimation of the probabilities of scenarios, i.e., P .S D si / of (5.1). A sample list of such events, or simply risk factors, is presented and categorized in Table 5.1 This list is nonexhaustive; hence, planners will need to adapt it to their own supply chains, which may require identification of additional risk factors. In order to simplify risk factor identification, we recommend decomposing the supply chain into five subcomponents: supply, transportation, production, storage, and demand. Risk identification should capture human-related risk factors (such as strikes, sabotage, accidents, absenteeism, and errors), infrastructure-related risk factors (such as tools and process changes, telecommunication disruptions, electricity disruptions, or even organizational accidents), and risk factors that may directly cause a disruption of any of the subcomponents (such as downtime and loss of efficiency). In addition, externalities (i.e., risk factors that are neither within the control

2 For example, Deleris et al. (2004) define the supply chain performance at the level of a firm as the number of products manufactured within a given timeframe, a scenario as a set of events that disrupts the operations of some of the production plants within a network, and then use a productmix model to measure the effect of the scenario on the value of the process to the firm.

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Table 5.1 A sample list of events and variables relevant to supply chains Category Examples Operational/Technological Forecast errors, component/material shortages, capacity constraints, quality problems, machine failure/downtime, imperfect yields, efficiency, process/product changes, property losses (due to theft, accidents, etc.), transportation risks (delays, damage from handling/transportation, re-routing, etc.), storage risks (incomplete customer order, insufficient holding space, etc.), budget overrun, emergence of a disruptive technology, contract terms (minimum and maximum limit on orders), communication/IT disruptions Social Labor shortages, loss of key personnel, strikes, accidents, absenteeism, human error, organizational error, union/labor relations, negative media coverage (reputation risk), perceived quality, coincidence of problems with holidays, fraud, sabotage, malfeasance, decreased labor productivity Natural/Hazard Fire, storm, flood, monsoon, tornado, hurricane, earthquake, epidemic Economy/Competition Interest rate fluctuation, exchange rate fluctuation, commodity price fluctuation, price and incentive wars, bankruptcy of partners, stock market collapse, global economic recession Legal/Political Liabilities, law suits, governmental incentives/restrictions, new regulations, lobbying from customer groups, instability overseas, confiscations abroad, terrorism, war, tax structures, customs risks (inspection delay, missing data on documentation)

of the firm nor always directly related to its supply chain but affect the outcome values, such as natural disasters, tax structures, or customs inspection time) have to be considered. Risk identification also implies defining the dependence relationships among the identified risk factors, their realizations (the values that they can take), and their probability distributions. To represent risk factors and their relationships to one another, we rely on an influence diagram. An influence diagram is a visual tool strictly equivalent (i.e., homomorphic) to a decision tree, which provides a high-level description of the spectrum of scenarios that lead to system failure. It displays uncertain events or random variables from the planner’s perspective (with oval nodes), decisions (with square nodes), and the value measure V (with an octagonal node) (For a sample influence diagram, please refer to Fig. 5.3). The arrows between the nodes represent probabilistic dependencies (Howard and Matheson 1984). The diagram neither represents the information about the realizations of the variables nor their probability distributions, and is completed by a table of the realizations and probability distributions of the events and variables represented by each node. Note that a time period needs to be associated with the diagram so that all marginal and joint probabilities are described for that time period. The choice of the realizations of the risk factors and the associated distributions is essential for the risk assessment and risk management steps. Indeed, the diagram alone provides only a qualitative, yet informative, description of risk.

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To direct the choice of distributions, information can be obtained from historical data, when available and relevant, from surrogate data, and also from experts’ opinions. This last source of information is especially appropriate for events that have never happened in the past. Further decomposition of risk factors can be useful. For instance, to estimate the probability of a strike at a manufacturer during any given year, one can consider factors such as the well-being of employees, the power of unions among such employees, and their propensity for negotiating versus going on strike. Once complete, the structure of the influence diagram may appear obvious to an outsider; yet, the process of building an influence diagram can be long and cumbersome because each event has to be carefully defined by its realizations and their probabilities. The factors included and those omitted in the diagram have a significant influence on the quality and relevance of the analysis, so this step needs to be addressed carefully. To structure the influence diagram, informal discussions with in-house experts (from all fields: supply, transportation, production, storage, design, marketing, and sales) followed by an appraisal of the resulting model by outside experts and third parties (e.g., CMs, insurers, suppliers, and carriers) are required. Including all these functional groups, and better yet several representatives from each functional group at different organizational levels, is crucial since it is quite unlikely that any single functional group or planner within each group possesses the information required to understand all the inherent risks of supply chains. Finally, as a sanity check, the results should be compared against global data, if available, and common sense. A central difficulty encountered in modeling supply chain risk arises from the fact that supply networks are usually complex and have a dynamic configuration. Therefore, the supply chain itself may exhibit a considerable variation within a practical analysis horizon; e.g., tier–n suppliers may change, product configurations may evolve over time, demand may shift from one region to another. The possibilities are staggering, implying that risk can be conditionally sensitive to an unmanageable number of factors. That is why steps 1 and 2 play a crucial role in our framework and should be revisited many times before effective risk quantification and management can be implemented.

5.4.3 Step 3: Risk Quantification A common approach to dealing with uncertainty in supply chains is to consider the mean and variance of a certain performance measure. However, while two supply chain structures may have the same performance in terms of expected value and variance, the tails of the loss distributions may differ by orders of magnitude, making one structure much less attractive than the other. Thus, the key is to estimate the full distribution of various performance measures, including the distributions’ tails in addition to mean and variance. The third step of our framework, the risk quantification step, aggregates the description of risks from the previous step into a

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quantitative assessment. It corresponds to the estimation of (5.1). The aggregated risk is thus described by the probability distribution of the chosen performance measure V . The estimation of the probability of a scenario, P .S D si /, is based on the aggregation of the probabilities of risk factors that comprise it, carefully accounting for the possible dependence relationships. The estimation of the value given a specific scenario, P .V jS D si /, is based on the model of the supply chain. Once these two terms of (5.1) are estimated, the risk can be characterized. Given the size and intricacies of supply chains, however, an analytical characterization may not be possible. Hence, the use of simulation rather than an analytical aggregation may be necessary, and this is the path we choose to follow.3 The overall risk is described by the probability distribution of the value to the firm, V . Because distributions are not necessarily intuitive, data visualization4 may be needed to communicate risk, including: (a) Summary statistics, including mean, standard deviation, minimum, maximum, and median of a performance measure; (b) Boxplots, which are graphical representations of summary statistics; (c) Probabilities of exceeding relevant thresholds; (d) Conditional means, representing the expected value of V for a specific set of scenarios. Once the overall risk is quantified through the probability distribution of V; the factors that are critical to the performance of the firm need to be identified. To capture the weak links of the supply chain, we use scenario analysis, which involves obtaining the distributions of the performance measures conditional on chosen variables or events. The comparison of these distributions with the base case thus brings additional insights into the behavior of the system, the current bottlenecks, and the effectiveness of risk management options by yielding an understanding of the main contributors of risk.

5.4.4 Step 4: Risk Management The results of the simulation provide a risk measure of the supply chain per time unit as captured by probability distributions of the chosen value measure V . This information can then be used to identify management decisions and actions, which represents the last step of our framework. First, risk reduction alternatives need to be identified and analyzed. Second, managers have to decide whether or not the benefits justify the cost of switching to each of these alternatives. 3 Others have followed the same path. For instance, Hicks (1999) describes a four-step method based on both simulation and optimization aimed at supply chain strategic planning. In his method, simulation is used to describe the dynamic behavior of a given supply chain structure and to assess the benefits of supply chain policies, such as inventory policies. Ingalls (1999) describes a simulation-based tool for supply chain analysis implemented at Compaq, which incorporates demand forecast errors. 4 Research in the domain of data visualization has been ongoing in the field of statistics, notably with Tukey (1977) and Tufte (1983).

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Table 5.2 Qualitative comparison of mitigation approaches Risk reduction strategy Options/Redundancy Reinforcement Diversification Shortcoming

Coupled systems

Risk Coupled systems displacement

Supply chain example

Building extra production capacity

Investing in upscale IT servers

Warning systems

False positive and false negative Splitting production Strike monitoring and partner’s (freight) financial between two health manufacturing monitoring sites (carriers)

Planners can address the problem of risk reduction either at the supply chain structure level (e.g., by adding a second supplier for a single-sourced subassembly) or at the component level (e.g., by working with the current supplier to improve his/her processes). As displayed in Table 5.2, we identify four broad families of mitigation approaches designed to reduce supply side risk (i.e., supply, transportation, production, and storage-related risks). Options are understood here in the financial sense of real options. Their aim is to secure the (partial) availability of the supply chain as a backup in case of disruption. In this context, one can consider adding redundancy as a full-scale option. Planners should ensure that their option will be available should an adverse event occur. Therefore, the option has to be decoupled from the adverse events that can cause the disruption of the regular supply chain. Reinforcement increases the robustness of a system, making it less vulnerable to adverse events. For example, preventive maintenance is a way to keep the system operational by preventing the deterioration of its elements. While using reinforcement, however, one should consider the possibility of risk displacement. For instance, one can create temporary capacity problems by improving the timeliness of delivery of components. Diversification increases the reliability of a supply chain by decoupling some of the chain’s elements, hence limiting the occurrence and magnitude of an adverse event to a portion of the supply chain. Finally, the objective of warning systems is to provide planners with leadtime so that they can react to the potential of adverse events. Effective supply chain risk management is generally achieved by combining these approaches into a coherent set. These modifications affect either the probability distribution of the uncertain factors or the impact of those uncertainties on the performance measure (and sometimes both). So far we have not mentioned the use of financial instruments, such as insurance, as a risk mitigation strategy. Financial instruments ensure that the firm is partially compensated if the supply chain is interrupted, but do not create a more robust or resilient supply chain. Such financial mitigation tools are thus especially effective to protect the firm from catastrophic events (i.e., the tail of the loss distribution) and they are part of supply chain risk management. By effectively addressing supply chain risk at the system level, a company is more likely to obtain a better insurance coverage.

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The risk mitigation strategies that we discussed above would lead to alternative supply chain structures, the decision among which is equivalent to a comparison of probability distributions. This task is especially difficult when V is multidimensional; thus, in PRA, it is customary to rely on expected utility decision analysis (e.g., Raiffa 1968), which is a normative framework based on the von Neumann– Morgenstern axioms of rationality (von Neumann and Morgenstern 1947). These axioms lead to choosing the alternative that maximizes the expected utility of the decision maker. Utility, in contrast to value, captures the risk preferences of the decision maker. Often utility is evaluated as a function of the value measure V . The main challenge of this approach is the need for explicit elicitation of a utility function. Although single-attribute utility functions are commonly elicited, multiattribute utility functions can be difficult to assess (Keeney and Raiffa 1976). Beyond using expected utility decision analysis, supply chain planners may rely on a simple comparison of a few chosen statistics from the probability distribution of V to help them make their decisions, as we discuss more through the case study.

5.5 Case Study of Pilot Application The proposed framework was developed, in part, in collaboration with a Silicon Valley based high-tech company. In order to produce several hundred stock keeping units (SKUs), the company operates a global supply network, which spans manufacturing sites in Europe, Asia, and North America. Prior to our collaboration, the company built a tool that helped planners configure alternative supply chain structures by estimating performance measures such as cost, leadtime, and inventory for a given supply chain structure. Those measures were based on expected values; thus planners could not evaluate the extent to which two solutions led to different levels of variability in performance. The company was ready to advance this tool further by incorporating risk. In particular, managers at the company were concerned by the overall vulnerability of their supply network, notably in terms of its strategic development: Is the geographical structure of the supply network adequate, or is it too concentrated in some regions of the world, such as Southeast Asia? Is the number of business partners that comprise the supply network appropriate, or should it be increased or reduced? Where does the risk reside in the supply network? Would an alternative supply network structure better serve the company’s purposes? Figure 5.2 displays the part of the supply chain for Product Family A (PFA) that we focus on in the case study. This supply chain is composed of the following three processes: sourcing, subassembly, and transportation to the final assembly location. After manufacturing the components and assembling them in Southeast Asia, the company ships the subassemblies to the USA, where the final assembly and sales and marketing take place. The performance measures chosen by the company are leadtime (reflecting customer satisfaction) and cost per unit (reflecting financial performance). Under ideal conditions, the total leadtime is 8.5 days and the total cost per unit is $435.10. The goal of the risk analysis in this case study is to obtain the distribution of cost and leadtime.


Subassembly

Transportation Customs

Sourcing

117

Components

Customs

5

Ground

Subassembly

Transportation to subassembly site Asia

Legend

Air / Ocean

Transportation to assembly site North America

Fig. 5.2 The supply chain structure for PFA

We define leadtime, denoted by VL , as the time elapsed between product order and its arrival at the assembly site. Thus, the total leadtime is the sum of the sourcing leadtime, the subassembly leadtime, the transportation leadtime, and an additional variable that represents the delays incurred when a disaster forces the company to use an alternative location or route: VL D Lsourcing C Lsubassembly C Ltransport C Ldisaster :

(5.2)

Total cost per unit, denoted by VC , includes material cost, manufacturing cost, and transportation cost: VC D Csourcing C Csubassembly C Ctransport :

(5.3)

Note that VC does not have a component for disasters; i.e., the additional costs brought to the company by disasters are estimated separately. This is because cost and time measures are not symmetric in the way they affect firms in times of disasters. Insurance contracts tend to cover the financial losses associated with catastrophes, but rarely (or imperfectly) compensate firms for the associated loss of customer goodwill. Also note that although inventory and overhead costs are a crucial part of total costs, in this pilot study the company’s managers chose not to consider them. The inclusion of these costs would imply allocating them across all products, which can be achieved by estimating all losses at an aggregate level for a given time period and renormalizing the outputs based on the number of units produced during that same time period. Figure 5.3 displays the output of the risk identification task (Step 2) at the company. After several meetings with the company’s supply chain planners, we identified several risk factors that affect PFA’s supply chain. In order to simplify the presentation and analysis, we identify two subcomponents of the supply chain: sourcing and subassembly (which includes risks associated with manufacturing at the subassembly site and sourcing-related risks) and transportation (which includes risks associated with transportation and storage activities, including customs). Some events, such as natural disasters and geopolitical instability, may affect the supply chain as a whole and are represented as externalities.

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Component Sourcing Problems

Forecast Variability Electronic Data Interchange Outages

Leadtime and Cost Sourcing

Contract Manufacturer Management Qualifications Contract Manufacturer Capacity

Fiscal Quarter Tools and Process Changes

Leadtime and Cost Sub-Assembly Yield Variability

Contract Manufacturer Bankruptcy

Time of Year

Engineering Change Orders

Beginning of Peak Season Monsoon Season

Leadtime and Cost per unit

Carrier Rate Damage and Loss

Rebuild Decision

Destination Delay

Import Clearance Time Hurricane Season

Leadtime and Cost Transportation

Destination Trucking Arrival Delay Origin Delay

Origin Trucking Export Clearance Time Natural Disaster Labor Strike

Leadtime Disaster

Geo-political Instability

Fig. 5.3 Influence diagram for the supply chain performance of PFA (We removed the decision nodes (square nodes) to improve the readability of the diagram. In this case, the company’s decisions were limited to the choice of subassembly location.)

At first, Fig. 5.3 may appear difficult to decipher. The best way to read the diagram is to go from right to left, from the performance of the supply chain to the events that affect those performance measures. An influence diagram visually resembles a root cause diagram (fishbone diagrams). Arrows, however, do not necessarily imply causality but indicate that the likelihood of the state of a child node is influenced by the state of its parents’ nodes. Furthermore, the underlying information supporting influence diagrams and root cause diagrams is not similar: in the influence diagram, probability distributions have to be associated with each node. Figure 5.3’s value resides in the fact that all the risk factors that affect the supply chain performance are represented on the same visual support. The diagram underlines common root factors that affect several activities at once. In the case of the company, seemingly unrelated problems were in fact correlated through seasonality (i.e., “Time of Year” node at the leftmost end of the figure). Such a diagram serves as an effective communication tool across different supply chain functions, often organized as silos and unaware of the ripple effects of their problems and decisions on the performance of the supply chain as a whole. We briefly describe some of the nodes that appear on the diagram. Several nodes, such as yield variability for the subassembly process, origin trucking, customs clearance time and destination trucking for the transportation process, represent the operational variability of supply chain processes. We incorporate the Asian monsoons

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because they affect the probability of packaging damage and increase both leadtime and cost. We also incorporate production forecast accuracy because underestimation causes insufficient capacity at the subassembly site, which eventually leads to either delay or increased costs. We account for the manufacturing peak season in Asia (generally August to October), mainly because of its effect on transportation costs from Asia to North America. Finally, to better model component shortage, we differentiate between commodity and complex (sole-sourced) components. Given the short timeframe of the analysis, we assume that some external risk factors, such as the state of the worldwide economy, political stability, and the characteristics of the product do not change significantly. For instance, we consider political disruptions such as war, revolution, political unrest, but we assume that the probability of such an event is constant over the time horizon. Similarly, we assume that component costs are stable. These would not be valid assumptions had we looked at a longer time horizon. Over time, probabilities of external events change and component costs tend to decrease in the high-tech industry. As this was the company’s first attempt to use PRA, the major roadblock was the estimation of the probabilities of the events in the influence diagram. Even though past data can be used to estimate the probabilities of specific events (e.g., forecast errors are usually good candidates for the use of historical data), in our case study, time and budget constraints did not allow us to gather data. Therefore, except for a few base case parameters (e.g., carrier rate and transformation leadtime), we relied on expert opinion to estimate the probabilities. After numerous discussions with planners, we concluded that using simple distributions, such as triangular distribution, binomial distribution, or beta distribution, would be a plausible first step toward establishing a probability database for the company. The choice of triangular distribution is motivated by the ease of assessment. A triangular distribution is fully specified by only three parameters: the minimum value, the most likely value, and the maximum value of the variable. We found those three parameters to be intuitive to supply chain experts. Most of the delays were represented either by triangular distributions, or when the shape of the triangular distribution was inappropriate to reflect the variance of the variables, through scaled beta distributions. A beta distribution is useful because it allows for a variety of different shapes, thereby lending itself to sensitivity analysis. A binomial distribution is the natural distribution for events that may either occur or not (Boolean events). Punctual events, such as the occurrence of damage, sourcing problems, EDI outages, natural disasters, and CM bankruptcy were all represented by binomial distributions. Finally, some risk factors (such as the quality of management or the accuracy of forecasts) were represented by discrete distributions with three degrees: low-base-high. We do not suggest that those should be the only families of distributions to be used in supply chain risk analyses. Those were the ones that we used for practical purposes in this case study. As the modeling of inputs relies heavily on the choice of parameter values, we performed a sensitivity analysis on the parameters in order to assess the extent to which the results were dependent on the modeling choices and to calibrate the model inputs. Tables A.1 and A.2 (in the Appendix) report the changes in average cost and

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leadtime from varying parameter values to ˙50% of their base values. Table A.1 presents the results for cost and leadtime-related parameters and Table A.2 pertains to probability parameters for events that have binomial distribution. Material and manufacturing costs are the most sensitive parameters, which should be expected, since these costs account for a large part of the total cost. These results, however, are not of serious concern, since these parameters are based on actual data from the company and a deviation of ˙50% is quite unlikely. The situation is similar for time-related parameters. Transit time and manufacturing time (together representing more than 50% of the average leadtime) lead to significant changes in the results but, again, they are based on data from the company and should not vary so widely. Overall, the sensitivity analysis shows that our results are not very sensitive to probability estimations. Only one parameter (the probability that the component for which there are sourcing problems is complex) results in noticeable (although small) variation in leadtime and cost. Thus, the distribution and its parameters for this particular event should be under close observation and should be among the first candidates for a detailed probability estimation analysis. For the PFA supply chain, we implemented the model described by (5.2) and (5.3) in Excel, and we further relied on @Risk to obtain the distribution of leadtime and cost based on the description of the risk factors presented in the previous step. We repeated the simulation 20,000 times (each repetition corresponds to a single order for a product in PFA) with different initial seeds to eliminate the possible benign, or extreme, effect of any individual seed value. Specifically, for each repetition of the simulation, the conditions under which this order is processed, manufactured, and transported are sampled based on the distributions of the risk factors, accounting for dependencies. When all the risk factors are attributed a value (whether they occur or not and their effect if they occur), we can compute cost and leadtime for that order. Figure 5.4 reports the resulting marginal distributions and summary statistics for leadtime and cost per unit. Observe that both distributions are very tight, suggesting that the two performance measures vary only slightly around their base value (i.e., 8.5 days for leadtime and $435.10 for per unit cost), the distribution of leadtime being comparatively wider than the cost per unit distribution. For the cost distribution, the variance is small but the maximum value is large: about $860 per unit. This means that few unlikely events affect the cost, but when they occur, their effect is significant. Although not extremely visible on the graph, the probability that cost per unit exceeds $435.10/unit (which is the ideal unit cost of the system) is 10%, and that it exceeds $730 is 1%. In terms of leadtime, the company used to quote 11 days for that part of the supply chain. This quote is in fact quite conservative, as a large proportion of the shipment takes less than 10 days to arrive. Note also that the probability that leadtime exceeds 11 days is 19.5%, and that it exceeds 13 days is 14%. Figure 5.4 represents only marginal distributions. However, an analysis of the results reveals that the two performance measures are highly correlated with a correlation coefficient of 0.76. Figure 5.5 displays the scenario analysis that we performed in order to identify the weak points of the supply chain for PFA. To that end, we generate six classes

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Distribution for Leadtime

Distribution for Cost

0.300

0.060

Mean=448.2149

Mean=10.96072 0.250

0.050

0.200

0.040

0.150

0.030

0.100

0.020

0.050

0.010

0.000

0.000

0

20

40

60

80

350

460

Leadtime in days

Leadtime (in days) Transportation Production Cost per unit (in $) Transportation Material Manufacturing

570

680

790

Unit Cost in Dollars

Minimum 7.61 4.56 3 398.02 10.1 342.57 45

Median 8.48 5.31 3 435.1 10.1 380 45

Maximum 78.97 13.76 58.5 860.71 11.87 786.81 90

Mean 10.96 5.37 5.55 448.21 10.27 392.8 45.01

Std. Dev. 5.99 0.48 5.91 53.10 0.28 52.76 0.81

Fig. 5.4 Marginal distributions and summary statistics of leadtime and cost per unit

Cost

Leadtime

No Tools or Process Changes

No Underforecasts

No Sourcing Problems

Complex Sourcing Problems

Mediocre Management

Disaster

Base Case 10

20

30

40 Values

50

Fig. 5.5 Scenario analysis for leadtime and cost

60

70

400

500

600 700 Values

800

900

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of scenarios. The first three represent an increased “load” on the system: (1) disasters (natural and geopolitical) (2) management-related problems, and (3) complex sourcing problems. The last three represent favorable conditions: (4) no sourcing problems, (5) no underforecasts, and (6) no tools or process changes. The box plots correspond to the conditional distributions; i.e., for each scenario, they display the results of the runs where the respective scenario is realized. The most obvious result from the box plots shown in Fig. 5.5 is that complex sourcing problems greatly affect both of the performance measures by introducing volatility and shifting the distributions to the right. What is less striking, yet informative, is that tool and process changes have no significant effect on the leadtime. This conclusion is supported by the fact that the base case distribution and the distribution for the “no tool and process changes” scenario are very similar. In addition, we observe that the variability of the leadtime is reduced when there are neither sourcing nor forecasting problems. A surprising observation is that the maximum leadtime for scenarios where disasters occur is smaller than the maximum leadtime in the base case. This observation weakly suggests that disasters are not the main source of the extremely long leadtimes in our case study and is consistent with practice. Disasters usually trigger BCBs [Business Continuity Plans] that have well-defined worst-case action procedures.5 Having said that, we caution the reader to be careful in interpreting the maximum values, which are very sensitive to the sampling process. In the case where the maximum values are important for the analysis, it is crucial that the simulation study be repeated many times before a conclusion can be reached. The company identified an alternative supply chain structure for its product, as shown in Fig. 5.6. This alternative structure simply entails replacing the

Transportation

Components

Customs

Subassembly

Customs

Sourcing

Legend Ground

Subassembly

Air / Ocean

Transportation to subassembly site

Transportation to assembly site North America

Fig. 5.6 Description of an alternative supply chain structure for PFA

5

In the base case, the maximum leadtime occurs as a result of mediocre management faced with a significant shortage of a complex component at the time of engineering changes on the subassembly site. In the disasters scenario, the maximum leadtime is achieved due to some engineering changes on the lines and a large labor strike.

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Table 5.3 Comparison of alternative & current supply chain structures for PFA (Positive values reflect an increase moving from the current to the alternative structure) Alternative – Current Minimum Average Maximum Standard deviation Difference in leadtime 56.08 2.30 61.88 7.67 (in days) Difference in unit cost 386.13 42.45 512.40 74.04 (in dollars)

Table 5.4 Comparing two performance measures of alternative and current supply chain structures for PFA: Percentage of scenarios in each of the four categories Cost Alternative > Current Alternative Current Leadtime Alternative > Current 13.5% (9.4 days and $127) 0.09% (3.89 days but $101) Alternative Current 80.31% (2:8 days but $42.5) 6.1% (21 days and $159)

subassembly in Southeast Asia with a subassembly in North America (not in the USA) in order to further reduce the leadtime variability. A plant in North America, while exposing the supply chain to different natural disasters, also implies more reactive subassembly, smoother customs transits, and shorter transportation legs, thus constituting reinforcement over the current supply chain structure. Table 5.3 displays the summary statistics of the differences in performance measures for the current and alternative supply chain structures. The decision involves a cost versus leadtime trade-off. We quantify the trade-off as follows: the alternative to the current option is 2.3 days shorter on average, but the average cost increases by $42.45/unit. Because the value measure V is bidimensional, the comparison of the distributions is cognitively difficult. In order to make the planners’ decision process easier, Table 5.4 provides yet another comparison between the two supply chain structures. We divide the system performance into four categories: (a) the alternative structure is better on both accounts, (b) the current structure is better on both accounts, (c) the alternative structure has a shorter leadtime but higher costs, and (d) the current structure has a shorter leadtime but higher costs. We characterize the trade-off in each category in terms of average leadtime difference and average cost difference. These statistics are useful to gauge the robustness of the decision. Depending on the planner’s objectives and risk tolerance, joint probabilities on two performance measures can be calculated to compare the alternative structures under extreme events and to make the comparison even more robust. For example, the planner may choose to compare between the two supply chain structures the probability that the leadtime will exceed 14 days and cost will exceed $750. We note that such comparisons should be accompanied by statistical tests to make sure that the differences are significant.

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5.6 Summary and Lessons Learned from the Case Study When we started our pilot study, risk analysis was a new activity for the company. Therefore, at times we were constrained in our modeling by the lack of precise and consistent information about the behavior of the supply chain. However, we laid the foundation for the framework, which is likely to foster future implementations. Despite apparent complexities, the framework enriched the company’s decisionmaking process, hence, the company has chosen to use the method presented here and has implemented it. Currently, they are working on developing tools for the integration of risk assessment and management into their day-to-day supply chain management. The process will then be used to include risk as a criterion in supply chain design. The supply chain manager of the company stated that this research “was extremely valuable in creating a foundation for [the company’s] approach to supply chain risk management, and is well on its way to becoming integrated in [the company’s] normal operating processes.” The results of the analysis show that the risk of a unit cost increase is quite limited as unit cost is protected through contracts with both the carriers and the suppliers. The company’s managers initially thought that cost, as defined here, would be one of the key factors. Unit cost, however, does not by itself constitute an adequate, insightful performance measure because it does not include all cost components, such as inventory-related costs. Managers should carefully define meaningful performance measures that incorporate the hidden costs within their supply chain (e.g., insurance, inventory). When quantifying the variability in leadtime, we observed that different actors within the corporation had opposite perceptions. While the planner responsible for transportation described a very controlled process where delays seldom occur (and when they occur, are of short duration), another planner felt that transportation delays were plaguing the performance of their supply chain. This observation is consistent with a general observation of the management of complex systems: various people have access to different portions of information and risk perception discrepancies often occur.6 A team of actors with different levels of responsibility needs to be in charge of the risk analysis, from the identification of risk factors to probability assessments. Such a process not only generates more realistic estimates, but also provides a forum for communication, which enhances the validity and the value of the method. Through our pilot study, the managers were able to attach a cost to the companygenerated risk factors, such as forecast accuracy. This information supports a more constructive and well founded discussion among different units in charge of supply

6

Risk perception is quite critical, especially when reliance on expert opinion is necessary due to lack of data to estimate the event probabilities. For example, a recent cover story of Time magazine explores Americans’ faulty risk perceptions (Time, November 26, 2006). Although we have not discussed it in this chapter, there is a prolific literature on risk perception. For a quick reference, we refer the reader to Slovic (1987).

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chain management. Furthermore, planners thought that the orders of magnitude of both the likelihood of the occurrence and the severity of some adverse events are as important to them as a precise quantification of the risk. Thus, from the managers’ perspective, the outcome of a risk assessment process should be to know where the major elements of the risk reside within the supply chain and their orders of magnitude, rather than numerical precision on the probability of losses.

5.7 Conclusion and Future Research Directions To the best of our knowledge, few modeling frameworks that quantitatively assess risks are available to guide and ease the task of planners in their analysis of integrated supply chains. Most of the current modeling approaches rarely allow for partial failures in subsystems. In this chapter, we provide a systematic approach that seeks to enhance this process. This research constitutes a further exploration of the application of decision and risk analysis methods to supply chain operations. The suggested framework permits identification of points of friction in day-today operations, such as recurrent delays in transportation, customs, or component sourcing. These types of inefficiencies are seldom addressed by management, as they are perceived as normal parts of doing business. Yet, operational variability is as much a component of risk as are catastrophic events. Our analysis emphasizes that risk analysis is not solely about managing disasters, for which most firms are already prepared, but also about operational variability. While we describe our framework in general terms, the application that we present is limited to a part of the supply chain of a single product. An important aspect of the analysis of risk in supply chains is to understand the interactions among facilities and among products. There are no technical difficulties in applying the framework to a more complex supply chain with several products. For instance, Deleris and Erhun (2005) have examined the risk of a supply network with 5 products and 26 locations in four echelons (3 critical supply sites, 5 subassembly sites, 9 assembly sites, and 9 distribution centers). However, for a larger scale implementation, keeping the same level of detail requires a significant amount of time in terms of modeling the supply chain, identifying the risk factors, as well as determining the probabilities. We caution the reader that keeping the analysis manageable with a spreadsheet may necessitate a change in the granularity of the framework and the scope of the supply chain analyzed. It is apparent from our description that the analysis of risks in supply chains is still an art rather than a systematic process. In recent years, there have been several efforts in the direction of categorizing risk factors and providing checklists to supply chain planners. The next step in that direction would be to provide supply chain decision makers with informational support about the probability distributions of risk factors. One approach is to rely on a database of such risk factors, which would, in addition, ensure consistency across the company about the various risk

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assessments performed. This database of risk factors should represent a collective effort by experts in relevant areas and should not solely be a matter of data collection and processing. It should also provide a global qualitative and quantitative description of those factors. To be truly valuable, this database would need to be updated on a regular basis. One of the possible limitations of such a database is to account for dependencies among the risk factors. While theoretically possible, conditional probability tables grow exponentially with the number of conditioning variables and may therefore be difficult to encode and manipulate. At a more conceptual level, there is also a need to understand whether some structures (be they in terms of facility structure or in terms of product routing) are simply more robust than others by design, regardless of the load that they take. Risk analysis provides the probability of a value vector V . The difficulty is then in making use of this information for efficient decision support. This reveals two other directions for further research related to supply chain risk management. The first is how to provide supply chain decision makers with tools to efficiently compare several joint probability distributions. This partially entails determining the tradeoffs among performance measures such as cost, leadtime, and inventory levels, or in other words, building efficient risk measures. The second direction is about the communication of the risk to higher levels of management; i.e., how to convey the results of the analysis to decision makers who may be unaccustomed or unwilling to consider probability distributions. It is critical to remember during this exercise that risk is in the eye of the beholder. Different companies, even different planners within a company, may have different tolerances to risk. Therefore, risk communication should be flexible enough to enable these different parties to have effective input in the decision process. Studies show that materials managers routinely spend a third to half their time trying to resolve supply chain disruptions. Forty percent of managers resolve lastminute supply issues based on habit and gut feel rather than through local or corporate-level business objectives (“Cost of Supply Chain Disruptions Estimated in The Six-to-Seven Figure Range,” Supply & Demand Chain Executive, November 9, 2004.). Therefore, it is important to empower managers with the ability to understand risk drivers and the disruptive role that they play in supply chains. Decision support tools to that effect will be invaluable. Acknowledgements This study was funded in part by NSF grant #NSF/CAREER-0547021. We gratefully acknowledge partial support from GM/SU Collaborative Lab and the company in the case study. To protect the company, all data and information provided here are either publicly available or have been sufficiently disguised without removing the essence of the situation and the results. We thank all parties at the company for allowing us to use this information. We also acknowledge the constructive comments provided by Debra Elkins (GM) and Professor Elisabeth Paté-Cornell (Department of Management Science and Engineering at Stanford University). Last, but not least, we are indebted to four anonymous referees whose suggestions improved the chapter considerably.

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Appendix Table A.1 Sensitivity analysis for cost parameters Parameter Base material costa Base manufacturing costa Carrier base ratea Cost of origin trucking leg Cost of destination trucking leg Cost of export customs clearance Cost of import customs clearance Cost multiplier from bankruptcy of CM Percentage increase in transportation cost during peak season Duration of export customs clearance Duration of origin trucking leg Transit timea Duration of import customs clearance Duration of destination trucking leg Time increase from rebuilt Maximum effect (time multiplier) of mediocre management Delay caused by simple component sourcing problems Delay caused by bankruptcy of CM Delay caused by political instability in Southeast Asia Delay caused by political instability in the USA Delay caused by a disaster in Southeast Asia given only transportation is disrupted Delay caused by a disaster in Southeast Asia given plant destroyed Delay caused by natural disaster in the USA Manufacturing timea Note: a denotes parameters based on data

Cost Low 43.8% 5:0% 0:5% 0:2% 0:2% 0:1% 0:1% 0.0%

High 43.8% 5.0% 0.5% 0.2% 0.2% 0.1% 0.1% 0.0%

Time Low 0% 0% 0% 0% 0% 0% 0% 0%

High 0% 0% 0% 0% 0% 0% 0% 0%

0.15

0.0%

0.0%

0%

0%

0.25 day

0.0%

0.0%

1:1%

1.1%

1 day 3 days 0.25 day

0.0% 0.0% 0.0%

0.0% 0.0% 0.0%

4:3% 13:2% 1:1%

4.3% 13.2% 1.1%

1 day 5 days 1.5

0.0% 0.0% 0.0%

0.0% 0.0% 0.0%

4:4% 0:1% 0:7%

4.4% 0.1% 0.7%

2 days

0.0%

0.0%

0:8%

0.8%

15 days 27 days

0.0% 0.0%

0.0% 0.0%

0.0% 0.0%

0.0% 0.0%

2 days

0.0%

0.0%

0.0%

0.0%

2 days

0.0%

0.0%

0.0%

0.0%

27 days

0.0%

0.0%

0.0%

0.0%

2 days

0.0%

0.0%

0.0%

0.0%

3 days

0.0%

0.0%

13:7%

13.7%

Base value $380/unit $45/unit $4.1/unit $2/unit $2/unit $1/unit $1/unit 2

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Table A.2 Sensitivity analysis for probability parameters Parameter Probability of inspection at export clearance Probability of inspection at import clearance Probability of transportation damage during monsoon season Probability of transportation damage outside of monsoon season Probability of rebuild given damage Probability of mediocre management quality Probability of overforecast Probability of accurate forecast Probability of incurring a sourcing problem when forecasts are underestimated and there are no EDI problems Probability of incurring a sourcing problem when forecasts are not underestimated and there are no EDI problems Probability of incurring a sourcing problem when forecasts are underestimated and there are EDI problems Probability of incurring a sourcing problem when forecasts are not under estimated and there are EDI problems Probability of EDI problems Probability that the component affected is “complex” given a sourcing problem Probability of engineering change order during manufacturing Probability of tools and process changes during manufacturing Probability of manufacturing capacity issues during quarters 1–2–3 Probability of manufacturing capacity issues during quarter 4 Probability of yield problems during manufacturing Probability of bankruptcy of CM Probability of political instability in Southeast Asia Probability of political instability in the USA Probability of natural disaster in Southeast Asia Probability of destruction of plant given disaster in Southeast Asia Probability of disruption of transportation system given disaster in Southeast Asia Probability of natural disaster in the USA Probability of labor strike in the transportation industry

Cost Low 0.0% 0.0% 0.0%

High 0.0% 0.0% 0.0%

Time Low 0.0% 0:1% 0.0%

High 0.0% 0.1% 0.1%

0.0%

0.0%

0.0%

0.0%

0.2 0.1 0.2 0.2 0.2

0.0% 0.0% 0.2% 0.2% 1:0%

0.0% 0.0% 0:2% 0:2% 0.9%

0:1% 0:2% 1.4% 1.4% 4:8%

0.1% 0.3% 1:4% 1:4% 4.5%

0.1

0:3%

0.3%

1:5%

1.6%

0.5

0:2%

0.1%

0:6%

0.6%

0.1

0.0%

0.0%

0.0%

0.0%

0:1% 1:5%

0.0% 1.4%

0:3% 5:9%

0.1% 5.7%

0.05

0.0%

0.0%

1:8%

1.7%

0.03

0.0%

0.0%

1:2%

1.4%

0.05

0.0%

0.0%

0:2%

0.3%

0.5

0.0%

0.0%

0:8%

0.8%

0.003614

0.0%

0.0%

0:2%

0.2%

1/50 years 1/50 years

0.0% 0.0%

0.0% 0.0%

0.0% 0.0%

0.0% 0.0%


0.0% 0.0%

0.0% 0.0%

0.0% 0:1%

0.0% 0.0%

0.2

0.0%

0.0%

0.0%

0.0%

0.3

0.0%

0.0%

0.0%

0.0%


0.0% 0.0%

0.0% 0.0%

0.0% 0.0%

0.0% 0.0%

Base value 0.005 0.02 0.04 0.01

0.038776 0.4

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¨ Erhun F (2006) Managing demand uncertainty with dual supply contracts. Working Yazlali O, paper, Department of Management Science and Engineering, Stanford University, Stanford, CA 94305 Zsidisin GA, Panelli A, Upton R (2000) Purchasing organization involvement in risk assessments, contingency plans, and risk management: an exploratory study. Supply Chain Manag Int J 5(4):187–197

Chapter 6

A Practical Multi-Echelon Inventory Model with Semiconductor Manufacturing Application Kaan Katircioglu and Guillermo Gallego

6.1 Introduction Semiconductor manufacturing is an operationally complex, financially capital intensive business. While companies try to keep up with technology, they try to manage their operations effectively by increasing their capacity utilization, improving manufacturing yields, and reducing cycle times and inventory levels. Manufacturing cycle times can be quite long depending on the technological complexity of wafer designs. Yields can be unpredictable, particularly during the initial phases of manufacturing new designs. Due to speedy introduction of new designs, product lifecycles tend to be short making demand highly volatile. The lean manufacturing practices of their customers make life even more difficult for semiconductor manufacturers because such practices require suppliers to provide very high levels of inventory availability. In addition, the significant fixed costs incurred in building semiconductor plants require high capacity utilization to keep average unit costs down. The desire to keep minimal excess capacity and provide high inventory availability to customers makes it difficult to keep inventory levels low. All these issues combined result in a serious supply and demand planning challenge for semiconductor manufacturers. In such difficult situations, the risk of inventory obsolescence can be significant. Although semiconductor manufacturing involves many complex iterative operations, at a high level there are four major stages in the manufacturing process (see Fig. 6.1). These are fabrication, wafer test, packaging, and testing. Wafer fabrication operation makes silicon-based wafers that have a few hundred chips on them. This process may take several weeks and varies by wafer design. Wafer test involves a wafer cutting and testing operation that can take a few days and transforms wafers into individual chips. Packaging essentially consists of bond and assembly operations. In this final stage, chips are wired and bonded to substrates. Finally, packaged chips go through a testing operation. Packaging and testing operations can each

K. Katircioglu () IBM T. J. Watson Research Center, Yorktown Heights, NY 10598, USA e-mail: [email protected]


133

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K. Katircioglu and G. Gallego

Wafer Stock

Testing

Packaging

Wafer Test

Fabrication

Device Stock

Module Stock

Tested Module Stock

Fig. 6.1 A simplified view of major stages in semiconductor manufacturing

take a few days, and sometimes over a week depending on various factors. Packaged and tested chips are usually called modules and are essentially end products ready for use in various applications. Not all modules can perform up to the intended original design performance level. A good manufacturing operation must keep the fraction of well performing modules high. Because the yields and cycle times in these operations are uncertain, manufacturing and inventory planning have to be done very carefully to prevent stock-outs. There are numerous published books on semiconductor manufacturing processes. Van Zant (2000) and Wolf (2004) can be cited among the relatively new ones that provide a concise view. There are several papers that address the practical business and operations issues in semiconductor manufacturing in more detail. Collins (2003), Kilgore et al. (2002), Kodama (2003), Lee (2001), and Navas (2003) can be cited among such work. Modeling a semiconductor manufacturing operation clearly requires a multiechelon perspective. There is extensive work in the inventory control literature on multi-echelon models since the very early years of the field. The work of Clark and Scarf (1960) is one of the earliest milestones of modeling and solving multiechelon problems. Among the key contributions since then are Federgruen and Zipkin (1984), Muckstadt (1973), Muckstadt et al. (1980), Sherbrooke (1986), Rosling (1989), and Schwartz and Schrage (1975), and Chen and Song (2001). More recently, there is some work on developing simple heuristics to solve multi¨ echelon problems including Shang and Song (2003) and Gallego and Ozer (2005). Gallego et al. (2007) have useful bounds on distribution systems. The work of Van Houtum and Zijm (1991) and Van Houtum et al. (1996) are among recent work worth mentioning. There are several books that focus on multi-echelon models that can be referenced as well. Some examples are the very early classical reference book by Hadley and Whitin (1963), the early work by Schwartz (1981), later work by Graves et al. (1993) and Sherbrooke (1992), and more recent work by Gershwin et al. (2003). Many multi-echelon models emphasize minimizing the sum of inventory carrying and backlogging costs. In practice, the use of service constraints is very common due to difficulty in measuring backlogging costs. Among some studies that address multi-echelon problems with service constraints are Rosenbaum (1981), Ettl et al. (2000), Simchi-Levi (2004), and Cohen et al. (1988). For multi-echelon systems, see also Boyaci and Gallego (2001) and Boyaci et al. (2003).

6

A Practical Multi-Echelon Inventory Model

135

There are several applications that employ some form of multi-echelon models in solving semiconductor manufacturing problems. Some examples are Brown et al. (2000), (2001), Braun et al. (2003), Chong et al. (2004), Gallego et al. (2002), (2005), Kapoor et al. (2005), Lendermann et al. (2003), Lin et al. (2002), Lyon et al. (2001), Piplani and Puah (2004), Suman and Harker (2004), Wang et al. (2003), (2004), Ovacik and Weng (1995), and Xu et al. (2003). Much of the published work assumes yields are either not an issue, or that they are deterministic so that the adjustments to manufacturing quantities are trivial. A review of production planning and scheduling models can be found in Uzsoy et al. (1992). Another review is provided in Yano (1995). In many practical manufacturing problems, yields are uncertain. In such cases, adjustments to manufacturing quantities are not simple and a clever modification of optimal basestock levels is needed. There is much fewer work addressing random yields. Among those are Bitran and Leong (1992a), Bitran and Dasu (1992), and Bitran and Leong (1992b). Here, we are interested in developing a practical model that can be used to calculate basestock policies in all stages of a series of manufacturing operations when manufacturing operations are subject to random yields and a service constraint has to be met at the end stage. Our objective is to seek a heuristic solution that can be calculated easily and has good performance so that it can be put to use in many practical problems. We will start with simple assumptions and show how the solution can be extended to more realistic cases where manufacturing operations are subject to random yields and random cycle times.

6.2 A Multi-Echelon Model with a Service Constraint Consider a manufacturing operation that consists of n stages in series and at the end of each stage some inventory is held. We will index the stages of the manufacturing such that j D 1 represents the first stage and j D n represents the last stage, which receives and fulfills the customer demand. The manufacturing operation in stage j takes lj amount of time, and it has a random yield of Yj . All inventory in stage j costs hj per unit per period and we have h1 < h2 < ::: < hn . We will denote customer demand in period t by Dt and assume that they are i:i:d . We will denote the total demand from period t1 to period t2 by DŒt1 ; t2 if it is inclusive of demand in period t2 and use DŒt1 ; t2 / if it is not inclusive. The objective is to minimize long run average inventory holding costs subject to a probability of no-stockout constraint in the last stage. This service constraint applies strictly in each period. The inventory in each stage is managed by using an echelon basestock policy. We will denote the echelon basestock policy in stage j by sj . The way an echelon basestock policy works is as follows. We monitor the echelon inventory position at stage j (i.e., all inventory in stage j and later stages C total quantity being manufactured in stage j – backlogged customer demand) for every j . Whenever the echelon inventory position drops below sj , we trigger an order to bring it up to sj .

136


For every echelon base stock policy, there is an equivalent local base stock policy defined by Sj D sj sj C1 ; (6.1) where snC1 D 0. Now we will derive the inventory cost in each stage and service constraint in stage n. Note that in stage j , an inventory replenishment action will impact customer service no earlier than lj C C ln periods later since this is the remaining lead time to complete the manufacturing of a product in stage j . Therefore, planning in period t requires that we meet the service constraint in period t C l1 C C ln C 1 through inventory policies applied in all stages of manufacturing starting in period t. Let us assume without loss of generality that echelon inventory position in period t in stage j is exactly sj . Let us define tj D t C l1 C C lj for j D 1; ::; n; and t0 D t: Total demand during periods tj 1 through tj 1 is DŒtj 1 ; tj /, which is essentially demand during lead time lj , starting in period tj 1 . Note that the DŒtj 1 ; tj /’s represent demands over nonoverlapping time intervals, so if period demands are independent then so are demands over different lead times. At the beginning of period tj 1 , the echelon basestock policy dictates that we bring the echelon inventory position in stage j to sj . This requires adequate amount of manufacturing starts in stage j . However, due to insufficient processed inventory in stage j 1, the supplier to stage j , we may not be able to start manufacturing the required amount. We will denote this shortage in stage j 1 by Bj 1 . Another way of looking at Bj 1 is that it is the backlogged manufacturing requests from stage j to stage j 1. Due to this shortage, we can only bring the echelon inventory position in stage j to sj Bj 1 . This is equivalent to bringing local inventory position to Sj Bj 1 . Now, lj periods later at the beginning of period tj all work-in-process inventory in stage j is completed and added to processed inventory. In other words, the entire local inventory becomes available to satisfy the next stage’s needs. During this time, we also receive a total customer demand of DŒtj 1 ; tj /. Then, at the beginning of period tj , the inventory on hand in stage j will be Ij D .Sj Bj 1 DŒtj 1 ; tj //C :

(6.2)

Similarly, at the beginning of period tj , the backlog in stage j is Bj D .Bj 1 C DŒtj 1 ; tj / Sj /C :

(6.3)

Here, B0 D 0 since the outside supplier to stage 1 is assumed to have infinite supply. In summary, we want to implement an echelon basestock policy sj in stage j but due to possible supply shortage of Bj 1 in stage j 1, we are forced to implement

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137

an echelon basestock policy of sj Bj 1 , which is random. In effect we end up implementing a randomized basestock policy. This complication does not exist in a single echelon problem with infinite supply. From (6.2), the inventory remaining in stage j at the end of period tj is .Sj Bj 1 DŒtj 1 ; tj /C : The total inventory cost in all stages as a function of local basestock policies is C.S1 ; S2 ; : : : ; Sn / D

n X

hj .Sj Bj 1 DŒtj 1 ; tj /C :

j D1

The customer service (i.e., the probability of no stock outs in stage n in period tn ) is given by P fDŒtn1 ; tn C Bn1 Sn g: We want to find the set of local basestock policies S1 ; : : : ; Sn that minimize the expected cost in the system and achieve a no stock-out probability of at least ˛. That is, min EC.S1 ; : : : ; Sn / D

S1 ;:::;Sn

s.t.

Pn

j D1

hj E.Sj Bj 1 DŒtj 1 ; tj /C

(6.4)

Bj D .Bj 1 C DŒtj 1 ; tj / Sj /C ; j D 1; : : : ; n P fDŒtn1 ; tn C Bn1 Sn g ˛:

Note that due to the nested structure of Bj ’s the objective function does not have a closed-form expression in general. In addition, the no stock-out probability in the service constraint depends on basestock policies in all stages. These make the problem difficult to solve.

6.2.1 A Simple Heuristic ¨ Gallego and Ozer (2005) developed a heuristic for calculating near-optimal echelon basestock levels in all stages when the unmet customer demand is fully backlogged at a cost of p per unit per period and there is no service constraint. They use a key ratio to calculate the echelon basestock policy at stage j. This ratio is given by rj D

n X kDj

hk lk

X n

lk :

kDj

Then, for stage j they suggest using an echelon basestock policy sj that satisfies P fDŒtj 1 ; tn sj g D .p C hj 1 /=.p C rj /:

(6.5)

138


We will make use of this heuristic by converting our service constrained problem to a problem with backlogging cost and no constraints. If the last stage was the only stage in manufacturing, then the optimal solution to the problem with service constraint would be a basestock policy that satisfies P fDŒtn1 ; tn Sn g D ˛: Similarly, the optimal solution to the problem with holding and backlogging costs but no service constraint would be P fDŒtn1 ; tn Sn g D p=.p C hn /: Now, if we define an implied backlogging cost by p D ˛hn =.1 ˛/

(6.6)

both solutions are equivalent. We will use this implied backlogging cost and adopt (6.5) as a heuristic solution to the service constrained problem in (6.4). From (6.5) and (6.6), our echelon basestock policy in stage j satisfies P fDŒtj 1 ; tn sj g D

˛hn C .1 ˛/hj 1 : ˛hn C .1 ˛/rj

(6.7)

The above basestock policy is easy to calculate. However, semiconductor manufacturing operations are subject to random manufacturing yields. In the next section, we need to adjust the above formula to reflect random yields.

6.3 Modified Heuristic for Random Yields When there are uncertain yields in manufacturing, there is no guarantee that inventory positions can be replenished to the planned basestock levels. Thus, there needs to be additional safety stock that can provide protection against such uncertainty. Now, we will show a way of adjusting the echelon basestock levels for random yields. Typically, yields apply to lots in manufacturing. A unit of inventory being processed in any stage is subject to the cumulative yields of all subsequent stages until manufacturing is completed. For instance, a unit being processed in stage j will be subject to random yields of Yj ; Yj C1 ; : : : ; Yn . Similarly, a unit already processed in stage j and sitting in inventory will be subject to random yields Yj C1; : : : ; Yn until it completes its manufacturing. Let us define cumulative yield by YOj D

n Y kDj

Yk :

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139

We assume Yj ’s are independent random variables. The following formulae can be used to calculate the expected value and variance of cumulative yields. E.YOj / D

n Y

E.Yk /;

(6.8)

kDj

Var.YOj / D Var.YOj C1 /.Var.Yj / C E2 .Yj // C E2 .YOj C1 /Var.Yj /;

(6.9)

where Var.YOn / D Var.Yn /. This assumption may not be valid in manufacturing systems where yield performance at a stage depends on previous stages. However, yields in the distinct stages of semiconductor manufacturing we focus on here (i.e., fabrication, wafer testing, packaging and final testing) are mainly determined by the precision of the equipment involved in their own processes unless there are chip design weaknesses that create pervasive yield problems throughout all stages in manufacturing. In such cases, yields will be correlated and the above formulae will require updating. Now, we need to calculate the variance of total supply in stages j through n. Note from (6.2) and (6.3) that due to potential backlogs not all the supply in inventory position sj can reach to the last stage by time tj . Some inventory may still remain in process since each stage triggers replenishment orders according to their own basestock policies. To simplify things, we will assume that all of the inventory in echelon stock sj will be available to satisfy demand in period tj . We just need to calculate the variability of this inventory due to uncertain yields. When echelon basestock policy sj D Sj C C Sn is implemented, different quantities are subject to different cumulative yields. Work-in process inventory in stage j is subject to a cumulative yield of YOj , whereas inventory on hand in stage j is subject to YOj C1 . To simplify calculations, we will assume that all the inventory in local basestock Sj is subject to a cumulative yield of YOj . This will slightly overestimate the total variance of supply, and hence the safety stocks. When calculating the supply variances, it is important to express lot sizes in all stages in gross amounts (i.e., unadjusted for yields). Let Qj be the fixed lot size in stage j . When we release a lot of size Qj in to stage j we expect to receive E.YOj /Qj at the end of stage n, the last stage. Let Nj be the total number of lots in period t in stage j . This is a random number with expected value E.Nj / D

Sj : O E.Yj /Qj

Therefore, when all lots that are being processed in stage j in period t are added to the finished goods inventory lj C C ln periods later, their contribution to the total net finished goods inventory will be Sj D

Nj X i D1

YOij Qj :

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Here for all i D 1; :::; Nj , we assume YOij ’s are independent and identically distributed random variables equivalent in distribution to YOj . The expected value and variance of S j are E.S j / D E.Nj /E.YOj /Qj D Sj ; Var.S j / D Var.Nj /E2 .YOj /Qj2 C E.Nj /Var.YOj /Qj2 : If the independence assumption is questionable as discussed earlier, the variance calculation can be modified using covariances. For new products, the identical distribution assumption is reasonable only within short periods of time. As the manufacturing process for the new product matures, there will be a shift in distribution since yields are expected to improve due to the increasing learning curve. When manufacturing process becomes mature, the learning curve becomes more flat and yield distributions become more steady. The number of lots can be random for a number of reasons. The integer multiples of lot size Qj may not always bring the inventory position to the desired basestock level forcing us to add one more lot than usual; there may not be enough supply to start manufacturing the desired number of lots; or for various reasons (such as machine breakdowns, order priorities) manufacturing might delay releasing lots. If manufacturing plans are executed regularly with no significant delays in lot releases, and daily demand volume is high enough making lot sizes small compared to basestock levels, this randomness will be small. We assume that the impact of this variance to the variance of S j is negligible. So, ignoring the first term in the variance of Nj , we will use E.S j / D Sj ; Var.S j / ' Sj Qj

Var.YOj / ; E.YOj /

where E.YOj / and Var.YOj / are given in (6.8) and (6.9). The above was the contribution of stage j inventory to total finished goods inventory lj C C ln periods later. We also need to consider the contribution of subsequent stages. When they are all added, the total contribution to finished goods inventory net of cumulative yields will be sj D

n X

Sk:

kDj

Note that S k ’s are independent random variables since they all belong to nonoverlapping intervals and we assumed that the yield in each period is independent of yields in other periods. Therefore, we have E.sj / D sj ; n X Var.YOk / : Sk Q k Var.sj / D E.YOk / kDj

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Now, we need to calculate the echelon basestock levels that account for random yields. We replace sj by sj in (6.7) since sj no longer represents the actual inventory available, which is given by the random variable sj . That is, P fDŒtj 1 ; tn sj g D

˛hn C .1 ˛/hj 1 : ˛hn C .1 ˛/rj

This is equivalent to P fZj sj g D where Zj is given by

˛hn C .1 ˛/hj 1 ; ˛hn C .1 ˛/rj

(6.10)

Zj D DŒtj 1 ; tn sj C sj :

We assume yields and demands are independent of each other, which is a reasonable assumption (although in some cases high demand levels might cause strains on manufacturing, thus impacting yields). Then, we have E.Zj / D E.DŒtj 1 ; tn /; Var.Zj / D Var.DŒtj 1 ; tn / C Var.sj /: Note that when yields are not random (6.10) is equivalent to (6.7). Since the variance of Zj depends on Sj through Sn , solving (6.10) for sj is not straightforward. As a practical approach, we will use the Sj ’s that are obtained by solving (6.7) to calculate the variance of Zj . Then, we can solve (6.10) for sj easily. Since the Zj ’s are sums of a number of random variables, a normal approximation can be used, especially in cases where manufacturing lead times are long.

6.3.1 Implementation of Basestock Policies To implement the basestock policies discussed above, we need to calculate the number of lots to release in each stage of manufacturing in each period. Since we chose to measure basestock policies net of cumulative yields, we assume that inventory position IPj is also measured net of cumulative yields. Let Ij and Wj represent the processed inventory and work-in-process inventory amounts, respectively, in stage j at the beginning of period t just after lots are released to manufacturing. Then, at the end of period t, the echelon inventory position in stage j will be IPj D

n X kDj

Ik C

n X kDj

W k Dt :

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The echelon basestock policy, sj , dictates that we release lots in to manufacturing in stage j so as to bring IPj to sj . However, if the minimum lot size is greater than 1, we may not be able to achieve sj exactly. Therefore, we assume that the inventory replenishments are done so as to bring the echelon inventory position IPj to a minimum possible level at or above sj . This ensures that the service target is met. In other words, the number of lots to release to manufacturing at the beginning of period t C 1 must be n n oo mj D max 0; roundup .sj IPj /=.Qj E.YOj // ; where the function roundupfxg gives the integer part of real number x C 1. Although the above number of lots ensures meeting the service target, applying it strictly each period will overachieve the service target in every single period. The alternative is to use a number that achieves sj as close as possible from either above or below. This way we can still achieve the service target on average, although not in every period. The number of lots in this case will be n n oo mj D max 0; rounddown .sj IPj /=.Qj E.YOj // C 0:5 ; where the function rounddownfxg gives the integer part of real number x. If acceptable, the latter quantity is recommended because it will give lower inventory levels. In the following two sections, we will comment on hot to use the heuristic when lead times are random, and when demands and yields are nonstationary random variables. Then, in Sect. 6.4, we will present our test results for the heuristic.

6.3.2 Random Lead Times The model formulation we used so far does not allow random lead times. However, the solution can still be used as an approximation provided that we make an adjustment for the mean and the variance of the demand during lead time in stage j , and use the same solution as a practical and simple alternative. If demand in each period is iid, the following expressions can be used to calculate the mean and variance of the demand during lead time: 0 E.DŒtj 1 ; tn / D @

n X kDj

Var.DŒtj 1 ; tn / D

n X kDj

1 E.lk / C 1A E.Dt /; 0

Var.lk /E2 .Dt / C @

n X

kDj

1 E.lk / C 1A Var.Dt /:

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6.3.3 Nonstationary Demand and Yields For problems with nonstationary demand, the model can be used one period at a time if service constraint is applied to each period strictly as we discussed. This is because the problem for each period is separable when the service constraint applies to every single period strictly. In this case, the demand mean and variance of the lead time demand will have to be updated in each period for each stage since their distribution will be changing. In addition, since the basestock policies will also be changing, the replenishment quantities will have to reflect changes in the basestock policies as well as compensating for inventory consumptions by the arrival of customer demands. Sometimes the optimal basestock levels may have to decrease from one period to another. This may happen when one or more of the demand mean or variance decreases. In such cases, inventory position in a period may be higher than the optimal basestock level, which implies that no manufacturing should be done in that period. Furthermore, since service targets will also be overachieved in such cases, one may want to lower the service targets for subsequent periods to bring the average service to the target level. This can be useful thing to do, particularly toward the end of product lifecycle, where the risk of inventory obsolescence is high. Nonstationary yields can also be modeled using the approach we have here. Since we took an arbitrary period t as a reference and built the formulation, the formulations can be used as they are provided that service constraint applies to each period strictly. One caveat, of course, is that the yields belonging to different periods will have different distributions. The supply variance calculations must be modified to take this into account in every period.

6.4 A Numerical Study and Practical Considerations We conducted a numerical study to test the performance of the heuristic. Table 6.1 shows the input data for the example problem we created. The range of input values covers typical semiconductor manufacturing operations.

Table 6.1 Input parameters for the numerical study Parameter Medium Service target (probability of no stock-outs) Lead time (fabrication) in days Lead time (wafer test) in days Lead time (bond and assembly) in days Lead time (module test) in days Demand variability (std/mean) Yield variability (std) Yield mean

0:95 25 5 6 7 0:6 0:10 0:70

Min

Max

0:79 10 2 2 2 0:1 0:00 0:45

0:99 40 8 10 12 1:1 0:20 0:95

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We assumed that the values added in each stage of the manufacturing are $5, $3, $2, $1 for fabrication (1), wafer test (2), bond and assembly (3), module test (4) operations, respectively. This implies that the holding costs used in each stage are h1 D 5h, h2 D 8h, h3 D 10h, h4 D 11h, where h is the holding cost rate for a dollar worth of inventory. We analyzed the following six different cases: 1. Set all parameters to their medium values and vary service target between the minimum and maximum (see Fig. 6.2a). 2. Set all lead times to their minimum values and all other parameters to their medium values. Vary service target between minimum and maximum (see Fig. 6.2b). 3. Set all lead times to their maximum values and all others to their medium values. Vary service target between minimum and maximum (see Fig. 6.2c). 4. Set all parameters to their medium values and vary demand variability between the minimum and maximum (see Fig. 6.3). 5. Set all parameters to their medium values and vary yield variability between the minimum and maximum (see Fig. 6.4). 6. Set demand variability and yield variability to their maximum values and all others to their medium values. Vary service target between minimum and maximum (see Fig. 6.5). For each case, the results can be seen in the respective figures. Service levels are calculated using simulation. The minimum and maximum service levels in the figures refer to 95% confidence interval for sample mean. Although in some cases the target service falls outside of the confidence intervals, a great majority of the cases are satisfactory. Service Performance (MEDIUM lead times) 100% 99% 98% 97% 96% 95% 94% 93% 92% 91% 90% 89% 88% 87% 86% 85% 84% 83% 82% 81% 80% 79%

81%

83%

85%

87%

89%

91%

Service Target

93%

95%

97%

99%

Minimum Service Average Service Maximum Service Service target

Fig. 6.2 (a) Service (i.e., probability of on time delivery) performance of the heuristic as a function of service target for average manufacturing lead times

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Service Performance (MIN lead times) 100% 99% 98% 97% 96% 95% 94% 93% 92% 91% 90% 89% 88% 87% 86% 85% 84% 83% 82% 81% 80% 79%

81%

83%

85%

87%

89%

91%

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Service Target

Service Performance (MAX lead times) 100% 99% 98% 97% 96% 95% 94% 93% 92% 91% 90% 89% 88% 87% 86% 85% 84% 83% 82% 81% 80% 79%

81%

83%

85%

87%

89%

Service Target

91%

93%

95%

97%

99%


Fig. 6.2 (continued) (b) Service (i.e., probability of on time delivery) performance of modified heuristic as a function of service target for short manufacturing lead times. (c) Service (i.e., probability of on time delivery) performance of modified heuristic as a function of service target for long manufacturing lead times

It is clear from Fig. 6.2a–c that the longer the lead time the wider the confidence intervals for the service performance. Nevertheless, the average service performance still fluctuates around the target in all cases. This suggests that as the lead times get longer the service performance of the heuristic does not get systemically

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100% 99% 98% 97% 96% 95% 94% 93% 92% 91% 90% 89% 88% 87% 86% 85% 84% 83% 82% 81% 80% 10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

110%


Demand Variability

Fig. 6.3 Service (i.e., probability of on time delivery) performance of modified heuristic as a function of demand variability Service Performance 100% 99% 98% 97% 96% 95% 94% 93% 92% 91% 90% 89% 88% 87% 86% 85% 84% 83% 82% 81% 80% 0%

2%

4%

6%

8%

10%

12%

Yield Variability

14%

16%

18%

20%


Fig. 6.4 Service (i.e., probability of on time delivery) performance of modified heuristic as a function of yield variability

away from the target service levels. It is also evident that as the service target gets higher, the length of the confidence interval tends to decrease. In today’s competitive market conditions as many manufacturers feel more pressure to maintain high service targets, it is reassuring to see that the heuristic delivers more accurate service

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Service Performance (MAX demand variability, MAX yield variability) 100% 99% 98% 97% 96% 95% 94% 93% 92% 91% 90% 89% 88% 87% 86% 85% 84% 83% 82% 81% 80% 79%

81%

83%

85%

87%

89%

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91%

93%

95%

97%

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Fig. 6.5 Service (i.e., probability of on time delivery) performance of modified heuristic as a function of service target for high demand variability and high yield variability

performance when service targets are high. Figure 6.5 shows that this is consistently so even when we have highly uncertain demand and manufacturing yields show high variations. Figure 6.3 shows that demand variability does not impact the service performance of the heuristics with any observable statistical significance. It is interesting to see that for low demand variability, the heuristic may slightly lower service performance than the target. On the contrary, it is pleasing to see that it performs well under highly uncertain demand. Similarly, Fig. 6.4 indicates that the level of yield variability does not impact the performance, although it is possible to observe a slight underachievement of service target (with no statistical significance) when yield uncertainty increases above 10%. One important issue to note is the lot size. In all these cases we listed above, we used a lot size (net of cumulative yields) equal to mean daily demand in all stages. The performance of the heuristic is still good when we use lot sizes of up to 3 days worth of demand. For larger lot sizes, performance starts deteriorating. In problems with high demand volumes where the lot sizes are not more than a few days of demand, the heuristic can be used safely. The performance should also depend on the inventory holding costs in each stage. We reversed the values added in stages and used $1, $2, $3, $5, respectively. The performance was similar. We must mention that some of the assumptions we made in deriving the heuristic can be relaxed. For instance if the demands in different periods are not independent or identically distributed, one can calculate the variance of the lead time demand by using covariances. Similarly, if the yields of different manufacturing lots are not

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independent or identically distributed, then the yield variance of all lots in a period can be calculated using yield covariances of the lots in that period. Another practical consideration is that when the yield and demand distributions are not stationary, then the base stock policies recommended by the heuristic should be updated frequently – every period if necessary. In some cases, updated base stock policies can be lower than the current inventory position. When this happens, since it is not possible to lower the inventory position, one can simply stop introducing new lots to the manufacturing line until inventory position comes below the recommended base stock level. Inability to bring inventory position down does not harm service performance but rather overachieves it by keeping more inventory in the system than desired. If the demands and or the yields do not show extraordinary fluctuations, such cases should not happen very often. Finally, manufacturing lead times can often be decreasing since many companies constantly strive to increase the speed of manufacturing and increase asset utilization based on the best practices such as lean manufacturing and six sigma. In such cases, although the heuristic can be used to calculate the base stock levels, it is not smart enough to modify the base stock levels based on the anticipated reductions in lead times. This is because it is myopic; it does not use any information beyond the total manufacturing lead time. If the changes in manufacturing do not increase the uncertainty in the system (since unexpected things can happen with new programs) shorter lead times will generally require lower base stock levels. If the lead time reductions happen gradually and slowly as they become visible to the heuristic, it can lower the safety stock levels with acceptable speed if updated often enough. However, if the lead time reductions happen abruptly – due to sudden changes such as putting different manufacturing practices in place, or introducing additional manufacturing capacity on a given date – the heuristic will not be able to react fast. In such cases, one practical approach can be to run the heuristic based on the anticipated reduced lead times, and start adjusting the base stock levels gradually downward. If the lead times are expected to increase for some reason – such as scheduled equipment maintenance or demand exceeding capacity hence causing manufacturing backlogs – the same approach can be used to adjust safety stocks upwards.

6.5 Conclusions The model presented here can be implemented in semiconductor manufacturing and similar manufacturing operations, where random yields play an important role in inventory planning. In some cases, there may be additional complexity in the manufacturing operation. For instance, there may be capacity limitations. In such cases, multiple periods will need to be solved numerically to allow building inventory ahead of anticipated capacity shortages. This may make it hard to use simple heuristic solutions. However, numerical methods can be used to solve the underlying dynamic programming problem.

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In practice, companies also use different measures of customer service such as fill rate or on-time delivery to customer request date. For such constraints, the basic constructs of the model we have here can still be used. However, one needs to develop and test different heuristics as ours is developed specifically for the probability of no stock-out constraint. In addition, in some manufacturing operations (in packaging operation for instance), there may be other components that are assembled together. Including the manufacturing operations of these components in the model will introduce an assembly node to the problem. In such cases, the ideas that we have here can be used provided that the assembly node is handled with care. Rosling (1989) has a method of how to convert problems with assembly operations to problems with stages in series. This method can be used first to convert the problem, then solution developed here can be implemented. The use of build-to-order practices is very common in semiconductor industry especially for custom products for understandable reasons. Such policies can be handled using the method we presented. Basestock policies with negative safety stocks can effectively be used to model build-to-order practices. One way of looking at build-to-order practice is that it is a model where advanced information about the ¨ demand is available. Further work on this can be found in Gallego and Ozer (2003) and (2001).

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Chapter 7

A Mechanism Design Approach for Decentralized Supply Chain Formation Dinesh Garg, Y. Narahari, Earnest Foster, Devadatta Kulkarni, and Jeffrey D. Tew

In this chapter, we describe a category of supply chain formation problems where the supply chain planner or the Central Design Authority (CDA) is faced with the decision of choosing a partner or service provider for each supply chain stage so as to meet delivery targets and schedules at minimum cost. We first look into the case, where the CDA has access to all the relevant information required to solve this problem. Such a supply chain formation problem with complete information becomes a plain optimization problem in a centralized framework. Since it is quite impractical for the CDA to have access to all the information, we next consider the incomplete information case. In this setting, the individual managers of the supply chain stages are not loyal to the CDA but are rational, intelligent, and autonomous entities always pursuing maximization of their individual payoffs and not necessarily revealing their true private values. The supply chain formation problem now becomes a mechanism design problem followed by an optimization problem. Our specific contribution is to show that Vickrey–Clarke–Groves (VCG) mechanisms provide a natural and compelling model for such problems. We propose a decentralized framework to solve the underlying mechanism design problem. We illustrate our approach with the help of an example of forming a three stage distribution process for a typical automotive supply chain.

7.1 The Supply Chain Formation Problem Supply chains provide the backbone for manufacturing, service, and e-business companies. The supply chain is a complex, composite business process comprising a hierarchy of value-delivering business processes. In today’s global supply chain environment, very few firms have complete control over the entire supply chain. Even in cases where the firm owns significant portions of its entire supply chain, different

D. Garg () Department of Computer Science and Automation, Indian Institute of Science, Bangalore, India e-mail: [email protected]


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segments are likely to be owned by different organizations. This is because in recent years, the business world has realized that business efficiency can be improved and risk can be minimized if the firms focus on building their core competencies, outsource the peripheral business processes, and partner with companies possessing complementary strengths. This puts the companies in a better competitive position. This paradigm shift in business strategy has given rise to the formation of supply chain networks and emergence of intermediaries, such as third-party logistics providers, contract manufacturers, and electronic marketplaces in almost all industries. Thus, a modern supply chain network can be viewed as a collection of independent companies, possessing complementary skills and integrated with streamlined material, information, and financial flows that work together to meet market demand. Many of these networks are controlled by Original Equipment Manufacturers (OEMs) or channel masters, who own the brand of the end product and select manufacturing and logistics partners in the supply chain network based on characteristics, such as market requirements, the capabilities, efficiency, reliability, delivery time, location of the partner, and the total cost of order fulfillment. It is no longer enough to be merely the best-of-breed manufacturer or contract manufacturer, it is also critical to partner with best-of-breed companies for other supply chain functions, such as component manufacturing, logistics, maintenance, testing, etc (Viswanadham and Gaonkar 2003). In view of this situation, selecting the appropriate partner for each supply chain stage, which is not owned by the firm itself is an extremely important strategic decision making problem for modern industrial supply chains. At an abstract level, we can define the supply chain formation problem as a problem of determining the best mix of partners for each supply chain stage, which is not owned by the firm itself (Walsh and Wellman 2003). This problem occurs more frequently in industries where the product lifecycle is very short, and the underlying supply chain is dynamic in nature which requires its configuration to be changed more frequently. The IT supply chain is a very good example of a dynamic supply chain. Manufacturing firms also face the problem of supply chain formation but less frequently than IT firms. The supply chain formation problem has many avatars depending upon the nature of the industry, business goals and objectives, and constraints such as market demand, delivery time, budget, etc. In this chapter, we will be focusing on a special class of supply chain formation problems where the decision facing the supply chain planner (or the CDA) is to choose a partner for each supply chain stage that ensures delivery targets and schedules to be met at minimum cost.

7.2 Centralized Versus Decentralized Supply Chain Formation We first note that any strategic, tactical, and operational decision-making problem in the realm of supply chains can be classified into two categories. The first category includes the decision-making problems, where the decision maker has

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access to all the information required to make the decision. We call such problems decision-making problems with complete information. The other category includes decision-making problems, where the information that is required to make the decision is not available at a central place but is distributed across the supply chain. We call such problems decision-making problems with incomplete information. The second category of decision-making problems can further be divided into two subcategories. The first subcategory corresponds to problems where it is possible for the decision maker to gather all the required information by just making a request to the individual entities holding different pieces of this information. In other words, these are the problems where although the information is distributed across the supply chain, the entities that own various pieces of the information, are loyal to the decision maker. The second subcategory includes the problems where the entities that are holding the various pieces of information, are rational and intelligent agents who understand the value of this information and hence strive to get the maximum mileage out of it. Thus, we have the following three categories in which we can classify any given strategic, tactical, and operational decision making problem: 1. Decision-making problems with complete information. 2. Decision-making problems with incomplete information. (a) With loyal entities. (b) With rational and intelligent entities. The supply chain formation problem with complete information and the supply chain formation problem with incomplete information but loyal entities can both be cast as optimization problems. We call such problems centralized supply chain formation problems. However, the supply chain formation problem with incomplete information and rational and intelligent entities is a much more difficult problem. We call such problems decentralized supply chain formation problems. These problems can be cast in a game theoretic setting. In this chapter, we deal with the decentralized supply chain formation problems and develop a game theoretic framework to solve the problems, based on the theory of VCG mechanisms.

7.3 Related Work We can classify the relevant literature on the supply chain formation problem into two categories. The first category includes papers, which make an implicit assumption of complete information, that is, the supply chain planner or decision maker has access to all the relevant information required to solve the problem. The second category includes papers, which assume incomplete information, that is supply chain planner does not have access to all the information. Our contribution falls into the second category.

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7.3.1 Complete Information Case There is a significant amount of literature on forming two-stage supply chain networks. This literature basically focuses on the problem of supplier selection faced by the manufacturer. In all this literature, the implicit assumption is that the manufacturer has all the information regarding suppliers, which is required to select optimal suppliers. Thus, they basically formulate the supplier selection problem as an optimization problem. Weber and Current (1993) discuss a multi-criteria analysis for vendor selection. They develop a model for minimizing total cost, late deliveries, and supply rejection, given the infrastructure constraints and constraints imposed by the company’s policy. Pan (1989) presents a simple Linear Programming (LP) model that can be used to determine optimal order quantities among suppliers subject to specific quality, lead time, and service requirements from the buyer. Chaudhry et al. (1993) consider the problem of vendor selection where buyers need to choose order quantities with vendors in a multi-sourcing network. Narsimhan and Stoynoff (1986) present a model for optimizing aggregate procurement allocation, keeping in mind contract requirements, supplier capacities, and economic manufacturing quantity-related constraints. The interested reader might find Weber et al. (1991) useful for a comprehensive classification of publications on vendor selection criteria. Recently, Viswanadham and Gaonkar (2003) have developed a mixed-integer programming model for integrated partner selection and scheduling in an Internetenabled dynamic manufacturing network environment. Once again the authors have assumed that all stakeholders in the supply chain (SC) share information on their capacities, schedules, and cost structures. Garg et al. (2004; 2006) have developed a four-step approach to solve the supply chain partner selection problem for six sigma supply chains.

7.3.2 Incomplete Information Case The available literature in this category can be summarized as follows. Walsh (2001), Walsh and Wellman (2003), and Walsh et al. (2000) address a

particular form of the supply chain formation problem. They consider the supply chain environment where the partners are rational and autonomous. They view the supply chain as a directed, acyclic graph, .V; E/. They define V D G [ A, where G is the set of goods and A D C [ ˘ is the set of agents, composed of consumers C , and producers ˘ . Edges, E, connect agents with goods they can use or provide. There exists an edge < g; a > from g 2 G to a 2 A when agent a can make use of one unit of g, and an edge < a; g > when a can provide one unit of g. When an agent can use or provide multiple units of a good, separately indexed edges represent each unit. A consumer c wishes to acquire one unit of a

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particular good and obtains value vc for doing so. A producer can produce a single unit of an output good conditional on acquiring some fixed set of zero or more input goods. must acquire each of its inputs, and in addition incurs cost to provide its output. For every subgraph of this network, the authors define the value of the subgraph as sum of value of all the consumers in the subgraph minus the sum of production costs of all the producers in the subgraph. The objective is to find a maximum value subgraph in which flow of material satisfy the balance conditions. They have proposed a market equilibrium-based protocol for finding such a subgraph in a decentralized fashion. Karabuk and Wu (2005) have addressed a different version of the supply chain formation problem, namely the capacity planning allocation problem at a major U.S. semiconductor manufacturer. In their model, there is a Product Manager (PM), who is responsible for a certain product and privately owns the demand information through regular interaction with the customers. Capacity-allocation is carried out by the corporate headquarters (HQ), which allocates manufacturing capacity to product lines based on demand information reported by the PMs. They show that PMs have an incentive to manipulate demand information to increase their expected allocation, and that a carefully designed coordination mechanism is essential for HQ to implement the optimal allocation. To this end, they design an incentive scheme through bonus payments and participation charges, which elicits private demand information from the PMs. Golany and Rothblum (2006) argue that decentralized decision-making in supply chain management is quite common, and often inevitable. For example, decentralized supply chain formation is essentially a decentralized decision-making problem. A typical decentralized way of decision-making is known to result in inefficient Nash equilibrium outcomes, and optimal outcomes that maximize the sum of the utilities of all agents need not be Nash equilibria. The authors demonstrate through several examples of supply chain models how linear reward/penalty schemes can be implemented so that a given optimal solution becomes a Nash equilibrium. Fan et al. (2003) propose a new approach in developing a supply chain information system with a decentralized decision-making process. In particular, they study the incentive structure in the decentralized organization and design a market-based coordination system that is incentive aligned, i.e., it gives the participants the incentives to act in a manner that is beneficial to the overall system. They also prove that the system monotonically improves the overall organizational performance and is goal congruent. The common thread that runs across all these papers is the assumption that entities in a typical supply chain environment are rational, intelligent, and autonomous. Each of these papers starts with an overall system-wide objective, which the CDA wishes to achieve. The objectives of the individual entities of the supply chain do not necessarily match with the system-wide objective. Therefore, the rational entities try to mislead the CDA, say, for example, by reporting untruthful information to

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the central authority. In each of these papers, such a problem is solved by suggesting an incentive scheme for the supply chain entities, which would ensure that the entities report their true values.

7.4 Contributions and Outline of the Chapter In this chapter, we focus on the following supply chain formation problem with incomplete information: The supply chain planner or the CDA is faced with the decision of choosing a partner or service provider for each supply chain stage so as to meet delivery targets and schedules at minimum cost. The individual managers of the supply chain stages are rational, intelligent, and autonomous entities always pursuing maximization of their individual payoffs without necessarily revealing their true private values. We formulate this supply chain formation problem as a mechanism design problem followed by an optimization problem. Our specific contributions include: To show that the Groves mechanisms (Mas-Colell et al. 1995) (more popularly

called as VCG mechanisms) would provide a natural and compelling mechanism design framework for such problems. To propose a decentralized framework to solve the underlying mechanism design problem. To illustrate the efficacy of the proposed approach through a case study of forming a three-stage automotive distribution network. The sequence in which we progress in this chapter is as follows. In Sect. 7.5, we describe a simple example to illustrate the supply chain formation

problem that we are interested in. In Sect. 7.6, we first suggest three dimensions along which this simple example

can be generalized to make the problem more realistic. Next, we discuss the generalization along the first two dimensions. In Sect. 7.7, we show that the aforementioned supply chain formation problem can be cast as an optimization problem and propose a centralized framework to solve the underlying optimization problem. This centralized framework relies on the assumption that all the entities in the supply chain are loyal. In Sect. 7.8, we argue that in practice, the loyalty assumption is unlikely to be true and entities in today’s global supply chain network can be naturally modeled as rational, intelligent, and autonomous agents who try to maximize their own payoffs. In Sect. 7.9, we show that the supply chain formation problem induces a noncooperative game with incomplete information among the individual stage managers of the supply chain and, therefore, the supply chain formation problem can be modeled as a mechanism design problem (Cachon and Netesssine 2005; Fan et al. 2003; Garg et al. 2005). In particular, we show that the VCG mechanism

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provides a natural and appealing way to model the supply chain formation problems. Motivated by this, we propose a decentralized framework to solve the underlying mechanism design problem. In Sect. 7.10, we illustrate the proposed decentralized framework with the help of an example of forming a three-stage vehicle distribution chain at a major automotive manufacturer. In Sect. 7.11, we conclude the paper by summarizing its contributions and providing several avenues for future work.

7.5 A Simple Example of the Supply Chain Formation Problem In this section, we describe a simple example of the supply chain formation problem. The example basically captures the Order-to-Deliver (OTD) process at General Motors (GM) in North America. Figure 7.1 provides a high level view of the OTD process at GM North America. The OTD process starts with receipt of the customer

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Fig. 7.1 Vehicle Order-to-Delivery (OTD) process at General Motors (GM), North America

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orders and ends with delivering the vehicles to the customers. GM has a dedicated subunit, known as the OTD department, which acts as an interfacing entity among Vehicle Sales, Service, and Marketing (VSSM), Manufacturing, and Outbound Logistics. The role of the VSSM department is to gather different kinds of customer orders which include orders from dealers as well as from end-customers. The VSSM, in consultation with the OTD department, plans the schedule for producing these orders at GM-owned manufacturing facilities. The finished orders are delivered to a third party owned outbound logistics facility in order to ship them to the appropriate dealers/customers.

7.5.1 Outbound Logistics Planning Problem Let us focus on the outbound logistics operation that is actually owned by a third party. A typical GM vehicle is transported first by truck, then by rail, and finally by truck to the dealer. Thus, the outbound logistics operations at GM can be viewed as a three-echelon supply chain. Let us assume that for a particular destination, the shipment time required for one unit of the vehicle at each of these three stages is given by X1 ; X2 , and X3 . Figure 7.2 depicts this scenario. We assume that each shipment stage is in the custody of a department manager, who has an idea about the delivery cost and the delivery performance of different transportation service providers available for that particular stage. Let us assume that these three managers report to an OTD manager managing the OTD process. Let us assume that the OTD manager requests the manager of each stage to gather the quotations from the prospective partners in the market who can offer the service required for that particular stage. Table 7.1 summarizes the quotes that were reported by the managers of stages 1, 2, and 3 to the OTD manager. With the help of this table, the OTD manager can generate Table 7.2 that summarizes all possible values for the end-toend outbound logistics time Y D X1 C X2 C X3 and the corresponding outbound logistics cost C . Now, based on factors such as the level of delivery performance expected from the outbound logistics operations and its allocated budget, the OTD manager can choose an appropriate combination of partners for these three stages. For example, if the OTD manager thinks that the vehicles should be shipped within 30–32 days, then one possibility is to choose partner P11 for stage 1, partner P21 for stage 2, and partner P33 for stage 3. In such a case, the OTD manager would assign

Fig. 7.2 A three-echelon vehicle distribution supply chain at GM North America

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A Mechanism Design Approach for Decentralized Supply Chain Formation Table 7.1 Quotes of partners for stage 1, 2, and 3 as managers of stage 1, 2, and 3 to the OTD manager Estimated shipment Partner Id time X1 (days/vehicle) P11 4 P12 5 6 P13

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reported by the Estimated cost C1 ($/vehicle) 20 15 10

Partner Id P21 P22

Estimated shipment time X2 (days/vehicle) 20 22

Estimated cost C2 ($/vehicle) 100 85

Partner Id P31 P32 P33

Estimated shipment time X3 (days/vehicle) 6 7 8

Estimated cost C3 ($/vehicle) 35 25 20

Table 7.2 End-to-end lead time vs. delivery cost tradeoffs Estimated cost Estimated shipment time Supply chain Y D X1 C X2 C X3 C D C1 C C2 C C3 (days/vehicle) ($/vehicle) configuration .P11 ; P21 ; P31 / .P11 ; P21 ; P32 / .P11 ; P21 ; P33 / .P11 ; P22 ; P31 / .P11 ; P22 ; P32 / .P11 ; P22 ; P33 / .P12 ; P21 ; P31 / .P12 ; P21 ; P32 / .P12 ; P21 ; P33 / .P12 ; P22 ; P31 / .P12 ; P22 ; P32 / .P12 ; P22 ; P33 / .P13 ; P21 ; P31 / .P13 ; P21 ; P32 / .P13 ; P21 ; P33 / .P13 ; P22 ; P31 / .P13 ; P22 ; P32 / .P13 ; P22 ; P33 /

30 31 32 32 33 34 31 32 33 33 34 35 32 33 34 34 35 36

155 145 140 140 130 125 150 140 135 135 125 120 145 135 130 130 120 115

a target of 4 days, 20 days, and 8 days to the managers of stage 1, 2, and 3, respectively. Also, the OTD manager would approve a budget of $20 for stage 1, $100 for the stage 2, and $20 for stage 3 against each vehicle that is being shipped from the plant to the end customer. This would result in a total cost of $140 for shipping one vehicle from the plant to the end customer.

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7.6 Generalization of the Outbound Logistics Example Note that the problem of the OTD manager is fairly simple in the previous example as there are only three stages and the shipment time of a partner at each stage is deterministic. For such a scenario, the problem of the OTD manager involves the following exercise. The OTD manager needs to get three tables, one from each stage manager, each of them summarizing the cost and delivery time trade-offs at the individual stages. After receiving these three tables, the OTD manager needs to create a fourth table, which brings out the trade-offs between end-to-end outbound logistics cost C and end-to-end outbound logistics time Y . Using this table, the OTD manager can effectively form the supply chain. A plethora of questions is in order at this point: 1. What would happen if there are more than three stages (say n stages) in the supply chain? Will the design problem of the OTD manager be as simple as this? 2. In practice, the shipment time is stochastic rather than deterministic. Thus, what would happen if the shipment time of each partner at each stage of the supply chain is stochastic rather than deterministic? 3. What would happen if managers of the stages are not loyal to the OTD manager but they are rational and intelligent in the sense of trying to maximize the welfare of their individual stages instead of the overall organizational welfare? In such a situation, it is not surprising if: The stage managers report untruthful cost versus lead time curves to the OTD

manager, that is they report untruthful tables (because doing so may help them improve their own individual welfare). The stage managers do not coordinate with each other (i.e., they do not obey the instructions given by the OTD manager). The above questions lead to generalizing the three-echelon supply chain formation problem in three different directions. Our objective in this chapter is to originate a general framework for addressing the supply chain formation problems, which can take care of all the issues raised above. First, in Sect. 7.6.1, we take up questions 1 and 2 and try to generalize the three-echelon model along these two dimensions. Next, in Sect. 7.8, we consider question 3 where we generalize the framework along the third dimension.

7.6.1 Generalization to Multi-Echelon Supply Chains with Stochastic Lead Times Let us generalize the paraphernalia developed above by considering an n-echelon linear supply chain with stochastic lead times. Look at Fig. 7.3, which shows a make-to-order serial supply chain consisting of n stages. Let us assume the following:

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X1~ N ( μ1 , σ ) 1

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Fig. 7.3 A linear multi-echelon supply chain with stochastic lead time

1. For each stage of the supply chain, there exist multiple partners in the market who can offer the services required at that stage. 2. At each stage, the time taken to deliver one unit of the order by each available vendor is stochastic in nature.1 3. Xi represents the delivery time for one unit of the order at stage i . Xi is assumed to be Gaussian distributed with mean i and standard deviation i .

1

We will keep using the terms cycle time, lead time, and delivery time interchangeably.

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4. Candidate partners for stage i may offer different values for the mean i , standard deviation i , and delivery cost of one unit of the order vi . The delivery cost vi typically depends on i and i . 5. The manager for stage i has access to information such as the cost of each partner and the corresponding values of i and i offered by the partner. The manager uses this information to come up with an aggregated cost function vi .i ; i / for stage i . This aggregated cost function can be obtained, for example by performing a polynomial curve fitting over the available quotations from the various partners for stage i . Such an aggregated cost function vi .i ; i / will capture the cost versus delivery performance trade-offs across different partners available for stage i . 6. The delivery times Xi 8 i D 1; : : : ; n of the various stages in the n-echelon supply chain are independent random variables and there is no time elapsed between end of process i and commencement of process i C 1 8 i D 1; : : : ; n 1. As a consequence, the end-to-end delivery time Y of an order is Y D

n X

Xi :

i D1

P It can be noted immediately that Y is normally distributed with mean D niD1 i P and variance 2 D niD1 i2 . In view of the above assumptions, we can show that the three-echelon example discussed earlier is a special case of this generalized model. In the three-echelon example: n D 3. i D 0 8 i D 1; 2; 3. Xi D i 8 i D 1; 2; 3. Truck shipment is the first stage and the delivery time X1 is equal to 1 , which can take values 4, 5, or 6 days. Rail shipment is the second stage and the delivery time X2 is equal to 2 , which can take values 20 or 22 days. Truck shipment is the third stage and the delivery time X3 is equal to 3 , which can take values 6, 7, or 8 days. v1 .1 ; 1 / is the shipment cost of one vehicle at stage 1, which is given by

v1 .4; 0/ D 20I v1 .5; 0/ D 15I v1 .6; 0/ D 10: v2 .2 ; 2 / is the shipment cost of one vehicle at stage 2, which is given by

v2 .20; 0/ D 100I v2 .22; 0/ D 85: v3 .3 ; 3 / is the shipment cost of one vehicle at stage 3, which is given by

v1 .6; 0/ D 35I v1 .7; 0/ D 25I v1 .8; 0/ D 20:

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7.6.2 Delivery Performance of Multi-Echelon Supply Chains with Stochastic Lead Times Now, let us assume that a CDA who is managing the whole supply chain, for example the OTD manager in the three-echelon example, wants to ensure a certain level of delivery performance of the supply chain. Although there may be many ways of defining the delivery performance, here we consider a particular definition of the delivery performance. We assume that the CDA’s target is to deliver the orders to the respective customers within ˙ T days of receiving the order. We refer to as the target delivery date and T as the tolerance. We also define L D . T / to be the lower limit of the delivery window and U D . C T / to be the upper limit of the delivery window. The CDA measures the delivery performance of the supply chain in terms of how precisely and accurately the orders are being delivered to the customers within the delivery window ˙ T . In other words, what fraction of the orders is being delivered within the delivery window ˙ T , which is a measure of the precision, and how closely these orders are being delivered to their target delivery date , which is a measure of the accuracy. In view of this definition, we can say that the delivery performance of the supply chain depends on the mean and standard deviation of the end-to-end delivery time Y . Because mean and standard deviation depend on the values of mean i and standard deviation i of the delivery time for each stage of the supply chain, there may be many ways in which the CDA can choose the partner for each stage of the supply chain so that the corresponding set of parameters .i ; i /i D1;:::;n will result in mean and standard deviation , which would render the desired level of delivery performance. However, there might be some way of choosing the partners for each stage of the supply chain so that the end-to-end delivery cost is minimized while achieving the desired level of delivery performance. Thus, the problem of the CDA is an optimization problem whose objective is to minimize end-to-end delivery cost subject to delivery performance constraints. Before we can formulate such an optimization problem, we need to develop the concept of delivery performance in a more formal way so that we can write these constraints. We do this using the process capability indices Cp ; Cpk and Cpm , which are popular in the areas of design tolerancing and statistical process control (Kane 1986; Kotz and Lovelace 1998; Garg et al. 2006). The three indices Cp ; Cpk , and Cpm for the end-to-end delivery time Y can be defined in the following manner: T U L D 6 3

(7.1)

min.U ; L/ d D 3 3

(7.2)

T U L D p ; 6 3 2 C b2

(7.3)

Cp D Cpk D

Cpm D

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where b D j j. The following relations among these three indices can be easily verified (Garg et al. 2006). Cp Cpk 0

(7.4)

Cp Cpm 0

(7.5)

j j b D T T 2 Cpk 1 D C 1 : 9Cp2 Cp

Cpk D Cp .1 k/ where k D 1 2 9Cpm

(7.6) (7.7)

Next, we define an important quantity – the Delivery Probability (DP) of the endto-end delivery time Y . This will capture the notion of precision of the deliveries. Definition 1 (Delivery Probability (DP)). The probability of delivering an order within a specified delivery window ˙ T is known as delivery probability (DP) of the end-to-end delivery time Y . One can verify that DP of Y can be expressed in terms of its capability indices Cp and Cpk in the following way (Garg et al. 2006). DP D ˚ 3Cpk C ˚ 6Cp 3Cpk 1;

(7.8)

where ˚.:/ is the cumulative distribution function of standard normal distribution. Looking at expression (7.8), it is easy to see that a unique (Cp ; Cpk ) pair results in a unique DP. For example Cp D Cpk D 1:548350 results in DP of 1 3:4 106 . In the literature, having such a DP is referred as six sigma delivery performance (Garg et al. 2004; Harry 1987; Harry and Stewart 1988). It has been shown in the literature that DP of the end-to-end lead time Y gives an estimate of how precisely the orders are being delivered to the customers with respect to the delivery window ˙ T . Similarly, the index Cpm gives an estimate of how accurately the orders are being delivered to the customers with respect to the target delivery date . For a detailed account of these concepts, please refer to Garg et al. (2006; 2004). Thus, being an indicator for accuracy of the deliveries, we prefer to refer to the Cpm of the process as Delivery Sharpness (DS). Definition 2 (Delivery Sharpness (DS)). The index Cpm measures how closely the orders are being delivered to the target delivery date and is known as delivery sharpness (DS) of the end-to-end delivery time Y . Thus, the CDA can use the two indices, DP and DS, to measure the end-to-end delivery performance of a supply chain.

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7.7 A Centralized Framework for Supply Chain Formation Problems Now, we describe the framework for addressing the CDA’s problem – How do we form an n-echelon supply chain in the presence of stochastic lead times so that endto-end delivery cost is minimized and at the same time minimum mandated levels of DP and DS are attained. This would essentially answer questions 1 and 2 raised in Sect. 7.6.1. We call this framework centralized because all the information that is required to solve the CDA’s problem can be made available at some central place. This will become clear once we illustrate this framework. In the centralized framework, CDA first invites each stage manager to submit its cost function vi .i ; i /. Here, the implicit assumption is that each stage manager is loyal to the benevolent CDA and will be truthful. Therefore, each stage manager honestly submits its true cost function to the CDA. In the context of the three-echelon GM vehicle distribution supply chain, the OTD manager can be viewed as the CDA and the manager of each of the three shipment legs can be viewed as the stage managers (players). The design exercise of the CDA is driven by the following vision: “The delivery performance of the chain as a whole should not fall below specified levels of DP and DS, and at the same time the organization’s welfare should be maximized (i.e., end-to-end delivery cost must be minimized).” After receiving the true cost function from each stage, the CDA needs to solve an optimization problem that will minimize the end-to-end delivery cost while ensuring the specified levels of DP and DS. The solution of this optimization problem gives the optimal values for the design parameters .i ; i /, which are communicated back to the respective stage managers by the CDA. The CDA also allocates the budget Ki D vi .i ; i / for the manager of stage i 8i D 1; 2; : : : ; n. Depending on the amount of the budget Ki and the target parameters .i ; i / that are assigned to stage i , the manager of stage i now figures out the best partner who can offer this service within the specified limit of the budget. This is shown in Fig. 7.4. In the case of the three-echelon supply chain example, the OTD manager collects the cost functions in the form of tables from all the three stage managers and minimizes the outbound logistics cost while ensuring that an order is delivered within 30–32 days. The optimal values for the design parameters turn out to be i D 0 days 8 i D 1; 2; 3 and 1 D 4 days, 2 D 20 days, and 3 D 8 days, which are communicated back to all the three stage managers, based on which, each stage manager figures out the best partner for that stage. The generic optimization problem faced by the CDA can be formally defined in the form of known parameters, decision variables, objective function, and constraints as follows.

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Known Parameters The following parameters are known to the CDA: 1. The delivery window .; T /. 2. The minimum mandated levels of DP and DS for end-to-end delivery time (Y ). We have already seen that DP and DS can be completely characterized by the process capability indices Cp and Cpk . Therefore, an equivalent way of expressing the same minimum mandated levels of DP and DS is via specifying the lower bounds on the values of Cp and Cpk , say, for example, Cp p and Cpk q. 3. Delivery cost function vi .i ; i / per unit order submitted by the manager of stage i .

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Decision Variables The decision variables are optimal means i and optimal standard deviations i of each individual stage i (i D 1; : : : ; n).

Objective Function and Constraints As stated already, the objective of the CDA is to minimize the end-to-end delivery cost and the constraints are specified in terms of minimum expected levels of DP and DS (or equivalently Cp and Cpk ) on end-to-end delivery time. Thus, the problem formulation becomes: Minimize n X

vi .i ; i /

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i D1

subject to

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T Cp D qP p n 2 3 i D1 i Pn P min U i D1 i ; niD1 i L D q qP n 2 3 i D1 i i ; i > 0 8i:

(7.10)

(7.11) (7.12)

An interesting special case of the above optimization problem is known as Variance Pool Allocation (VPA) problem. In the VPA problem, the mean delivery time i of all the partners available for each stage i is the same and is also known to the CDA. However, for each stage i , different partners may offer different variability in the delivery time of the service they are offering and hence also charge different amounts for the service. Therefore, the delivery cost function vi .i ; i / is now functions of i alone. In the VPA problem, we assume that delivery cost function vi .i / has the following form (7.13) vi .i / D ai 0 C ai1 i C ai 2 i2 : The function vi .i / is a nonnegative and nonincreasing function of i , that is not all the three coefficients ai 0 ; ai1 ; ai 2 are positive. We assume that manager of each stage i submits its cost function essentially in the form of three coefficients: .ai 0 ; ai1 ; ai 2 / to the CDA2 . Thus, the problem of the CDA is to decide the optimal variance i for each stage i so that the following goals can be met: 2

In most of the practical cases, getting values of the coefficients ai0 ; ai1 ; ai2 is not an easy task for the stage manager.

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The parameters .i ; i / result in specified levels of DP and DS or an equivalent

minimum levels of Cp and Cpk , say for example Cp p and Cpk q.

The supply chain formation which would emerge out of this solution will have

minimum possible delivery cost. One way of solving the VPA problem is to express the constraints in terms of the decision variables and then apply the Lagrange multiplier method.

7.8 A Decentralized Framework for the Supply Chain Formation Problem A critical assumption while working with the centralized framework is that the manager of each stage is loyal to the benevolent CDA in the sense that he or she obeys the CDA’s instructions and is totally honest about his or her cost functions. However, in most real-world situations, the manager of each stage of a supply chain is typically rational, intelligent, and autonomous and hence cares only about maximizing his own department and employees’ welfare rather than the welfare of the overall supply chain. Therefore, it is not surprising if the managers of individual stages report untruthful cost functions to the CDA (because, in their perception, doing so may help them improve their own individual utilities). On the contrary, the CDA typically sets forth an overall organization wide (centralized) goal and wants all the stage managers to plan and operate in a way that is aligned with this goal. The differing local interests lead to inter-firm as well as intrafirm conflicts and, hence, all the managers (rational, intelligent, and autonomous agents) try to outwit one another to maximize individual goals. Such game theoretic behavior of the selfish managers does not comply with the CDA’s plan. In other words, any strategic, tactical, and operational decision made by the central authority might, in general, not be accepted by the managers of all the stages. In such a situation, the centralized framework cannot be used and one needs to look for an alternative scheme. For example, consider the case of the three-echelon GM vehicle distribution supply chain. It is quite possible that the stage managers do not report true cost functions to the OTD manager due to many reasons. Now, if the OTD manager just goes by the face value and solves the optimization problem based on the reported cost functions, then he or she might not actually solve the problem optimally. Although, he or she may manage to achieve the desired level of delivery performance, he will not realize the optimal cost. Thus, we need to have an alternative approach to tackle such scenarios. The basic problem here is that the critical information is required by the CDA to form the supply chain is not centrally available but is distributed across the whole supply chain. Moreover, it is not even possible for the CDA to get hold of this information easily because each piece of the information is being held by a selfish and intelligent stage manager who understands the value of this information and, hence, tries to get maximum mileage out of it. Thus, the distributed nature of the information makes the problem harder and the centralized framework cannot tackle such

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a situation. The problem identified here is similar to the problem of eliciting truth from a set of selfish agents. Therefore, this problem falls precisely in the domain of mechanism design. The problem faced by the CDA is to design an incentive compatible mechanism to elicit true types of the stage managers. Here, the true type of a stage manager corresponds to the true cost function which is the private information of the manager. In view of this critical observation, we propose a new framework, namely decentralized framework for addressing such situations. This framework is founded on the theory of mechanisms design. A similar line of attack can be traced to recent articles by Kutanoglu and Wu (2000; 2002; 2005; 2006).

7.9 Supply Chain Formation as a Mechanism Design Problem For the sake of illustration, we consider the VPA problem discussed earlier in Sect. 7.7. In a typical VPA problem, the mean delivery time i of all the partners available for each stage i is the same and is known to the CDA. However, for each stage i , different partners may offer different variabilities in the delivery time of the service they are offering and hence charge different amount for the service. Thus, the problem of the CDA is to decide the optimal variance i for each stage i so that the following goals can be met The parameters .i ; i / result in a specified levels of DP and DS or an equivalent

minimum levels of Cp and Cpk , say, for example, Cp p and Cpk q.

The supply chain formation which would emerge out of this solution will have

minimum possible delivery cost. Recall that the implicit assumption made in Sect. 7.7 while defining the VPA problem was that cost functions vi .:/ are known to the respective stage managers and all the managers are loyal to the CDA and reveal the true cost function if the CDA asks them for it. However, here we assume that vi .:/ is the actual cost function of stage i , which is known only to the manager

of stage i . The managers of all the stages are rational, intelligent, and autonomous. There-

fore, on receiving a request from CDA, the i th stage manager reports a cost function wi .:/ to the CDA, which need not be the same as vi .:/. This is shown in Fig. 7.5. It is easy to see that because of the above assumption, the CDA now faces the following two problems: 1. True Cost Revelation (Elicitation) Problem: The first problem that the CDA faces is to estimate the true cost function vi .:/ for each stage i , which is the private information of manager of stage i . 2. Computing the Optimal Variance i : After knowing the true cost vi .:/ of each stage i , the next problem that the CDA faces is to compute the optimal variance i for each stage i .

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SUBUNIT 1

ial

ter

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wn−1(σn−1) In−1 wn (σn )

In

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SUBUNIT n vn(σn)

Fig. 7.5 The idea behind decentralized design paradigm

In this section, we show that the VPA problem essentially becomes that of designing an incentive compatible direct revelation mechanism in a quasilinear environment. VCG mechanisms are well known to be dominant strategy incentive compatible direct revelation mechanisms in quasilinear environment (Mas-Colell et al. 1995). Motivated by this, we start exploring VCG mechanisms as a possible approach to solving the CDA’s problem. We begin by defining the following notation: Di D Set of attainable values for standard deviation i I D D D1 D2 : : : Dn I D .1 ; 2 ; : : : ; n / D A standard deviation profile of the stagesI 2 DI

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vi W Di ! R D True cost function (actual type) of manager of stage i I wi W Di ! R D Reported cost function (reported type) of manager of stage iI Vi D Set of all the possible types of manager of stage i I Wi D Set of all the possible reported types of manager of stage i I V D V1 V2 : : : Vn I v D .v1 ; v2 ; : : : ; vn / 2 V I W D W1 W 2 : : : W n w D .w1 ; w2 ; : : : ; wn / 2 W Vi D V1 : : : Vi 1 Vi C1 : : : Vn vi D .v1 ; : : : ; vi 1 ; vi C1 ; : : : ; vn / 2 Vi Wi D W1 : : : Wi 1 Wi C1 : : : Wn wi D .w1 ; : : : ; wi 1 ; wi C1 ; : : : ; wn / 2 Wi : We make the following assumptions regarding the actual cost functions (actual types) and reported cost functions (reported types) of the stage managers: 1. The i can take values from the set .0; i , that is Di D .0; i . 2. The actual type of the manager of stage i is of the following form vi .i / D ai 0 C ai1 i C ai 2 i2 8i 2 .0; i ; where we assume that vi .i / is a nonnegative and nonincreasing function of i , that is all the three coefficients ai 0 ; ai1 , and ai 2 are not nonnegative simultaneously. 3. For each stage i , it is possible to obtain an interval for each one of the three coefficients ai 0 ; ai1 , and ai 2 . That is ai 0 2 Œai 0 ; ai 0 I ai1 2 Œai1 ; a i1 I ai 2 2 Œai 2 ; ai 2 :

4.

5. 6.

7.

These intervals are such that choosing the coefficients from them will always result in a cost function vi .i / that is nonnegative and nonincreasing. Also, for a given type vi .i /, values of all the three coefficients lie in these intervals. The previous assumption enables us to view the type set Vi as Œai 0 ; ai 0 Œai1 ; ai1 Œai 2 ; ai 2 , which is a compact subset of R3 . The set V can now be viewed as a compact subset of R3n . Each element of such a compact subset of R3n represents a unique type profile of all the stage managers. Wi D Vi 8 i D 1; : : : ; n. Let i be a algebra over the type set Vi (a compact subset of R3 ). Let be a algebra over the set V (a compact subset of R3n ), which is generated by the algebras i . We assume that P is a probability measure over the algebra and hence .V; ; P / forms a probability space. We call P as common prior distribution

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over type profiles of the stage managers and assume that it is common knowledge among all the stage managers. Let 4V represent the set of all the probability measures that can be defined over the measurable space .V; /. 8. Let the manager of each stage i have a belief function pi W Vi ! 4Vi , where 4Vi represents the set of all the probability measures that can be defined over the measurable space .Vi ; i /. The belief function pi describes the conjecture of the manager of stage i about the types of other managers conditioned his own type. That is, for any possible type set i , where i Vi and i 2 i , pi .:j i / will be a probability measure over the measurable space .Vi ; i / and will represent what the i t h stage manager would believe about the other stage managers’ type sets if his own type set were i . Recall that we have assumed common prior distribution P is known to the managers of all the stages. Therefore, the manager of each stage can compute its belief function as follows Z

Z dP .v1 ; : : : ; vn /

pi . i j i / D

vi 2i vi 2i

Z

Z

; dP .v1 ; : : : ; vn /

vi 2Vi vi 2i

where all the integrals are in the Lebesgue sense. Having set up the mathematical paraphernalia required, we now illustrate that the VPA problem in the presence of rational, intelligent, and autonomous stage managers can be modeled as a direct revelation mechanism where the CDA is the social planner and the stage managers are the agents. The various components of the direct revelation mechanism can be defined in the following manner.

7.9.1 Outcome Set X An outcome x is a vector .i ; Ii /i D1;:::;n , where: i is the allowed variability in the delivery time at stage i . This means, the

manager for stage i needs to choose a partner who can offer the service with variability less than or equal to i . Ii is the total budget sanctioned by the CDA for the manager of stage i . The manager is supposed to procure the service for stage i within this budget. Any surplus funds can be used by the manager and the department. On the contrary, if there is any deficit, the manager needs to mobilize the extra funds from own department. In such a case, the set of feasible outcomes is X D f.i ; Ii /i D1;:::;n ji 2 Di ; Ii 2 Rg :

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7.9.2 Utility Functions of Stage Managers ui .:/ The Bernoulli utility function of a manager for stage i can be given, for x D .i ; Ii /i D1;:::;n , by ui .x; vi / D Ii vi .i /:

7.9.3 Social Choice Function f .:/ The general structure of the social choice function for this case is f .w/ D .i .w/; Ii .w//i D1;:::;n ; 8 w 2 V: However, we know that the CDA wants to achieve a desired level of delivery performance at minimum cost so the social choice function that the CDA wishes to implement is the following f .w/ D .i .w/; Ii .w//i D1;:::;n ; 8 w 2 V;

(7.14)

where .i .w//i D1;:::;n is the solution of the following VPA problem Minimize n X

wi .i /

(7.15)

i D1

subject to

Cpk

T p Cp D qP n 2 3 i D1 i P P min U niD1 i ; niD1 i L D q qP n 2 3 i D1 i i > 0 8i:

In view of the above definition of .i .w//i D1;:::;n , it is easy to see that the social choice function f .w/ is allocatively efficient. If the budgets Ii .w//i D1;:::;n are chosen properly, then it is possible that the social choice function f .:/ becomes dominant strategy incentive compatible in which case no manager will have an incentive to report an untruthful cost function. Thus, the problem of the CDA will be solved if we can come up with an allocatively efficient and dominant strategy incentive compatible social choice function.

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Recall that the VCG mechanisms (Vickrey 1961; Clarke 1971; Groves 1973) are allocatively efficient and dominant strategy incentive compatible. Therefore, we can say that the above allocatively efficient social choice function will be dominant strategy incentive compatible if the CDA computes .Ii .w//i D1;:::;n by using the VCG payment scheme. It is easy to verify that the VCG budget Ii for stage i can be given by the following relation: X wj .i .w// 8 w 2 V; (7.16) Ii .w/ D ˛i j ¤i

where .1 .w/; : : : ; n .w// is the optimal solution of the VPA problem (7.15). ˛i is some constant, which can be assumed to be the same for all the stages and can be used to normalize the value of Ii so that it has some meaningful value. Thus, we can say that if the CDA uses the social choice function (7.14), where he or she directly asks the stage managers to report their cost functions and then uses the reported cost functions w1 ; : : : ; wn in order to compute: Target variability .i .w//i D1;:::;n by solving the VPA problem (7.15). Budget Ii .w/ by using the VCG payment scheme given by (7.16)

then by Groves Theorem (see Proposition 23.C.4 in (Mas-Colell et al. 1995)), we can claim that no manager will find it worthwhile to report untruthful cost function and all the managers will indeed report their true cost functions.

7.10 A Case Study: Forming an Automotive Vehicle Distribution Network To illustrate the decentralized framework, we revisit the problem of forming a threeechelon GM vehicle distribution supply chain shown in Fig. 7.2. We assume that the manager for each stage is rational and autonomous and hence may not report the true cost function to the OTD manager. The objective behind the current experiment is to show that invoking the VCG incentive scheme will enable the OTD manager to elicit true cost structure from each manager. This will in turn help the OTD manager in choosing an optimal partner for each leg of the outbound logistics supply chain. We make the following assumptions: 1. The OTD manager has an ideal target of delivering a vehicle to the dealer on .30 ˙ 5/th day counting from the day it is ready for shipping at the plant. 2. Recall there are three shipment legs in the journey of a vehicle from plant to dealer. We call these the first leg (truck), the second leg (rail road), and the third leg (truck). For each leg, there are alternate transportation service providers. We assume that there are ten alternate service providers for each leg. 3. The mean shipment time is the same for all the ten service providers for each of the three legs and the CDA is aware of this fact. The mean shipment time for the first, second, and third legs is 4 days, 21 days, and 7 days, respectively, which is known to the CDA.

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4. For each leg, the variabilities in the shipment time and the shipping cost vary across alternate service providers. This is private information of the manager for that leg and is known neither to the OTD manager nor to the remaining two managers. Tables 7.3, 7.4, and 7.5 show the private information available with the individual managers that we will use in our study. The data in these tables is generated synthetically; however, we have been careful in generating this data so as to represent the real-world scenario as closely as possible. Note that we

Table 7.3 Private information of the manager for stage 1



Partner Id P11 P12 P13 P14 P15 P16 P17 P18 P19 P110

Standard deviation in shipment time 1 (days) 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 2.25 2.50

Estimated cost C1 ($/vehicle) 20.0 15.0 12.0 10.0 09.0 08.0 07.50 07.25 07.0 07.0



Estimated cost C2 ($/vehicle) 105 70 55 45 40 35 32 30 29 28



Estimated cost C3 ($/vehicle) 35.0 27.0 22.0 19.0 18.0 16.0 14.5 13.5 13.0 12.5

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are assuming that the shipping costs at each stage depend only on variability and not on the mean of the shipping times. The study can, however, be generalized without much difficulty to the case, where shipping costs depend on mean as well as variability. 5. The shipment time at each leg is normally distributed for all the service providers. Moreover, the shipment times at the three legs are mutually independent. 6. The CDA wishes to choose the service provider for each stage in such a way that values Cp 1:8 and Cpk 1:08 are attained for the end-to-end shipment time Y , which is equal to the sum of the shipment times of the three legs. These values of Cp and Cpk ensure that DP 0:9995 and DS 0:75. 7. The manager of leg i .i D 1; 2; 3/, uses his private information (as provided in Tables 7.3, 7.4, and 7.5, respectively) to compute the true cost function as a quadratic function vi .i / D ai 0 C ai1 i C ai 2 i2 using the least square curve fitting method to compute the coefficients ai 0 ; ai1 , and ai 2 . For the present instance, these functions turn out to be: v1 .1 / D 022:638 16:0171 C 4:01512 v2 .2 / D 231:085 68:6242 C 5:75822

v3 .3 / D 052:255 29:8273 C 5:63632 :

We now apply the VCG incentive scheme for the above case study. For the sake of completeness, we start with the centralized framework and then take up the decentralized framework.

7.10.1 Centralized Framework As discussed earlier, in the centralized framework, each manager submits the true cost function and CDA just solves the single optimization problem. For the above case study, the OTD manager needs to solve the following optimization problem Minimize KD

n X

vi .i /

i D1

D 305:978 .16:01751 C 68:6242 C 29:8273 / C.4:01512 C 5:75822 C 5:63632 / subject to 12 C 22 C 32 D

T2 d2 25 D 2 2 29:16 9Cp 9Cpk

i > 0 8 i D 1; 2; 3:

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Using the Lagrange multiplier method, it is easy to show that the above optimization problem has the global minimum at 1 D 0:201835764 days 2 D 0:828360820 days

3 D 0:361107815 days: The optimal costs for the different legs and the total optimal cost turn out be: v1 .1 / D 019:568758 $/vehicle v2 .2 / D 178:190292 $/vehicle v3 .3 / D 042:219166 $/vehicle

K D 239:978216 $/vehicle:

7.10.2 Decentralized Framework Let us first compute the budgets Iit and net payoffs ui .:/ to the managers in the case when each of them reveals the true cost function, that is wi .:/ D vi .:/I 8 i D 1; 2; 3. Assuming ˛1 D ˛2 D ˛3 D 250 $/vehicle, it is easy to see that I1t D ˛1 .v2 .2 / C v3 .3 // D 029:590542 $/vehicle I2t D ˛2 .v1 .1 / C v3 .3 // D 188:212076 $/vehicle

I3t D ˛3 .v1 .1 / C v2 .2 // D 052:240950 $/vehicle

u1 .1 ; 2 ; 3 ; I1t ; I2t ; I3t ; v1 / D I1t v1 .1 / D 10:021784 $/vehicle

u2 .1 ; 2 ; 3 ; I1t ; I2t ; I3t ; v2 / D I2t v2 .2 / D 10:021784 $/vehicle u3 .1 ; 2 ; 3 ; I1t ; I2t ; I3t ; v3 / D I3t v3 .3 / D 10:021784 $/vehicle: Now for the purpose of convincing the managers that reporting the true cost function is a dominant strategy, we calculate the budgets Ii and net payoffs ui .:/ to the managers in the case when each of them reveals the untruthful cost function, that is wi .:/ 6D vi .:/I 8 i D 1; 2; 3. For this, we assume that the following cost functions are submitted by the managers: w1 .1 / D 25 101 C 512

w2 .2 / D 240 652 C 622 w3 .3 / D 55 253 C 632 :

It is easy to verify that wi .i / vi .i /I 8i I 8i D 1; 2; 3. In such situation, the CDA solves the following assignment problem:

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Minimize KD

3 X

wi .i /

i D1

D 320 .101 C 652 C 253 / C .512 C 622 C 632 / subject to 12 C 22 C 32 D

T2 d2 25 D 2 2 29:16 9Cp 9Cpk

i > 0 8i D 1; 2; 3: The solution of the above optimization problem results in 1 D 0:135086395 days 2 D 0:854962796 days

3 D 0:328831845 days:

The CDA can use the above values to compute the incentives for the managers which turn out to be: I1 D ˛1 .w2 .2 / C w3 .3 // D 13:758827$/vehicle I2 D ˛2 .w1 .1 / C w3 .3 // D 178:831637 $/vehicle I3 D ˛3 .w1 .1 / C w2 .2 // D 37:446436 $/vehicle: Now the manager for each leg computes his/her own payoff in following manner: u1 D I1 v1 .1 / D 6:788761 $/vehicle u2 D I2 v2 .2 / D 2:208727 $/vehicle

u3 D I3 v3 .3 / D 0:598155 $/vehicle:

Thus, we see that if the cost function reported by managers is untruthful, then the payoff for the managers is less. This can be verified by choosing any untruthful cost function for any subset of the stage managers. Thus, reporting the true cost functions is the best strategy for the stage managers in the decentralized framework.

7.11 Summary In this chapter, we have shown how mechanism design could be used in a natural and effective way to model and solve the supply chain formation problem with incomplete information. We first formulated the supply chain formation problem

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with complete information as an optimization problem and proposed a centralized framework to solve the underlying optimization problem. Next, we argued that the entities in today’s global supply chain networks are not loyal but are rational, intelligent, and autonomous and always try to maximize their own payoffs. This fact motivated us to model the supply chain formation problem (with incomplete information) as a mechanism design problem followed by an optimization problem. In particular, we showed the use of the well-known VCG mechanisms in modeling and solving a real-world supply chain formation problem in a very effective way. We proposed a decentralized framework to solve the underlying mechanism design problem. We illustrated our approach with the example of forming a three-stage vehicle distribution process at GM North America. There are plenty of opportunities for further investigation. In what follows, we suggest a few possible future directions: 1. The optimization problem that we solved was formulated as a VPA problem. In particular, the mean value of the lead time promised by alternate service providers in a given echelon was assumed to be the same. This assumption was made only to simplify the presentation in the paper. The assumption could be easily relaxed in the following way. At each echelon (that is, stage), let the candidate service providers specify their costs as a function of mean and variance of lead time. The echelon manager will now compute his/her own cost function using a suitable curve-fitting scheme. The use of Groves mechanisms will make it a best response for each echelon manager to reveal his/her cost function truthfully to the CDA. 2. Imagine a situation where the CDA has a budget constraint on forming the supply chain. In such a case, the VCG approach may not work because it is quite possible P the total payment to be made by the CDA to all the managers, which is equal to niD1 Ii , exceeds the budget available with the CDA. In such a situation, one can look for an alternative mechanism known as dAGVA mechanism due to d’Aspremont and Gerard-Varet K (1979) and Arrow (1979). This mechanism results in lower cost for the CDA however, truth telling is not a weakly dominant strategy equilibrium for the stage managers but is a Bayesian Nash equilibrium (which would mean that truth revelation is a best response for each manager whenever the other managers are also truthful). 3. One can also investigate optimal mechanisms for this problem on the lines of Myerson’s optimal auction design (Myerson 1981). Such a mechanism will result in minimum possible cost for the CDA, while ensuring Bayesian Nash incentive compatibility and individual rationality. 4. In practice, the stage managers may not fully understand the idea behind the VCG incentives. Therefore, it is quite possible that even though the CDA is following the VCG incentive scheme, the stage managers still report untruthful cost functions. In such a situation, one can think of designing an iterative scheme, which will help the stage managers learn that truth telling is indeed their best response strategy.

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References Arrow K (1979) The property rights doctrine and demand revelation under incomplete information. In: Boskin M (ed) Economics and human welfare. Academic, New York Cachon G, Netesssine S (2005) Game theory in supply chain analysis. In: Simchi-Levi D, Wu D, Shen (eds) Supply chain analysis in the ebusiness area. Kluwer Academic, Dordecht Chaudhry SS, Frost FG, Zydiak JL (1993) Vendor selection with price breaks. Eur J Oper Res 70:52–66 Clarke E (1971) Multi-part pricing of public goods. Public Choice 11:17–23 d’Aspremont C, Gerard-Varet K L (1979) Incentives and incomplete information. J Public Econ 11:25–45 Fan M, Stallaert J, Whinston A (2003) Decentralized mechanism design for supply chain organizations using an auction market. Inf Syst Res 14(1):1–22 Garg D, Narahari Y, Viswanadham N (2004) Design of six sigma supply chains. IEEE Trans Autom Sci Eng 1(1):38–57 Garg D, Narahari Y, Foster E, Kulkarni D, Tew JD (2005) A Groves mechanism approach to decentralized design of supply chains. In: IEEE conference on E-commerce technology, IEEE CEC 2005, pp 330–337 Garg D, Narahari Y, Viswanadham N (2006) Achieving sharp deliveries in supply chains through variance pool allocation. Eur J Oper Res 171(1):227–254 Golany B, Rothblum U (2006) Inducing coordination in supply chains through linear reward schemes. Nav Res Logist 53(1):1–15 Groves T (1973) Incentives in teams. Econometrica 41:617–631 Harry MJ (1987) The nature of six sigma quality. Technical report, Motorola Inc., Motorola University Press, Schaumburg, IL 60196-1097 Harry MJ, Stewart R (1988) Six sigma mechanical design tolerancing. Technical report, Motorola Inc., Motorola University Press, Schaumburg, IL 60196-1097 Kane VE (1986) Process capability indices. J Qual Technol 18:41–52 Karabuk S, Wu D (2002) Decentralizing semiconductor capacity planning via internal market coordination. IIE Trans 34(9):743–759 Karabuk S, Wu D (2005) Incentive schemes for semiconductor capacity allocation: A game theoretic analysis. Prod Oper Manag 14(2):175–188 Kotz S, Lovelace CR (1998) Process capability indices in theory and practice. Arnold, London Kutanoglu E, Wu D (2000) Schedule selection game: A game-theoretic analysis for incentive compatible scheduling. In: Proceedings of 9th industrial engineering research conference, IIE, Cleveland, OH Kutanoglu E, Wu D (2006) Incentive compatible, collaborative production scheduling with simple communication among distributed agents. Int J Prod Res 44(3):421–446 Mas-Colell A, Whinston MD, Green JR (1995) Microeconomic theory. Oxford University Press, New York Myerson RB (1981) Optimal auction design. Math Oper Res 6(1):58–73 Narsimhan R, Stoynoff LK (1986) Optimizing aggregate procurement allocation decisions. J Purch Mater Manag 22(1):23–30 Pan AC (1989) Allocation of order quantity among suppliers. J Purch Mater Manag 25(3):36–39 Vickrey W (1961) Counterspeculation, auctions, and competitive sealed tenders. J Finance 16(1):8–37 Viswanadham N, Gaonkar RS (2003) Partner selection and synchronized planning in dynamic manufacturing networks. IEEE Trans Rob Autom 19(1):117–130 Walsh W (2001) Market protocols for decentralized supply chain formation. PhD thesis, School of Business, University of Michigan Walsh WE, Wellman MP (2003) Decentralized supply chain formation: A market protocol and competitive equilibrium analysis. J Artif Intell Res 19:513–567

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Walsh WE, Wellman MP, Ygge F (2000) Combinatorial auctions for supply chain formation. In: Proceedings of second ACM conference on electronic commerce, EC, Minneapolis, Minnesota, pp 260–269 Weber CA, Current JR (1993) A multiobjective approach to vendor selection. Eur J Oper Res 68:173–184 Weber CA, Current JR, Benton W (1991) Vendor selection criteria and methods. Eur J Oper Res 50:2–18

Chapter 8

Procurement Network Formation: A Cooperative Game Approach T.S. Chandrashekar and Y. Narahari

In this chapter, we are interested in a procurement network formation problem. We present a case for modelling the procurement network formation problem as a shortest path cooperative game. We investigate recent results in shortest path cooperative games and their implications to the procurement network formation problem. We then enhance the model for procurement network formation by incorporating asymmetry in the information that agents have. Specifically, we model the procurement network formation problem as a shortest path cooperative game with incomplete information. We point out the incentive compatible core as an appropriate solution concept for this category of games. We then review the current state of the art on the topic of incentive compatible core, pose a conjecture and end with some directions for future work.

8.1 Introduction Within the context of business-to-business electronic commerce, focus has been on the automation of the procurement process right from the days of Electronic Data Interchange (EDI). More recently, emphasis, both in research and application, has shifted to harnessing the power of auction-based optimization models to automate supplier selection and contracting decisions (Chandrashekar et al. 2006). While this focus has been fruitful in advancing the state of the art for some areas of application within business-to-business transactions, practitioners have been reluctant to embrace purely auction-based methods for settling procurement contracts across the board. Specifically, they see the utility of auction-based methods only when (a) there are a large number of suppliers for the items being sourced and (b) the focus is on choosing suppliers in the immediately preceding echelon in the supply chain. First, in many procurement scenarios, practitioners have to work with a limited number of supply options. This forces them to favor negotiations with suppliers as a tool for settling contracts rather than just single-sided auctions. Bajari et al. (2003) in

T.S. Chandrashekar () Department of Computer Science and Automation, Indian Institute of Science, Bangalore, India e-mail: [email protected]


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fact find that the absence of a large supplier base is a positive determinant in favour of negotiations as a preferred mechanism for settling contracts with suppliers. To reinforce this point, they provide anecdotal evidence from the construction, medical procurement, timber procurement, automotive, and semi-conductor industries. To be fair, auctions are often used as a first-level elimination tool to identify suppliers with whom further negotiations can be carried out. Second, alliance formation among supply chain partners is becoming fairly commonplace because procurement organizations are often explicitly involved in making decisions related to the supplier’s supplier in order to plan capacities, monitor incoming quality, etc. For instance, in one of the leading global automotive suppliers that the first author has worked with, not only do the OEMs explicitly indicate the raw material supplier from whom the items need to be sourced but also negotiate raw material prices on behalf of the supplier. Nagarajan and Soˇsić (2005) provide many examples of alliance formation among suppliers and assemblers. For instance, Symbol Technologies, a leading scanner manufacturer, has formed a strategic partnership with Paxar/Monarch, a leader in bar code labeling. Together, they have become a major supplier of kits continuing a bar code scanner and a labeler to retailers. In another example, Stallkamp (2005) cites an example of Delphi and Lear, both major automotive suppliers, displaying interest in forming strategic alliances with suppliers of wiring harnesses, carpets and molded plastic with the view of becoming major cockpit suppliers to the automotive OEMs. These examples point us to a vector of development, wherein we could expect that major procurement organizations would be interested in explicitly designing the flow of material through what one may call the procurement network. In this chapter, we are interested in exploring ways for automated formation of such procurement networks. Although such decisions are inherently multi-objective and multi-attribute in nature, we believe that there is a lot to be gained by focusing on the single attribute problem in terms of understanding the issues involved, the modelling approaches, and the solution concepts therein. We could then build on this understanding to point the way towards a generalized approach to the automated design of procurement networks.

8.1.1 Focus and Outline of the Chapter The focus of this chapter is to (1) show how the procurement network formation problem can be modelled as a cooperative game in both the complete and incomplete information settings and (2) review some of the existing results in cooperative game theory and see how they relate to the procurement network formation problem. The structure of the chapter is as follows. In Sect. 8.2, we review the various techniques that have been used to address the problem of automated formation of procurement networks in particular and supply chain networks in general. This section provides the motivation for us to model the procurement network formation problem as a shortest path cooperative game. We introduce this as the Procurement

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Network Formation (PNF) game in Sect. 8.3. In Sect. 8.4, we study the complete information version of the PNF game. Here, we discuss some of the results that are germane to the PNF game, and also present a numerical illustration. In Sect. 8.5, we extend the analysis to PNF games with incomplete information. Here, we explore the scenario where agents are assumed to know only their own private information and not that of the others. In many real-world cases, this is a more natural way of modelling the problem. In Sect. 8.6, we summarize the discussion by providing pointers to future work that is required before the models can be moved from theory to practice.

8.2 Extant Literature The problem of PNF has been variously addressed by researchers under the topic of supply chain formation and hence we use these terms interchangeably in the review of literature. Broadly, the methodological approach to address this problem has followed two tracks: the first one uses techniques based on competitive equilibrium analysis (Walsh and Wellman 1999, 2003) and the second makes use of auction technology (Babaioff and Nisan 2004; Walsh et al. 2000; Walsh 2001; Babaioff and Walsh 2005; Jiong and Sadeh 2004; Collins et al. 2002; Collins 2002; Fan et al. 2003).

8.2.1 Competitive Equilibrium Models In the competitive equilibrium analysis of (Walsh and Wellman 1999, 2003), the authors address the supply chain formation problem from the perspective of a thirdparty market maker. This market maker is interested in constructing the supply chains of multiple end customers given the fact that they could potentially share some common suppliers. They construct a network of possible supply chain configurations and use a price-directed search for a feasible supply chain configuration, i.e. one that maintains material balance and profitability. The price-directed search leads to what is essentially an approximate competitive equilibrium, which is in line with standard results in competitive equilibrium theory for indivisible goods. They describe two distributed protocols called SAMP-SB (Simultaneous Ascending (MC1)st Price with Simple Bidding) and SAMP-SB-D (SAMP-SB with decommitment) to find the equilibrium prices along with the supply configuration. To do this, they run a series of simultaneous two-sided auctions for each of the goods to be produced/consumed in the supply chain. The competitive equilibrium that they reach allows them to guarantee that no single active agent in the formed network will find it beneficial to move away from the recommended solution. This equilibrium-based approach is predicated on the assumption that there are a large number of agents who are part of the network and hence the scope for strategic behaviour by the agents is limited. While it may be true that a large number of suppliers need to be

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coordinated along the supply chain, in many real cases, we find that there are a limited number if not just one supplier for a particular commodity. This leaves ample scope for strategic behaviour and hence cannot be assumed away.

8.2.2 Concurrent Auction Models In Babaioff and Nisan (2004), the authors consider a linear supply chain, where a commodity market exists for each of the goods – both final and intermediate, that can be traded. The supply chain formation protocols that are discussed essentially build on Double Auction (DA) rules, where buyers and sellers submit asks and bids simultaneously. The protocol as such governs the construction of supply and demand curves at each of the markets (supply, intermediate, and final). In each market, after the supply and demand curves are constructed, a DA rule is invoked to finalize the traded quantities and the prices. The characteristics of the supply chain formation protocol are obtained from the characteristics of the underlying DA rule that is used in market making. If the DA rule is strategy-proof, efficient, individually rational but not budget balanced, then so is the supply chain formation protocol. This happens when the DA payment rule is modeled after the Vickrey–Clarke–Groves (VCG) payment rule. With such a rule, the market maker subsidizes the formation of the supply chain. If, however, the DA rule is strategy-proof, budget balanced and individually rational but is not ex-post efficient, then these characteristics are reflected in the supply chain formation protocol too. This is achieved by using a Trade Reduction (TR) rule (first pointed out by McAfee (1992)) or a variant of it. The authors present two randomized (DA) rules, which are essentially randomizations between the VCG rule and the TR rule. With these rules, the supply chain formation protocol is able to achieve ex-post individual rationality and incentive compatibility along with budget balance and suffers from only a slight loss of efficiency. This idea is further extended in Babaioff and Walsh (2005) to more general supply chain structures, where two or more goods may be used for making a single good.

8.2.3 Combinatorial Auction Models Another approach to modelling the supply chain formation problem has been to use combinatorial auction/exchange technology. By using combinatorial auction/exchange technology, some of the problems related to uncoordinated action across the supply chain are avoided; specifically situations where intermediaries are alloted input materials but are not alloted contracts for outputs are avoided (deadlocks). In Walsh et al. (2000), the supply chain formation problem is modelled for single-minded supply chain agents (buyers and sellers), where single minded refers to the fact that the agents are interested in one particular bundle of goods alone. The agents submit bids for bundles of goods to a central auction/exchange. The

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auction/exchange then solves a combinatorial problem and indicates the allocations. Without invoking Vickrey payment rules, it is quite evident that the agents would resort to strategic bidding. However, the analysis of bidding behaviour to obtain closed-form solutions corresponding to Bayesian–Nash equilibrium turns out to be difficult. So, it is assumed that agents will bid so as to garner a share of the expected surplus. This is done by choosing an averaged bid based on Monte Carlo simulations of the costs of other producers, its own costs, and the valuations of buyers. Experimental results on a variety of networks show that when the surplus is large then the combinatorial auction/exchange technology for supply chain formation outperforms the SAMP-SB protocols. In addition, it has the added advantage of not selecting deadlocked networks. In Fan et al. (2003), the supply chain formation problem is modelled as a multicommodity flow problem. Standard linear programming techniques, such as LP decomposition, are the usually used techniques to solve such problems. However, since the cost functions are under the control of agents who solve the decomposed problems, the authors argue that a more natural way to solve the problem is to use a combinatorial auction-based protocol. Here, the supply chain agents bid for bundles of goods that they are interested in. The combinatorial auction is constructed to minimize the cost of forming the supply chain while meeting the demands of the customer. The authors make the following key assumptions in their approach to the problem: (1) all customer orders are expected to be serviced. In reality at any given instance, a supply chain wishes to pick the set of orders to be serviced and the required network of supply partners so that the surplus of the supply chain is maximized. (2) the resources are considered to be divisible. This enables them to compute prices for individual goods, and (3) the number of agents is assumed to be very large. This allows them to argue that since the benefits that the agents may expect to get by indulging in strategic bidding is very small, there is very little incentive for them to actually do so, and hence they would be better off reporting their true valuations to the auction mechanism. However, most supply chain formation problems in reality need to be addressed within a context, where there are a limited number of agents available for each of the goods that need to be acquired or transformed through the supply chain.

8.2.4 Summary of Extant Literature and Contributions To summarize the current state of the art and the emerging needs in supply chain formation, we have identified the following key issues: In all the literature that we have seen to date, the approach is to solve the supply

chain formation problem from the perspective of a third-party market maker. In our view, it would be beneficial to view the supply chain formation from the perspective of a supply chain leader, i.e. one who orchestrates the supply chain, since this is in line with trends in actual business.

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In addition, supply chain practitioners and researchers have long argued that a

supply chain is created and sustained when it is able to create a surplus value and share this surplus with its supply chain partners. These two points when taken together imply that models for procurement network formation should explicitly include the aspect of surplus sharing, where the market maker neither subsidizes nor profits from the trade. In facilitating such trade, the assumption that agents will bid truthfully because of competitive forces as is assumed in Walsh et al. (2000); Walsh and Wellman (2003); Fan et al. (2003) may be unrealistic to expect since at many levels in the supply chain, we are faced with monopolistic or oligopolistic situations where the agents can gain from indulging in strategic behavior. So, models for supply network formation must explicitly take into account the possibility of strategic action by suitably engineering the incentives to promote truthful behaviour. Motivated by these gaps in current research and insight into the domain, we believe that the procurement network formation problem may be explicitly analyzed as a surplus sharing network formation problem, where we provide for strategic behaviour by the agents who own parts of the network. Problems in surplus sharing and cost sharing are replete with ideas from cooperative game theory. Accordingly, our contributions in understanding and formulating the PNF problem are as follows: 1. We cast the PNF problem as a shortest path cooperative game. We examine two scenarios for this game: (1) the PNF game with complete information and (2) the PNF game with incomplete information. 2. We discuss two models for the PNF game with complete information. We examine the core as a solution concept for this game and explore conditions for its nonemptiness in both models. 3. We then extend the analysis to PNF games with incomplete information. Here, we review the current state of the art in using the interim incentive compatible core as an analogous solution concept, and point out the important directions in which further research is required.

8.3 Procurement Network Formation as a Shortest Path Cooperative Game To illustrate the PNF problem, we consider a linear supply chain through which a product can be manufactured. For example, consider an automotive assembler who wishes to acquire a subassembly required for assembly in a car. The assembler attaches a certain value to this acquisition. Assume that the assembly goes through three distinct stages of value addition – (1) casting manufacture, (2) machining and (3) transportation. In each of these stages, there may be multiple suppliers, each with their own costs of manufacture and supply. Given the options available at each stage of manufacture, a feasible network may be constructed as shown in Fig. 8.1, where each path in the network indicates one possible way of acquiring the subassembly.

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Fig. 8.1 A sample procurement network

Each arc in the path is owned by an agent who incurs a certain cost for the use of the arc. So, the length of the path is given by the summation of costs of each of the arcs in that path. A surplus is created when the path length is less than the value attached to the product by the buyer. This surplus is maximized when we can find a shortest path through the network. In line with the earlier discussion, we seek to divide the surplus among the agents of the shortest path and possibly the assembler in a fair manner. If we assume that both the valuation of the buyer and the costs of the arcs are completely known, then the problem may be modelled as a surplus sharing shortest path cooperative game with complete information. For this class of games, the core is a well-accepted solution concept. It is based on the premise that a group of agents can cooperate and agree upon a coordinated set of actions which results in an outcome that is both feasible and can be enforced. For the simpler transferable utility version of the cooperative game, this outcome creates a certain value which is to be shared among the cooperating agents in a fair manner. Finding a division of this value so that it is in the core signifies stability in the procurement network that will be formed. Unfortunately however, the core may sometimes turn out to be empty. So, finding conditions under which the core is nonempty is a pertinent issue. So far, in our discussion of the PNF problem we have assumed that cost and valuation information of all agents in the network is commonly known to everyone. In practice, however, this information is almost always incompletely known, in the sense that each agent is only aware of his or her own costs and valuations but not that of the others in the network. It is then appropriate to view the PNF problem as a shortest path cooperative game with incomplete information. A solution to this game should then be a rule that takes the costs and valuations reported by the coop-

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erating agents as input and indicates an outcome. The outcome indicates the path to be selected and the share of the surplus that each of the agent gets. So, the solution to the game consists of a rule for allocations of surplus value rather than the allocation of surplus value per se (Wilson 1978). Such a rule is also called a state contingent contract in cooperative game theoretic literature or a mechanism in mechanism design theory and welfare economics. For the purpose of this chapter, we will use these terms interchangeably. Conceptually however, a mechanism should be seen as a way to implement a rule or a state contingent contract. In identifying such a state contingent contract, we would be naturally well placed if it were to satisfy the following requirements: First, the state contingent contract should be:

– Incentive Compatible: A state contingent contract is incentive compatible if the agents maximize their expected utilities by revealing their true costs and valuations. Depending on the equilibrium achieved by truthful bidding, an incentive compatible mechanism is qualified as Bayesian Nash incentive compatible or dominant strategy incentive compatible. – Efficient: A general criterion for evaluating a state contingent contract is Pareto efficiency, meaning that no agent could improve its allocation without making at least one other agent worse off. When the utility functions take a special form (such as quasi-linear form Mas-Collel et al. 1995), Pareto efficiency implies allocative efficiency i.e. the total utility of all the winners is maximized. – Individually Rational: A state contingent contract is individually rational if its allocations do not make any agent worse off than had the agent not participated in the contract. – Budget Balanced: A state contingent contract is said to be weakly budget balanced if the sum of monetary transfers between the buyer and the sellers is non-negative while it is said to be strongly budget balanced if this sum is zero. Second, for the solution to be fully analogous to the solution of the PNF game

with complete information, we would like the state contingent contract that is chosen to be in an appropriately defined core of the cooperative game with incomplete information. This will also ensure stability of the network.

8.4 The PNF Game with Complete Information A non-trivial PNF game with complete information arises when it is feasible for agents to cooperate and find a path through the network so that a positive surplus is created. This surplus is to be divided among the cooperating agents in a fair manner. Here, we discuss models for two scenarios: In the first, we assume that the agents who can provide the shortest path know the valuation of the assembler for acquiring the commodity and they will bargain over division of the surplus that

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Table 8.1 Notation for the Procurement Network Formation (PNF) game with complete information G Directed acyclic graph V Set of vertices of G E Set of arcs/edges in G Origin vertex vo vt Terminal vertex c.e/ Cost of using edge e I Set of agents who own arcs in G S A subset of agents from I P A shortest path from vo to vt in G A shortest path from vo to vt in G owned by agents in S PS C.P / Cost of path P C.PS / Cost of path PS owned by agents in S b Valuation of the buying agent for a path from vo to vt vb Characteristic function of the cooperative game with valuation b of the buying agent Cooperative game with set of agents I and characteristic .I; vb / function vb C.vb / Core of the game .I; vb /

can be created. Hence, only the agents who provide the path are game theoretic agents and they will receive a share of the surplus that arises from a transaction. We present results for this class of games and interpret them in the context of the PNF problem (see Table 8.1). In the second scenario, we assume that the assembler too has bargaining power and can negotiate with the arc-owning agents to get a share of the surplus. This is then a typical market scenario where a bargaining model includes the assembler also as a game theoretic agent. The analysis of this model gives us an insight into the kind of results that we may expect to obtain for the more realistic incomplete information version of the problem.

8.4.1 Model 1: Arc Owners are the Only Game Theoretic Agents As indicated earlier, different possible ways of forming the procurement network may be captured as a directed graph G D .V; E/ with V as the set of vertices, two special nodes vo (origin vertex) and vt (terminal vertex), and E V V as the set of edges. With each of the edges e 2 E is associated a number c.e/ representing the cost of the arc. A path P in the network connecting the two special nodes vo and vt is a collection of edges f.v Po ; v1 /; .v1 ; v2 /; : : : ; .vt 1 ; vt /g from E. The cost C.P / of the path P is given by ti D1 c.vi 1 ; vi /. The path P is a shortest path from vo to vt if there is no other path PO such that C.PO / < C.P /. Now, assume that each of the arcs is owned by an agent i who belongs to a finite set of agents I . That is o W E ! I such that o.e/ D i implies that the arc e is owned by agent i . For any

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path P , we denote the set of owners of the path P as o.P /. We assume that if the product is processed through the path P , then it costs C.P / and the assembler is willing to compensate the arc owners with a value b. So, the surplus from such a transaction, is b C.P /, which is to be shared among the agents who own arcs in the network in a fair manner. Now suppose that a coalition of agents S I can process the subassembly only through arcs owned by them. So, we let PS be the shortest path owned by agents in S and its cost is C.PS /. We can now describe the shortest path cooperative situation as .I; vb /, where I is the set of agents and vb is the characteristic function given by:

vb .S / D

8 < b C.PS / :

0

: :

if S owns a path that is profitable in the network, i.e. C.PS / < b otherwise

(8.1)

for every S I , where C.PS / is the length of the shortest path owned by coalition S . A shortest path game is now given in the characteristic form .I; vb /. A solution to this game is an allocation of the surplus to the agents such that it is in the core (C.vb /). Recalling the discussion of the core in Sect. 8.3, where we said that an allocation in the core is such that no subset of agents finds it more profitable to move away from and negotiate an outcome separately, we can formally state it as follows: ( ) X X I xi D vb .I /; xi vb .S /; 8S I : C.vb / D x 2 R jxi v.i /; 8i 2 I; i 2I

i 2S

(8.2) From Scarf’s (1967) result, we know that the core of a cooperative game is nonempty if and only if it is balanced. A cooperative game is said to be balanced, if the following conditions hold: A collection C D fS1 ; S2 ; : : : ; SK g of subsets S I is a balanced collection if

P we can assign non-negative numbers ıS such that S2CIS3i ıS D 1 for every i 2 I. A game is said to be balanced if for every balanced collection, we have P SI ıS :vb .S / v.I /. The question now is whether the core of this game is non-empty so that an allocation in the core of the game may be found.

8.4.1.1 Key Results A key point to note in this model is that the valuation of the buying agent is completely known and the division of the surplus is only happening between the agents

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who own arcs in the network. In this model, the notion of shortest veto agents plays an important part and is defined as given below: Definition 1. For any shortest path cooperative game situation, a shortest veto player (s-veto player) is a player who owns at least one arc of every shortest path in the cooperative game. Fragnelli et al. (2000) provide two conditions under which the core is non-empty for this class of shortest path cooperative games. These are given by the propositions below. Proposition 1. For any shortest path cooperative game .I; vb / either it is balanced for all b 2 Œ0; C1/ or there exists B 2 Œ0; C1/ such that .I; vb / is balanced if and only if b B. Proposition 2. For any shortest path cooperative game .I; vb / with a set of s-veto agents S , if x is an allocation such that xi 0 for some i 2 I n S then x 62 C.vb /. Proposition 3. For any non-trivial shortest path cooperative game .I; vb / with a set of s-veto agents S , we define a new game .S; vQ b / with the set of s-veto agents Q as the S players and a characteristic function vQ b given by vQ b .;/ D 0 and vQ b .S / D v.SQ .I nS // for every non-empty SQ S . The core of the game is non-empty if and only if the following two conditions hold: (1) S is non-empty and .S; vQ b / is balanced. (2) Every profitable path in the cooperative game contains an arc owned by an s-veto agent. Proposition 1 above indicates a result that is fairly counterintuitive, i.e. when the surplus increases, the stability of the game can be lost. Ideally, we should expect that when the surplus increases there is more for everyone to share and hence all agents should be satisfied in staying in the coalition. However, since the agents are utility maximizers, when the surplus increases, if there are opportunities for this increased surplus to be disproportionately divided, then agents who would gain from this would embrace strategies that would allow them to do so. This can then result in instable allocations and hence non-emptiness of the core. Propositions 2 and 3 together imply that the solution to a PNF can be stable if and only if the s-veto suppliers occupy positions in the feasible network that are indispensable to the formation of the procurement network. We now present a numerical example based on Fig. 8.1 to demonstrate the results.

8.4.1.2 An Example for Model 1 Consider the example as given in Fig. 8.1. The labels on the nodes indicate the owner of the arc originating from that node and the label on the arc indicates the cost of using the arc. The set of agents I here are indicated by the labels on the nodes.

196 Table 8.2 Characteristic value function

T.S. Chandrashekar and Y. Narahari vQb .T 2/ vQb .C 3/ vQ.T 2; C 3/

0 0 0.5

We assume that the assembler is willing to pay $3.5 for a path through the network. By inspection, the set of s-veto agents S is S D fT 2; C 3g. The cooperative game defined on the set of s-veto agents is given by the characteristic function in Table 8.2 and an allocation in the core may be found by solving the following linear programming problem. Minimize xT 2 C xC 3 subject to: xT 2 C xC 3 0:5 xi 0, 8 i 2 fT 2; C 3g. One possible solution to the above LP and hence an allocation of surplus value is given by .xT 2 D 0:25; xC 3 D 0:25/. In fact, any allocation that splits the surplus value between agents T 2 and C 3 with all other agents in the network receiving zero allocations is in the core of the game. This is because only agents T 2 and C 3 have the bargaining power to use the competitive forces between M 2 and M 3 to reduce their allocation to zero. The example above demonstrates clearly how the power structure in the network plays itself out in forming the shortest path and dividing the surplus value. We have, in this model however, not considered the bargaining power of the buying agent. We incorporate this in the next model.

8.4.2 Model 2: The Arc Owners as Well as the Buyer are Game Theoretic Agents When the buying agent is also included as an active agent in the network, the surplus of the transaction is to be divided among the agents who own arcs i D 1; 2; : : : ; n as well as the buying agent .n C 1/. We designate an allocation of the surplus as .x1 ; x2 ; : : : ; xn ; x.nC1/ /. An allocation of the surplus that is in the core of the game can be found if non emptiness of the core is guaranteed. This can be shown through simple linear programming techniques. Given below in Table 8.3 is a linear programming formulation of the core allocation problem and it is dual. The primal variables xi for all i 2 I [ f.n C 1/ correspond to core allocations and the dual variables are ıS for all S I [ f.n C 1/g. Now, from Scarf’s (1967) theorem, we know that a game has a non-empty core if it is balanced. It is easy to verify that the above game is balanced using the following facts: (1) only coalitions that contain the buying agent can create surplus value and (2) the value created by any coalition cannot be more than the value that is created by the grand coalition. Using these facts it can be shown that for any balanced

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Table 8.3 Primal and dual formulations for the core of the PNF game Primal problem Dual problem PnC1 P Minimize iD1 xi Maximize SI [f.nC1/g ıS v.S/ subject to: subject to: P P i2S xi v.S/, 8S I [ f.n C 1/g S3i ıS D 1, 8i 2 I [ f.n C 1/g

P collection of subsets of the agents, we have SI ıS :vb .S / v.I /, which is the condition for balancedness of the game. Hence, this game has a non-empty core. For the example in Fig. 8.1, when the buying agent is also considered as a game theoretic agent, it can easily be shown that .xnC1 D xT 2 D xC 3 D 0:167/ is an allocation in the core. In fact, any allocation that strictly divides the surplus among these three agents in any manner is in the core of the game. The core of this version of the shortest path cooperative game with complete information when the buying agent is also part of the game has been shown to be non-empty without taking recourse to the notion of s-veto agents. We capture this as Remark 4 below. Remark 4. For any shortest path cooperative game where the arc owners as well as the buying agent are game theoretic agents, the core is non-empty. This should provide a hint as to what we should expect to find as a solution to the incomplete information version of the problem. That is, we should explore the possibility of designing a state contingent contract that does not depend upon a priori listing of agents with all the bargaining power in the network. We explore this question in the next section.

8.5 The PNF Game with Incomplete Information In the previous section, we have considered models where the valuation and the costs are completely known to all the agents in the network and cooperation is achieved when agents can agree upon an appropriate division of the surplus. In many cases however, cost and valuation information is only privately known to individual agents. Cooperation in this context means that agents need to agree upon a state contingent contract as discussed in Sect. 8.3. To model this situation, we begin by setting out the basic notation required and then present the structure of the PNF game with incomplete information (see Table 8.4).

8.5.1 The Model We consider a directed graph G = .V; E/ with V , the set of vertices, E the set of edges and two special nodes vo (source) and vt (target). We let n D jEj be the number of edges. Each edge in the network is assumed to be a commodity that is owned by an agent. For expositional clarity, we assume that the number of agents

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T.S. Chandrashekar and Y. Narahari Table 8.4 G V E vo vt n I C

S ei eij eQi ti Ti tS TS XS .ts / XS X ui .x; t / S ˘ ˘S

Notation for the PNF game with incomplete information Directed acyclic graph Set of vertices of G Set of arcs/edges in G Origin vertex Terminal vertex Number of edges Set consisting of the buying agent and agents who own arcs in G Set of all possible coalitions of I A subset of agents from I Initial endowment vector of agent i Component of initial endowment vector ei indicating the ownership of resource j by agent i Final endowment vector of agent i Type of agent i Set of all possible types of agent i Vector of types of agents in S Set of vector of types of agents in S Set of all possible outcomes for coalition S when type vector is tS Set of all possible outcomes achievable by coalition S Set of all possible outcomes Utility of agent i for outcome x when the type profile is t Randomized mechanism for coalition I Randomized mechanism for coalition S Set of all randomized mechanisms for coalition I Set of all randomized mechanisms for coalition S

is equal to the number of edges in the network. However, any analysis that follows can be extended to cases where agents own multiple edges. We therefore let I D f1; 2; : : : ; n; n C 1g be the set of all agents, where 1; 2; : : : ; n are edge owning agents and agent .n C 1/ is the buying agent.

8.5.1.1 The Notions of Endowment Vectors and Information Types We assume that each agent i 2 I has an initial endowment that indicates the edge(s) owned by him/her and the amount of monetary resource that he/she currently holds. Formally, we represent this with a vector ei 2 f0; 1gn 0, we can always reduce it to zero by increasing Q to QCq, and thereby increase the objective value. Similarly, when vS > vM , we must have Q D 0 in the optimal solution. Hence, combining the two cases, we have the following objective, for the integrated supply chain: maxQ GI .Q/WD.r C pM m/Q.r C pM max.vM ; vS //EŒQDC pM ; (9.7) from which the optimal order quantity QI follows as: QI D F

1

r C pM m : r C pM max.vM ; vS /

(9.8)

Note that in the integrated supply chain, the first two equations in (9.1) reduce to one: m max.vM ; vS /:

(9.9)

That is, as there is no w0 in the integrated supply chain, we assume w0 D m. Also note that, in general, we have QI Q0 . Observe that the objective function in (9.7) relates to the objective functions in (9.2) and (9.4) as follows: NV GI .Q/ GM .Q/ C GSNV .Q/:

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Since QI maximizes the left-hand side, we have NV GI .QI / GM .Q0 / C GSNV .Q0 /:

(9.10)

That is, the profit of the integrated system dominates the sum of the manufacturer’s profit and the supplier’s profit. Channel coordination is achieved when (9.10) holds as an equality, i.e., when the supplier and the manufacturer make decisions individually in a decentralized manner, but the sum of their individually maximized profits is equal to that of the integrated supply chain. Generally, the combined expected profits of the two parties, when decisions are made individually, are lower than that of the integrated supply chain. This fact is known as the double marginalization effect (see, e.g., Cachon 2004a). We will show how the double marginalization effect can be minimized by adopting a profit-sharing model in Sect. 9.5.

9.4 The Option Model We shall focus here and the next section on the call option model, as the put option can be related to the call option through a parity relationship established in Sect. 9.6. The call option works as follows. At the beginning of the period, the manufacturer places an order of quantity Q, paying a price of w0 for each unit. Note w0 is exogenous and cannot be changed after the order is placed. In addition, the manufacturer can also purchase from the supplier q (call) option contracts, at a cost of c per contract. Each option contract gives the manufacturer the right (but not the obligation) to receive an additional unit, at a cost w (exercise price of the option), from the supplier at the end of the period after demand is realized. Under this arrangement, the supplier is committed to producing the quantity Q C q. The supplier can salvage any unexercised options at the end of the period at a unit value of vS . Clearly, this call option includes as a special case the existing practice of adding quantity flexibility to supply contracts, which will allow the buyer to order additional units, at a premium (corresponding to the option exercise price), up to a certain limit (corresponding to the number of option contracts), after the initial order is placed. (In the case of the put option, the manufacturer will have the right to sell, i.e., return, to the supplier any surplus units, up to q, at the exercise price, after demand is realized. The put option generalizes the existing practice of buy-back contracts, as discussed in Sect. 9.6.) We shall assume that the relations, c C w w0 ;

c C vM w0 ;

r C pM c C w;

(9.11)

always hold. If the first inequality is violated, it would cost less to buy a unit via option than to place a regular order, which would make the regular order useless. If the second inequality is violated, i.e., if w0 vM < c, then the option plan is never worthwhile, since buying a unit up front and (in the worst case) salvaging it

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later is less costly than buying an option contract. As to the third inequality, consider the case of pM D ws r, where ws is the unit price from the spot market. Then r C pM c C w reduces to ws c C w; otherwise, the spot market will make the option plan superfluous. The determination of c; w; Q, and q is the result of the supply contract negotiation or bargaining process between the supplier and the manufacturer. This bargaining process can be modeled as a Stackelberg game, in which the supplier is the Stackelberg leader, meaning that the supplier will optimize its own profit when it decides on c and w while the manufacturer has to accept the prices offered and optimize its decision on Q and q based on these values. We further assume that both parties are rational, self-interested, and risk neutral (expected value maximizers). In the next two sections, we study the optimal decisions of the manufacturer and the supplier, respectively.

9.4.1 Manufacturer’s Order Decisions The manufacturer’s decision variables are .Q; q/, so as to maximize the total expected profit: GM .Q; q/ WD rEŒD ^ .Q C q/ C vM EŒQ DC wEŒ.D Q/C ^ q pM EŒD Q qC w0 Q cq D pM C .r C pM w0 /Q C .r C pM w c/q .r C pM w/EŒQ C q DC .w vM /EŒQ DC : (9.12) In this case, the total supply is at most Q C q (hence the terms weighted by r and pM ), and the q option contracts cost cq up front, plus w for each one exercised after demand is realized (hence the term weighted by w). Note that if we let q D 0, then the above reduces to the base model in (9.2). We can write the above objective as: GM .Q; q/ D pM C .r C pM w0 /Q C .r C pM w c/q Z QCq Z Q .r C pM w/ F .x/dx .w vM / F .x/dx: 0

(9.13)

0

It can be shown that (9.13) is concave w.r.t. Q and q. Hence, we use its first-order conditions to obtain 1 c C w w0 ; (9.14) QDF w vM

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q D F 1

r C pM w c r C pM w

Q DW QQ Q:

217

(9.15)

For the above to be well defined, in addition to the relations assumed in (9.1) and (9.11), we require c C w w0 r C pM w c ; w vM r C pM w which reduces to: .r C pM vM /c C .w0 vM /w .r C pM /.w0 vM /:

(9.16)

Proposition 1 (Properties of the Manufacturer’s Objective Function): The objective function in (9.13) is (jointly) concave in .Q; q/. Consequently, the manufacturer’s optimal decisions on .Q; q/ are as follows: (i) if (9.16) holds as a strict inequality, then the optimal Q and q follow (9.14) and (9.15); (ii) otherwise, the optimal Q D Q0 in (9.3) and q D 0. In case (ii) of Proposition 1, the manufacturer has no incentive to adopt the option model. The expected profit of the manufacturer and the supplier in this case is the same as in the newsvendor model. Remark (Properties of the Manufacturer’s Optimal Decision): There are two special cases of Proposition 1 (ii) that warrant special attention: Q undefined in (9.15). However, substituting .c; w/ D .0; r CpM /, which makes Q this into (9.13) reduces the latter to the newsvendor objective function. Hence, the optimal solution is Q D Q0 and q D 0. .c; w/ D .w0 vM ; vM /, which makes Q undefined in (9.14). Again, substituting this into (9.13) makes the objective interchangeable in Q and q. Hence, the optimal solution in this case is either Q D Q0 and q D 0 or Q D 0 and q D Q0 , or any point in between. Proposition 2. The manufacturer’s optimal decisions, .Q; q/, satisfy the following properties: (a) Q is increasing in .c; w/, q is decreasing in .c; w/, and QCq is also decreasing in .c; w/. (b) Q Q0 Q C q, where Q0 is the newsvendor solution in (9.3). Furthermore, the expected profit of the manufacturer is decreasing in .c; w/. Compared with the base case (no flexibility), the manufacturer’s expected net profit is no less in the flexibility model. It is strictly higher if the inequality in (9.16) holds as a strict inequality.

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9.4.2 Supplier’s Pricing Decisions The supplier’s objective function is given by max GS .c; w/ WD w0 Q C cq m.Q C q/ C wEŒ.D Q/C ^ q CvS EŒq .D Q/C C D .w0 m/QC.cCvS m/qC.wvS /

(9.17) Z

QCq

FN .x/dx:

(9.18)

Q

Note that if and when Q D Q0 and q D 0, i.e., the manufacturer takes the newsvendor solution, then the supplier’s profit also becomes equal to that in the newsvendor model: GS D .w0 m/Q0 D GSNV .Q0 /: In addition, the decision variables .c; w/ must satisfy the following constraints, in view of (9.11) and (9.16): c w0 vM ;

(9.19)

c C w w0 ;

(9.20)

.r C pM vM /c C .w0 vM /w .r C pM /.w0 vM /:

(9.21)

Note that the last inequality in (9.11) is superseded by the stronger (9.21), since cCw

c.r C pM vM / C w r C pM ; w0 vM

where the first inequality follows from r C pM w0 (refer to (9.11)), and the second inequality is (9.21). The supplier treats .Q; q/ as functions of .c; w/. Specifically, .Q; q/ will follow the optimal solutions from the manufacturer’s model in (9.14) and (9.15). Note that if the supplier knows that the manufacturer uses a Gaussian model to forecast demand, then knowing the manufacturer’s order decisions .Q; q/ is equivalent to knowing the demand distribution – the two parameters of the Gaussian distribution, its mean and variance, are uniquely determined by Q and q via (9.14) and (9.15). We rewrite the objective function in (9.18) as follows: Z QCq F .x/dx GS .c; w/ D .w0 m/Q C .c C w m/q .w vS / Q

D .w0 w c/Q C .c C w m/QQ .w vS /

Z

Q Q

F .x/dx: Q

(9.22)

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Taking partial derivatives of the objective function w.r.t. c and w, we have: @GS D q C Œw0 w c C .w vS /F .Q/Qc0 @c Q QQ c0 ; CŒc C w m .w vS /F .Q/ @GS D q C Œw0 w c C .w vS /F .Q/Qw0 @w Z 0 Q Q CŒc C w m .w vS /F .Q/Qw

(9.23)

Q Q

F .x/dxI

(9.24)

Q

where Qc0 , QQ c0 , Qw0 , and QQ w0 denote the partial derivatives of Q and QQ w.r.t. c and w. d Let f .x/ WD dx F .x/ denote the probability density function of demand (whenever it exists). From (9.14) and (9.15), we have Qc0 QQ c0 Qw0 QQ w0

D Œ.w vM /f .Q/1 ; Q 1 I D Œ.r C pM w/f .Q/ 0 N D Qc F .Q/; Q D QQ c0 FN .Q/:

Substituting the last two equations into (9.24), we have @GS D q C Œw0 w c C .w vS /F .Q/FN .Q/Qc0 @w Q FN .Q/ Q QQ c0 C Œc C w m .w vS /F .Q/ Since

@GS @c

Z

Q Q

F .x/dx: Q

D 0 implies

Q QQ c0 D qŒw0 w cC.w vS /F .Q/Qc0 ; (9.25) ŒcCw m .w vS /F .Q/ we have Z QQ @GS 0 Q Q CqF . Q/ F .x/dx: DŒw0 w c C .w vS /F .Q/ŒFN .Q/ FN .Q/Q c @w Q Furthermore, from (9.14), we have w0 w c C .w vS /F .Q/ D .vM vS /F .Q/: Hence,

@GS @w

D 0 takes the following form:

Q .vM vS /F .Q/ŒF .Q/

" F .Q/Qc0

Q C qF .Q/

Z

Q Q

Q

# F .x/dx D 0:

(9.26)

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Note that when vM vS , the first term on the left-hand side above is non-negative (Qc0 0 follows from Proposition 2), and so is the other term. Hence, when S vM vS , we have @G @w 0. Consequently, the supplier will prefer a w value as large as possible, only to be constrained by the inequality in (9.21). However, if this inequality holds at an equality, we know the manufacturer will forego the options, leaving the supplier with no additional profit beyond the newsvendor model. Hence, the supplier will set the c value close to zero, and the w value just slightly below the spot price r C pM . This way, the left-hand side of (9.21) is slightly below to its right-hand side. Proposition 3. Given the demand distribution, the supplier’s optimal decision .c; w/ follows the two equations in (9.25) and (9.26) when vM vM , channel coordination is achieved at .c2 ; w2 /. (v) When vO D vM and vS < vM , we have Z GI GM S D .vM vS /

QI

F .x/dx: Q0

Note that when vS ¤ vM it is crucial where leftover inventory is salvaged. It will be impossible to achieve the integrated supply chain’s profit if the option contract results in some stock being salvaged at unfavorable terms. When vS > vM channel coordination is achieved if the manufacturer does not place a firm order and buys QI option contracts. In case (v), where vS < vM , any leftover inventory is salvaged at a lower price than in the integrated supply chain whenever realized demand is between Q and Q C q. However, the supplier cannot give sufficient incentive for the manufacturer to place a firm order of size QI . Since channel coordination is achieved on the whole line segment (9.28) if vS D vM , the coordinating contract is not unique; rather, a continuum of coordinating contracts exists, resulting in different profit improvements for the manufacturer and the supplier.

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Proposition 5. Suppose as in Proposition 4, the supplier’s decision .c; w/ falls on the line in (9.28). Then, the expected profit of the manufacturer is decreasing in w and increasing in c; whereas the supplier’s expected profit is increasing in w and decreasing in c. Proposition 5 corresponds to Theorem 7 of Lariviere (1999), where the results are established in the context of buy-back contracts. Two interesting observations can be made. First, option contracts on the line segment (9.28) do not depend on the demand distribution, just like coordinating buy-back contracts in Lariviere (1999). Second, the manufacturer prefers greater flexibility with larger values of q, whereas the supplier’s interest is just the opposite. In other words, each party wishes, quite naturally, to avoid as much as possible the risk of keeping excess stock. When of vS D vM , a continuum of contracts exists that achieves channel coordination. This result, along with Proposition 5, further explains the phenomenon alluded to in Proposition 3, i.e., why the supplier wants to push c as low as possible so as to capture all the additional profit, but cannot quite set c to zero since this would revert to the no-flexibility newsvendor solution. From the last two propositions, we know that when the supplier’s decision falls on the line in (9.28), the manufacturer’s expected profit is guaranteed to be no worse than its newsvendor solution, since the worst for the manufacturer happens at the end point .c1 ; w1 / D .0; rM C pM /, where it opts for the newsvendor solution. The same, however, cannot be guaranteed for the supplier. Its worst case happens at the end point .c2 ; w2 /, which corresponds to, letting Q D 0 and q D QQ D QI in (9.22), Z GS .c2 ; w2 / D .c C vS m/QI C .w vS /

QI

D .m vS /FN .Q0 /QI C .w vS / Z .m vS / Z D .w m/

QI

0 QI

FN .x/dx

0

Z

QI

FN .x/dx

0

FN .x/dx C .w vS /

Z

QI

FN .x/dx

0

FN .x/dx

0

.w m/E.D/: When Q0 E.D/ D , a very likely scenario, then the above expression is dominated by the supplier’s newsvendor profit, .w0 m/Q0 . Therefore, to synthesize the above discussion, we propose that the supplier and the manufacturer work out an agreement to share their total expected profit GM S as follows: the supplier receives ˛GM S and the manufacturer receives .1 ˛/GM S ,

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with ˛ 2 .0; 1/ being a parameter agreed upon by both parties, which must be such that both parties do no worse than the newsvendor solution. That is, ˛GM S GSNV

and

NV .1 ˛/GM S GM :

(9.29)

From the above, we have the following boundary conditions on ˛ NV G NV GMS GM WD ˛u ˛ ˛l WD S : GMS GMS NV One reasonable way to determine ˛ is to require ˛=.1 ˛/ D GSNV =GM or,

˛D

GSNV : NV GSNV C GM

That is, each party receives the profit improvement proportionate to its profit in the newsvendor model. This will guarantee that ˛ satisfies the constraints in (9.29), NV since GM S GSNV C GM : To summarize, under this profit-sharing scheme, given the choice of ˛, the four decision variables, .c; w/ and .Q; q/, are determined by the following four equations: Equation (9.28) that relates c and w. Equations (9.14) and (9.15) relating Q and Q C q D QI to .c; w/. Equation GS D ˛GM S , where GS follows (9.18) and GM S follows (9.27).

Figure 9.1 confirms that channel coordination can be achieved when vS D vM . Here, the sum of optimal profits of the supplier and the manufacturer equals the profit of the integrated supply chain, and the two parties only need to decide how to split the total profit. Note that the ˛ value is determined through negotiation. As long as the ˛ value is within the feasible range where both parties are better off using the option solution than using the newsvendor solution, the two parties can always find the corresponding option price c and exercise price w such that the desired profitsharing scheme will be realized, i.e., the supplier’s expected profit equals ˛GM S and the manufacturer’s .1 ˛/GM S . When vS > vM , channel coordination can still be achieved if the feasibility condition (9.29) is relaxed. This means one can maximize the expected total supply chain profit GM S such that GM S D GI , at the price of the supplier’s expected profit falling below its newsvendor profit GSNV : Figure 9.2 illustrates such a case. However, given (9.29), channel coordination cannot be achieved in this case simply because the supplier has no incentive to do so. However, the two parties can still decide an ˛ such that both of them will be better off than using the newsvendor solution. When vS < vM , Fig. 9.3 shows that there is a gap between the expected profit of the integrated supply chain GI and the expected total profits of the two parties GM S . In this case, channel coordination simply cannot be achieved, even with the supplier’s profit falling below its newsvendor value.

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Fig. 9.1 Profit sharing model with vS D vM

Fig. 9.2 Profit sharing model with vS > vM

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5000

0.75

4000 0.7

Profit

3000 0.65 2000

0.6 1000

0

0.55

1 Supplier

2 Mfg

3

4

option price c

Supplier(NV)

Mfg(NV)

5

6 ISC

7 Total

Fig. 9.3 Profit sharing model with vS < vM

9.6 Put Options: The Put-Call Parity In the put option model, the manufacturer, in addition to the up-front order quantity Q with unit price of w0 , purchases q put option contracts at a unit price of p. Each such contract gives the manufacturer the right to return (i.e., sell back) to the supplier a surplus unit after demand is realized, at the exercise price of w. The supplier in this case is committed to producing the quantity Q and to taking back up to q units. As before, the supplier can salvage any returned units at a unit value of vS . Note that the put option contract is a generalization of the buy-back contract. With the buy-back contract, the supplier will buy any leftover units back from the manufacturer after demand is realized. This is equivalent to associating with every unit of the up-front order quantity Q a put option contract (at no additional charge), with the exercise price being the buy-back price. To distinguish the put option from the call option, below we shall write the manufacturer’s decision variables in the two models as .Qp ; qp / and .Qc ; qc /. The manufacturer’s objective is to maximize the following expected profit: GM .Qp ; qp / WD rEŒD ^ Qp C vM EŒQp qp DC C wEŒ.Qp D/C ^ qp pM EŒD Qp C w0 Qp pqp D pM C .r C pM w0 /Qp pqp .rCpM w/EŒQp DC .w vM /EŒQp qp DC:

(9.30)

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The supplier wants to maximize the following objective function: GS .p; w/ WD w0 Qp C pqp mQp wEŒ.Qp D/C ^ qp CvS EŒ.Qp D/C ^ qp C D .w0 m/Qp C pqp .w vS /ŒE.Qp D/C E.Qp qp D/C :

(9.31)

It turns out that the put option model relates directly to the call option model analyzed in the earlier sections through the following parity relation: Proposition 6. Suppose the following relations hold: c p D w0 w;

(9.32)

and Q p D Q c C qc ;

qp D qc :

(9.33)

Then, the objective functions in (9.30,9.31) of the put option model are equal to the objective functions in (9.12,9.17) of the call option model: GM .Qp ; qp / D GM .Qc ; qc /;

GS .p; w/ D GS .c; w/:

(9.34)

Applying the above proposition, the solutions to the put option model can be summarized as follows: Proposition 7. In the put option model, the optimal decisions for the manufacturer are: 1 r C pM w0 p Qp D F ; (9.35) r C pM w p : (9.36) qp D Qp F 1 w vM Consequently, the relations in (9.32) and (9.33) hold. In addition, the optimal decisions for the supplier follow those in the call option model, with the variables .c; w/ changed to .p; w/ following the parity relation in (9.32), with .Qc ; qc / replaced by .Qp ; qp / via (9.33). Note that the parity relation in (9.32) has the same form as the put-call parity of financial options, specifically, European options on stocks paying no dividend, with w0 being the stock price at time zero and w being the exercise price; refer to Hull (2002). (Here, we have ignored the discounting of the exercise price, which is paid at the end of the period, to time zero.)

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Also note that with the parity in (9.32), the inequalities (9.11) that characterize the parameters in the call option model change to the following, which now govern the parametric relations for the put option: p 0;

w p vM ;

r C pM w0 C p:

The inequality (9.21) takes the following form in the put option model: .r C pM w0 /w .r C pM vM /p .r C pM w0 /vM : Also, since c > 0 we deduce from (9.32) that w p w0 : Similarly, based on (9.32) and (9.33) and following Proposition 7, we can derive the supplier’s decisions for the put option by modifying the solutions in the call option models. The supplier’s optimal decision .p; w/ follows the following two equations, provided vM < vS : Œw0 Cp m.wvS /F .Qp /Qp0 D qp CŒp .w vS /F .Qp qp /.Qp0 qp0 /; and .vM vS /F .Qp qp /ŒF .Qp / F .Qp qp /.Qp0 qp0 / " # Z C qp F .Qp /

Qp

F .x/dx D 0I

Qp qp

where Qp0 and qp0 denote the partial derivatives of Qp and qp with respect to p: Qp0 D Œ.r C pM w/f .Qp /1 ;

qp0 D Qp0 Œ.w vM /f .Qp qp /1 :

When vS vM , the optimal .p; w/ is a point inside the feasible region with w just below the spot price. Furthermore, a profit sharing scheme similar to that described in Sect. 9.5 for call options can be designed for put options as well.

9.7 Concluding Remarks We have developed an option model to quantify and price a flexible supply contract by which the buyer, in addition to a committed order quantity, can purchase option contracts and decide whether or not to exercise them after demand is realized.

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We have considered both call and put options, which generalize several widely practiced contracting schemes such as capacity reservation and buy-back/return policies. We focused on (a) deriving the optimal order decision of the buyer, in terms of both the committed order quantity and the number of option contracts; and (b) the optimal pricing decision of the seller in terms of both the option price and the exercise price. We have shown that the option contracts shift part of the buyer’s risk due to demand uncertainty to the supplier. The supplier, in turn, is compensated by the additional revenue obtained from the options. We have also shown that a better alternative to the two parties’ individual optimization is for them to negotiate a mechanism to share the profit increase over the no-flexibility contract, and that this profit sharing may achieve channel coordination. Combining the call and put options, we can readily extend our models to construct a flexible contract that will allow the manufacturer (buyer) to purchase both call and put options, with quantities qc and qp , respectively, in addition to the upfront quantity Q. This way, the manufacturer can acquire up to qc more units should the realized demand be higher than Q, or return to the seller up to qp units if the demand turns out to be lower than Q. Thus, the manufacturer will have to decide on three variables, .Q; qc ; qp /. The supplier, in turn, will have four decision variables: call and put option prices, c and p; and the two exercise prices, wc and wp . We have not addressed the issue of risk profiles associated with the two parties’ decisions. For instance, although the supplier is the main beneficiary of the option model, the improvement is in terms of expected profit, whereas in its newsvendor solution, the profit is deterministic, i.e., there is no risk involved. Hence, it is important to characterize the risk associated with the additional profit the supplier can expect from the option model. This can take several forms, such as the variance of the profit, or the probability that the profit will exceed that of its newsvendor solution. These will be the subject of further studies. Acknowledgments David Yao’s work was undertaken while he was an academic visitor at IBM T.J. Watson Research Center, and was also supported in part by HK/RGC grant CUHK4173/03E.

References Akella R, Araman V, Kleinknecht J (2005) B2B markets: Procurement and supplier risk management in E-business. In: Supply chain management: models, applications, and research directions. 33–66. Kluwer Academic Publishers Barnes-Schuster D, Bassok Y, Anupindi R (2002) Coordination and flexibility in supply contracts with options. Manuf Serv Oper Manag 4(3):171–207 Bassok Y, Srinivasan R, Bixby A, Wiesel H (1999) Design of component supply contracts with forecast revision. IBM J Res Develop 41(6) Cachon G (2004a) Supply chain coordination with contracts. In: Graves S, de Kok T (eds) Handbook of operations management. North-Holland Cachon G (2004b) The allocation of inventory risk in a supply chain: Push, pull, and advance purchase discount contracts. Manag Sci 50(2):222–238

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Cachon G, Lariviere M (2005) Supply chain coordination with revenue-sharing contracts: Strengths and limitations. Manag Sci 51(1) Cachon G, Netessine S (2004) Game theory in supply chain analysis. In: Simchi-Levi D, Wu D, Shen M (eds) Handbook of quantitative supply chain analysis: Modeling in the ebusiness era. Kluwer Dong L, Zhu K (2007) Two-wholesale-price contracts: Push, pull, and advance-purchase discounts contracts. Manuf Serv Oper Manag 9(3):291–311 Eppen G, Iyer A (1997) Backup agreements in fashion buying: The value of upstream flexibility. Manag Sci 43:1469–1484 Ertogral K, Wu SD (2001) A bargaining game for supply chain contracting. Working paper, Lehigh University Farlow D, Schmidt G, Tsay A (1995) Supplier management at sun microsystems. Case Study, Graduate School of Business, Stanford University Golovachkina N, Bradley J (2002) Supplier-manufacturer relationship under forced compliance contracts. Working Paper, Johnson Graduate of Management, Cornell University Hull JC (2002) Options, futures, and other derivatives, 5th edn. Prentice Hall, Upper Saddle River, NJ Lariviere MA (1999) Supply chain contracting and coordination with stochastic demand. In: Tayur S, Ganeshan R, Magazine M (eds) Quantitative models for supply chain management (Chapter 8). Kluwer Academic Mart´ınez-de-Albéniz V, Simchi-Levi D (2009) Competition in the supply option market. Oper Res 57(5):1082–1097 Shi D, Daniels R, Grey W (2004) Managing supply chain risks with derivatives. IBM Research Report RC 23139, Yorktown Heights, NY Tsay A, Lovejoy W (1999) Quantity flexibility contracts and supply chain performance. Manuf Serv Oper Manag 1(2):108–111 Wu DJ, Kleindorfer P, Zhang J (2002) Optimal bidding and contracting strategies for capitalintensive goods. Eur J Oper Res 137:657–676

Chapter 10

Build-to-Order Meets Global Sourcing: Planning Challenge for the Auto Industry Melda Ormeci Matoglu and John Vande Vate

10.1 Introduction Auto manufacturers today face many challenges: The industry is plagued with excess capacity that drives down prices, international competitors are seizing share at both ends of the market and consumers are well informed about options and prices. All these factors combine to heighten competitive pressures, squeeze margins, and leave manufacturers struggling to increase revenues and market share. A nearly universal strategy in the ensuing battle for market share and survival has been to increase product offerings in terms of both models and options. Long gone are the days when a black Model T, or a black Hongqi for that matter, was the only choice. Today’s increased product variety, however, complicates operations and confounds demand forecasting. Correcting the inevitable forecast errors with discounts and rebates not only erodes manufacturers’ margins, but also damages brand image. Consumers now expect to get less than they wanted and to pay less for it. As competitive pressures increase, companies look overseas for new markets and low cost suppliers, elbowing their way into high-growth developing markets with new international assembly operations and sourcing more and more components from distant low-cost suppliers. Offshore suppliers and international assembly operations bring long and variable lead times that complicate demand forecasting, production planning and supply. Under pressure to keep inventories lean, manufacturers often find needed supplies are still at sea and end up expediting parts to keep production lines running. It is no wonder that Womack and Jones (1996) said, “Oceans and lean production are not compatible.” Incompatible or not, both lean production and transoceanic supply are here to stay and manufacturers are left with the daunting task of managing this “inherent incompatibility.” We address this challenge from the perspective of a global build-to-order (BTO) auto manufacturer. BTO is an attractive strategy for dealing with increased

M. Ormeci Matoglu () Faculty of Economics and Administrative Sciences, Ozyegin University, Istanbul, Turkey e-mail: [email protected]


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product variety and reducing finished goods inventories, defined as “the practice of building customized or standard products as they are ordered and shipping them directly to customers, instead of building-to-forecast and shipping from inventory” (Anderson 2004). Today, many auto manufacturers, such as BMW, Toyota Scion, Renault, Mercedes-Benz, use BTO to some extent. As BTO meets global sourcing, we observe that some familiar tools for managing domestic supply can improve international supply, but have yet to be effectively exploited. In other cases, traditional approaches come up short or are fatally flawed. For example, we find that increasing the frequency of shipments, a fundamental strategy of lean production long employed in improving local supply, also reduces inventory and risk for international supply. However, improving the accuracy and detail of demand signals, while still important, loses much of its impact in the face of long and variable lead times. Proprietary studies we carried out for a European auto manufacturer show that as things stand in the global auto industry even the Herculean feat of doubling forecast accuracy would reduce inventory and expediting costs by less than 10%. In fact, our analysis strongly suggests that reducing the level of demand detail communicated to distant suppliers can simultaneously improve their quality of service and reduce their cost in providing it. Chen et al. (2000) reach a similar conclusion and observe that updating demand information periodically results in a higher variance in the orders placed. We reach this counterintuitive conclusion through a new shipping policy, called “Ship-to-Average,” which ships the same quantity (based on the long-term average forecast) each time and adjusts this quantity only when inventory drifts out of prescribed ranges or the forecasted average rate of demand changes. Ship-to-Average is much easier to implement than currently accepted “Ship-to-Forecast” policies that slavishly follow detailed demand signals and, in the process, unnecessarily amplify the bullwhip effect, creating wild swings in capacity requirements on both the suppliers and the transportation providers. In Sect. 10.2, we elaborate the challenges that auto manufacturers face today. We focus on the resulting product proliferation as the manufacturers target smaller and smaller segments of the market in an attempt to maintain and grow market share. We discuss the shortcomings of traditional push systems when faced with a wide variety of product offerings and observe how BTO helps manage this variety. However, BTO simply shifts the challenge from forecasting finished goods demand to forecasting individual component demands. We look at how variability in part usage and the trend toward global outsourcing in the auto industry affect forecast accuracy and the value of improving it. Finally in Sect. 10.3, we propose first step strategies for managing the inherent incompatibilities in automotive supply chains.

10.2 Refining the Challenge Definition Auto manufacturing is a capital-intensive industry. Developing a new vehicle can cost $1 billion and a new assembly plant to produce the vehicle typically costs another $1–3 billion. To be price competitive, manufacturers must spread these capital

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investments over large volumes, but vehicle lifecycles are shortening and the number of different models on the market is growing. Manufacturers often look for new sales volume in overseas markets. North American manufacturers, such as Ford and GM, for example, have long had operations in Europe and are rapidly ramping up operations in Asia, especially in China and India. European manufacturers are also highly involved in foreign operations. Manufacturers from Japan and Korea have made deep inroads in the USA and are taking a second run at Europe; Chinese manufacturers are poised to follow. Globalization has been both a blessing and a curse. While it has opened up new markets, it has also brought new competitors. The big three in North America (GM, Ford, and the Chrysler unit of DaimlerChrysler) have lost more than 20% of their share in the USA, primarily to Japanese and Korean competitors, in the past two decades. In 1965, the Big Three accounted for 95% of all vehicles sold in North America. Today, that figure has fallen to only 58.5% and will continue to decline. In fact, some industry analysts forecast the Big Three’s market share will fall to 50% by 2008. Japanese brands alone now account for 30.6% and the more recent entrants from Korea already account for 4.1% (Office of Aerospace and Automotive Industries (2005)). Now, Chinese manufacturers are poised to enter the market with lower costs and prices as Chinese auto assembly workers typically work for as little as $2/h including wages and benefits, compared to $22/h in Korea and nearly $60/h in the USA (Bremner and Kerwin (2005)). For example, the Chinese manufacturer, Chery, is aiming at the premium end of the market but with prices 30% below those of its rivals (Dyer and Mackintosh (2005)). EU manufacturers face a similar threat. While initial efforts by Japanese manufacturers in Europe were not as successful as in the USA, Japanese brands are the largest external players in European markets today. In the passenger car market, Japanese and Korean brands’ market share climbed from 11% to more than 17% between 1990 and 2005 (ACEA (2006)). American manufacturers also compete in the European markets, but mostly through their European branches and hence brands. As auto manufacturers increase their global presence by opening new plants in new or emerging markets, they contribute to the significant overcapacity already present in the industry. Globally, it is estimated that in 2005, the industry had enough idle capacity to produce an additional 18 million cars (PWC (2006)) – equivalent to almost 33 times the annual production of the largest assembly plant in North America (the Smyrna, TN Nissan plant makes around 550,000 vehicles per year). There is clearly a mismatch between capacity and demand.

10.2.1 Product Proliferation In the race for market share, almost every manufacturer has pursued smaller and smaller segments of the market with more and more models and options. Even a mass-market auto manufacturer such as Ford Motor Company offers a dizzying array of products. Consumers can choose from among 23 models of Ford vehicles

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and a variety of options for any given model, so there are several million possible configurations to choose from. For example, among the 5 different Ford Escape models (XLS manual, XLS automatic, XLT automatic, XLT sport, and limited automatic) consumers can choose:

Either front-wheel drive or four-wheel drive. A 2.3L or 3.0L engine. 4-speed or 5-speed transmission. From nine exterior color options, three interior colors, four wheel options, two choices of tires, four options of electronics, and four options of seats. Various combinations of five special option packages representing 32 different possibilities. Various combinations of four different upgrades representing a further 16 options. All told, these options lead to something like 70 million different configurations of the Ford Escape. As an extreme example, BMW offers its customers an essentially infinite number of products: 1032 different vehicle configurations by its own estimates. Just the 7-Series with more than 350 model variants, 175 interior trims, 500 options, and 90 standard colors, represents 1017 possible configurations. That is nearly 17 million different configurations for each man, woman, and child on the planet. To put these astronomical figures into context, consider this: the Spartanburg, SC plant produced a quarter million Z3s (the predecessor of Z4) before it produced two that were identical. BMW is so convinced that this huge variety of options drives additional sales that they even retain options with very low (lower than 1%) uptake rates. The success of this strategy is evidenced by the fact that BMW sales surpassed Mercedes in the years after the introduction of the “Customer-Oriented Sales and Production Process” (KOVP), BMW’s BTO system. Similar strategies have been widely adopted across the industry, though the success of certain Japanese brands, such as Lexus, one of the BMW’s strongest competitors, serves as a strong counterpoint to the argument.

10.2.2 From “Push” to BTO The traditional “push” systems in which manufacturers build to forecast and meet customer demand from available finished goods inventory are struggling to keep up with the new challenges of product proliferation. Can Ford Motor Company, for example, really expect to accurately forecast demands for each of the 70 million variants of the Escape? The company only sold about 183,000 Escapes in 2004. In such a crowded market, even forecasting total annual sales for the model is a challenge. When forecasts are wrong, manufacturers are forced to offer significant incentives to sell remaining inventories. In the USA alone, automakers are estimated to

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have spent $60 billion in rebates in 20041 , with more than 90% of all cars sold having some form of incentive. The Big Three spent more than $4,500 per vehicle on incentives that year and even popular Japanese brands that long shunned the practice succumbed to the inevitable pressure: “Toyota’s incentives in all forms were up 31.6% in 2004, reaching a level of over $3,100 per vehicle. Nissan’s incentives were up 26.0% to almost $2,000 a vehicle and Honda increased incentives by 79.5% to almost $2,000 a vehicle” (Office of Aerospace and Automotive Industries (2005)). Despite these big incentives, however, no one is really satisfied with the end result. Manufacturers give up margin and consumers have to settle for what is available, not necessarily what they really wanted. It is no wonder so many manufacturers are moving toward BTO systems that convert orders to products without holding any finished goods inventory. For BTO strategies to be successful, it is essential to have short order-to-delivery lead times. Typical lead times for a custom-built vehicle range from 6 weeks to 10 weeks, while customers expect their vehicles within 2–3 weeks. As a result, only a minority of vehicles in the market are custom built, and manufacturers end up holding up to 100 days of sales in the market place (Miemczyk and Holweg (2004)). Especially, in the USA, customers rely mainly (more than 90% of sales (Miemczyk and Holweg (2004))) on the build-to-stock model and expect vehicles (close to what they really wanted) in 2 or 3 weeks (i.e., the time required to transfer it to their dealer). In contrast, European customers rely more heavily on BTO and accept lead times of several months, but expect exactly what they ordered. To get consumers and dealers to accept lead times of 2–3 months, some manufacturers allow them to change their choices within this period. For example, BMW’s Customer-Oriented Sales and Production Process (KOVP) allows a customer to change his or her order up to 6 days before the vehicle is produced. In fact, a customer may change major specifications such as the engine, transmission, color, or optional equipment within days before the vehicle is assembled, without affecting the agreed upon delivery date. And customers exercise this flexibility: BMW responds to more than 120,000 change requests every month. This flexibility also allows BMW dealers to meet individual customer requirements more quickly. Typically, dealers place an order for a basic vehicle in advance, and make changes to that order as customer demand takes shape. In many cases, the dealer can offer a customer exactly the vehicle he or she wants within a 2 or 3-week window. A side benefit of this flexibility is that, more often than not, customers tend to upgrade to more expensive options such as navigation systems, xenon lights, and electronically adjustable comfort seats, etc., as the delivery date approaches. Thus, by allowing its customers flexibility, BMW is not only able to get those orders earlier – generally months in advance – but it also enjoys enhanced revenues from the resulting upgrades.

1

The US Bureau of Economic Analysis estimates “Final sales of motor vehicles to domestic purchasers” at $518 billion in 2004.

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10.2.3 Capacity and Variability In the auto industry production, capacity represents a major capital expense and production labor is skilled, highly organized, and expensive. Consequently, companies rely on a variety of strategies to smooth demand. In fact, an assembly plant’s daily production is set to a takt time, e.g., a vehicle every 50 s. Changes in production are accomplished by speeding or slowing the takt time, adding or reducing shifts or shutting down the facility for a period of time during the year. The latter two adjustments are very crude indeed and are planned far in advance. Reductions to the takt time translate immediately into increases in the labor cost per vehicle produced and so are again only made reluctantly and as part of a broader plan to manage capacity. Thus, over significant time periods the production rate in terms of vehicles produced per day, per shift, and even per hour is quite constant. Significant deviations from this rate are rare and usually the result of quality or supply problems. Although an assembly plant typically produces vehicles at a remarkably constant rate, the composition of these vehicles can vary widely in terms of both the options they require or – as manufacturers move to more flexible lines, in terms of the mix of models produced. For example Hyundai’s plant in Montgomery, AL builds Sonata sedans and Santa Fe SUVs on the same line. That plant is designed to accommodate as many as four different models simultaneously. Similarly, Honda’s plant in East Liberty, OH produces cars and light trucks on the same assembly line, while Ford’s flexible plant in Chicago is capable of building eight models off two platforms and the Dearborn plant nine vehicles off three platforms. This flexibility helps spread capital costs and risk. As a consequence, even though automobile assembly plants make the same number of vehicles every day, demand for the components that go into those vehicles is increasingly unpredictable. Apparently, modifications in plant design are ahead of modifications in vehicle design: while manufacturers are driving down the number of “platforms” progress in achieving commonality in parts across models has been slow. In fact, it is not unusual in the industry to see usage of a part vary by more than 70% from one day to the next. BTO, which was intended to free manufacturers from the tyranny of poor forecasts, simply shifts the problem from forecasting finished vehicle demands to forecasting demands for components.

10.2.4 Global Sourcing A major trend in the auto industry is globalization. Auto manufacturers not only produce vehicles around the world, but they also increasingly rely on low-cost overseas suppliers to do it. For example, U.S. manufacturers import parts from Mexico, Brazil, and now China and India. Sourcing from offshore suppliers is not limited to the original equipment manufacturers. This trend is visible even among first tier suppliers such as Cummins International (engine parts), Delphi and Visteon (www.ibef.in/industry/autocomponents.aspx). As a result, in the past

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decade, China’s exports of auto components have increased more than eightfold and U.S. and European imports of auto components have doubled (See Fig. 10.1). However, many companies supply international operations from domestic markets for reasons of quality, economics, and sometimes out of social obligations. EU manufacturers, for example BMW, Volvo, Daimler Chrysler, and PSA, supply operations in USA, Brazil and increasingly China with parts from European suppliers. The decision to use domestic markets to supply international operations is based on several factors. Having already invested in the tooling in Europe, the volumes in this hemisphere may not support replicating those costs. Furthermore, the quest for quality or desires to protect proprietary processes sometimes prevents the use of untried new overseas suppliers. Likewise, existing relationships with suppliers and social responsibility issues all play a part in the decision to supply international operations from home markets. Hence, imports of auto components into countries such as China are on the increase along with the export volumes of EU countries supplying them (See Fig. 10.2).

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10.3 Rethinking Familiar Tools and New Models The wide variety of product offerings and commitments to rapid order fulfillment and near-zero inventories create demands for individual parts that vary wildly from day to day. Combined with global sourcing – hence, long and variable lead times – supplying individual parts becomes a very complex process, often forcing manufacturers to expedite parts to keep production lines running. Managing this “inherent incompatibility” between lean production and transoceanic supply is a daunting task for manufacturers. In this section, we observe that increasing the frequency of shipments – a fundamental strategy of lean production long employed in improving local supply – also reduces inventory and risk for international supply. In other cases, traditional approaches come up short or are fatally flawed. For example, improving the accuracy and detail of demand signals, while still important, loses much of its impact in the face of long and variable lead times. Finally, we look at a new shipping strategy, Ship-to-Average, which relies less on the inevitably erroneous forecasts that accompany international supply and instead focuses on longer-term trends in demand. In fact, our analysis strongly suggests that reducing the level of demand detail communicated to distant suppliers can simultaneously improve their quality of service and reduce their cost in providing it.

10.3.1 Increasing Frequency Auto manufacturers have long recognized the value of frequent deliveries from local suppliers: more frequent shipments mean smaller shipments and smaller shipments mean less inventory. More frequent shipments also mean less time between shipments and so less risk of interrupted supply – if something goes wrong with one delivery, another is not far behind. Frequent deliveries are a cornerstone of lean production that Toyota has exploited to the fullest. The carmaker brings parts into its plants more than once an hour. Can these same ideas improve international supply? Hourly shipments may not be feasible, but increasing the frequency of shipments and reducing the risk of expediting is still a viable approach. Today, most manufacturers make what amounts to weekly shipments for internationally sourced parts. This means that on average they must carry not only a large safety stock to protect against delays in delivery or sudden increases in demand, but also half a week’s supply as cycle stock. Proprietary studies we carried out for a European auto manufacturer provide valuable insights about the impact of shipping frequency (as well as some other tools and policies mentioned in following sections). In these studies, using the characteristics of the manufacturer’s original demand and rolling forecast data, we created an extensive test bed of realistic data. We similarly analyzed the statistical properties of shipping delays and lead times. We tested each of the different approaches via simulation using the resulting test bed of data on forecasts, demand and shipping delays.

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Fig. 10.3 Cost savings through increased frequency of shipments

The results of our analysis, depicted in Fig. 10.3, suggest that increasing the frequency of international shipments can offer significant savings both in inventory and in expediting costs. Here, we ignore the transportation costs, other than expediting, since over all the same amount of parts have to be shipped, and ocean shipping costs are generally incurred per container shipped. Increasing the frequency of shipments reduces the number of containers on each vessel, but does not increase the number of containers shipped each year. Our analyses, based on the characteristics of a European manufacturer supplying assembly operations in the USA, show that doubling the frequency of shipments from once-per-week to twice-per-week simultaneously reduce in-plant inventories by 14% and expediting costs by 7%. Moving to three shipments per week reduces in-plant inventories by 25% and expediting costs by 9%. Increasing frequency, by itself, has no effect on in-transit inventories, but since the total costs decrease, in-transit inventories as a share of total avoidable costs in Fig. 10.3 increases slightly as we increase the frequency of shipments. The ocean container lines’ sailing schedules force manufacturers that source parts from overseas to work under a periodic review system since, regardless of when orders are placed, they can only be delivered when vessels are scheduled to arrive. Scarf (1960) established the optimality in this setting of (s, S) policies, which at each time period either place an order to bring the inventory level up to level S

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or, if the order quantity is too small and inventory levels are already greater than s, place no order at all. As a consequence, although the time between orders is relatively constant, the order quantities can vary widely. The risk of stocking out in an order cycle depends on the variability of demand in a period determined by both the time between orders and the lead time (Silver et al. (1998)). Increasing the frequency of orders reduces the time between orders and so reduces the risk of stocking out in an order cycle (assuming we hold safety stocks constant). However, increasing the frequency of orders increases the number of order cycles in a year and so means we face this reduced risk of stocking out more often. The overall impact of increased frequency on safety stock depends on the balance between these two factors. In our experience, the benefits of reducing the risk of a stock out in each cycle outweigh the costs of additional cycles even for frequencies beyond those achievable with ocean shipping schedules. Increasing frequency only reduces inventory and expediting costs if the shipments are mixed. Simply increasing the number of vessels used has little or no impact beyond spreading the risks if individual components are still shipped once per week. In other words, shipping containers on three different vessels each week is little different from shipping them all on a single vessel if each part number is only on one vessel. To realize the savings, each component has to ride on each vessel and achieving this may require more packaging flexibility and increase handling costs. In our study, the inventory and expediting savings more than compensate for any extra handling. Increasing the frequency of shipments from local suppliers typically increases transportation costs because it reduces capacity utilization of transportation or requires a larger number of smaller vehicles to make the deliveries. This is not generally the case for international shipments, or at least the impact on transportation costs is less pronounced. This is because, with the exception of a few high-volume suppliers that ship direct, most internationally sourced components are already consolidated for international packaging. Thus, while increasing the frequency of these shipments will generally increase transportation costs between the supplier and the consolidation center, it has little effect on the international transportation costs. Ocean shipping costs are generally incurred per container shipped, so shipping 100 containers on three different vessels costs essentially the same as sending 300 containers on a single vessel each week. Unfortunately, sailing schedules make it difficult for companies to increase international shipment frequencies. The carriers all try to set sail at the end of the week, so their vessels are not idled at port over the weekend. The impact: 90% of the fastest 30% of services from Hamburg to Charleston and 80% of the fastest 30% of services between Hong Kong and Long Beach are scheduled to arrive between Friday and Sunday. To achieve higher frequencies, shippers are forced to use services out of and into alternative ports. While this certainly complicates the logistics, it has the added benefit of reducing the risks of disruptions in the case of a port strike or a hurricane.

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10.3.2 Improving Forecasts As manufacturers struggle to manage the “inherent incompatibility” between lean inventories and long lead times, their first reaction is typically directed at improving the accuracy of demand forecast through investments in information technology. They develop advanced forecasting models in order to capture the nature of demand and invest in new sales and operations planning software to get a better handle on their supply chain. While in general there is a consensus that improved forecast accuracy has positive impact, there is no clear single method to achieve it, or even identify the extent to which it can be achieved. In any case, everyone agrees it is impossible to eliminate forecast errors altogether. While forecast accuracy is important and efforts to improve it should not be abandoned, this avenue offers little prospect for resolving a significant portion of the “inherent incompatibility.” Demand variability for option parts is an inherent component of the flexibility manufacturers must allow consumers (and dealers) in order to get orders far enough in advance. Hence, a certain amount of forecast inaccuracy is inherent in BTO. There are four ways to reduce the demand variability engendering forecast errors: 1. Reduce the number of options: Low volume parts are the most difficult to forecast and managing them takes the same effort if not more than high volume parts. Eliminating the least popular options improves forecast accuracy for the more popular ones and reduces the complexity of managing supply. 2. Increase the “frozen horizon”: Freezing orders earlier fixes the production schedule further in advance, allowing the manufacturers to work with shorter forecast horizon i.e. closer to the actual demand. 3. Source from local suppliers: Reducing lead times significantly improves forecast accuracy. 4. Exploit postponement strategies: By delaying the point of product differentiation as close as possible to actual order information and waiting until actual order signals are received to complete the products, postponement offers manufacturers the flexibility they need to efficiently produce customized products. Toyota effectively exploits postponement to provide U.S. customers with almost unlimited customization of the Scion even though that model is built in Japan. Scion customers in the USA can chose from over 40 different options, leading to 240 different versions of the vehicle – more than 3,000 different versions for each person in the USA. However, the vehicles produced in Japan are standardized; distinguished only by transmission (automatic or manual) and color (there are 6 choices). These standardized vehicles are customized to order in the U.SA and delivered within 5–7 days. Toyota’s postponement strategy for the Scion would be difficult to implement for higher end vehicles with fewer standard features and more complicated option offerings that can only be added during production, but that does not mean postponement is not a viable strategy for complex subassemblies such as wiring harnesses and cockpits.

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Unfortunately, none of these options is particularly attractive or available to manufacturers. Furthermore, with the exception of the Scion, these “solutions” are contrary to the trends in the industry. Our studies strongly suggest that even if we could improve forecast accuracy significantly, it would have limited impact. In fact, our studies indicate that the Herculean accomplishment of halving forecast errors would only reduce inventory and expediting costs by less than 10%. In our studies, using a history of orders and rolling forecasts similar to those of a European auto manufacturer, we artificially improved the forecast accuracy and evaluated the impact on inventory and expediting. Figure 10.4 illustrates that even significant improvements in forecast accuracy yield only relatively small improvements in inventory and expediting costs. In fact, our studies indicated that cutting forecast errors in half – from 75% to about 37% – reduced inventory and expediting costs by less than 10%. The reason: Poor forecasts are the scapegoat for all the excess inventories, stock outs, and premium freight charges. But they simply do not deserve the blame, at least not by themselves. Inaccurate forecasting is just one factor contributing to the problem. The other culprit is lead time variability. Accurate demand forecasts can tell you how much you will need, but if lead times are unreliable you are still left with the question of when to ship, so it arrives when you need it.

Fig. 10.4 Inventory and expediting costs at different forecast error levels

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The mean and variance of lead time demand when lead times are stochastic are modeled by the following well-known expressions, where L and L are the mean and standard deviation of lead time, respectively, and D and D are the mean and standard deviation of demand over a single time period. .lead time demand/ D D L 2 2 .lead time demand) D L D C 2D L2 :

Hence, it is clear that both the mean and the variance of lead time play an important part in managing lead time demand. Increased variability in lead times increases variability in lead time demand, making it more challenging to manage supply. When lead times are long (as in the case with manufacturers working with transoceanic suppliers), the risks of stocking out from one order cycle to the next are dependent since the intervals in question overlap, and thus, more challenging to quantify; one motivation for our reliance on simulation studies. Lead time variability has been on the rise, driven by increasingly violent and unpredictable weather (the 2005 Atlantic hurricane season, with 28 nameable storms is the most active season on record, surpassing the 1933 season’s 21 (National Oceanic & Atmospheric Administration (2005))) and the general strain on the global transportation system. North America has a growing port capacity problem, and the resulting congestion is affecting global supply chains negatively. Over the last 20 years, container volumes in North American ports have grown at an average annual rate of 7%, but port capacity has not kept pace with volume growth (Maloni and Jackson (2005)). A study by the National Chamber Foundation of the U.S. Chamber of Commerce pointed out that most major North American ports are already operating at or near full capacity and will have significant capacity deficits by 2010 (National Chamber Foundation of the U.S. Chamber of Commerce (2003)). Port congestion leads to unpredictable delays, causing manufacturers to increase inventory levels and adjust supply networks to minimize the risk of stock outs and shutdowns. A 2004 survey by Logistics Management magazine revealed that respondents were experiencing average delivery delays of 6.5 days. Approximately 43% of the respondents reported 8 or more days of delay, while 53% reported delays between 3 and 7 days (Levans (2005)- See Fig. 10.5). Even if manufacturers’ ability to predict what they need improves, their ability to predict when it will get there is declining. The principal impact of improved forecasts is to reduce the need for expedited shipments. The auto industry with its heavier, lower value parts, typically reserves airfreight for emergencies. Hence, increasing forecast accuracy principally reduces the already small airfreight costs and so has a relatively small impact overall. A parallel study with a telecommunications equipment manufacturer, however, indicated that improved forecast accuracy can have significant impact on companies that rely heavily on airfreight and expedited shipments to meet customer orders.

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Fig. 10.5 “How much delay are you experiencing in delivering your products due to West Coast backup?” (Levans (2005))

10.3.3 Ship to Average Traditional methods for managing supply rely on detailed forecasts. Each order quantity is based on forecasted demand over the period the order is intended to cover: a week’s forecasted production for weekly shipments, a few days’ forecasted production with more frequent shipments. These methods suffer from two obvious difficulties: First, as we have seen, actual production quantities over periods as short as a week or a few days vary widely; second, forecasts of these quantities are remarkably inaccurate. The result: Traditional methods magnify the bullwhip effect and send international suppliers and logistics service providers on a wild goose chase after phantom peaks in demand. We propose a simpler, and it turns out, more effective strategy for managing supply, which we call Ship-to-Average. The idea is to keep order quantities constant, adjusting them only when inventory drifts out of prescribed ranges or the average rate of demand changes. Ship-to-Average offers several advantages: First, since changes in the order quantity are the exception rather than the rule, Ship-to-Average policies reduce the effort involved in managing supplies. Second, since order quantities are consistent and reliable, suppliers and service providers can more efficiently plan production and manage labor requirements. Finally, the manufacturer can count on more consistent and reliable shipment quantities and no longer needs complicated calculations to determine whether it is necessary to expedite parts. Ship-to-Average policies do use forecasts to calculate fixed order quantities, but ignore the details in these forecasts and instead focus on longer-term trends. Figure 10.6 shows the relationship between forecast errors and the period of demand forecasted in the case of a European manufacturer forecasting demands

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Fig. 10.6 Forecast errors covering different periods

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for an option-driven component at a North American plant 30 days in advance. Figure 10.7 compares the stability of the forecasts averaged over different time periods. As expected, forecast accuracy and stability improve significantly with the length of the period forecasted. While daily forecasts are inaccurate and swing wildly, weekly, monthly and quarterly forecasts are increasingly accurate and stable. Ship-to-Average policies ship to the more stable and accurate average of forecasts covering longer periods. Although there is no theoretical proof that Ship-to-Average is a best strategy in general, Ormeci et al. (2008) proved that, in the case of zero lead times, policies of this form are optimal. Initial studies based on the characteristics of a European auto manufacturer suggest that Ship-to-Average policies are at least as effective as the current Ship-to-Forecast strategies in terms of average inventory and expediting costs. In many cases, Ship-to-Average simultaneously reduces total avoidable cost, expediting costs and the variability in order quantities, and the reductions in expediting costs and order variability are significant (60% and higher reductions for expediting; 50% and higher reductions for order variability) with no increase in

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total avoidable costs. So far, we have always been able to identify a Ship-to-Average policy that significantly improves order stability without increasing total avoidable cost when compared to the best Ship-to-Forecast policy. Ship-to-Average policies produce significantly more stable order patterns, which simplify the suppliers’ task of managing labor and capacity. Consider the example of a large, high-value option-driven part shipped from Europe to a North American plant. Adopting the standard that a change of more than 10% in successive order quantities creates planning and scheduling challenges for the supplier, we found that while the best Ship-to-Forecast policy exceeded this limit more than 60% of the time, our Ship-to-Average policy exceeded it less than 15% of the time with the same total avoidable cost. That additional constancy and predictability helps suppliers better manage their resources and realize savings that should eventually be reflected in piece prices.

10.4 Conclusions One of the most important challenges facing BTO auto manufacturers is the wide variety of product offerings, commitments to rapid order fulfillment and near-zero inventories creating demand for individual parts that vary wildly from day to day, while supplying these parts globally with long and variable lead times. Increasing frequency is a simple and efficient tool to improve performance in terms of inventory costs and the risk of stocking out. However, given current carrier schedules increasing frequency is far from being a simple task. Besides, since with traditional Ship-to-Forecast policies each order quantity is based on forecasted demand over the period, the order is intended to cover more frequent shipments rely on more detailed forecasts, which are both less accurate and more variable. As a consequence, more frequent shipments can magnify the bullwhip effect unnecessarily. In the auto industry, attempts to redress this by improving forecast accuracy face daunting challenges and even if they succeed beyond all expectations will have little impact on the problem. We introduce a new type of policy as an alternative way to deal with variable lead times and inaccurate forecasts. Our Ship-to-Average policy performs at least as well as the more complicated Ship-to-Forecast policies, and also significantly smoothes the order quantities suppliers must manage. Theoretical studies (Ormeci et al. (2008)) indicate this form of policy is optimal in the case of zero lead times, simulation studies based on the parameters of actual component demands at the North American operations of a European manufacturer provide further support and suggest Ship-to-Average policies are at least as good as traditional Ship-toForecast policies in terms of inventory and expediting costs. These results are still new and much work is left to do. We are currently working to identify conditions that especially favor Ship-to-Average policies and develop analytic tools for quickly finding optimal policy parameters and assessing the long-term impacts of implementing them.

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Acknowledgements The authors thank their contacts at the North American operations of a European auto manufacturer for their tireless and patient support and encouragement. The authors also want to recognize the efforts and contributions of Thomas Drtil, Stefan Lier, Matthias Pauli and Claus Reeker, Master’s students who contributed to the evaluation of the Ship-to-Average policy.

References ACEA, European Automobile Manufacturers Association (2006) Historical series: 1990–2006: New PC registrations by manufacturer Anderson DM (2004) Build-to-order and mass customization. CIM, California Bremner B, Kerwin K (2005) “Here Come Chinese Cars”. Business Week, May 26 Chen F, Drezner Z, Ryan JK Simchi-Levi D (2000) Quantifying the Bullwhip effect in a simple supply chain: the impact of forecasting, lead times, and information Manag Sci 46(3):436–443 Dyer G, Mackintosh J (2005) “Next for the West Are Cars ‘Made in China’.” The Financial Times, 1 June India Brand Equity Foundation (n.d.). Auto Components. Retrieved from http://www.ibef.in/ industry/autocomponents.aspx. (Accessed on 21 February 2006) Levans MA (2005). “Tight Squeeze.” Logist Manag 44(1):45–50 Maloni M, Jackson EC (2005) North American container port capacity: an exploratory analysis. Transport J Summer: 1–22 Miemczyk J, Holweg M (2004) Building cars to order: what does it mean for inbound logistics operations? J Bus Logist 25(2):71–197 National Chamber Foundation of the U.S. Chamber of Commerce (2003) Trade and Transportation, a Study of North American Port and Intermodal Systems National Oceanic & Atmospheric Administration, U.S. Department of Commerce (2005) http:// www.noaanews.noaa.gov/stories2005/s2540b.htm OECD (2005), “SITC Revision 3”, International Trade by Commodity Statistics (database). (Accessed on 24 April 2006) Office of Aerospace and Automotive Industries, Manufacturing and Services, International Trade Administration, US Department of Commerce (2005) The Road Ahead 2005 for the U.S. Auto Industry Ormeci M, Dai JG, Vande Vate J (2008) Impulse control of Brownian motion: the constrained average cost case. Oper Res 56(3):618–629 PWC AutoFacts Light Vehicle Capacity Data (2006) Scarf H (1960) Optimality of (S, s) policies in the dynamic inventory problem. Mathematical methods in the social sciences 1959. Proceedings of the First Stanford Symposium. In: Arrow KJ, Karlin S, Suppes P (eds.) Stanford University Press Silver EA, Pyke DF, Peterson R (1998) Inventory management and production planning and scheduling, 3rd edn. Wiley, New York Womack JP, Jones DT (1996) Lean thinking. Banish waste and create wealth in your corporation, pp. 224. Simon & Schuster, New York

Chapter 11

Practical Modeling in Automotive Production Jonathan H. Owen, Robert R. Inman, and Dennis E. Blumenfeld

11.1 Introduction We all want to make a difference. We all want our work to enrich the world. As production planners, we have a great opportunity. Since the industrial revolution, production planning has enabled industry to extract the most from the era’s manufacturing technology. Henry Ford’s assembly line, with its associated production planning, dramatically improved production efficiency. While production planning continues to advance productivity thereby enhancing society’s prosperity and quality of life, the benefits from production systems modeling are often not realized in practice. As noted by the editors, there is a widening gap between research and the needs of industry. The cause is not that the models are not sophisticated enough to capture the complexities of the real world. Neither is it that there is a lack of technology transfer. From our experience in industry, the gap arises from underdeveloped modeling. Underdeveloped modeling is diverting us from making a bigger difference. To impact production, we need models that can be put into practice. Not necessarily simple, but actionable. If a firm cannot act on a model, the model (and its associated solution methodologies) will not enhance the firm’s performance. The authors admit to straying from this advice themselves. But we have learned that models implemented gratify the most. Therefore, we propose that the most fruitful future research direction is practical modeling. In our terminology, practical production modeling is modeling that has a positive impact on actual production; underdeveloped production modeling rarely influences the factory floor. Successful implementation, practical modeling’s supreme goal, requires that the model must represent an actual industrial problem. In addition, the model must comprehend what information is available, how the analysis can be accomplished, how the model will be used, and importantly – the user’s objectives and skill set. In our experience, passing all these implementation hurdles requires an iterative refinement process.

J.H. Owen () General Motors Company, Global R&D, 30500 Mound Road, Warren, Michigan 48090, USA e-mail: [email protected]


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Problem-driven modeling steers the model’s insights or solutions toward the real problem. In contrast, technique-driven modeling has become more common. It is natural for experts in a particular technique to view problems – such as production planning problems – as opportunities to exercise their technique. While often extremely challenging, these problem statements miss the mark. Technique-driven modeling steers the model’s insights or solutions down that technique’s road – which may detour around the real problem due to the technique’s capabilities. Once we state the problem in terms of a particular technique, the solution will undoubtedly involve that technique. As Volkema (1995) puts it, “A problem’s formulation determines the solutions it will produce.” Miser (1993) warns us to avoid the corrupting lie of a poorly stated problem. Ackoff (1979) decries operations research’s immersion in techniques, whereby “practitioners decreasingly took problematic situations as they came, but increasingly sought, selected, and distorted them so that favoured techniques could be applied to them.” Daskin (2006), the 2006 president of INFORMS, sums it up writing that “We need to change the paradigm from that of solving models to that of solving problems.” The regrettable consequence is that much of our technique-driven effort misses the point and goes unused. Or worse, our technique-driven efforts are implemented and potentially make matters worse. Accepting an optimal solution to a techniquedriven model of production planning can lead us to a terribly inferior solution to a real-world situation. Therefore, to maximize our impact, we argue for more industrial-problem-driven modeling. Problem-driven modeling begins with understanding the problem. But beginning with the industrial problem is insufficient; we must adhere to it by verifying that whatever model we build represents reality. The validation step, which assesses whether the model’s behavior is sufficiently like the real world, ties the modeler to the real world. If a model fails to be validated, can we rely on the solution’s validity? In practice, prior to implementation, a model must pass a gauntlet of tollgates. Since the chance of passing all tollgates on the first pass is very low, practical production modeling requires iteration. The production planning issues facing industry today are not textbook problems; instead they are complex, ill defined, and intertwined with related challenges. Iterative modeling is the learning process whereby the modeler builds understanding. Most textbook modeling processes (see, e.g., Dilworth (1989), Tersine (1985), and Winston (1987)) begin with a real-world problem and include steps to verify the model and modify the model until it can approximate the real world. Although we cannot provide a closed-form recipe for practical modeling, Fig. 11.1 outlines a rough process that highlights the role of iteration. Steps 1–3 in Fig. 11.1 represent the initial abstraction and model development. Step 4 displays a first check of whether or not we can collect or estimate the necessary data. If not, we need to revise the model accordingly. After verifying that we can get the data, we need to verify that the model reflects reality (Step 5). Once the model represents the real system, we may not be able to solve it; if we cannot, we need to iterate to alter

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1. Abstract aspects of real world.

Poorly defined real-world production planning issue.

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2. Draft analytic model of real world.

3. Exercise analytic model.

No

4. Can we get necessary data for model? Yes

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5. Can model be analyzed in timely manner? Yes

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6. Does model replicate observed behavior? Yes

No Yes

12. Address No another problem.

11. Can we improve the model?

10. Make real world decisions.

9. Study model’s behavior.

7. Does model satisfy the decision maker? Yes

8. Analyze model.

Fig. 11.1 Practical production modeling is an iterative process

the model so that we can analyze or optimize it with its realism (Step 6). Once the model represents reality and can be analyzed, we need to review it with the decision maker (Step 7). This review often uncovers new issues, priorities, constraints, objectives, or implementation concerns that require further modeling iteration. Impatient modelers may skip these validation steps and proceed directly from Step 3 to 8 – oftentimes leading to models that have little impact. Although frustrating and time consuming, iteration eases implementation. As Pidd (1999) puts it, “model building may feel like muddling through.” Yet, practical modeling must be iterative modeling. Underdeveloped modeling is the bottleneck constraining our contribution. More practical modeling can unleash our impact. We have sketched a modeling process that incorporates two keys into practical modeling, which are to be industrial problem driven, and to iteratively revise the model so that it represents reality and can be implemented. The rest of this chapter provides our insights into practical modeling annotated with examples of successful and unsuccessful implementation attempts in the automotive industry. The following sections discuss implementation challenges, system description and problem abstraction, model consistency and modular design, and the relationship between models and solution methodologies.

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11.2 Understanding the Challenges of Implementation For our models to be used in practice, we must first understand what data is available or could be collected. Generally in industrial production, data is scarce and costly. However, as modelers we quite happily introduce variables with multiple subscripts and superscripts to account for various situations. Every new subscript or superscript adds another dimension to the model – and requires an order of magnitude more data for implementation. We have seen what can happen in practice in such cases. Instead mn for every i , j , k, m, n of collecting data to estimate the values of input variables xijk (i D 1; 2; : : : ; j D 1; 2; : : : , etc.), users often enter a single value for the whole set just to complete all the required fields in the input data file, and the model becomes meaningless. For instance, one component plant production planning model required demand variability data for every component produced. Since this data was not readily available and was time consuming to compile, a universal estimate of the coefficient of variation was used for all components – grossly overestimating the demand variability for some of the components. The error was noticed only after the model specified unreasonably high safety stock inventory levels for certain components. Had users not checked whether results from the model were realistic and consistent with their intuition, the problem may have gone undetected. While the source of the problem was quickly identified in this instance, the process of troubleshooting dataintensive models is usually cumbersome and time-consuming, since it is unknown at the start whether the fault lies with the underlying assumptions, the analysis, or the data. Not only is data scarce, it is often of uncertain quality. Production facilities involve a vast range of operations, and collected data sets are often incomplete and contain errors or inaccuracies. If a model requires huge amounts of data, discrepancies in the data are more likely to go undetected. A modeler cannot assume that every input variable will have a reliable value when the model is used. If the scarcity and uncertain quality of data were not enough, in practice the flow of data is often constrained by organizational boundaries. Many functions guard their data as treasure, and only reluctantly provide it to outsiders. And if they do, they may not explain the data, its limitations, or its implicit assumptions. To maximize the likelihood of successful implementation in the presence of limited, costly, uncertain, and hard-won data, we should investigate what data is needed when designing the model. In general, we should develop models with parsimonious data requirements. Data is necessary but not sufficient for implementation. To design an implementable model, it is critical to understand the decision maker’s sphere of influence and the organizational and policy barriers that the user must live with. Such awareness is especially important in production planning, where the supply chain links many different players, each with its own responsibilities and objectives. On the one hand, optimizing a subsystem in isolation can harm the system overall. On the other hand, taking the total system’s perspective and attempting to optimize the whole integrated system will almost certainly be unimplementable because of competing

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organizational goals. Hence, the challenge is to work within the customer’s sphere of influence while improving overall system performance. In supply chain analysis, for example, even basic models for the trade-off decisions among production setup, inventory, and transportation costs cut across organizational boundaries. A manufacturing plant that serves a network of destinations typically bears the cost of production setups and inventory at the plant, in order to enable batch production cycling and load makeup for outbound shipping. Figures 11.2a and 11.2b illustrate this for a simple example of regular setups and shipments to meet deterministic demand for one destination. Figure 11.2a plots the cumulative production and shipments at the manufacturing plant for the destination. The vertical distance between the two curves represents the total inventory of parts in the plant for that destination over time. However, as shown in Fig. 11.2b, this inventory is composed of two separate inventories, one for production cycling and the other for shipping. The production cycling inventory depends on setup frequencies which the plant management controls, but the outbound shipping inventory depends on shipment frequencies, which are typically determined by decision-makers at the destinations. (In cases where it is the plant that determines the shipment frequencies, the destinations incur the corresponding inbound shipping inventory costs at their end – costs that are then not under their control). Hence, the objectives for production and for logistics are often at loggerheads. Ideally, we would model the total system. But if this degrades any of the decision makers’ performance metrics, they may resist even if the solution is best for the overall supply chain. In one instance, a decision maker asked for help in reducing finished goods inventory at an automotive component plant. Attempting to take a total system’s perspective, we wanted to simultaneously consider the finished component inventory at the component plant, the component inventory at the assembly plant, and the logistics mode and schedule from the component plant to the assembly plant. Unfortunately, the assembly plant and logistics were controlled by different decision makers with conflicting objectives. Consequently, to permit implementation, we resorted to limiting the scope of the model to the inventory and production planning within the component plant’s control, while obeying the constraints imposed by the assembly and logistics supply chain elements. In general, we need to recognize that decision variables may not fit nicely within organizational boundaries. Another implementation challenge is to anticipate how the model and its results will be used. Some applications require solutions for direct implementation. Others require solutions to illustrate trade-offs for policy making. The output of a model and how it is communicated depend on the user’s objectives. They also depend on the user’s technical knowledge and understanding to interpret results. In some cases, a decision-maker wants a comprehensive software package that can address a range of different production issues. In many cases, however, all that is desired is a set of simple and approximate analysis tools that the decision-maker can readily incorporate into a standard spreadsheet to quickly evaluate production planning alternatives. Making the effort to thoroughly understand the actual needs of the user facilitates successful implementation.

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a Cumulative Number of Parts

Production

I(t) Time Between Production Setups

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I(t) = Inventory of parts at plant for given destination at time t Time t Average Demand

b Cumulative Number of Parts

Production

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Shipment Departure

Time Between Production Setups

I2(t)

Time Between Shipments Production Lot Size

I(t) = I1(t) + I2(t) where

I1(t) = production cycling inventory I2(t) = shipping inventory Time t

Fig. 11.2 (a) Cumulative production and shipments of parts at the plant for a given destination. Vertical distance I.t / between production and shipment curves represents total inventory of these parts at the plant at any instant t . (b) Cumulative plots with average demand rate for parts superimposed. Total inventory I.t / of parts at the plant for given destination is made up of two separate components: production cycling inventory I1 .t / is determined by setup frequency and shipping inventory I2 .t / is determined by shipment frequency

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Using simulations only: Simulations of many alternative system designs Time Start of Production

Concept Approval

Using analytical models: Principles and formulas

Targeted simulations of few promising designs Time

Concept Approval

Start of Production

Fig. 11.3 Analytical models based on simple mathematical relationships between key system parameters help reduce time for production system evaluations

The appropriate level of modeling detail (depending on users’ needs) is particularly important for production system design. Within the large automotive companies, there are many different types of production systems, depending on the manufacturing processes and types of products. Design of these production systems is a complex task that often requires many time-consuming detailed simulations. Often there is insufficient time to simulate in detail all alternatives before the start of production, and the final design may well be based on incomplete analyses. We found that the development of principles and analytical models based on simple mathematical relationships provided the tools designers needed to quickly identify efficient proposals at the initial design stage. Detailed simulations could then be focused on a small number of promising design alternatives in time for the start of production (see Fig. 11.3). For our models to be implemented, we need to know what data is available, how the user fits into the overall system, and how the results will be used. Given these implementation challenges, the remainder of this chapter addresses how problem-driven modeling can take into account the practical implications of production planning in an industrial setting. We focus first on the approach to model formulation.

11.3 System Description and Problem Abstraction Effective production planning requires careful description of the production system to formulate a suitable model. With production systems becoming ever more complex, it is tempting to formulate ever more complex models as well. However, in

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our experience, we have found that for acceptance by practitioners, it is even more important in the case of complex systems to ensure that our models are straightforward enough to be clear and meaningful to the user. Users appreciate simple models that capture the essence of complex systems. In fact, in industry one of the modeling’s key benefits is to clarify the issues and relationships to build the practitioner’s insight. The purpose of a model is not to reproduce all the complexities of a real-world system, but to capture the main characteristics of the system and obtain a useful simplified representation (Michalewicz and Fogel 2004). The model should be an abstraction that can be used to gain insight into the system’s behavior. Obtaining the right level of abstraction requires observing the system, identifying the key parameters, and establishing the basic mathematical relationships. This is the approach that has been successfully used in the natural sciences for centuries, and the same basic methods of research have been recognized as important in guiding operations research and management science (Hall 1985). As early as the 1300s, a scientific principle of parsimony was established, known later as Occam’s Razor (Shugan 2002), that entities should not be multiplied beyond necessity. In more recent times, Albert Einstein is said to have advised scientists to make things as simple as possible, but no simpler. These principles apply just as well to production planning modeling, where simplified abstractions are important for practical problem-solving for complex systems. For example, while Conway et al. (1988) modeled work in process in serial production lines, they observed that making the model as simple as possible, while still exhibiting the behavior in question, was crucial to building understanding and to broadening the model’s applicability. Practitioners designing and operating production systems typically want more than numerical answers to specific problems based on a specific set of parameter values. They desire an understanding of why a system behaves the way it does, what are the most important parameters, what practical alternative solutions are available in addition to the optimal solution, and how sensitive solutions are to changes in operating conditions. For production systems modeling, this need for practical insight into system behavior has long been recognized. When computers were first being developed, Hamming (1962) pointed out that the purpose of computing is insight, not numbers. Geoffrion (1976) applied this thinking to the use of mathematical programming, and showed several examples of simple models in production facility location that provide basic understanding. More recently, Hopp and Spearman (1996) used fundamental mathematical relationships to establish practical models and principles for successful production system management. The success of their book Factory Physics demonstrates the power of a systematic scientific approach to enhancing an understanding of system behavior. An example of such modeling is in the design of a conveyor at an automotive component plant to carry jobs from an upstream production line to a set of parallel test stations for inspection and testing. Designers requested help in estimating the conveyor capacity needed to avoid blocking upstream jobs, given random arrivals of jobs and random testing times. Using simple analytical results for a G=G=m

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queue (e.g., Hopp and Spearman (1996) and Sakasegawa (1977)) and classic heavy traffic approximations by Kingman (1961), we obtained an approximate formula for the required conveyor capacity in terms of just a few basic parameters: the number of parallel test stations, mean and variance of job interarrival times, and mean and variance of station testing times. The formula allowed the designers to obtain quick estimates for different parameter values as part of their evaluation of design alternatives. Simple models also help in assembly line design. At the initial design stage, there is often no detailed data on individual workstations. In such situations, basic analytical models of idealized systems with identical workstations and buffers provide designers with general guiding principles on the behavior of a line. The models give throughput estimates that require only the line length, buffer size, and station processing characteristics. Simple formulas have been derived for idealized systems composed of reliable workstations with variable processing times, as in manual operations (e.g., Hunt (1956), Buzacott (1990), and De Kok (1990)), and unreliable workstations with fixed processing times, as in automated operations (e.g., Buzacott (1968) and Blumenfeld and Li (2005)). These formulas have been used to develop design principles that help designers with decisions on buffer sizing, parallel vs. serial operations, and work content allocation. In addition, formulas in closed form do not require any iterative procedure, and thus are particularly well suited for interactive analysis (e.g., in spreadsheets as embedded calculations). Figure 11.4 shows an example Java tool that was deployed to support production system design evaluations; it includes sliding scales for certain parameters, so that users can immediately see the impact of changes in parameter values on performance curves. For manufacturing engineers, tools like this are extremely useful for understanding the relative sensitivity of certain design parameters and developing intuition on the system’s dynamics. Basic models and principles are useful in different stages of production planning, from early system design to the day-to-day operations. For production system operations in the automotive industry, an important problem is the sequencing of jobs to be built on an assembly line. Job sequencing involves conflicting objectives, depending on the particular tasks performed at different workstations along the assembly line. Some tasks, such as automobile painting, incur changeover costs. For these tasks, it is preferable to batch jobs with similar attributes. Grouping jobs of the same color reduces paint color changeovers, thus saving on material and other setup costs, and making it easier to maintain paint quality. Other tasks involve extra resources. For such tasks, it is preferable that jobs requiring the same task be spaced evenly among other jobs to balance the workload over time. An example of this type of task is the installation of a particular option on certain automobiles. Since some tasks favor grouping jobs together while other tasks favor spacing jobs evenly, the optimal job sequence is difficult to determine. Rather than using mathematical programming techniques or complex heuristics, which would provide little insight, research efforts first focused on modeling the separate effects of grouping and spacing, and then on analyzing the interdependence between these two responses to sequencing (Burns and Daganzo 1987). This approach led to basic principles that provide practical sequencing guidelines.

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Fig. 11.4 Simple analysis tools based on fundamental models allow engineers to quickly evaluate key sensitivities while developing production system design alternatives

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Jordan and Graves (1995) introduced the principle of chaining for achieving production flexibility for a group of plants manufacturing a set of products. Two plants are considered linked if they share a product. If one of these plants shares another product with a third plant, then these three plants form a chain. Jordan and Graves showed that limited flexibility has peak benefits when all the plants are linked forming a complete chain. Even though most plants will only need to produce two different products, if the plants form a chain the benefits are almost the same as if every plant could produce every product. In this context, chaining reduces lost sales because the system of plants can shift production of linked products to make room for high demand products. This principle of chaining has had a major impact on production in the automotive industry. The need for careful model formulation is a lesson we learned while working with material control managers on the development of a decision support tool for supply chain analysis. Production in the automotive industry involves shipping a large variety of material, parts, components, and finished products over a vast network of suppliers, stamping plants, component plants, assembly plants, and dealers. To operate such a network effectively requires many intertwined decisions on production scheduling, routing strategies, shipping frequencies, and shipping modes (such as railcars or trucks). In working with the decision makers, we found it necessary to begin with basic models requiring minimal data and only a few key parameters, each with a clear physical meaning, and use the models to gain an understanding of the interdependence between production, transportation, and inventory costs (Burns et al. 1985; Blumenfeld et al. 1985, 1987). Basic models were formulated that provided insight to explore the basic trade-offs between competing costs. Using these models as building blocks, we were able to construct a decision support tool that the managers found easy to understand and apply, and that allowed them to test the results against their intuition. In developing this tool, the challenge was to simultaneously optimize routing strategies and shipping frequencies over an entire network. For a general network of many origins and destinations and transfer points, this would have required a largescale mathematical programming application, with no guarantee of a solution and no transparency in the results. In the specific problem presented to us, however, the network consisted of many destinations but only a few origins and one consolidation terminal. This allowed us to decompose the problem into subnetworks, one for each origin, obtain optimal solutions for the subnetworks analytically, and simply enumerate solutions over the few origins to optimize the entire network. The specific solution method was much simpler to apply than a general optimization procedure, the results were easier to understand, and it gave the users better control to evaluate alternative strategies. If we had used technique-driven or solution-driven modeling, not only would we have compromised the applicability of the results, but also would have settled for an inefficient solution methodology. Key to decision tool development is the focus on industrial-problem-driven modeling. Specific solution techniques to optimize the model can be considered after the model formulation stage. Even if simple, a custom-designed model for the industrial problem at hand will generally outperform models designed for different

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problems. Industrial-problem-driven modeling can provide a model that dominates technique-driven modeling in both simplicity and accuracy. An example of a simple industrial-problem-driven model that was more accurate than complex technique-driven modeling is parts inventory control in an automotive assembly plant. While there were many existing inventory models, none exactly fit the peculiar circumstances of parts inventory in assembly plants operating under new production control rules. It was customary for automotive assembly plants to operate with at least 2 weeks of vehicle orders locked in a frozen schedule, so that the plants’ material requirements could be sequenced in advance. With the introduction of a more responsive pull system for customer orders from the plants, a level shipping schedule within each week for material deliveries from suppliers to plants, and longer supply chain lead times, materials cannot be sequenced as before. Plants must hold material inventory as a buffer against variations in production. This industrial problem had the following three key characteristics not found together in existing, even very complex existing, models: finite maximum demand for parts in a day determined by the finite line rate, a gargantuan stockout cost (because missing only one of the thousands of parts needed would stop the entire plant), and very limited data (because in industry, data is expensive). To determine appropriate inventory levels under these conditions, we developed simple models that led to formulas requiring only the minimal existing data (Inman et al. 1997; Inman 1998). The modeling approach took into account the daily variability in demand for the different parts, given the fixed maximum production. Using the ratio of a part’s average daily usage to its maximum demand as the average penetration of the part, we constructed simple binomial distribution models for the variations in demand from day to day, and derived formulas for setting the inventory of each part to ensure a high service level. The only data needed were the total daily production, shipment frequency and lead time for part deliveries to the plant, and each part’s average daily usage. The simplicity and practical realism of the models were keys to user acceptance, and the formulas were readily implemented.

11.4 Consistency and Modular Design Although we find simple models to be powerful, this simplicity cannot be achieved by ignoring the environment in which a proposed model and the results it yields will be used. Decisions are never made in isolation; practical models must provide insight while simultaneously fostering a consistent understanding of the system across process and organizational boundaries. Two related challenges of practical modeling are (1) to determine the appropriate scope, i.e., the model’s problem formulation and the processes and decisions that it comprehends, and (2) to understand how the model will be used in the organization’s broader decision-making context. How modelers approach these two challenges – from a technique-driven or a problemdriven perspective – will often determine the ultimate value and influence of the resulting models.

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In coping with these challenges, it is worthwhile to pursue modeling approaches that allow the problem to be decomposed. Where feasible, such problemdecomposition approaches permit the problem-driven modeler to choose the appropriate scope without being confined to the limits of tractability for a single analysis technique; instead, the modeler maintains the flexibility to apply different analysis techniques to different aspects of the problem. Ideally, the entire problem can be analyzed through a set of self-contained modules, where the outputs of some modules provide inputs for others, and collectively to the overall system model. In this case, each module may represent a particular subsystem, for example, or consider only a single performance measure. Each module’s inputs and outputs may be relatively simple, providing the transparency needed for user understanding and acceptability. For example, consider the challenges of operating an automotive sheet-metal stamping plant. Stamping plants use one or more press lines to produce a multitude of part types, and typically supply multiple vehicle assembly plants. There are several critical decisions for each part type manufactured on a press line, including:

The batch size for production lots. Inventory levels and reorder points for triggering pull signals. Part storage layout and location. The number of racks and containers.

One modeling approach would be to develop a single monolithic model to consider all of these (and other) decisions simultaneously. Beyond the practical challenge of developing a cohesive integrated representation of such a complex system, and the technical challenge of analyzing such a model directly, a practitioner is always faced with the cold reality that a model’s shortcomings will inevitably be exposed through use. These shortcomings will require the analyst to revisit the assumptions built into the model, to question the relationships among different performance criteria and decision variables, and to adjust both the problem representation and the analysis approach. So even if a single model of the system can be developed, the practical challenges of maintaining the model and interpreting its results, and gaining acceptance of what to a user appears as an incomprehensible black box, are often barriers to successful implementation. An alternative approach is to start by considering possibilities for decomposing the problem into smaller, manageable pieces. Focusing on how different decisions relate to one another, from both a logical and business process perspective, often reveals opportunities for decomposition. In the case of our stamping plant, the decisions mentioned above were addressed through separate analysis modules, including: Lot Sizing Inventory Module – determines how frequently to set up presses and how large of a batch to produce for each part type. This module also determines the safety stock inventory needed for each part to account for uncertain demand and the number of racks or containers needed for storage, based on trade-offs among setup, inventory, and container costs, subject to production capacity and floor space constraints.

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Part Storage Layout Module – determines how to fit inventory and containers into available plant floor space while adhering to existing standard stock rotation policies and container storage methods. The storage layout decisions include practical considerations, such as sufficient aisle space for accessing parts and storage space for empty as well as loaded containers. Part Storage Addressing Module – reduces material handling distances for moving parts between press lines, inventory locations, and shipping depots. It determines where the inventory of each part should be stored to ensure efficient plant operations, based on production times, frequency of line transitions, and material handling requirements. Staffing Analysis Module – determines staffing levels and equipment requirements for material handling and maintenance. These decisions account for the resources needed for work by direct and indirect crews. Labor Zone Module – assigns material handling and maintenance crews to work areas. The assignments are based on objectives of workload balancing, response times, and labor utilization. Piece Cost Analysis Module – allocates operational costs (e.g., floor space, equipment, etc.) across part types and calculates total unit piece cost using relevant outputs from the previous modules. Each of these modules, taken individually, has a well-defined scope and addresses only a subset of the decisions. For this reason, the results produced by any single module can be explained by analysts and readily understood by plant-level decision makers. However, it is important to note that these modules were designed to work together (Fig. 11.5), with a conceptual view of plant operations that is consistent from one module to the next. Consequently, the outputs of one module may serve as inputs to others (e.g., the production schedule of the lot size module determines the inventory storage requirements for the part layout module), and the collective set of modules may be used in combination to provide managers with a system-level perspective for making operational decisions (e.g., by evaluating the capacity of the plant to accept new work). In this example, the set of modules are used together to prioritize across different operation improvement initiatives affecting line transition schedules, labor planning, material handling, and throughput. A basic modeling framework was constructed for the decisions in each module, and further developed to include constraints encountered in practice. For example, the Lot Size Module required critical decisions on the production lot sizes for each part, reorder points to trigger pull signals, inventory safety stock levels, and the numbers of racks and containers. To address these decisions, we began simply with the standard economic production quantity (EPQ) model, and then extended the model to account for practical floor space and machine utilization constraints, and production setup times and costs (Kletter 1994). In our experience with users, we found that decomposing the real problem into separate modules, each with limited scope, helped to maintain the goals of simplicity and transparency. Also, when there was a need to extend the original modeling

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Part to Production Line Allocation Module

Lot Sizing Inventory Module • • • •

Reorder Points Lot Sizes Safety Stocks Number of Racks and Containers

Part Storage Layout Module • Storage Method • Floor space allocation

Part Storage Addressing Module • Storage Locations

Staffing Analysis Module • Direct and Indirect Labor Requirements • Equipment Requirements

Labor Zone Module • Assignment of Crews to Work Areas

Piece Cost Analysis Module Fig. 11.5 Model decomposition allows analysis of an automotive sheet-metal stamping plant through a set of separate (yet consistent) modules

framework, the required changes and additions could be made just to those modules affected, leaving the other modules intact. This decomposition approach offers two major advantages. It allows the modelers and users to easily know where changes are made as the overall model evolves and improved versions are developed. It also facilitates testing and validating any extensions, since experiments can be performed on the modules independently.

11.5 Models and Solution Methodology As modelers with highly specialized technical expertise, too often our natural tendency is to develop production-planning models with a particular analysis technique in mind. While this may be beneficial in cases where the chosen approach is particularly well suited to the problem – because it allows us to quickly develop a useful solution to the target problem – it can also lead to overspecialization of the model, artificially limiting its usefulness in other application contexts. In contrast, problemdriven modeling allows researchers to consider the real issues and to prioritize them based on their impact on the practitioner’s objectives, data availability, and required

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degree of accuracy. It encourages two-way communication between the researcher and the user in model formulation, so that models can be revised based on user feedback. These benefits are generally not possible with technique-driven modeling. For example, using software based on a library of optimization algorithms, without regard to actual production planning issues in practice, can generate misleading results; the standard algorithms are likely to be based on assumptions that do not comprehend the objectives and constraints of specific production system scenarios. By separating the solution methodology from the model formulation, one can avoid forcing a technique on a problem and artificially confining the scope of the model to the chosen technique’s limits of tractability. Such decoupling also allows different analysis approaches to be employed, depending on the users’ requirements and the context in which the solutions are implemented. To illustrate, consider a system developed for productivity performance and throughput improvement analysis at General Motors (Alden et al. 2006). The basic design of the system separates manufacturing-system modeling from throughput analysis, much like languages such as AMPL (Fourer et al. 2003) separate mathematical modeling from optimization solvers such as CPLEX, MINOS, or OSL. Conceptually, this system consists of the following three key components (see Fig. 11.6): Performance-Estimation Modules: Given a manufacturing-system model as input, these modules produce a variety of system- and station-level performance statistics, including measures of hourly throughput and work-in-process, as well as block and starve time for individual stations.

User Interface

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User Interface

Fig. 11.6 The throughput analysis callable library decouples system modeling from solution techniques and provides embedded throughput-analysis capabilities for multiple, domain-specific user interfaces

C-MORE Callable Library

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Optimization Modules: These modules address manufacturing-system design and operation problems characterized by user-specified performance objectives and constraints. They rely on the performance-estimation modules to analyze design alternatives and the impact of system modifications. They include scenario-analysis modules in which the feasible set is typically constrained to include only a finite set of user-specified model variations. Modeling Isolation Layer: This component provides the means for constructing manufacturing-system models and interfacing with the performance-estimation and optimization analysis modules. Although individual performance-estimation and optimization modules embody particular analysis techniques, the modeling isolation layer is generic in the following sense: its modeling constructs and interface mechanisms are not tied to any particular analysis module. For example, the isolation layer provides objects and methods for constructing a model by describing the physical components (e.g., workstations, buffers) and operational characteristics (e.g., job-flow routing, control policies) of a manufacturing system, without regard to the requirements of a particular analysis technique (e.g., queuing analysis or discrete-event simulation). Instead, it is left up to each individual analysis module to determine its suitability for a given model based on the model’s characteristics and limitations of the underlying technique; the analysis module may then provide feedback to the user (through channels provided by the modeling isolation layer) indicating whether or not the model was analyzed or alerting the user to any simplifying assumptions or model modifications (e.g., for purposes of tractability). By decoupling modeling from analysis, we can avoid the pitfall of scope limitation inherent in technique-driven modeling. In most cases, users construct manufacturing-system models without concern for (or direct knowledge of) details of the analysis techniques that will be used; instead, their focus is at the level of the modeling interface inputs and outputs. The complexity of the specific model constructed and the user’s analysis requirements determine which performance-estimation and optimization modules will be used. The system relies on the individual modules to report their suitability for analyzing a given model. However, an additional collection of model analysis modules is underdevelopment to independently validate models and automatically select performance-estimation or optimization analysis modules based on the model’s characteristics, the analysis capabilities of available modules, and the user’s performance requirements; Fig. 11.7 shows a more detailed view of the software system’s architecture with these additional modules highlighted. The separation of modeling from analysis has two practical advantages beyond promoting problem-driven modeling. First, it allows the decoupled development of user interfaces, modeling capabilities, and analysis modules. Second, it enables the modeling and analysis capabilities of the library to be extended over time to meet evolving user needs through the expansion of the modeling isolation layer and the addition of new analysis modules (see Fig. 11.7). Both of these features are critical for business implementation and ongoing development.

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. . .

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Fig. 11.7 The throughput analysis software architecture design supports future extendibility; new or refined analysis techniques are implemented as pluggable analysis modules, and modeling capabilities are expanded by extending the application programming interface (API) of the modeling isolation layer. The highlighted model analysis modules are used to validate constructed models and determine suitability for use with available performance-estimation and optimization modules

The ability to separate modeling from analysis required very careful design of the modeling isolation layer. The development of the generic modeling capability was an iterative process requiring a working knowledge of current and envisioned automotive manufacturing systems, an appreciation of the end-user’s modeling and analysis needs, and an understanding of the likely techniques and their bounds of tractability. Without careful design, it would have been easy to overspecialize the modeling constructs to a particular application or to a particular analysis technique, which would have severely limited the system’s value. By providing modeling constructs suitable for representing automotive manufacturing lines, the system promotes consistency across the organization and allows different user groups – each with their own interests – to use identical analysis capabilities while sharing and reusing system models. For example, the models developed by engineers designing a future manufacturing line are consistent with those used later by plant personnel to conduct continuous improvement activities based on the line’s actual performance data, even though the two user groups access the throughput analysis tools through different user interfaces. This greatly facilitates corporate wide implementation of the system, which is currently in use in many assembly plants for identifying bottlenecks, allocating buffers, and estimating the impact of system changes on throughput performance. The modeling isolation layer provides a common language for manufacturing-system modeling and throughput analysis across organizational, process, and regional boundaries, and thus enables the fast transfer of common best practices among users worldwide.

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11.6 Conclusions As production planners, we can amplify our impact via industrial-problem-driven modeling. In the field of engineering design, designers concern themselves first and foremost with functionality, and then consider design-for-manufacturability or design-for-quality. Modeling is analogous to engineering design. Expert product designers consider functionality and quality, manufacturability, cost, and style. Analogously expert modelers should consider how the model that they are designing will be implemented and impact operations (functionality and quality), how to solve the model (manufacturability), how to get the data (cost), and how appealing and understandable the model is to the decision makers (style). (And should not the appropriate judge of a model’s style be the users of the models, not other modelers? If so, a stylish model is one that users understand and find attractive, not one that is mathematically elegant.) Technique-driven modeling focuses on how to solve the model and often neglects how the model will be implemented and used, how to get the data, and how appealing the model will be to users. This paper advocates iterative modeling that focuses on how the model will be implemented and will drive operations, what data is available, and on how well the model will appeal to the users. In addition to letting an industrial problem drive modeling, and iterating until the model represents reality, has data available, is solvable, and satisfies the user, we have found keys to make models more effective. First, we need the appropriate level of abstraction, and following the principle attributed to Albert Einstein, we should make our production planning models as simple as possible, but no simpler. Other keys are consistency and modular design, separating the model from the solution algorithm, and thoroughly understanding the decision making context. By taking the initiative to work on real-world problems, and following these keys to practical modeling, we can turbo-charge our work’s contribution to society. Many experienced researchers (such as Pollock 1976; Cross and Moscardini 1985; and Powell 1995) advocate teaching modeling as an art. Powell argues that modeling is best learned by doing – as one learns art. We wholeheartedly agree. At the same time, we must avoid taking the art metaphor too far. The power of practical modeling lies in its application – not in its elegance. Models should not be displayed for admiration in a museum – they should be used. To say that there is no single correct model is not to say that any model will do. Let us focus on production planning problems actually facing industry. Let us design practical models that can be implemented. Let us make a difference.

References Ackoff RL (1979) “The future of operational research is past.” J Oper Res Soc 30(2):93–104 Alden JM, Burns LD, Costy T, Hutton RD, Jackson CA, Kim DS, Kohls KA, Owen JH, Turnquist MA and Vander Veen DJ (2006) “General Motors increases its production throughput.” Interfaces 36(1):6–25

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Blumenfeld DE, Burns LD, Diltz JD, Daganzo CF (1985) “Analyzing trade-offs between transportation, inventory and production costs on freight networks.” Transport Res 19B(6): 361–380 Blumenfeld DE, Burns LD, Daganzo CF, Frick MC, Hall RW (1987) “Reducing logistics costs at General Motors.” Interfaces 17(1):26–47 Blumenfeld DE, Li J (2005) “An analytical formula for throughput of a production line with identical stations and random failures.” Math ProblEng 2005(3):293–308 Burns LD, Daganzo CF (1987) “Assembly line job sequencing principles.” Int J Prod Res 25(1):71–99 Burns LD, Hall RW, Blumenfeld DE, Daganzo CF (1985) “Distribution strategies that minimize transportation and inventory costs.” Oper Res 33(3):469–490 Buzacott JA (1968) “Prediction of the efficiency of production systems without in-ternal storage.” Int J Prod Syst 6(3):173–188 Buzacott JA (1990) “Abandoning the moving assembly line: models of human op-erators and job sequencing.” Int J Prod Syst 28(5):821–839 Conway R, Maxwell W, McClain JO, Thomas LJ (1988) “The role of work-in-process inventory in serial production lines.” Oper Res 36(2):229–241 Cross M, Moscardini AO (1985) Learning the art of mathematical modelling. Ellis Horwood Ltd., Chichester and Wiley, New York Daskin MS (2006) “Models vs. Problems.” OR/MS Today 33(4):6 De Kok AG (1990) “Computationally efficient approximations for balanced flow-lines with finite intermediate buffers.” Int J Prod Syst 28(2):401–419 Dilworth JB (1989) Production and operations management:Random House, New York pp. 77 Fourer R, Gay DM, Kernighan B (2003) AMPL: A modeling language for mathematical programming. Brooks/Cole-Thomson Learning, Pacific Grove, CA Geoffrion AM (1976) “The purpose of mathematical programming is insight, not numbers.” Interfaces 7(1):81–92 Hall RW (1985) “What’s so scientific about MS/OR?” Interfaces 15(2):40–45 Hamming RW (1962) Numerical methods for scientist and engineers. McGraw-Hill, New York Hopp WJ, Spearman ML (1996) Factory physics: foundations of manufacturing management. Irwin/McGraw-Hill, New York Hunt GC (1956) “Sequential arrays of waiting lines.” Oper Res 4:674–683 Inman RR (1998) “In-plant material buffer sizing for just-in-time systems in the automotive industry.” Research Publication R&D-8860, General Motors R&D Center, Warren, Michigan Inman RR, Bhaskaran S, Blumenfeld DE (1997) “In-plant material buffer sizes for pull system and level-material-shipping environments in the automotive industry.” Int J Prod Res 35(5):1213–1228 Jordan WC, Graves SC (1995) “Principles on the benefits of manufacturing process flexibility.” Manag Sci 41(4):577–594 Kingman JFC (1961) “The single server queue in heavy traffic.” Proc Camb Phil Soc 57(1): 902–904 Kletter DB (1994) Determining production lot sizes and safety stocks for an auto-mobile stamping plant. Master’s thesis, Massachusetts Institute of Technology Michalewicz Z, Fogel DB (2004) How to solve it: Modern heuristics. Springer, Berlin Miser HJ (1993) “The easy chair: avoiding the corrupting lie of a poorly stated problem.” Interfaces 23(6):114–119 Pidd M (1999) “Just modeling through: a rough guide to modeling.” Interfaces 29(2):118–132 Pollock SM (1976) “Mathematical modeling: applying the principles of the art studio.” Eng Educ 66:167–171 Powell SG (1995) “The teachers’ forum: teaching the art of modeling to MBA stu-dents.” Interfaces 25(3):88–94 ı Sakasegawa H (1977) “An approximation formula Lq Š ˛ ˇ .1 /.” Ann Inst Stat Math 29A:67–75

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Shugan SM (2002) “Marketing science, models, monopoly models, and why we need them.” Market Sci 21(3):223–228 Tersine RJ (1985) Production/operations management: concepts, structure, and analysis. NorthHolland, New York Volkema RJ (1995) “Creativity in MS/OR: Managing the process of formulating the problem.” Interfaces 25(3):81–87 Winston WL (1987) Operations research: applications and algorithms. Duxbury, Boston

Chapter 12

Why Is It So Hard to Build and Validate Discrete Event Simulation Models of Manufacturing Facilities? Seth A. Fischbein and Edward Yellig

12.1 Introduction Discrete event simulation modeling methods are widely used to evaluate the performance of manufacturing systems. These models provide factory management the ability to test different manufacturing methods and operational policies before a factory is built, or before significant changes are made to a facility. These policies could include testing of proposed manufacturing methods, layout methodologies, production equipment assignment strategies, equipment maintenance and repair policies, operator staffing scenarios, and factory automation configurations. Using these models, it is possible to evaluate the performance or predict future performance based on a detailed sensitivity analysis of these different operating variables. This ability has proven to be invaluable, especially if the analysis is performed before major investment decisions are made or prior to finalizing factory designs. Consequently, discrete event simulation modeling continues to be an enabling capability for performance evaluation of manufacturing systems. Despite these advantages and benefits, building a valid simulation model of a factory has proven to be extremely complex and difficult. Once a model has been built and validated, keeping that model validated over the lifecycle of the factory is even more complex and difficult. The reasons for this complexity can be broadly categorized under the following headings:

Dynamics of the manufacturing environment. Resolution (level of detail) needed for valid models. Data and operational complexity of manufacturing systems. Domain expertise and practical modeling skills of the modeling engineer. Need for ongoing or frequent customer education of simulation capabilities/limitations.

In this chapter, we address the preceding categories from the standpoint of a highly complex manufacturing facility. This includes production flows that are very S.A. Fischbein () Intel Corporation, Mailstop CH3–113, 5000 W. Chandler Blvd., Chandler, AZ 85248, USA e-mail: [email protected]; [email protected] K.G. Kempf et al. (eds.), Planning Production and Inventories in the Extended Enterprise, International Series in Operations Research & Management Science 152, c Springer Science+Business Media, LLC 2011 DOI 10.1007/978-1-4419-8191-2 12,

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re-entrant, involve many hundreds of units of production equipment, large and diverse quantities of work-in-process (WIP), and complex operational policies that define the way production is planned and executed in the factory. We also propose solutions to overcome some of the issues posed by these challenges.

12.2 Dynamics of the Manufacturing Environment Conventional discussions on discrete event simulation categorize model features into static and dynamic types. It is convenient to classify the transient (time variant) entities – production lots moving through the system, seizing and releasing resources, changing equipment status – in the dynamic category. In some instances, these discussions consider product mix as a dynamic component, but beyond that, very little is treated as such. In reality, however, very little in a modern manufacturing facility is ever static. When multiple parameters are varied over time, discrete event simulation modeling tends to become extremely complicated.

12.2.1 Production Equipment In a typical academic modeling example, production equipment sets are fixed and therefore static. Perhaps, the modeled factory’s management team might elect to purchase a piece of equipment, but the toolset is rare if ever considered to be dynamic beyond the distributions for interarrival times of WIP to the equipment, duration of maintenance events, and perhaps service times. In real factories, there are production transitions, human-induced effects on equipment, and large variability in demand, as well as programs to continuously improve the performance of the equipment, quality and cost. The time during which a production process increases from a startup-production phase to a high volume manufacturing (HVM) phase is sometimes known as a ramp. This is a period that, by its very nature, is highly variable in many important aspects. During this period, the equipment may be the most dynamic component of the facility as the equipment is installed, tested, brought up to specification, modified, and also used for engineering experiments to certify the equipment qualified for production. Despite all of this activity (which is frequently undocumented or only documented locally and offline), production still must go on and customer demands must be met. Further, all of these dynamic effects result in interactions that thwart traditional production planning methods such as those that utilize spreadsheet calculations. The component of the process ramp that is sometimes known as the learn rate is a variable that accounts for many different time-dependent factors. For instance, operators have a learn rate while steadily improving maintenance of new equipment. Process or product engineers have a learn rate managing statistical process control as they gain a better understanding of the parameters that have significant effect on

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the health of a product as it moves through the factory. These rates (and others) can be tracked and modeled as time-varying efficiency factors on maintenance, rework rates, engineering hold times, etc. However, they present yet another set of model input vectors that obfuscate trend detection via simulation. Demand for a product(s) varies over time. A frequent practice in factory cost control is to “bag” equipment, i.e., temporarily cease the use of particular tools in an underutilized toolset. In this “bagged” condition, the equipment still remains installed on the production floor, but is not used in manufacturing. This reduces consumables, headcount, facilities, and maintenance costs with an expected trade-off in reduced capacity to absorb line variability. However, the decision processes used to determine when to bag or unbag equipment are rarely documented, and are typically complex heuristics that are open to interpretation, on-the-fly modification, and violation by manufacturing supervisors. This presents great challenges in implementing these somewhat chaotic rules in a simulated environment. Another critical element often encountered on the manufacturing floor and tooinfrequently codified in simulation models is continuous improvement. Continuous improvement of the manufacturing process is critical to improve factory efficiency, and projects occur regularly to address cost, quality, and equipment performance. If simulation models are built with constant input values while these values are changing in the real factory, the model outputs will often be invalid. However, determining which parameters must be included and the rate at which values change is not an easy task. In a factory where hundreds of production tools are involved, keeping a simulation model abreast of these changes can be a monumental effort.

12.2.2 Labor The authors have found that accurate modeling of labor and its interactions with both WIP and equipment can be critical to successfully predicting factory performance with any accuracy. While it is impossible to model human decision-making perfectly, it is important for the model to comprehend the competition for operator resources that are observed when WIP, equipment, and facilities all require service. Unfortunately, modeling labor at an appropriate level requires comprehending the training (skill) level of the operators and technicians who keep the factory and the equipment running. Skills-training databases are often incomplete, standalone systems that suffer from significant update lag compared to the online and real-time systems that track equipment availability or WIP movement. In addition, factory personnel often receive training to expand or improve their skills, thereby increasing their efficiency in performing operations, maintenance, or moving WIP through the system. Thus, inclusion of learning rates in models could prove beneficial in increasing labor model accuracy. A further dynamic factor in factory performance is the degree of cross-training and cross-coverage that occurs in the factory at any given time. Cross-training (defined as the extent to which operators or technicians are capable of running or servicing different equipment types) can be captured in a simulation model at the

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same level of detail at which regular training is modeled. However, it is the “human glue” of cross-coverage (i.e., filling in temporary gaps in headcount allocation by moving individuals to wherever shortages exist) that permits a real factory to perform and provide lower output variability than any reasonable simulation model. While in some cases it is technically feasible to create optimization programs and assignment algorithms to assist in reallocating labor in a simulation model, such models are approximations, with assumptions that can fail to capture the nuances of a real system. It is also very likely that frequent use of such systems in simulation could unacceptably degrade model runtime performance. One final note about the difficulty of labor modeling is that factories almost always have a last resort – when the in-house technical staff cannot solve a problem with equipment, typically the vendor that sold the equipment can fly in a “specialist” to fix the problem. The corporate recordkeeping for these specialists may be very detailed in some ways for contractual reasons, but almost assuredly that data is housed in systems that are neither synchronized with nor related to other factory data.

12.2.3 Internal Factory Dynamics In the authors’ experience, simulations are best used for relative comparisons of policies and decisions in order to detect trends due to a change or changes in a system. Multiple trend-inducing factors cannot easily be modeled simultaneously – as mentioned earlier, multiple factors confound results and it is difficult to comprehend which parameter had the greatest impact on model output response in an environment containing tens of thousands of input data values. There has been some significant research (Schruben 1986) in correlating output sensitivity to input parameters, but this has yet to be put to practical use in environments with thousands of potential parameters to which the model might be sensitive. In addition, runtime issues and system hysteresis likely preclude this type of approach in very large and complex models. In discrete part manufacturing, the single-product factory is quickly becoming a thing of the past. With globalization, increasing market diversity, consolidation of manufacturing resources to benefit from economies of scale, and other factors, the variety of different products types (known as product mix) that must be manufactured at any particular facility is a constantly increasing trend. The result is that many different products are constantly being introduced, produced in high volumes, and phased out simultaneously all in the same facility. Simulations to understand factory performance must include these changing demand profiles to accurately predict the effects of system capacity changes. Unfortunately, with every infinitesimal shift in product mix it is highly likely that the shift itself will affect system capacity due to differing product-level processing times, product routes, and product maturity (i.e., is the product “easy” to manufacture or “difficult”? Product maturity can greatly impact the amount of rework, analysis, quality control measures, etc.). Thus, over the course of a short-period simulation, demand and factory output

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may not converge – a product could very well be ramping or de-ramping during the simulated period, no matter how short that period is. This is just one source of variability and frequently a confounding factor in experimentation and analysis. The combinations of these factors can be so confounding; in fact, it can be argued that requests for ramp analysis should generally be declined. Given the nature of simulation as a trend detection tool, isolating a trend in behavior when inputs are constantly trending themselves is nontrivial. When building simulation models, the process route (or manufacturing sequence) is generally fixed, but on the floor, inspection operations utilize “skip rates” based on product and process health. As a result, inspection sampling rates are dynamically dictated by how well the process is performing. This is but a single example of process flow impacts. Capturing behavior like this is often required to achieve a valid model. A recent survey of this volatility showed that a manufacturing floor in a large volume factory exceeded 2000 changes over a 3-week period. Keeping a simulation model updated with this large number of modifications is impossible to do, unless the models are automatically synchronized with the real factory. This requires that the data systems in the real factory automatically track these changes. Product rework flows are often a major problem in a modern manufacturing facility, and unless the manufacturing execution systems closely track and archive the rework frequencies and locations in the flow where these occur, modeling rework is untenable. Adding further convolutions are the paths of the defective material, and the resulting quality control schemes that determine where and whether the reworked material rejoins sibling or parent lot(s) and proceeds to the subsequent steps in the manufacturing flow. Often the modeler must invest significant time to determine whether data is even available for modeling these flows accurately. In summary, there is never a timeframe (or period) where a leading edge factory is ever in a steady state – it is always either ramping up on a new product or ramping down an older product. Thus, new production equipments are being installed or older unused equipments are being de-installed frequently. Equipment availabilities are always changing because leading edge equipment performance is greatly dependent on product quality. Furthermore, equipment needs to be constantly qualified for new products. All of these factors lead to highly variable equipment downtime. There are frequent changes in production sequences arising as a result of new products being introduced. The frequency of inspection or quality control operations are constantly changing based on the volume and health of the manufactured products. Sampling rates change over time as do equipment dedication (or allocation) schemes, i.e., which equipment is assigned for which operation step. In a modern manufacturing facility, change is not only continuous, it is accelerating.

12.2.4 External Factory Dynamics Nearly a decade’s worth of high volatility in the world economy and the manufacturing industry provides an excellent backdrop for a discussion of the impacts of external forces on the validity of a manufacturing simulation model.

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Market demand, geopolitical issues, equipment technology, workforce profile, raw materials costs, energy costs, and many other factors all play parts in the challenges of production planning. Manufacturing simulation models are typically employed for both strategic and tactical purposes to support planning decisions. When the very foundations upon which these models are based are themselves highly variable, the task of model validation can seem overwhelming. As a brief example, capacity requirements generated by a simulation may be used to guide future equipment purchases. Orders for equipment may be placed with an expectation of a specific delivery schedule. Should circumstances change, reducing or increasing equipment needs between the time of order and the time of delivery, some parties in the transaction may face significant losses as a result. The solutions to mitigating this risk are neither trivial in nature nor easily implemented. A critical component is a market analysis and/or demand predicting organization that uses the latest technologies and best-known methods to evaluate probable scenarios and then submits the most likely to occur. Based on this information, it then becomes incumbent on the simulation analyst to assess the ramifications of those potential demand profiles. Accompanying these, the analyst must include contingency plans to manage changes should demands shift from one scenario to another. All too often the analyst will focus on a single scenario without taking into account the other possibilities. The result of this can be excessive capital expenditures for unneeded equipment, or an inability to meet market demand – thus enabling the competition.

12.3 Resolution (Level of Detail) Needed for Valid Models Deciding at what level of detail or resolution to build and validate manufacturing models is often problematic. If one constructs the model at too high (or abstract) a level, the model can neither reach the potential it was intended to achieve nor provide answers of value to the customer. Conversely, too much detail and complexity will force the analyst into the difficult task of gathering real factory data at the required resolution. In addition, high complexity can lead to extremely long model runtimes, which severely curtails the number of experiments that can be investigated and the number of replications that can be performed for the purposes of variability reduction. If the modeling effort is one at a “module” level or “functional area” level, smaller changes in tool behavior drive greater relative output characteristics that could be impossible to capture without requisite granularity in simulation design. This creates another issue – once the module-level analysis is complete, questions that can only be answered within the framework of a full-factory model may arise. The modeler’s decision to “black box” major portions of the process can easily result in the loss of major factory effects due to small variations in equipment performance.

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Fig. 12.1 A hierarchy of simulation modeling illustrating applicability of levels of detail to specific analysis goals

12.3.1 Model Resolution The degree of accuracy required is a hierarchy that closely matches that of a manufacturing plant. For example, a model may be extremely accurate at the overall factory cycle time level but not at the individual operation cycle time level. Determining the degree of accuracy that is required at each step in the process flow is dependent entirely upon the question being asked of the simulation and the interactions between the various levels within the model. Further, consider the question “When will lot X complete processing in the factory given conditions Y ?” If only 10% of the factory operations are impacted by Y; and greater accuracy is needed at factory-level output metrics than at operation-level ones, the analyst need only focus detailed efforts on those 10% of the total operations (Fig. 12.1). It is critically important for the analyst to comprehend not only the immediate impacts of the level of detail decision but also the future impacts. Development time, reusability, model runtime, and a host of other consequences will result from the decision on intended model resolution. All of these factors must be balanced with the need to provide an analysis tool that will assist in providing a useful answer.

12.3.2 Model Runtime Determining what type of model to build has a sizable effect on the granularity of the system. Will it be a full-factory model? If so, there are reasons to dispense with highly detailed tool-level behaviors, for example. Model verification takes

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significantly longer when finer details are considered when task selection is occurring, and hence the least detail possible tends to be best at driving on-time delivery of results. In addition, as more detail is added, model run times become excessive. By the time the modeler produces results the factory customer could use, the production issues may have come and gone, and the original problem (the one the model was originally meant to address) no longer exists! This is a common frustration experienced in the modeling world. Often the deciding factor in determining the detail level of the model is the availability and granularity of the source data required to drive the model. As detailed later in this chapter, sometimes the data needed to populate a highly detailed simulation model is either inaccessible or simply does not exist.

12.3.3 Modularity There are many situations where analysis demands modular models that can coexist with other modeling tools in the analyst’s tool box. This can be particularly frustrating for the developer because the model resolution requirements may be dictated by the overall modeling “framework” that is/will be established, even though that detail may not be required in a standalone model. One way of managing this issue is to treat model and/or framework development as a true software engineering endeavor. Gathering and assessing requirements is potentially the most critical phase, as it must not only set the stage for developing a model to meet current needs but also provide a foundation on which to address future analysis requests. These potential future requests can provide a strong enough influence on design direction that modularity will be ensured. The ability to anticipate and identify these future requirements is one trait that distinguishes a superior simulation analyst. Finally, utilizing a standard software development process can help ensure well-understood behavior among the various components of the system.

12.3.4 Validating Models Another frequent challenge is the level of validation accuracy that the customer expects. The customer may demand 95% accuracy for all major metrics (cycle time, utilization, output, etc.) or 80% for all metrics except one – and expect 100% for that measurement of interest. An important factor to consider is whether the customer has real data that can be compared to model results so that realistic comparisons can be made on accuracy. It has been stated that significant detail is required to achieve a representative model of a complex factory. The production equipment’s processing run-rates and

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the availabilities of tools are key driving components when developing a basic simulation model, but are not enough to validate a model. Other intricate details, such as the time required for the first piece of the WIP to complete processing through the equipment (also called first unit delay), product mix, tooling used, equipment qualification schemes, automated transport system delivery times, spoilage, or timedependent process segment limits (constraints defined by manufacturing process complexity) are critical to make progress toward validation. Better understanding of this variability is required to develop a model of sufficient robustness to achieve a valid framework and begin experimentation and exploratory analysis. When validating a simulation model, the prime consideration is to determine when the model is valid. First, validation criteria must be defined in the initial stages of the analysis, while taking into account the granularity of the model needed to answer the customer’s questions. Assumptions must be clearly stated, accepted by the customer, and documented in the early stages of model development. The final concern with model resolution is the amount of detail required for customer confidence in the model. Making major decisions based on the model outputs requires the trust and belief that the appropriate level of diligence has been applied to making the model as accurate as possible. The authors have found a higher-than-expected level of validation is required for acceptance of simulation models and modeling results. This is due to the inherent understanding of how static models work versus the nearly universal lack of understanding of discrete event simulation. The expected levels of accuracy and precision in validation are inversely proportional to the level of comfort in this “new” modeling methodology. These expectations are unprecedented compared to those for any static modeling framework, such as a spreadsheet. This raises a concern about the validity of static spreadsheet-based models that are often used in most firms.

12.3.5 Deployed Vs. Disposable Models One of the earliest decisions made on model detail is determining what level of model accuracy will meet customer acceptance. If the simulation runs are to be performed by the simulation customer, then a highly customized and polished modeling tool with a comprehensive lifecycle from development through implementation, continuous improvement, and enhancement is necessary. Simply put, simulation is generally hard and must be treated from a software engineering perspective to ensure usability in the long term. In addition, to ensure that future customer needs are met, the model must be designed with a level of flexibility that will enable the insertion of future revisions and enhancements. Each revision cycle takes time, effort, and money that could prove onerous enough to lose customer interest. Alternatively, sometimes a “disposable” simulation model (generally singlepurpose and not intended for reuse or integration with a larger model) is all that is required to provide a solution. In this case, the model developer is likely to act

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in the capacity of the analyst as well. In these situations, the modeler must not only foresee the detailed requirements, but also perceive what kinds of follow-up questions may be asked of the model. This allows construction of a “high-level” or “lean” model that provides detail only where necessary and is simplified as much as possible to avoid nonvalue-added development time. However, the experience level required of the developer/modeler to perceive these future questions is quite high. The goal must be to give the customer what he needs, when he needs it – not what he wants.

12.4 Data and Operational Complexity of Manufacturing Systems Inherent manufacturing complexity drives significantly longer simulation model development cycles. There is also an increasing expectation of simulation/simulator capability from factory management for understanding complex operational policies and the data associated with these actual floor policies.

12.4.1 Functional Complexity Modern factories tend to be highly automated with many data collection and reporting systems. In addition, there are many procedures and policies that are encoded into decision-making systems for the purpose of standardization and consistency of production. However, any system that allows even a modicum of human interaction (or interference) is subject to the situation where the operator, technician, or engineer solves a problem and fails to communicate that solution such that it later becomes encoded into the factory systems. This “latent knowledge” tends to grow over time as emphasis is placed on fixing new problems that arise rather than performing due diligence and codifying these new “best-known methods.” Thus, it is often the case that the policies of the entire factory are not understood by any single person in the factory, but for modeling purposes must be (often for the very first time) collected from several factory operations personnel and combined into a set of operational policies (model input data assumptions) to drive factory performance. The task of interviewing and developing a set of factory policies and procedures to the appropriate level of detail can take months, often requiring interviewing multiple individuals possessing a wide array of skills to find the true “keepers of the knowledge” at the appropriate level. Finding and being able to extract this knowledge from the various individuals is very demanding and time consuming. The next step – converting these factory operational policies into model execution logic within the simulator – becomes second nature after being understood by the simulation modeling specialist. The major problem is that if the tasks

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of data collection and model building take too long, the value of the simulation outputs is often lost – new complexities and problems surface over time, which could invalidate the assumptions and inputs previously gathered. When modeling a factory at a fairly high level (with minimal detail beyond simple operating policies and equipment quantities), it is rarely necessary to encode the detailed behavior at the equipment level into the simulator. However, in today’s competitive environment, with ever-increasing pressure to contain costs and expectations of greater performance per dollar invested, detailed understanding of the internal behavior within the production equipment which affects the overall factory performance is becoming more and more in demand. While this may seem simple, in reality it is very complex to do because often the logic of the detailed micro-flows within a unit of production equipment is locked in intellectual property owned by the maker of the equipment. For example, vehicle or robot routing logic to prevent deadlocking may be considered proprietary information and therefore unavailable. The two solutions available are to either purchase simulation software directly from the equipment vendor (however, this simulation software is generally incompatible with full-factory simulation environments) or to make gross assumptions based on limited observation. Implementation of functional complexities in a simulator is very difficult and often a higher-level abstracted view is taken to simplify the problem and meet the time constraints of the simulation project. One critical consideration is the actual development time of the functionality under proposal. Development time for simulation is directly proportional to the level of complexity of the logic. Often, it is best to apply a Pareto-like analysis and provide 80% of the functionality in 20% of the total development time to ensure that some functionality is available before the solution window of opportunity has passed and the factory has moved on to address the next issue. The only alternative is to anticipate the future issue in advance of the customer request. This allows the customer to be removed from the complexity and long development cycles. The ability to quickly model new features and new process issues puts the modeler in a highly satisfying position and provides an immediate boost in confidence of the modeling technique, in contrast to subjecting the customer to a long development cycle. This anticipatory model development is not without risk, and it requires the modeling resource to be engaged in future product and manufacturing process planning to make intelligent decisions on what capabilities to add to the modeling framework. Simulation models are often unable to comprehend inputs from the myriad realtime decision support systems that exist to aid operators on the production floor. An example might be a dynamic web page that changes contents in real time, and is used by operators to make continuous decisions about work prioritization due to customer demand. These systems often pervade the factory, but the feasibility of incorporating them into the simulation models inputs is very low due to the nondeterministic nature of their input data sources and the fact that different actions may be taken by different shifts based on operator knowledge and intuition when interpreting the real-time data.

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12.4.2 Poor and Often Changing Data Systems A perpetual dilemma in simulation modeling is validating the simulation model of a future manufacturing facility that is yet to be built. Typically, the sole option available to the modeler is to use a previously validated model that bears some resemblance to the proposed system, and make modifications as necessary to bring the model in line with anticipated future reality. However, there are many critical data that can cause significant difficulty in creating a useful model of a future facility. Assuming the modeler successfully implements and verifies a model of a future facility, validation must occur via methods similar to those employed in building the model. The model must be validated on criteria for which data is currently available and anticipated to change little in the upcoming factory. The options for validating the remaining, nonvalidated criteria are slim – often the modeler must consult with domain experts and engage their services to confirm the model is a reasonable representation, where the nonvalidated criteria are taken into account. One source of ambiguity (in a new factory) is not having any real knowledge of the future production equipment’s reliability and maintainability (R&M) metrics. As data systems change or are replaced or even created anew, historical (real) data becomes stale, incompatible, or is disposed of. Without historical sources of data from past systems, there is no information to permit validation of model accuracy. Relying on equipment suppliers’ specification sheets on tool R&M metrics has not proven to be useful in many facilities. A major reason is that the data provided by suppliers are best-guess estimates and average values. Simulation methods need probability distributions for maintenance, repair times, and the times between maintenance and repair times to provide reasonable estimates of future system performance. Tool usage methodologies specific to a factory can have significant impact on the realized variability of tool behavior. For example, equipment could be down twice a week for 3 hours each time for a process-induced setup, or down once per week for 6 hours. Both of these examples generate the same average availability, but completely different dynamics and impacts on factory performance. In addition, today’s capital equipment purchasing practices tend to rely on average availability metrics. In actuality, the variability of availability often has a much greater impact on factory performance. This attitude proliferates to the production floor to the point that availability is often reported as an average percentage for a time period (often in a spreadsheet). In many cases, the variability of availability is not even acknowledged as a component in factory performance. This leads to a culture that values average performance over consistent performance. For this and other reasons, factory behavior regarding preventive maintenance (PM) may not be well documented. As a result, best-known practices may be lacking. In some factories, for example, long duration PMs are delayed to the start of a shift to avoid the need for a major PM status communication across shifts. Many instances of “humaninduced batching” have been observed in action, where the maintenance personnel groups several PM actions for the sake of convenience. All these significantly add

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to the factory variability but are often hard to document. As a result, these sources of variability are not incorporated or require significant effort to incorporate into the simulation model.

12.4.3 No Two Tools in a Leading-Edge Factory Are Ever the Same Equipment sets are rarely created equal. Typically, there tends to be a specific tool or subgroup of tools that is unable to operate as well as the others in a fleet. No matter how much PM, qualification activity, monitoring, or adjustments are performed, these so-called “dog tools” fail more frequently, fall out of specification too easily, never achieve expected availability levels, or fail to perform predictably or generate high yields. The production personnel are aware of these underperformers, and as a result of this, perceived differentiation, operators avoid running product on them. It is difficult for simulation modelers to know about the existence of these kinds of real-life anomalies when building models and attempting to validate them. Without the benefit of lengthy visits to the production environment, modelers may tend to assume that most workloads are equally divided across all the equipment resources that are available, and this assumption generates a highly optimistic estimate of what the equipment set can do in reality. Thus, the model’s results will not converge with behavior observed in the real world.

12.4.4 Systems Do Not Track Data at Sufficient Granularity to Satisfy Simulation Needs Another cause of modeling difficulty is that manufacturing execution systems’ equipment tracking and WIP tracking subsystems are often not at the level of granularity needed for precise simulation model verification and validation. In addition, these systems allow human interference to recommended WIP schedules, where personnel make decisions that are not documented or well understood and vary widely from shift to shift. These decisions are neither analyzed at a later time nor captured in the system for future use. Finally, these policies are frequently designed by nonindustrial engineers, often without understanding of overall effects before implementation. For example, a WIP rule such as “least processing time remaining” has certain benefits in particular scenarios, but in others tends to be the worst possible decision to make in terms of overall factory performance. However, without actually simulating these policies, making changes is an “experiment” with multiple millions of dollars at risk should the policy perform badly. This can result in an unwillingness to permit modifications to existing policies even if new proposals are demonstrated to be effective in high precision simulation environments.

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12.5 Domain Expertise and Practical Modeling Skills of the Modeling Engineer Modeling cycle time must keep up or be ahead of changes in customer requirements for model change. In such a changing environment, how can validation occur? If modelers are not proactive, it becomes very difficult to stay ahead of the advances in complexity. Building deep modeling expertise requires sustained model maintenance and management commitment with dedicated modeling resources. As stated previously, simulation modeling is a complicated activity. In order to stay current as well as maintain valuable development skills, a simulation engineer needs to commit full-time to simulation-related work. Often, the engineer cannot convince management to support this – in some cases, the simulation process could be slow, monotonous, and frequently perceived by those ignorant of simulation to be of less value than other analysis techniques. The ideal simulation engineer needs to have a combination of strong software engineering skills and industrial engineering skills. In addition, the engineer developing the simulation must have a very good understanding of the manufacturing area (understand the domain very well) and the details that are being modeled so that they can accurately validate the results as model enhancements occur. This requires significant time spent on the manufacturing floor toward better understanding of the production processes, factory automation systems, equipment behaviors, and WIP management policies used. The simulation modeling skill set is difficult to find and foster. Those that do have an interest typically lack serious training in disciplined software engineering techniques, which can lead to credibility loss for simulation as an analysis tool. Simulation also requires extremely high detail orientation, as there are extensive data inputs necessary to make these models accurate and value added. Sifting through megabytes of input data to determine whether a particular parameter is correct requires great patience and investigative proclivity. None of these skills comes easily or without strong factory operations understanding and software engineering experience. It takes several months or even a year before a new simulation engineer can begin producing truly effective analyses and new modeling capabilities.

12.6 Need for Ongoing or Frequent Customer Education of Simulation Capabilities/Limitations “All models are wrong. Some are useful.” (Box 1979). The complexity of today’s factories precludes the use of intuition to make appropriate decisions. In the past, key factory decision makers often relied on latent knowledge, intuition, or gut feel. Today, they reject the notion that their decisions, while perhaps being very reasonably made with years of experience, could result in millions of dollars of wasted capital expenses if their conclusions were based on incorrect data and assumptions.

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The complexity of today’s manufacturing systems and the dynamic interactions that are possible makes it impossible to base decisions on intuition alone. Even the most senior factory managers cannot possibly digest or internalize the ever-changing requirements of the latest production technologies in a complex manufacturing environment. A senior factory manager most likely earned his/her stripes as a very successful manufacturing manager or supervisor. In the intervening years, the factory and its underlying technologies have most likely changed dramatically. Unless this person stayed very well connected to the details of daily changes of the evolving factory and technology transitions, making decisions based on intuition alone is often flawed. Consequently, simulation modelers face a continuous challenge convincing these critical experienced customers, especially when their simulation models are significantly different and the outputs are not close or aligned to the intuition of the senior stakeholder. Further, the hysteresis in large manufacturing lines and the human decisionmaking processes that are often in place to reduce variability frequently allow suboptimal, intuitive decisions to appear to work. To management, this is often sufficient proof that simulation models are unnecessary. The negative impacts of poor decisions are sometimes slow to manifest themselves and when they do, they can usually be blamed on some other change in the factory. Another challenge that many simulation modelers face is that they have difficulty communicating their results to manufacturing customers in a way that nonsimulation experts can understand and internalize. To be successful, they need to be cognizant of their audience and communicate in the same jargon as manufacturing personnel. They need to clearly articulate and explain in simple terms why a result looks the way it does. The customer must be made to believe in the model and its conclusions. Otherwise, modelers can become victims of the Not Invented Here (NIH) syndrome – they must be able to respond to the question “Who are you to tell me XYZ should be changed?”

12.6.1 Complexity of Today’s Factories Preclude Static Analysis Variability and the resulting emergent behavior created by dynamic interactions of sources that continue changing can have as much or greater impact than average behavior. The only tool known to utilize methods that comprehend multiple levels of variability is discrete event simulation. However, despite all these inherent advantages, many firms still avoid the use of simulation and insist on relying on spreadsheets to remain relevant – on the theory that it worked in the past and the state-of-the-art in manufacturing has not changed. While these static models have their uses for rough estimation for long-range planning purposes or in situations where minimal data is available (and therefore the effort in producing a simulation model would be a poor investment), even these models can still be highly sensitive to data quality and availability – and can be badly misused without the appropriate education.

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12.6.2 Existence of Biases Against Simulation Good simulation requires all the skills mentioned earlier. Unfortunately, simulation is frequently attempted in isolation, without customer and domain experts’ engagement, which usually leads to poor results. A modeling engineer who is not fully trained in the technology and is unaware of the pitfalls of simulation may attempt to use the simulation language he or she learned as a student in college and take on a problem that is orders of magnitude more complex than the simple questions they addressed in school. These efforts are inevitably doomed to failure, and it creates a bias against the tool, not the user or developer – the remaining legacy is that “simulation failed.” Usually, the customer is rarely willing to understand the root cause of the failure. Since some of these failures typically occur early in a modeling engineer’s career, the customer generally considers failure of the simulation concept as the reason why modeling failed, rather than that the problem was the technical competency of the people involved in the modeling effort. For years, spreadsheet models have been used for major decisions despite their known flaws. Since the numerous complexities of today’s manufacturing processes conspire to mask the flaws in such modeling practices, simulation tends to be either ignored or the target of incredibly high expectations. Because of this, simulation continues to be less well understood than it should be, yet it is often expected to be 100% correct. This perception is fueled by the fact that a great deal of input data is often needed to drive these kinds of models. In the authors’ experience, one failed simulation analysis can destroy the simulation modeling methodology of five highly valued analyses and several years of credibility development. Flawed spreadsheet analysis is quickly forgotten due to the quick turn around and the volume of analyses performed using spreadsheets.

12.7 Conclusion Discrete Event Simulation modeling can provide very advanced analysis capabilities in a state-of-the-art factory environment. There is a strong business case for providing this sort of functionality to allow factory operators, technicians, shopfloor management, and policy makers to make the highest fidelity decisions possible about a myriad of manufacturing issues. However, the barriers to building, validating, and then fully utilizing these capabilities are numerous and complex at many levels.

12.7.1 Business Processes The business processes in a manufacturing culture tend to be entrenched, resulting in several hurdles that a simulation engineer must proactively overcome before they

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begin to provide new analysis capabilities. Many organizations often institutionalize static decision support methods due to their simplicity, ease of management, and ease of cross/replacement training due to turnover in manufacturing decision support personnel. Data systems are designed and implemented with this level of flexibility in mind, and with less than optimal regard for the future potential for integration into unified decision support systems. Further, requests for analysis often assume that simulation analysis can be performed in the same time horizon as a lower-accuracy static model. Due to the steep learning curve to attain effective simulation development and usage, training budgets are often insufficient to provide the expertise necessary to perform analyses at the level required for effective decision support. These and several other similar factors continue to be daunting challenges standing in the way of pervasive proliferation of simulation analysis technologies.

12.7.2 Technical Challenges The technical complexities of simulation development and validation are frequent barriers to entry in modern manufacturing environments. System dynamics is often complicated, making model input data capture even more difficult. Customer requirements of the model often drive the depth of detail required by the analysis task. Complete data to drive these models are rarely easy to access, or are in the wrong format, or at inappropriate resolution, or sometimes nonexistent. This makes simulation modeling even harder. Despite these challenges, a validated simulation model can be extremely useful in supporting difficult decisions regarding capacity and policy analysis.

12.7.3 Perception Management via Constant Education The issues listed above are all surmountable in various degrees. However, accomplishing this requires support from factory management at all levels with a deep understanding of the capabilities, training, resourcing, systems, and time needed to build robust discrete event simulation models. This education is extraordinarily difficult in environments that lack a culture of standardization and data-driven decision making.

References Box GEP (1979) “Robustness in the strategy of scientific model building.” In: Launer RL, Wilkinson GN (eds.) Robustness in statistics. Academic, New York DeJong CD, Fischbein SA (2000) “Integrating dynamic FAB capacity and automation models for 300mm semiconductor manufacturing”. Proceedings of the 2000 winter simulation conference, 1505–1509

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Morrice DJ, Schruben LW (2001) “A frequency domain metamodeling approach to transient sensitivity analysis”. IIE Transactions 229–244 Pillai D, Bass E, Dempsey J, Yellig E (2003) “300mm full factory dynamic simulations for 90nm and 65nm IC manufacturing”. IEEE International Symposium of Semiconductor Manufacturing conference (ISSM), San Jose, CA Pillai D (2002) “Operations management challenges for 300mm production”. Modeling and Analysis of Semiconductor Manufacturing (MASM), Phoenix, AZ Roeder T, Fischbein S, Janakiram M, Schruben L (2002) “Resource-driven and job-driven simulations”. Proceedings of the 2002 international conference on modeling and analysis of semiconductor manufacturing 78–83 Schruben L (1986) “Simulation optimization using frequency domain methods.” Proceedings of the 1986 Winter simulation conference 366–369 Sohn B, Pillai D, Acker N (2000) “300mm factory design for operational effectiveness”. IEEE/ASMC Conference, SEMICON Europa. Munich, Germany Subramaniam B, Golla C, Lockwood C, Pillai D, (2003) “300mm equipment buffering strategy to maximize fab productivity”. IEEE International Symposium of Semiconductor Manufacturing conference (ISSM). San Jose, CA Yellig E, Fox D, Tag P (2002) “Maximizing value of factory automation: 300mm fab capacity.” Proceedings of the IEEE international symposium on semiconductor manufacturing – ISSM 271–274

Chapter 13

A Practical Approach to Diagnosing and Tuning a Statistical Forecasting System Ying Tat Leung and Kumar Bhaskaran

13.1 Introduction and Background 13.1.1 The Role of Demand Forecasting Most, if not all, commercial enterprises require some form of demand forecasting for financial and operations planning. For financial planning, a high level, aggregate forecast (e.g., in dollar value) of major product groups or geographies is sufficient. For operations planning, a more detailed forecast, such as forecast by product or even by product-location, is necessary. A manufacturing enterprise employing a make-to-stock strategy needs a demand forecast to plan what products and how much of each to build. A make-to-order manufacturer uses a demand forecast to plan the purchase of parts and materials and its production capacity. A retailer needs a demand forecast to determine how much of each product to stock at the different retail locations. Other service enterprises utilize a demand forecast to plan and locate their capacity (for both labor and equipment). We focus on the latter situation in this chapter, namely detailed, product level forecasts that drive the planning of a supply chain. Forecasting processes play a critical role in the demand planning function in a supply chain. Demand planning specifies how much demand we would like to satisfy, where, and at what selling price. It is an aggregation of forecasting, target marketing, and product-planning processes (see Fig. 13.1). The forecasting processes within demand planning use sales data, such as point-of-sale scanner data, customer and market segment information from target marketing, promotion plans, and product categories from product planning to generate demand forecasts. These dependencies also highlight the collaborative nature of the forecasting activity in an enterprise and typically involve many role players in a supply chain, such as the forecast analyst, the promotion planner, and the merchandizing manager. Forecasting processes are also supported Y.T. Leung () IBM Almaden Research Center, 650 Harry Road, San Jose, CA 95120, USA e-mail: [email protected] K.G. Kempf et al. (eds.), Planning Production and Inventories in the Extended Enterprise, International Series in Operations Research & Management Science 152, c Springer Science+Business Media, LLC 2011 DOI 10.1007/978-1-4419-8191-2 13,

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DEMAND PLANNING

Sales Data Customer Preferences

Forecasts

Target Marketing

Forecasting Product Categories; Promotion Plans

Replenishment Plan

Market Plan

Segment Bundles

Product Planning

Legend Process

Information Flow

Product Plan

Fig. 13.1 Business processes in demand planning

by demand-planning-related analytics, such as analysis of elasticity or sensitivity of sales to promotional events, analysis of demand for products that are considered slow movers, impact of clearance pricing, data mining of customer buying patterns, etc. The final demand forecasts from the demand planning function become a key input to the production planning function. In particular, the master scheduling process, in the production planning function, uses the product level forecasts (or higher level, more aggregate forecasts, depending on the structure of the product families), the material constraints from material planning processes, the capacity constraints from the capacity planning processes, and the current schedule status from the finite capacity scheduling process to formulate the production plan (see Fig. 13.2).

13.1.2 The Practice of Demand Forecasting and the Role of This Chapter A supply chain is often engaged in the production of thousands or even tens of thousands of products, where each individual product may account for only a small portion of the total revenue (Fildes and Beard 1992). Therefore, it is neither practical nor economical to spend a lot of effort in forecasting a single product, except for the few top-selling products dominating a significant fraction of the total business. A relatively inexpensive and efficient way to forecast the sales of each of a large number of products repetitively is to use an automatic forecasting system. Such a forecasting system can be based on heuristics [e.g., as in “focus forecasting” (Smith 1991)], statistical methods (e.g., time series, regression; see Makridakis et al. 1997, Chaps. 3–7), expert system-like rules, or a combination.

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PRODUCTION PLANNING Product Plan; Forecasts

Material Constraints

Supply Data

Legend

Material Planning

Material Availability

Master Scheduling Schedule Production Status Plan

Finite Capacity Scheduling

Capacity Constraints

Capacity Planning

Capacity Availability

Process

Information Flow

Production Schedule

Fig. 13.2 Business processes in production planning

The most popular approach, besides the use of judgmental forecasting (Sanders 1997), seems to be statistical methods. This is also evidenced by the large number of available software packages for statistical forecasting (Yurkiewicz 2004). Due to the large number of products involved, it is common that the system runs on a server computer and is part of a larger supply chain management system. In principle, these forecasting systems, once setup, require minimal human intervention to operate and generate a forecast. This, however, only means that little human effort is required to produce some forecast; whether the forecasts are accurate or suitable for use is a different issue altogether. In addition, over time, the forecast performance of such a system tends to deteriorate if they are not diagnosed and tuned in a continuing basis. In this chapter, we propose a practical approach to diagnose and tune the forecast performance of a typical statistical forecasting system under such an environment. Indeed, it has been shown, at least in one case of the Holt–Winters forecasting procedure, that an automatic version can be significantly improved by subjective modifications (Chatfield 1978). Commercial statistical forecasting systems typically come in two types – standalone, specialized software for statistical analysis including statistical forecasting, or as a module in an integrated supply chain management or enterprise resource planning system. Specialized forecasting software provides more in-depth functionalities to forecast an individual product. Most notably, many of such specialized packages provide the ability to find optimal parameter settings for the chosen forecasting model (see table of software survey in Yurkiewicz 2004 and other software surveys listed in Rycroft 1999). But because of their standalone nature, significant integration effort has to be spent in utilizing these packages to develop an overall system suitable for managing thousands of products on a

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daily basis. For this reason, they are not nearly as popular as forecasting modules of integrated supply chain management systems in our experience. We focus on the use of statistical forecasting within an integrated supply chain management or enterprise resource planning system. Supply chain management systems have gained wide acceptance in the last 20 years, covering a wide range of supply chain management activities, including sales forecasting, inventory management, distribution requirements planning, master production planning, materials requirements planning, and even shop floor scheduling and control. Due to the enormous scope and the necessary complexity of such a system, the system designer has no choice but to limit individual modules to a relatively small set of key functionalities. In forecasting, for instance, there is usually very little provision for diagnosis and tuning such as forecast error analysis or parameter optimization. The approach discussed in this chapter and the simple methods we have selected within the approach represent one way to fill this gap, providing the forecast practitioner with easily accessible and simple-to-use tools that can be utilized immediately. We note that such diagnostic and tuning activities should be carried out at regular time intervals, even if the products and the supply chain delivering them have remained largely the same. This is in contrast to other quantitative models used in manufacturing planning (e.g., production planning models), where the models themselves do not need to be maintained unless there is a change in the user requirements, product or supply chain structure. Many enterprises today have not budgeted for the model maintenance of forecasting systems (other than data maintenance) and perhaps erroneously conclude that their existing forecasting system no longer works and then unnecessarily invest in a new system. Through this chapter we hope to raise awareness of the importance of the analytic maintenance of a model-based forecasting system and to illustrate how it can be done easily with simple tools. Throughout the chapter, we use results of an example we have encountered in practice to illustrate the diagnostic and tuning tools. In this example, a manufacturing enterprise produces industrial and consumer items of a commodity and semicommodity nature. It adopts a make-to-stock strategy and utilizes a demand forecasting module of a commercially available supply chain management system. At any one time, there are more than 3,000 active products for which demand forecasts are made regularly. For each product, the system can be set up to use a user-chosen forecasting model or a model selected by the system based on selected error criteria. Once a month, the model is used to generate a demand forecast in monthly buckets for the next 12 months. We assume that this forecasting module will continue to be used in the future. The complex issues of whether there is much to be gained by switching to a specialized, standalone forecasting system, what kind of models should be considered, and how the system fits in the business process of forecasting are outside the scope of this work. (See, e.g, Chambers et al. 1971; Jenkins 1982.) Our objective is to improve the forecast performance of the system in use. To this end, we assume that the principal quantitative measure of forecast performance is the mean squared error of the one-period-ahead forecast of the recent past 12 periods, i.e., we attempt to minimize this mean squared error. A key reason to use the

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mean squared error is that the direct cost of the forecast error is inventory in the supply chain, since the root mean squared error is typically used to calculate the necessary safety stock to serve a demand point. The one-period ahead forecast is considered because the lead time to change the production plan is one period, so that longer range forecasts are less critical for production and inventory commitment. Note that we use the terms “sales” and “demand” interchangeably. Strictly speaking, we should be performing demand forecasting as sales may be influenced by the actions or constraints of the business itself.

13.1.3 Organization of This Chapter Diagnosing and tuning is one of a number of important issues in setting up and operating a statistical forecasting system. Section 13.2 provides a discussion of these other issues. Assuming that we now have a statistical forecasting system in operation, our objective is to improve its performance (in terms of forecast accuracy) using the following main steps: 1. Diagnosis of the overall system to identify what products have the most potential for forecast accuracy improvement (Sect. 13.3.1). 2. Diagnosis of the individual products identified in Step 1 to gain insights into the characteristics of the demand and the current forecast (Sect. 13.3.2). 3. Tuning of the individual products analyzed in Step 2 to improve their forecast accuracy (Sect. 13.4.1). 4. Tuning of the overall system using the results of Step 3 (Sect. 13.4.2).

13.2 Practical Issues in Using a Statistical Forecasting System A statistical forecasting system typically contains forecast procedures that determine how a user can interact with a forecasting model to generate a forecast. The “user” in this context refers to someone who is knowledgeable about the data and is using the mathematical methods contained in the statistical forecasting system to obtain forecasts. More importantly, the user is typically very knowledgeable about the manufacturing operation and the business, but is not necessarily an expert in statistics or mathematical modeling. More often than not, the users regard a forecasting model as a black box. How then do they use the system? In some cases, forecasting system vendors or system-integration consultants fine tune the software and set the parameters during installation, and provide users with canned recipes which they can then use to operate the system. In other cases, the users install the system using default parameters programmed into the system by the vendor and let the system work. Ad hoc methods are then devised as needed to adjust the parameters, often with unpredictable but significant consequences for supply chain

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performance. The system manuals accompanying the forecasting software generally offer very little help to the user. The software vendors encourage the formation of “user groups”, which allow users to exchange information. Such a forum is useful to resolve procedural issues but does not provide technical information on what parameters or which forecast models are relevant for the data at hand. The diagnostic and tuning procedures reported here were developed to help the user maintain the forecasting system so that forecasts of acceptable quality can be obtained throughout the life of the system. It is important to distinguish what is done here as part of the maintenance of a forecasting system and what is typically meant by “maintenance” from the system point of view. The latter includes maintaining the required data and the software itself. This is usually recognized as a significant effort and financial and human resources are allocated for it. Maintenance as referred to here is forecasting model maintenance, which is different and complementary to the system maintenance effort. At present, model maintenance is unfortunately not routinely budgeted as a regular activity. Consequently, the forecast accuracy decreases over time. In some cases, this will eventually lead to the perhaps unjustified conclusion that the system is no longer adequate; a new forecasting system is purchased and the cycle starts all over again. Regular model maintenance will help avoid premature abandoning of existing forecasting systems in which significant investments have already been made. It will also help direct efforts toward situations where more advanced models or methods are really needed while leveraging well-known and well-proven methods for most of the forecasts. Gung et al. (2002) discuss reasons and opportunities for new forecasting models. Model maintenance is covered in subsequent sections. Next, we discuss some issues that typically occur in the ongoing application and use of forecasting systems. Addressing these issues involve subjective judgments and governance policies associated with forecasting in supply chain management. Product lifecycles, especially short lifecycles, complicate the application of forecasting systems in at least two ways. First, the system has to support new product introductions. Complex, manual procedures to create a new product in the forecasting system can become too expensive very quickly. Second, to be able to obtain a reasonable forecast at the beginning of the product’s life, it is common to borrow the sales history of another “similar” product or that of the predecessor product. If the new product has significant new features or if the market has changed (e.g., with the entry of a new competitor), using the history of the predecessor product may be questionable. Finding a similar product is not trivial. Subjective judgment often plays an important role. The forecasting system should at least provide reasonable reporting or data visualization to support the user in making the necessary judgments. Even for mature products, market conditions can change over its lifecycle such that the historical sales data of the product need to be adjusted to reflect the changing conditions. For example, in a wholesale business, a significant customer is gained or lost; in a retail business, a new market segment may be gained through product

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bundling, or a new competitor has entered the market. Customized procedures to adjust the product history have to be devised based on the individual situation. Once again, subjective judgment plays an important role. Supply chains feature multiple products and/or multiple geographic regions (or customer segments). This is set up as a product hierarchy in forecasting systems to support various business processes including marketing, financial planning, and production planning. Different levels of the hierarchy are used for different purposes. For example, for financial planning one uses the forecasts at the top one or two levels (most aggregated); for detailed production planning, the forecasts at the bottom one or two levels (most detailed, say the bottom level being the SKU(Stock Keeping Unit)-location) are used. Clearly, the forecasts within the entire hierarchy need to be consistent, in the sense that the sum of the lower level forecasts is equal to the higher level. In one common approach, the forecasting systems derive the forecast for each product or each node in the hierarchy using an independent model, then enforce consistency through taking the upper level as the reference forecast and splitting that forecast based on the lower level forecasts. Fliedner (2001) reviews different approaches in forecasting a hierarchy. Such a hierarchy makes the forecasts of the different products dependent on each other, even though the underlying forecasting models for each product may be independent. A consequence is that when some products are changed (e.g., new product introduction or product discontinuation), the entire hierarchy has to be re-run from the beginning of time in the forecasting system. Besides the additional effort required for re-running the forecasts, the new forecasts will not be identical to the historical ones, causing potential confusion. For example, historical forecasts on which business decisions have been based cannot be traced back and analyzed.

13.3 Diagnosis of the Statistical Forecasting System Generally speaking, there are two situations where diagnosis and tuning of the forecasting system is necessary: 1. On a continual basis, the system should have some automatic way of monitoring the forecast errors. These methods are specifically developed for monitoring the forecasting system continuously and are sometimes included in commercial forecasting software. Automatic monitoring is a research subject in itself; we mention only a few well-known methods here. The most widely known methods for forecast monitoring include the cumulative sum tracking signal (Harrison and Davies 1964) and smoothed error tracking signal (Trigg 1964). Golder and Settle (1976) compare these two methods using simulation. Gardner (1983) provides a general treatment while Batty (1969), Chatfield and Yar (1988), and West and Harrison (1986) discuss monitoring of specific forecasting models. Once the forecast of a particular product has been identified as problematic using an automatic monitoring method, diagnosis and tuning for that product can be performed as discussed below.

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2. In addition to continuous monitoring, we propose that diagnostic and tuning of the forecasting system should be performed periodically, regardless of the results of the automatic monitoring schemes. Our experience shows that such an exercise, even with small-to-moderate efforts, often leads to significant improvement in forecast accuracy. This situation is the focus of Sects. 13.3 and 13.4.

13.3.1 Diagnosis of the Overall System The first question we ask of the performance of a statistical forecasting system is how it compares to the case of no forecasting. Even though an automatic forecasting system is used, time and other resources are needed to manage the input data and the system itself. Therefore, we expect some return on our investment. If we do not wish to perform any serious forecasting, one of the simplest and least expensive ways to obtain some input for planning is to use historical sales data of the recent past. For example, we can take the average and standard error of sales in the past 12 months as the forecast and the basis for safety stock calculation. Indeed, an informative way to evaluate the forecast performance is to perform such a comparison, using a histogram of the coefficient of variation (CV) of the historical sales and one-period-ahead historical forecasts, over the past 12 months, of all the products we are trying to plan. The CV of sales is estimated by its standard deviation divided by its average, as usual, while the CV of the forecasts is estimated by the root mean squared error of the forecasts divided by the average forecast. The cumulative plots of the same quantities indicate the percentiles. Figures 13.3 and 13.4 contain an example of these plots. Note that the data used in all the examples of this chapter were taken from a system in real life but were arbitrarily rescaled. From Fig. 13.3, we can see that the forecasting system did shift the distribution of CV to the left, showing a general improvement (reduction) over the raw sales. However, the shift is relatively small, meaning that the improvement is not large. This is confirmed in Fig. 13.4. Ideally, a good forecasting system should produce a steeper cumulative plot of forecast CV than that of sales.

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The above CV analysis can be repeated with deseasonalized sales data, if a relatively confident set of seasonal indices are available (say from historical use of the forecasting system). This will avoid picking up products with high sales CV but with a fairly predictable demand pattern (i.e., seasonality). Deseasonalization removes the part of the CV that is attributed to the predictable, “seasonal” pattern. In our example, the products are not highly seasonal in nature, so we rely on the CV analysis of the original data. We conclude from this analysis that there is potential to improve the overall accuracy of the forecasts. An important usage of such a CV analysis is that it shows a profile of the products in terms of their suitability or necessity of forecasting using a statistical model. Forecasting is an art that has almost limitless improvement opportunities. Given one’s finite resources, we want to dedicate them to areas that are most promising in terms of returns on our investment. The following describes how a sales CV analysis help focus our forecast improvement effort. The products with the highest sales CVs (outliers or the top few percentiles) are naturally difficult to forecast statistically with good accuracy. For these products, we may have to seek alternative methods of planning without relying heavily on sales forecasts. For example, if the sales volume of the product is low or the unit value of the product is very low, then one way to plan production is to follow a cyclic schedule with relatively large production runs (say, one production run per half year or a year). Then little to no safety stock will be necessary due to the large order size but the stocking cost is still low because of the low volume or low unit value. For products with small CVs of sales (e.g., less than 0.2 for raw sales data and less than 0.1 for deseasonalized data), there is no need to use a sophisticated statistical model for forecasting. For example, we can simply take the moving average of historical sales in the past several (up to 12) months. Under these conditions, any sensible forecasting model should give a reasonably accurate forecast. Even if we had made significant effort to forecast the sales statistically, the improvement in accuracy (in terms of reduction in CV) would not have been significant in practice. The advantages of using simple methods such as moving average are less computational effort and ease of maintenance.

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The remaining products after eliminating those with high and low CVs as mentioned are the ones for which we should try to tune the statistical forecasting models. They are most likely to result in significant forecast accuracy improvements with our effort; in other words, they give the best return of our investment in time and other resources. Excluding products with the largest or smallest sales CV as explained, we can start with products with the largest 20% of all forecast CVs and cross-check with our high volume or high dollar value products (class A products in a Pareto classification) and strategically important products. If there are products falling in both categories, these are the products with the most urgent need of forecast improvement. If there is no product in both categories, we can take the next lower 20% of all forecast CVs and so on.

13.3.2 Diagnosis of Individual Products Using the approach described in Sect. 13.3.1, we select the most promising products to be considered for forecast improvement. In this section, we perform a more detailed analysis of these selected products to further understand their sales and forecast characteristics. The forecasting modules of most supply chain management systems usually include some functionality for analysis of sales and forecasts. We will mention some of the typical functionality available which should be utilized per the software vendor’s recommendations. We then focus on approaches we have found useful; some of these are provided as part of a supply chain management system while others are not but can be found in standalone spreadsheet or statistical analysis applications.

13.3.2.1 Sales Analysis The emphasis of sales analysis in most supply chain management systems is more on operational reporting and management of sales, rather than on aiding sales forecasting. More advanced systems have on-line analytical processing (OLAP) capabilities for highly customizable data display and “drilling down.” These are useful for obtaining general insights into the business, such as what products sell the most, where and what time of the year they sell at the highest rate, etc. For a selected product, analysis of the historical sales data can give some insight into how a statistical forecasting model would behave. For example, if the historical sales is smooth, then we know a simple method such as the moving average would be quite accurate and a very sophisticated or computationally intensive model is not necessary. Clearly, sales data analysis is useful in the selection of statistical models also; but we restrict our attention to whether our forecasting system in use (assuming its existence) is adequate. Starting with a simple task, we consider the standard descriptive statistics of the sales data over the full historical horizon and the past 12 periods separately: sample

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mean, sample standard deviation, sample CV, sample mean absolute deviation, sample mean absolute deviation as a fraction of the sample mean, the sample quartiles, the sample minimum and maximum. Besides the obvious patterns these quantities indicate, they are very useful in seeing quickly whether the past data are reasonable. For example, if the second or the third quartiles show zeroes, then we know that over half of our data are zero. This may mean that we do not have much past data, or our past sales were indeed very sporadic. In the former case, we can shorten the historical data horizon and in the latter case find a model, which is particularly designed for intermittent demands (e.g., Johnston and Boylan 1996). In any case, the point of this exercise is to diagnose – to raise warning flags. The difference in the statistics between the full horizon and the recent past indicates the trend in the data pattern and whether our data has changed significantly so that a smaller horizon may lead to better forecasting results. These simple methods are also useful during diagnosis of the overall system in the following way. Instead of presenting screens of historical sales data across products to the analyst, a table of selected summary statistics, say the second quartile and difference of it between the full horizon (e.g., 36 months) and most recent past (e.g., 12 months), across products is shown. In this way, it is easier for the analyst to obtain general insight into the products and their relative behavior. Further, automatic warning procedures can be easily programmed to monitor a few summary statistic values. For an overview of the data, it is useful to plot the historical sales data as a function of time for visual detection of data pattern and the data’s general behavior. If the randomness of the data is not too high, then patterns such as seasonality and increasing/decreasing trends can be detected by visual inspection. Otherwise, an autocorrelation plot of the full data set and data of the recent past can help in detecting seasonality in the presence of noise. Figure 13.5 shows our implementation of the above sales data analysis in a spreadsheet. In this example, the CV is around 0.3, in the lower middle range among all the products forecasted. From the CV, mean absolute deviation, and the quartiles,

Descriptive statistics of sales per period over full horizon: Average Std deviation Coeff of variation Mean absolute deviation MAD / Average

31282.86 8646.83 0.2764 6458.70 0.2065

Minimum 18108 1st quartile 26301 2nd quartile 29268 3rd quartile 35162.25 Maximum 58298

Descriptive statistics of sales per period over last 12 periods: Average Std deviation Coeff of variation Mean absolute deviation MAD / Average

Fig. 13.5 Sales data analysis

34496.92 11469.75 0.3325 9036.24 0.2619

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the range of sales is not large (especially compared to some other products not shown). The sales plot does not reveal any obvious pattern. Comparing the statistics of the full horizon and the last 12 periods, they show a trend of increasing variability and a slight increase in average level. The autocorrelation graph over the entire horizon does not show any significant pattern in time lags of less than 10 (higher

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time lags are not reliable due to the short time series of 36 data points). For the last 12 periods, there is a relatively strong autocorrelation (around 0.6) at a time lag of 6 periods. Coupled with the increasing variability, this product may be showing an upcoming pattern. The overall conclusion is we should be able to generate a relatively accurate forecast, but it may be getting more difficult to achieve the same level of accuracy lately. Note that we made the above judgments through comparing the different products being forecasted. While it is difficult to pinpoint critical values of the statistics to make an absolute statement, it is much easier to select the better (or worse) ones from a given set. For brevity we have not shown the analysis of more than one product, but comparing the different products is a critical part of our approach. The tools provided are meant to help indicate the general pattern of the historical sales data, from which we may infer whether the class of statistical forecasting models we are using is appropriate (i.e., overkill or inadequate) or, moving back one step, whether we should use statistical forecasting for this product at all. Assuming that the answers to both questions are in the affirmative, we may move on to analyze the forecasts made by the automatic forecasting system for this product. 13.3.2.2 Forecast Analysis The forecasting module of practically all supply chain management systems offers some summary measures of forecast error such as mean absolute deviation, mean absolute percentage error, root mean squared error, etc. These should be used per the software vendor’s recommendations. Here, we describe a few additional approaches we have found useful. The underlying theme of our selection is relative comparison – the quantities under attention are compared to each other to help find the focus of our forecast improvement effort. A natural step is to consider a time plot of the historical sales and forecasts, together with the future forecasts, as shown in Fig. 13.6. (Future forecasts are forecast made for future time periods at the time when the forecasting model is run. Hence, there are no sales data for these forecasts.) The general forecast-vs-sales pattern in the recent past (e.g., past 12 periods) provides a visual representation of major forecast inadequacies, if any, such as bias, time lag, over-reaction, overdamping effects, or missing seasonality. In the example shown in Fig. 13.6, the forecasting model (a version of the Holt–Winters model) is doing relatively well in capturing the trend and cyclic pattern, but overshoots the peaks and valleys in a number of occasions (i.e., overreaction). Standard descriptive statistics such as the sample mean, sample standard deviation, sample mean absolute deviation, sample mean absolute deviation as a fraction of the average, sample quartiles, sample minimum and maximum of the recent past historical forecasts are useful for comparison with that of historical sales. For example, if the forecasting system is doing an equitable job, there should be a reasonably close match of the averages and standard deviations. Matching the average is particularly important since a yearly or other aggregate forecast is often used for financial planning purposes.

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One way to analyze the performance of the forecasting system is to see whether the forecasted change in sales from one period to the next matches well with the actual change. We use three tools for this purpose: the correlation coefficient, Theil’s U-statistic, and Prediction-Realization (P-R) diagram for the forecasted and actual changes in the recent past. The correlation coefficient measures, roughly, our forecasted changes against a perfect forecast which is the actual change. This measure gives some information on the absolute performance of our forecasts. However, a perfect forecast is unrealistic and we need an additional measure based on a more practical baseline. Theil’s U-statistic is one such measure. Let fi and si be the forecast and actual sales for period i , respectively. Theil’s U-statistic (Theil 1966, pp. 26–32) is defined as v u n1 uP 2 u .n 1/ .FPE APE / i C1 i C1 u i D1 u ; U Du n1 P t .APEi C1 /2 .n 1/ i D1

where FPEi C1 D APEi C1 D

fi C1 si si si C1 si si

is the forecasted relative change, and is the actual relative change.

This statistic essentially provides a comparison of the current forecasting method with the naive method of using the current period’s sales as the forecast for the next period. When we use the naive method, FPE D 0 and U D 1. On the contrary, if the forecasted changes are perfect (FPE D APE), then U D 0. If we are using a computationally intensive method of forecasting but getting a value of U close to (or larger than) 1, we are not getting very good value of our forecasting investment.

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The P-R diagram (see Fig. 13.7) offers a pictorial view of the forecasted vs. actual changes for one-period-ahead forecasts. (The forecasted change is the difference between the last period sales and the one-period-ahead forecasted sales.) Clearly, we would like all the points to lie in the first or third quadrant of the graph. A perfect forecast has all points lying on the y D x diagonal line. Points appearing in the second or fourth quadrant indicate that even the direction of change was predicted incorrectly in those instances. The P-R diagram can be repeated for forecasts any number of periods ahead that are of interest. To determine whether the forecasting model is missing significant patterns in the historical sales, we use two simple tools: a time plot of the forecast errors and a plot of the autocorrelation coefficient of the forecast errors. If the forecasting model is adequate, then the forecast errors are random noise with no particular pattern. In addition, the autocorrelation coefficients should be small, showing a random pattern across lags. The bottom half of Fig. 13.7 shows an example of these plots. In the example shown in Fig. 13.7, comparing the sales forecast statistics with that of the actual sales shown in Fig. 13.5 shows a reasonable match, except that the variability in sales may be under-forecasted. This is confirmed by both the P-R diagram and the forecast error plot, which show that the forecast tends to be too high when the demand goes down. However, the problem does not seem very serious. The correlation of the forecasted and actual changes in sales is very high (>0:9), showing a general agreement between forecast and actual changes. This is confirmed by the P-R diagram. Theil’s U-statistic is at 0.4, which is not too bad (especially in the set of products at hand). The autocorrelation shows that there is a fairly strong correlation of forecast errors for time lags of 3 and 6, hinting that the forecasting model may be missing some structure of the data. We also note that the above judgments were made through comparing the different products at hand. For brevity, we have not shown the analysis of more than one product, but comparing the different products is a critical part of our approach, which is to identify the most promising products for improvement. Note that the results of this section should be used together with that of Sect. 13.3.2.1 to determine whether the forecasting model of a selected product needs to be changed. If the sales analysis shows a highly variable sales history with no easily discernible pattern, then we can expect the forecasts to be relatively poor. While there are always opportunities for improvement given enough effort, Descriptive statistics of sales forecast per period over forecast horizon: Average Std deviation Coeff of variation Mean abs deviation MAD / Average

35535.54 7618.62 0.2144 6145.81 0.1729

Correlation of forecasted & actual changes Theil's U-statistic

Fig. 13.7 Forecast analysis

Minimum 1st quartile 2nd quartile 3rd quartile Maximum 0.9216 0.4021

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such a product is perhaps not the most productive one to be considered in a practical environment. On the contrary, if the sales history is relatively stable but the forecasts are relatively inaccurate, we should focus our improvement effort on this product.

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13.4 Tuning of the Statistical Forecasting System Based on an analysis discussed in Sect. 13.3, one can determine the candidate products for forecast improvement. When forecasts are poor, it is possible that the forecasting model is structurally inadequate, or that the parameters of the model are not set properly. Sometimes it is difficult to distinguish between the two cases, unless one happens to have a lot of experience with the particular type of forecasting model in use and knows its capabilities well. One (perhaps less than elegant) approach is to tune the parameters of the existing model regardless and see whether the forecasts are acceptable after tuning. If they are not, most likely the forecasting model itself has to be changed. To illustrate the tuning process, we describe a simple tool developed for tuning the parameters of a specific forecasting method, which is a version of the wellknown Holt–Winters model (see, e.g., Makridakis et al. 1997, p.161). This model is used for many products in the supply chain planning system under study. We note, however, that the concepts and the tuning process are generally applicable for many other statistical models. The Holt–Winters model represents a family of exponential smoothing models, a robust approach commonly used in practice. Exponential smoothing is featured in every one of the forecasting and supply chain planning systems available commercially (Yurkiewicz 2004). Sanders (1997) found from his survey of 350 US manufacturers that 26% of low sales (< $100 M) firms and 38% of high sales (> $500 M) firms used exponential smoothing. In an older but more general survey of 500 US corporations (not just manufacturers), Sanders and Manrodt (1994) reported that exponential smoothing was used by 13–20% of the respondents in forecasting up to 1 year ahead. Exponential smoothing has performed favorably in past forecasting accuracy empirical studies (Makridakis and Hibon 1979; Makridakis et al. 1982). Because of its fast speed of computation, it is popular among retailers who have to generate a large number of forecasts (up to millions) on a regular basis. For a comprehensive review of exponential smoothing and related issues, see Gardner (1985). In the remainder of this section, our discussion assumes the use of the Holt– Winters model. Our objective is to find the values of the model parameters such that the mean squared forecast error is minimized. It addresses the issue of choosing the smoothing parameters in a discussion of practical use of the Holt–Winters model in Chatfield and Yar (1988).

13.4.1 Tuning of Individual Products Although dedicated forecasting software packages usually provide some parameter optimization capability (Yurkiewicz 2004), the forecasting modules of many integrated supply chain management systems do not provide such capability or such capability is very limited. For example, they may only have built-in criteria for

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Fig. 13.8 Implementation of tuning method in a spreadsheet

parameter optimization such that the user cannot specify their own criterion. Choosing the appropriate criterion to measure forecast error is a nontrivial and important issue that affects the performance of the business (see, e.g., Lee et al. 1993). A custom-built parameter optimization tool, such as the simple tool described here, can accommodate criteria that are chosen deliberately for the business. The main parameters in the Holt–Winters model are the three smoothing factors (one for each of the permanent, trend, and seasonality components). We developed a simple three-option tuning process for the nonlinear optimization problem of minimizing the mean squared forecast error. (An implementation in a spreadsheet is shown in Fig. 13.8.) Each option is more refined than the previous one, producing an improved parameter set, but requires a larger amount of computational time. The options are: 1. Grid search. It simply performs a search over a grid formed by the feasible ranges of a given set of parameters and chooses the point, which yields the lowest mean squared error of the historical forecasts in the recent past (say 12 periods). We chose the feasible parameter set in the following way. The smoothing factors control the “responsiveness” of the forecasting model. When the factors are large (i.e., close to 1), the model quickly disregards past sales and reacts to recent changes in sales. The short-term forecasts may be more accurate, but are usually less robust in the long run. With large smoothing factors, the forecasts for a given future period generated in successive periods may be very different. This can create a certain degree of nervousness in the production plan driven by the forecasts, which causes problems for downstream execution.

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(Approaches to handle such nervousness have been devised, e.g., Kadipasaoglu and Sridharan 1995. Nonetheless, this adds another level of complexity.) An organization needs to determine the extent of plan changes its productiondistribution system can handle. For example, if the system is flexible because of investments in the production facilities, then we can use a larger smoothing factor to take advantage of the system flexibility. Otherwise, it may not be very useful to have highly adaptive forecasts while the production-distribution plans cannot be changed frequently. In the latter case, the organization pays the price of having higher inventory. In addition, the nature of the business influences whether we need a responsive forecast. When sales are relatively stable, a large smoothing factor may actually decrease the forecast accuracy because the forecast will be too sensitive to occasional fluctuations. It is therefore prudent that the range of the smoothing factors be chosen based on the knowledge of the business and the capability of the organization. The chosen range limits the grid search to a smaller set than the unit cube (which is the entire feasible set for the three smoothing factors). For our implementation, we used a maximum of 0.45 for all three factors. We judged this to be a reasonable maximum value of the smoothing factors after consideration of the issues mentioned above. We experimented with some models that used higher values and generally found them to be less satisfactory than those found by the grid search, in terms of forecast errors over the long run. The specific parameter sets for the grid search are then chosen by uniformly placing 125 points in the feasible cube. Obviously, a larger number of points will give better results but will require more computational time. For our implementation, we find, after some experimentation, that 125 seems to be a reasonable number in terms of trade-off between computational time and forecast error. Grid search, although somewhat old fashioned and rather cumbersome, is a general approach for parameter optimization in the Holt–Winters forecasting model (Chatfield and Yar 1988). 2. Grid search with fine tuning. This option first runs a grid search as described, then takes the selected parameter set from the grid search as a starting solution for a nonlinear optimizer to find the final parameter values. In our implementation, we used an equation solver with a nonlinear optimizer based on the GRG2 algorithm (Lasdon et al. 1978). 3. Multiple-start optimizer for global optimization. Because the mean squared forecast error function of the three smoothing factors is likely to contain many local extrema, we couple the nonlinear optimizer with multiple starting points to attempt global optimization. We run the optimizer to find a local minimum using each point selected for the grid search described in option 1 as a starting solution. The final parameter set is chosen to yield the least of all the local minima. From our experience, when the sales data were fairly smooth (e.g., the sales data for a large class of products in a national market), the multiple-start optimizer would produce the same solution as the grid search with fine tuning. In other words, the GRG2 algorithm was able to find the global optimum with a single starting point.

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13.4.2 Tuning of the Overall System In the previous section, we discuss how we can gain some insight about the behavior of the forecasting model and tune it for a single product. Now we need to find a way to tune the system as a whole. The simplest but the most cumbersome way to optimize the system is to perform the above tuning process for each product and then upload the resulting parameters to the forecasting system. As we are dealing with many products, this may not be practical to do on a regular basis. We propose two approaches which can be used as alternative schemes or as complements, as follows. First, we follow the usual approach of choosing the most important products for the business. For example, these include the top sellers (or class A products in a Pareto or ABC classification of sales), products with consistently high inventory levels in the past (which can be identified using a Pareto analysis of the total inventory), products strategically important to the survival or growth of the business (e.g., products with advanced proprietary technology), and products which consume the largest amounts of the most critical resources of the business. We then cross reference with the diagnostic results of Sect. 13.3.1 to find the most promising products among the important products. These top priority products should be optimized carefully and individually. A second approach is tuning by product groups. As discussed in Sect. 13.2, the forecasting system frequently utilizes a product hierarchy such that aggregate forecasts are at the higher levels in the hierarchy. For example, the root of the hierarchy could be the demand of an entire product family around the world, while the bottom level could be the demand of individual SKUs at each existing distribution center. Depending on the nature of the products and the physical processes in the supply chain, we should tune the forecasts at the levels that drive important planning or operational decisions. For example, if production capacity is a bottleneck at the time and therefore optimal production plans to maximize production throughput is important. The production plan may use a certain level in the forecast hierarchy as input. The forecasts at this level would be prime candidates for tuning. In addition, forecast parameters for each of the large families of products (i.e., the top levels in the forecast hierarchy) should be tuned individually. In this way, we are assured that the overall volume of the business is forecasted reasonably well, so that strategic decisions based on such aggregate numbers are well supported. For the rest of the products, we could classify them into a manageable number of categories according to the sales pattern of the products. (The actual number of categories will depend on the resources available for forecasting and the relative importance of the sales forecast to the business.) For example, we can group all products whose sales are known to be highly seasonal or, in the other extreme, very “flat,” products in the same stage in their lifecycles (new products, mature products, etc.), products which are by their nature often sold together, products sold according to some common industry practice. For each of the category, we choose a representative product and optimize its forecasting parameters as above. These parameter values can be used for all products in the same category. Performing such product

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clustering in a systematic manner is not trivial; however, it could potentially yield significant improvement in the forecast accuracy (Leung et al. 2002). Clustering approaches have also been used to estimate seasonal indices in time series forecasting (e.g., Bunn and Vassilopoulos 1993). In general, parameter tuning need not be performed very frequently. For the Holt– Winters model, Chatfield and Yar (1988) suggested updating the parameters every year or two. For the types of products in hand, we also feel that once a year is a reasonable interval. Many forecasting systems have built-in forecast error tracking, so that the system will issue a warning when its tracking criterion is violated. Even with such features, we still suggest that a yearly parameter tuning activity be used. This will ensure that we are getting the most of the forecasting system and will also serve as a forecast monitor as well.

13.5 Conclusions The use of automatic demand forecasting modules in the context of an integrated supply chain management system provides many advantages and is now widely adopted. Maintaining the forecasting system, however, takes more effort than routine data and software maintenance. It requires regular effort in maintaining the forecasting models throughout the life of the system and such activities need to be planned and budgeted for. In a practical environment, resources are always limited and we would like to spend the bulk of the resources on forecasts that are most promising in accuracy improvement. Hence, our approach in forecast maintenance starts with a diagnosis of the overall system (all products), then identifying a subset of products to be investigated more closely. Subsequently, the selected product forecasts are tuned and approaches for applying tuning to the overall system are discussed. We illustrate that forecast maintenance is not difficult but can bring substantial accuracy improvement in the overall system. In the example from our own practice discussed in the chapter, we were able to reduce the mean absolute forecast error at the product level from the 30% range to the 20% range, on the average, over about 3,000 products. The simple tools used to diagnose and tune the forecasts are easily accessible and easy to understand and use. They complement the statistical forecasting capabilities provided by a commercial supply chain management system. While the latest forecasting software packages may provide similar parameter optimization capabilities, we have not seen any that provides similar forecast diagnostic functions. Our work also accompanies the list of features described by Fildes et al. (2006) as desirable design points of a forecasting system. Depending on the programming capabilities of the supply chain management system, some or all of the tools developed here could be implemented within the system itself. At the very least, selected data from the system could be extracted and imported into a spreadsheet for analysis. The latter is what we have done in our own practice.

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References Batty M (1969) Monitoring an exponential smoothing forecasting system. Oper Res Q 20(3): 319–325 Bunn DW, Vassilopoulos AI (1993) Using group seasonal indices in multi-item short-term forecasting. Int J Forecast 9(4):517–526 Chambers JC, Mullick SK, Smith DD (1971) How to choose the right forecasting technique. Harvard Bus Rev, July-August 1971, 45–74 Chatfield C (1978) The Holt-Winters forecasting procedure. Appl Stat 27(3):264–279 Chatfield C, Yar M (1988) Holt-Winters forecasting: some practical issues. The Statistician 37:129–140 Fildes R, Beard C (1992) Forecasting systems for production and inventory control. Int J Oper Produc Manag 12:4–27 Fildes R, Goodwin P, Lawrence M (2006) Design features of forecasting support systems and their effectiveness. Decis Support Syst 42(1):351–361 Fliedner G (2001) Hierarchical forecasting: issues and use guidelines. Ind Manag Data Syst 101(1):5–12 Gardner ES Jr (1983) Automatic monitoring of forecast errors. J Forecast 2(1):1–21 Gardner ES Jr (1985) Exponential smoothing: the state of the art. J Forecast 4(1):1–28 Golder ER, Settle JG (1976) Monitoring schemes in short-term forecasting. Oper Res Q 27 (2, Part 2):489–501 Gung RR, Leung YT, Lin GY, Tsai RY (2002) Demand forecasting TODAY. OR/MS Today, December 2002 Harrison PJ, Davies O (1964) The use of cumulative sum (CUSUM) techniques for the control of product demand. Oper Res 12(2):325–333 Jenkins GM (1982) Some practical aspects of forecasting in organizations. J Forecast 1:31–49 Johnston FR, Boylan JE (1996) Forecasting for items with intermittent demand. J Oper Res Soc 47:113–121 Kadipasaoglu SN, Sridharan V (1995) Alternative approaches for reducing schedule instability in multistage manufacturing under demand uncertainty. J Oper Manag 13:193–211 Lasdon LS, Waren AD, Jain A, Ratner M (1978) Design and testing of a generalized reduced gradient code for nonlinear programming. ACM Trans Math Softw 4:34–50 Lee TS, Cooper FW, Adam EE Jr (1993) The effects of forecasting errors on the total cost of operations. Omega Int J Manag Sci 21(5):541–550 Leung YT, Levanoni M, Hu CW, Ramaswamy S (2002) Better demand models through data clustering. Presented at the INFORMS marketing science conference, Edmonton, Canada, June 2002 Makridakis S, Hibon M (1979) Accuracy of forecasting: an empirical investigation (with discussion). J R Stat Soc A 142:97–145 Makridakis S et al (1982) The accuracy of extrapolation (time series) methods. J Forecast 1:111–153 .Makridakis S, Wheelwright SC, Hyndman RJ (1997) Forecasting: methods and applications, 3rd edn. Wiley, New York Rycroft RS (1999) Microcomputer software of interest to forecasters in comparative review: updated again. Int J Forecast 15:93–120 Sanders NR (1997) The status of forecasting in manufacturing firms. Prod Inventory Manage J, Second Quarter, 1997, 32–36 Sanders NR, Manrodt KB (1994) Forecasting practices in US corporations: survey results. Interfaces 24(2):92–100 Smith BT (1991) Focus forecasting and DRP. Vantage Press, New York

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Theil H (1966) Applied economic forecasting. North Holland, Amsterdam Trigg DW (1964) Monitoring a forecasting system. Oper Res Q 15:271–274 West M, Harrison PJ (1986) Monitoring and adaptation in Bayesian forecasting models. J Am Stat Assoc 81(395):741–750 Yurkiewicz J (2004) Forecasting: predicting your needs. OR/MS Today, December 2004

Chapter 14

The Ongoing Challenge: Creating an Enterprise-Wide Detailed Supply Chain Plan for Semiconductor and Package Operations Kenneth Fordyce, Chi-Tai Wang, Chih-Hui Chang, Alfred Degbotse, Brian Denton, Peter Lyon, R. John Milne, Robert Orzell, Robert Rice, and Jim Waite

14.1 Introduction In the mid-1980s, Karl Kempf of Intel and Gary Sullivan of IBM independently proposed that planning, scheduling, and dispatch decisions across an enterprise’s demand-supply network were best viewed as a series of information flows and decision points organized in a hierarchy or set of decision tiers (Sullivan 1990). This remains the most powerful method to view supply chains in enterprises with complex activities. Recently, Kempf (2004) eloquently rephrased this approach in today’s supply chain terminology, and Sullivan (2005) added a second dimension based on supply chain activities to create a grid (Fig. 14.1) to classify decision support in demand-supply networks. The row dimension is decision tier and the column dimension is responsible unit. The area called global or enterprise-wide central planning falls within this grid.

14.1.1 Decision Tiers Demand-supply network (or supply chain) decisions in the semiconductor industry typically fall into one of four decision tiers (row dimension): strategic, tactical, operational, and response (dispatch). The categories are based on the planning horizon, the apparent width of the opportunity window, and the level of precision required in supporting the information. The first decision tier, strategic scheduling, is typically driven by the lead time required for business planning, resource acquisition, and new product introduction. This tier can often be viewed in two parts: very long-term and long-term. Here, decision makers are concerned with a set of problems that are 3 months to 7 years into the future. Issues considered include, but are not limited to, what markets the firm

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Tier 1: Strategic Tier 2: Tactical Decision Tiers

Tier 3: Operational "daily" Tier 3.5: Sub-daily Guidance Tier 4: Response

Fig. 14.1 Grid representation of enterprise-wide supply chain planning

will be in, general availability of tooling and workers, major process changes, risk assessment of changes in demand for existing products, required or expected incremental improvements in the production process, lead times for additional tooling, manpower and planning. The second tier, tactical scheduling, deals with problems the company faces in the next week to 6 months. Estimates are made of yields, cycle times (CTs), and binning percentages. Permissible material substitutions are identified. Decisions are made about scheduling starts or releases into the manufacturing line (committing available capacity to new starts). Delivery dates are estimated for firm orders, available “outs” by time buckets are estimated for bulk products, and daily going rates (DGRs) for schedule-driven products are set. The order/release plan is generated or regenerated, and (customer-requested) reschedules are negotiated. The third tier, operational scheduling, deals with the execution and achievement of a weekly plan. Shipments are made, serviceability levels are measured, and recovery actions are taken. Optimal capacity consumption and product output are computed. The fourth tier, real-time response system, addresses the problems of the next hour to a few weeks by responding to conditions as they emerge in real time. It also accommodates variances from availability assumed in the plan creation and commitment phases. Within the demand-supply network, real-time response is often found in two predominant areas: manufacturing dispatch (which assigns lots to tools) and order commitment (available to promise, or ATP). In manufacturing dispatch scheduling (DS), decisions concern monitoring and controlling of the actual manufacturing flow and instructing the operator what to do next to achieve current

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manufacturing goals. The goal of most ATP applications is to provide a commit date to a customer order as quickly as possible. Although it may not respond in real time, its goal is to modify the current match between assets and demand to provide a real-time commit to an order placed by a customer. Within semiconductor manufacturing, the decisions made across the tiers are typically handled by groups with one of the three responsibilities: establishing product demand, maintaining an enterprise-wide global view of the demand-supply network, and ensuring that subunits (such as manufacturing location, vendor, warehouse) are operating efficiently. Although ideally all planning would be central, in practice the level of complexity precludes this. Capacity or tool planning is a good example. At the enterprise level, capacity is modeled at some level of aggregation, typically viewing a tool set as a single capacity point. At the factory level, each tool, or potentially each chamber in a tool, is modeled.

14.1.2 Basics of Enterprise-Wide End-to-End Central Planning The activity of concern in this chapter, called “enterprise-wide end-to-end central planning,” falls into the second column in Fig. 14.1 (enterprise-wide global view) and straddles all the rows: strategic, tactical, and operational. This planning activity, which is a key requirement for successfully managing operations in any manufacturing industry, involves the coordination of supply and demand (current and future). In large-scale manufacturing systems such as semiconductor, this planning activity must handle the dual challenges of scope (complexity) and scale (size). In its simplest form, supply chain planning combines the below three “basic pillars” (Fig. 14.2): – Business processes and organizational structure – Data collection and storage mechanisms – Analytical or modeling methods to execute the following steps

The Firm Fig. 14.2 Three basic pillars for supply chain planning

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1. Create a demand statement 2. Gather and project assets to a decision point 3. Create an enterprise-wide end-to-end central plan by matching current and future assets with current and future demand (Fig. 14.3) 3.1 To generate 3.1.1 A projected supply linked with exit demand 3.1.1.1 Including projecting supply without demand, capacity utilization, and pegging; 3.1.2 Synchronization signals across the enterprise 3.1.2.1 Including starts (or manufacturing releases), target outs, due dates, ship plans, stocking levels, lot priorities, planned substitutions, capacity utilization, etc. 3.2 Typically, this is an iterative process 3.2.1 That consists of a set of model runs under different settings, such as 3.2.1.1 With and without capacity. 3.2.1.2 With and without new projected supply, 3.2.1.3 With and without new forecasted demand, 3.2.1.4 Where different runs occur at different times during the week. 4. Execute the plan, that is, 4.1 Send signals to each core enterprise organization (such as manufacturing, storage, vendors, etc.) 4.1.1 Converting this signal into more detailed guidelines for each organization, 4.1.2 Executing the detailed manufacturing or transport activities. 4.2 Send projected supply to ATP which 4.2.1 Handles incoming requests for product, 4.2.2 Makes tradeoffs or reallocation as needed and defined by the business rules. 5. Repeat This flow is summarized in Fig. 14.4. Step 3 is often referred to as the enterprise-wide best-can-do (BCD) matching, or the central planning engine (CPE). The core task of the CPE is to deploy modeling methods to match assets with demand across an enterprise to create a projected supply linked with demand and synchronization signals. Assets include, but are not limited to, manufacturing starts (or releases), work in progress (WIP), inventory, purchases, and capacity (manufacturing equipment and manpower). Demands include, but are not limited to, firm orders, forecasted orders, and inventory buffer. The CPE has four core components, which 1. Represent the (potential) material flows in production, business policies, constraints, demand priorities, current locations of asset, etc., and relate all this information to exit demand. 2. Capture asset quantities and parameters (CTs, yields, binning percentages, etc.).

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Fig. 14.3 Typical data inputs and outputs for enterprise-wide central planning engine

Fig. 14.4 Information and decision flows in a successful demand-supply network

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3. Search and generate a supply chain plan, relate the outcome to demand, and modify the plan to improve the match. 4. Display and explain the results. This is the focus of this chapter. Historically, the concept that an enterprise needs a reasonably tightly coupled central planning process to be successful in the market took hold in the middle 1990s (Shobrys 2003). Within the semiconductor industry, the work at Harris Semiconductor (Leachman et al. 1996) and Digital Equipment Corporation (Arntzen et al. 1995) certainly elevated awareness. The work by Tayur et al. (1998) noted the growing importance of quantitative models, and the work by Lee et al. (1997), Swaminathan and Smith (1998), and Lin et al. (2000) made it clear that real improvements in organizational performance were possible from centralized planning. To be clear, we are not saying the concept did not exist before and certain businesses had some success at centralized control. Glover et al. (1979) produced, at a minimum, a clear ancestor to today’s supply chain planning models (including using different models depending on whether strategic or tactical planning was being done). The ten ways that a material requirements planning (MRP) can fail (Woolsey 1979) still ring true today. Fogarty and Hoffman (1983) identified some core requirements. Duchessi (1987) identified the importance of a feasible production plan and the applicability of knowledge-based techniques to this problem. The papers by Hackman and Leachman (1989), Uzsoy et al. (1992, 1994), and Graves et al. (1995) clearly identified the emerging need and interest. Norden (1993) articulated the general trend in applying quantitative methods from well-structured operations to the more speculative aspects of strategy and policy formation. In some respects, little is new after Orlicky (1975). It is clear that the mid-1990s saw the “launch” of the supply chain management (SCM) industry, consisting of vendors, consultants, analysts, watch groups, specialized reports, and numerous internal efforts. The pace remains full speed ahead in 2007; see, for example, the supply chain operations reference (SCOR) model (www.supply-chain.org). Much of the work in SCM from the mid-1990s until 2004 or 2005 focused on the creation of a centralized planning process and a centralized data view of the status of the organization’s manufacturing and distribution activities. Less emphasis was placed on the third pillar of the SCM triangle (Fig. 14.2) – the decision model or the analytics – the enterprise-wide detailed level CPE. Certainly, basic matching engines were produced and used, but most relied on heuristics despite the promise shown by linear programming (LP) in the Harris work (Leachman et al. 1996) to handle the complexity of demand-supply networks in semiconductor manufacturing. It is true that if you do not have your processes and data in place, the ability of your “CPE” to handle business complexity does not matter. However in 2007, just having your data and processes in place is no longer enough. Firms now compete on analytics (Davenport 2006) and many of those complexities, such as alternative bill of materials (BOMs) or customer request and commit dates, are core competencies for firms and cannot be “leaned” away without damaging the firm’s competitive position. Kempf (2004) observed: “the tradition of referring to this supply-demand

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network as simply supply-chain grossly understates the actual complexity.” The ability to handle these complexities resulted in a more effective supply chain plan.

14.1.3 IBM Microelectronic Division’s SCM Transformation Between 1992 and 1999 (Promoting 2005; Fordyce 1998, 2001), the microelectronics industry went through a dual transformation in core technology and use (or market). On the technology side, chip size, speed, and versatility took quantum leaps. For example, IBM pioneered copper circuits, RISC-based CPU processors, silicon germanium and silicon-on-insulator technologies, and innovative insulation techniques for copper circuits. The market for microelectronic devices expanded from an initial base in computers to a wide range of products, such as cell phones, car security systems, advanced GPS-based trackers, greeting cards, and aids for the physically challenged. Microelectronic devices truly pervade the world. This dual expansion has transformed manufacturing from making large quantities of just a few parts to varying quantities of many different parts. To quantify this change, in the mid-1980s IBM Microelectronics had about 100 active part numbers (PNs) with demand; the current number is about 6,000. Like other industries, microelectronics is also under tremendous pressure to be more responsive to customers. Today’s IBM Microelectronics Division – a tightly coupled set of manufacturing facilities employing a centralized SCM process, which subweekly determines which orders can be met when for the entire Division and can respond rapidly to customer requests to place or change orders as well as manage what-if scenarios – was only a dream in 1994. In 1994, IBM manufacturing facilities located worldwide to produce wafers, devices, modules, and cards were semi-independent of one another at best. They supplied the parts for downstream manufacturing facilities for mainframes, workstations, printers, networks, and storage equipment. Their geographic linkages were strong. Typically, a facility in Europe would provide component parts for an IBM box plant in Europe. Supply chains consisted of individual manufacturing facilities linked directly with the box plants they supported with no concept of centralized control. Each manufacturing facility produced far fewer products and therefore managed far fewer products than it does today. During the early 1990s, IBM Microelectronics Division needed to compete in the business of supplying components to original equipment manufacturers (OEM), while continuing to provide the IBM box facilities with state-of-the-art (and often custom) components, reducing costs and improving customer (internal and external) satisfaction. To accomplish these goals, IBM Microelectronics had to transform from a loose confederation of manufacturing facilities into a unified division. A major reengineering effort was launched in 1994 to completely restructure the Division’s SCM applications (Lyon et al. 2001), covering processes, data, and planning models. Note that sometimes SCM is referred to as “customer order fulfillment” (COF).

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14.1.4 IBM’s Decision to Invest in a “Smart” Planning Engine & Supply Chain Efficiency Frontier After an extensive review in 1994 (Sullivan 2007), the IBM Corporation made a clear business decision in 1995 as part of its reengineering effort (Lyon et al. 2001; Sullivan 2005) that a “simple” planning engine would not meet the long-term needs of its microelectronics business. IBM chose to invest to build a “smart” CPE as it was a key to being at the frontier of “supply chain efficiency.” In macroeconomics and financial engineering, a well-established concept is the “efficiency frontier” (also called production possibility frontier). In macroeconomics, if an economy is operating below its efficiency frontier, an increase in one product as well as in other goods can be obtained by increasing overall economic efficiency. If an economy is operating at its efficiency frontier, any increase in one product requires reducing the quantity of all other goods produced in the same period. Obviously in practice, there are multiple dimensions to this curve which can be education, transportation, or medicine. In finance, a similar pattern exists between risk and return: more risk, higher return; less risk, less return. A similar concept can be applied in supply chains. As in macroeconomics, the efficiency frontier has multiple dimensions, consisting of inventory, on-time delivery (OTD), customer satisfaction, profit, revenue, stability, growth of market, etc. For illustrative purposes, we will have two dimensions (Fig. 14.5): (a) improvements in inventory performance (e.g., turns, excess, shelf life), and (b) improvements in OTD (e.g., meeting request date, finding acceptable commit date, meeting commit date).

Fig. 14.5 Two-dimensional efficiency frontier: Inventory and on-time delivery

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Fig. 14.6 Simple example of allocating supply to meet demand

Just as in macroeconomics, if a supply chain is at its efficiency frontier, improving inventory requires reducing OTD. As Shobrys and Fraser (2003) make clear, few supply chains are operating at their efficiency frontier. Therefore, the goal of each firm is to first shift the curve outward towards the efficiency frontier; then, the firm will navigate along the curve with great precision to the point that it believes best meets current market requirements. A smarter mathematical model (planning engine) can directly improve a firm’s performance. Let us look at the following simple example (Fig. 14.6). Assume a firm makes two products: Module 1 and Module 2, and both modules require Device 12 as their only component part. The time it takes to produce Module 1 and Module 2 is 10 and 4 days, respectively. We will assume that once a unit of Device 12 is allocated to either Module 1 or Module 2, work begins immediately to make the module. If 3 units of Device 12 are allocated to Module 1 on day 2, then 3 units of Module 1 are available to meet demand on day 12.D 2 C 10/. Similarly, if 4 units of Device 12 are allocated to Module 2 on day 6, then 4 units of Module 2 are available to meet demand on day 10.D 4 C 6/. The anticipated supply of Device 12 is as follows: Anticipated supply device 12 Day Supply Cum sup 00 20 20 02 30 50 06 10 60 10 20 80

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And the anticipated demand is as follows:

Order ID G A B H W C D N M L total

Part Module_2 Module_2 Module_2 Module_2 Module_2 Module_1 Module_1 Module_1 Module_1 Module_2

Demands Cycle time Priority 04 2 04 3 04 3 04 3 04 2 10 3 10 1 10 3 10 5 04 2

commit date 05 05 06 06 07 10 12 14 14 15

amt 15 8 2 2 4 10 15 5 15 4 80

The primary task of the CPE in this simple scenario is to allocate the anticipated supply of Device 12 to produce Module 1 and Module 2 so as to best meet demand and minimize inventory. Below, two different search mechanisms come up with different allocations. Table 14.1 is interpreted as follows. The first set of columns under “Demands” repeat the demand information. It is the order identifier, the type of part, the cycle time to produce the part, the priority of the demand, the commit date to meet this demand and the amount of the demand. The second set of columns under “Method 1” is the allocation of supply of device 12 to produce either Module 1 or Module 2 to meet demand. The row for demand D for Module 1 can be read as follows: 15 units from the supply of Device 12 available on day 00 are allocated to meet this demand. Fifteen units of Module 1 comes to stock (is completed) on day 10 and is allocated to demand D. Since the commit date for demand D is day 12 and the supply is available on day 10, the supply is 2 days earlier (this is the delta column). The OTD score is 0. The algorithm for scoring is: (a) if the demand is met on time or early, the OTD score is 0; (b) if the demand is late the smaller of the number of days late and 5 is divided by the demand priority (this caps the “days late” at 5 and weighs it inversely to the demand priority). The OTD scoring mechanism is designed only to be illustrative. Observe that method 2 improves both OTD (4:67 compared to 6:50) and inventory (39 compared to 78) in comparison with method 1, moving the firm closer its efficiency frontier. The “smarter” planning engine makes a direct impact on the firm’s performance. Below illustrates a third allocation where OTD is improved (4:33 compared to 4:67) but inventory is increased (43 compared to 39). This is an example of moving along the curve of efficiency frontier. The smart engine is able to follow the firm’s business rules and position the firm at the point on the efficiency frontier, where it currently believes is optimal. The planning engine institutionalizes the corporate rules and insures that they are known and followed (Table 14.2).

ID D G W L A B H C N M

Part Module Module Module Module Module Module Module Module Module Module

1 2 2 2 2 2 2 1 1 1

Cycle Priorit commit time y date 10 1 12 04 2 05 04 2 07 04 2 15 04 3 05 04 3 06 04 3 06 10 3 10 10 3 14 10 5 14 amt 15 15 4 4 8 2 2 10 5 15

alloc date 00 02 00 02 02 02 06 06 10 10

all amt 15 15 4 4 8 2 2 10 5 15 stk stk date amt delta 10 15 02 06 15 −01 04 4 03 06 4 09 06 8 −01 06 2 00 10 2 −04 16 10 −06 20 5 −06 20 15 −06 on time delivery score inventory days

otd score 0.00 −0.50 0.00 0.00 −0.33 0.00 −1.33 −1.67 −1.67 −1.00 −6.50 78

alloc date 02 00 02 06 02 02 02 06 10 10

all stk stk amt date amt delta 15 12 15 00 15 04 15 01 4 06 4 01 4 10 4 05 8 06 8 −01 2 06 2 00 2 06 2 00 10 16 10 −06 5 20 5 −06 15 20 15 −06 on time delivery score inventory days

Table 14.1 Different allocations resulted from different search mechanisms Comparing Two different Methods to Allocate Device_12 to the production of Module_1 and Module_2 Demands Method 2 Method 1

otd score 0.00 0.00 0.00 0.00 −0.33 0.00 0.00 −1.67 −1.67 −1.00 −4.67 39

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Table 14.2 Alternate allocations on the efficiency frontier Method 3 to allocate device_12 to module_1 and module_2 Method 3 Results Demands ordered by priority, date, qty ID D G W L A B H

Part Module 1 Module 2 Module 2 Module 2 Module 2 Module 2 Module 2

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amt 15 15 4 4 8 2 2

need date 02 01 03 11 01 02 02

alloc date 02 00 02 10 02 02 06

10

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00

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Priorit commit date y 1 12 2 05 2 07 2 15 3 05 3 06 3 06

all amt 15 15 4 4 8 2 2

stock date 12 04 06 14 06 06 10

stk amt 15 15 4 4 8 2 2 10

delta 00 01 01 01 -01 00 -04

otd score 0.00 0.00 0.00 0.00 -0.33 0.00 -1.33

-10

-1.67

5 04 15 -06 on time score inventory days

0.00 -1.00 -4.33 43

IBM’s conclusion was that a smarter planning engine translated directly into improved supply chain efficiency and consistent performance. It was this understanding and general sense of urgency that kept the supply chain transformation, including the work on the planning engine, on track. The result was the creation of the IBM Advanced Supply Chain planning team in January 1995, consisting of decision science and computer science professionals along with business analysts who had formerly had responsibility for operational planning. This team remains active today. One of this team’s key missions was to build and deploy a CPE that could handle the scope and scale of the demand-supply networks at the IBM Microelectronics Division. This planning engine would need to operate at PN, order number, and lot level detail, be completely data driven, run either as a batch job at night or by planners (not as the personal tool kit of operations research professionals), and provide support for operational, tactical, and strategic planning and scheduling. A key challenge within this mission was to blend LP with heuristics in a manner that was transparent to the planners. As we will describe in detail later, the complexity associated with managing demand-supply networks for semiconductor manufacturing makes it an ideal candidate for LP technology. However, the scale of the enterprise problem is too large to be solved by a single LP model. Simple heuristics such as greedy algorithms can handle the scale. But they fail to handle even such simple complexities as binning and typically underutilize expensive assets. To build a CPE, the IBM team developed and implemented a series of significant advances, including an LP model for binning optimization that could be invoked by a heuristic, an advanced heuristic, which provides highly effective decisions with regards to such items as demand priorities and binning (invoking the binning LP model), an advanced LP-based algorithm, which handles complex trade-offs and nonlinearities such as lot sizing and preemptive demand priorities, an overall solution structure, which dynamically blends both

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decision technologies and partitions the solution process in a manner that adds to the quality of the solution, and mechanisms to fully automate the process. The CPE is capable of solving enterprise-scale SCM problems with both function and speed. Its partitioning logic automatically classifies product structures and solves the most complex ones using large LPs, the moderately complex ones using small LPs, and the relatively simple ones using heuristics. The CPE achieves all this without any human intervention. The core of this work is described in the following sections. On the professional side, this work has been recognized by IBM and INFORMS (Lyon et al. 2001; Denton et al. 2006). Perhaps, the best testimony came from Jerry Dundon (senior VP SCM, Analog Devices, Inc.) after the IBM CPE was successfully installed at Analog: “Much has been written about improving supply chain operations through reducing demand volatility and improving forecast accuracy. We have put equal focus on understanding and modeling the much-neglected supply side of the equation. Having a robust, credible model of supply capability allows us to be more flexible and responsive to inevitable changes to demand” (IBM white paper 2006). The rest of this chapter proceeds as follows: basics of producing semiconductorbased microelectronic devices, overview of the core requirements of a CPE, challenges of complexity, description of the core solution mechanism in the IBM CPE, future work extending the big bang, and a conclusion.

14.2 The Basics of Producing Semiconductor-Based Microelectronic Components Semiconductor manufacturing is a complex process involving everything from growing silicon crystals, manufacturing silicon wafers upon which integrated circuits (ICs) are built, to the actual placement and soldering of chips to a printed circuit board (Sullivan 1990; Kempf 1994, 2004; Leachman et al. 1996; Denton et al. 2006). Typically, the core manufacturing flow is wafers to devices to modules to cards (Fig. 14.7).

14.2.1 Wafer Fabrication A wafer is a round, thin piece of silicon that looks like a CD. The goal of the wafer fabrication process is to build a set of ICs on the wafer surface according to a specific circuit design. The manufacturing process begins with cutting raw wafers from a silicon ingot. Next, circuit components (such as transistors or resistors) are sequentially built on the wafer surface and then interconnected to form ICs. At a high level, IC fabrication requires many repetitions of four essential steps: deposition (depositing special materials on the wafer surface), photolithography (forming

326 Fig. 14.7 Simple flow for production of semiconductor-based package parts

K. Fordyce et al. Wafer_2 cycle time = 60 days; start of BOM chain; one wafer makes 200 devices

Device_2 cycle time = 3 days; requires 1/200 unit of Wafer_2 to build

Module_2 cycle time = 8 days; requires 1 unit of Device_2 to build

Card_2 cycle time = 4 days; requires 2 units of Module_2 to build; end of BOM chain

circuit patterns by protecting only the deposited material which corresponds to circuit structures), etching (introducing chemical to remove the unprotected material), and ion implantation (modifying the wafer’s conductive properties to complete the building of circuit components). These four steps are repeated for as many times as the design requires, resulting in a three-dimensional, layered structure on the twodimensional wafer surface. In addition, toward the end of the fabrication, an iterative set of “metallization activities” occurs to connect the components. This repetitive use of the same core processes, and hence the same core equipment sets, is called “reentrant flow.” For complex designs, it may take three to six hundred individual steps to complete the entire IC fabrication process. Cycle times range from 30 to 130 days, and yield (percentage of quality wafers) can show significant variation. There is also wide variation in the demand patterns for wafers. A wafer that is a component of a video game has large demands over a 6- to 12-month period. In contrast, usually wafers that support specialty products have sporadic demands and those used in sensing and testing devices have stable demands. Those involved in computers or control units have moderate demands over a reasonable period of time and then fall into an end of life phase that must cover warranty issues. Demand is stated either as a finished wafer or a planned start. For all of the complexity and uncertainty in wafer fabrication, all enterprise planning engines have a simple representation of this manufacturing process: a few key capacity points and a limited number of purely serial decision points.

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Wafer_2 Start of BOM chain; cycle time = 30 days; one wafer makes 200 devices

30%

Device_2A On average, 60 out of the 200 devices on Wafer_2 (30%) sort to this part

45%

Device_2B On average, 45% of Wafer_2 becomes this part

25%

Device_2C On average, 25% of Wafer_2 becomes this part

Fig. 14.8 Simple binning or sorting situation from wafer to device

14.2.2 Wafer Testing and Device Generation (Binning) Once the circuits have been built on the wafers, they are tested to determine the resultant yield of operational circuits (good or bad, speed, power consumption, etc.) and tagged for reference. The wafers are then diced (cut) into individual units and sorted or binned based on the prior testing creating what is referred to as the device or “die bank.” This process is generally referred to as “binning” (illustrated in Fig. 14.8). Observe it is the device (not the wafer) that is packaged and placed into video games, cell phones, laptops, etc. A single wafer generates anywhere from 30 to 900 devices, and sometimes, the exact type of device is not determined until it is tested. That is, the testing not only determines whether the device is usable or not, but also determines its final part identity. Differences between devices from the same wafer typically occur as result of speed and power consumption. Wafer testing has a short cycle time (3 10 days) and involves a purely sequential process through test operations except for rework.

14.2.3 Modules The device step is followed by a sequence of assembly steps to mount devices onto a substrate (called wire bonding) and package to make a module. Packaging protects the fragile device inside and makes it suitable to be incorporated in other electronic products. These modules are further tested to determine electromagnetic and thermal characteristics. As with devices, in some cases the final identity of the part may not be determined until the test activities are done. Again, the cycle times are short compared to fabrication times. For some modules, a significant amount of complexity is introduced when the production process requires a sequence of packaging and testing steps and/or the

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K. Fordyce et al. Wafer_2 cycle cycletime time= =3030days, days,start startofofBOM BOM chain, chain,one onewafer wafermakes makes200 200devices devices 30%

25%

45%

Device_2C Device_2C

Device_2A

Device_2B

3030% % ofofWafer_2 Wafer_2sorts sorts to to Device_2A Device_2A

45% 45%ofofWafer_2 Wafer_2sorts sorts to to Device_2B Device_2B

P_1

25% of ofWafer_2 Wafer_2sorts sorts 25% to Device_2C to Device_2C

P_2

P_3

Module_2

Module_2

Module_A

cycletime time==6 6days, days,made made cycle viaprocess processP_1, P_1,consumes consumes via unit of of Device_2A Device_2A 11 unit

cycletime time==8 8days, days,made made cycle viaprocess processP_2, P_2,consumes consumes via unit of of Device_2B Device_2B 11 unit

cycletime time==9 9days, days,made made cycle viaprocess processP_3, P_3,consumes consumes via unit of of Device_2C Device_2C 11 unit

Alternative processes (methods) to make Module_2, i.e., Module_2 stocks from three production paths

Module_2 10% of Module_A Module_A 10 % of becomes Module_2 Module_2 becomes

10%

90%

Module_AA Module_AA

Location: VEND001

Card_2 cycle cycletime time==22days, days, madeatatvendor vendorVEND001, VEND001, made requires22units unitsofof Module_2 Module_2 requires

Location: VEND002

90% of of Module_A Module_A 90% becomes Module_AA Module_AA becomes

Card_2 Card_2 cycle cycletime time==44days, days, madeatatvendor vendorVEND002, VEND002, made requires22units unitsofof Module_2 Module_2 requires

Alternative Alternative vendors vendors to to make make Card_2, Card_2, i.e., Card_2 stocks from two vendors

Fig. 14.9 Alternative processes to make Module 2, Module Binning, and Card 2

same module can be made from different processes that may or may not consume the same component part (a.k.a. alternate build paths). Figure 14.9 provides an example of this type of complexity.

14.2.4 Cards The modules are eventually combined onto printed circuit boards to make cards. Finally, the cards are tested and those that pass are used in the assembly of a wide range of electronic products such as PCs, printers, and CD players. The cards are often produced at multiple vendor locations (Fig. 14.9).

14.2.5 Total Journey To create manageable enterprise wide supply chain models, we introduce some abstraction from the complex routing of lots through the production process. We project the continuous process onto a discrete set of PNs. We use a BOM that specifies the component(s) of each PN to generate a graphical representation of the components needed for finished products, the alternative paths to produce the same PN, and the binning or sorting activities that determine the final status of a PN (Fig. 14.10).

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The Ongoing Challenge: Creating an Enterprise-Wide Detailed Supply Chain Plan Finished Mod. X

Finished Mod. Y

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Device (Fast)

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30%

329

10%

Device (Untested)

= BOM = Alternate BOM = Binning

Wafer BEOL

other BEOL wafers

Wafer FEOL

other FEOL wafers

= Substitution Raw Wafer

Fig. 14.10 Graphical representation of a bill of material structure

Observe the variety (complexity) inherent within the total “journey from sand to module or card.” The first stage (fabrication of wafers and then testing and sorting into devices) has a long lead or cycle time, where the capital expense for equipment ranges from hundreds of millions to billions of dollars. There is no assembly at this stage; in fact, the number of PNs goes from a few to many. In stage two (modules) and three (cards), the cycle times are short and the process is assembly and test. Typically, more than one component is needed in the assembly process and there are multiple ways to produce the same part. The number of finished goods explodes compared to the original number of wafer parts. A manufacturing facility typically specializes in only part of the process. As an example of a common product flow: the wafer is produced at location A; it is tested and diced at location B; the module is assembled at location C1, C2, or C3; it is tested at location D1 and then sent to location E1 or E2 to be assembled onto a card or sent directly to a customer. Within this network, different PNs that support a wide range of customers are constantly competing for the same tool or machine capacity. The complexity extends to variability across time as well. Most of the planning parameters and business preferences are date or time effective. This includes cycle times, yields, capacity required, capacity available, binning percentages, manufacturing processes, and so on. For example, in Fig. 14.9 the cycle time to produce Module 2 might reduce from 8 to 5 days in 3 weeks, and the entire process might be eliminated in 30 weeks.

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To add to the complexity, customer or exit demand may occur at any level (card, module, device, wafer tested, and wafer untested), or be stated as required starts or allocation of capacity.

14.3 Overview of BCD Enterprise-Wide CPEs The BCD enterprise-wide detailed CPE is the control point for the flow of material or product within an organization, and focuses on how to best meet prioritized demand without violating temporal, asset (WIP and inventory), or capacity constraints. A CPE application minimizes prioritized demand tardiness and some aspects of cost, establishing a projected supply and synchronized targets for each element of the supply chain. The core of the CPE process is matching assets with demand, which refers to aligning assets with demand in an intelligent manner to best meet demands. The alignment or match occurs across multiple facilities within the boundaries established by the manufacturing specifications, process flows, and business policies. Assets include, but are not limited to, starts (manufacturing releases), WIP, inventory, purchases, and capacity (manufacturing equipment and manpower). Demands include, but are not limited to, firm orders, forecasted orders, and inventory buffer. The matching must take into account manufacturing/production specifications and business guidelines. Manufacturing specifications and process flows include, but are not limited to, build options, BOM, yields, cycle times, anticipated date on which a unit of WIP will complete a certain stage of manufacturing (called a receipt date), capacity consumed, substitutability of one part for another (substitution), the determination of the actual part type after testing (called binning or sorting), and shipping times. Business guidelines include, but are not limited to, frozen zones (no change can be made on supplies requested), demand priorities, priority trade-offs, preferred suppliers, and inventory policy. Many of the manufacturing specification and business guideline values will often change during the planning horizon (time effective). The creation of a CPE plan requires a solver (sometimes referred to as a model or an engine) with the following core features: 1. Method(s) to represent the (potential) material flows in production, business policies, constraints, demand priorities, current locations of asset, etc., and relate all this information to exit demand. 2. Capture asset quantities and manufacturing specifications (parameters). 3. Search mechanism(s) to generate a balanced supply chain plan, relate the outcome to demand, and modify the plan to improve the match. 4. Display and explain the results of the BCD match. The first task of any “BCD” CPE is to “flow material” and maintain a “feasible material flow” (see Graves et al. 1995 for a review of material flow control mechanisms). Simply put, the CPE must decide a sequence of manufacturing starts to produce finished goods, and for each start the CPE places into the plan, the required component

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parts and capacity must be available and the manufacturing activity must be permitted on that day (e.g., it is not a shutdown day). For example, in Fig. 14.7 if the CPE decides to manufacture 10 cards on day 10 to be completed on day 14 to meet a customer’s demand, then on day 10 it must have 20 modules and tool/equipment capacity for the manufacturing process to consume. Typically, the CPE handles this requirement either implicitly with material balance equations or explicitly with explosion and implosion. Explosion and implosion are the core processes of the CPE which either move work units (WIP or starts) forward (implosion) to project completed parts or backward (explosion) to determine starts required across the BOM supply chain following the appropriate manufacturing data such as cycle time, yield, capacity, and product structure. We typically use implosion to estimate what finished goods will be available to meet demand and explosion to estimate what starts are needed at what due dates to ensure meeting the existing demand on time. To review implosion and explosion, consider Fig. 14.7 again which represents a simple production flow. The first manufacturing activity is the production of Wafer 2. This manufacturing activity has a cycle time of 60 days, i.e., it takes on average 60 days to take a raw wafer and create a completed wafer with the part ID Wafer 2. The second activity is device production. Creating 1 unit of Device 2 requires 3 days and consumes 1/200th unit of Wafer 2. Module 2 consumes 1 unit of Device 2 and takes 8 days to produce. Finally, Card 2 consumes 2 units of Module 2 and takes 4 days to produce. Referencing Fig. 14.7, implosion can be illustrated with the following example. Manufacturing estimates that 4 units of Device 2 will be available or completed on day 10. This is called a projected receipt. If manufacturing immediately uses these 4 units to produce Module 2, on day 18 (10C“Module 2 cycle time” D 10C8 D 18) 4 units of Module 2 will be completed. Continuing the projection process, the 4 units of Module 2 are immediately used to create 2 units of Card 2, which will be available on day 22 (18 C “Card 2 cycle time” D 18 C 4 D 22). The implosion process enables manufacturing to estimate the future supply of finished goods. Again referencing Fig. 14.7, explosion can be illustrated with an example. To meet demand for 1 unit of Card 2 on day 20, the plant must have two (completed) units of Module 2 available on day 16 (20 “Card 2 cycle time” D 20 4 D 16). This generates an exploded demand of 2 units of Module 2 with a due date of day 16. To continue the explosion process, to produce the 2 units of Module 2, the plant must have 2 units of Device 2 available on day 8 (16 “Module 2 cycle time” D 16 8 D 8). Next, the device demand is exploded creating a demand for 2/200th units of Wafer 2 on day 5 (D 8 3). This exploded information creates the guidelines for manufacturing to meet existing demand. For example, the device department must start production of 2 units of Device 2 no later than day 5 to meet the demand for 1 unit of Card 2 on day 20. Since the cycle time to produce Wafer 2 is 60 days, it needs to have one already in production and close to completion. Within the explosion and implosion process is a method called “demand pegging.” This method links each allocation of an asset or creation of a start with either

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a specific exit demand, or, at a minimum, the demand class or priority (relative importance of demand) associated with the exit demand being supported. Using the explosion example described above, if the exit demand for 1 unit of Card 2 on day 20 has a demand class of 3, each exploded demand will carry that demand class with it. Therefore, the units of Module 2 that are started on day 8 will have a demand class of 3. Similarly, if 3 units of Card 2 are desired on day 20 for a customer with demand class 5, then 6 units of Module 2 to be started on day 8 will also have a demand class of 5. The total required starts picture on day 8 is 8 (2 C 6), with 2 units with demand class 3 and 6 units with demand class 5. If by chance, there is only enough capacity on day 8 to start 2 units of Module 2, they will be allocated to the more important demand (demand class 3).

14.4 Challenges of Scope and Scale “The great 20th century revelation that complex systems can be generated by the relationships among simple components” (Goldman 2004) applies to supply chain planning (and almost all aspects of planning, scheduling, and dispatch) (Little 1992). Although simply creating a feasible central plan which maintains material balance, observes date effectivity, obeys business rules, captures existing WIP and inventory, and does a rough job at meeting demand is by itself challenging, it is no longer sufficient for a firm to remain competitive. The failure to “create a more accurate assessment of supply” forces the firm to compensate with slack (Galbraith 1973) or inefficiencies that leaves it at a competitive disadvantage. The purpose of this section is to identify and describe some of the key challenges a CPE must handle to provide an accurate assessment of supply. By “handle,” we mean provide a mathematical representation of the individual characteristics, the system or relationships, and a method to search for an intelligent, if not provably, optimal solution. In the following paragraphs, we describe the challenges of demand class; simple binning; complex binning, substitution, and alternative bill of materials (BSA); lot sizing; sourcing; fair share; and commit date versus request date. Leachman et al. (1996), Kempf (2004), and Denton et al. (2006) also provide excellent reviews of the complexities/challenges within the production of semiconductor-based parts. Leachman focuses on binning and demand class; Kempf develops and explains the stochastic challenge (uncertainty in demand and supply); and Denton deals with demand class and lot sizing in detail.

14.4.1 Allocating Perishable and Nonperishable Assets Based on Demand Class A fundamental decision found in most aspects of planning and scheduling is deciding which demand “gets to go first?” That is, when more than one demand needs

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Module_1 CT = 10 days Supply Amt

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Fig. 14.11 Example of nonperishable asset allocation

either capacity (a perishable asset) or a component part (typically viewed as a nonperishable asset) and there is not sufficient supply to meet all immediate demand, the question is which demands get the asset and which have to wait. The CPE needs to allocate the asset based on the relative importance of the demand (indicated by demand class or priority) and the impact on delivering the finished goods on time. Figure 14.11 shows a simple example of allocating supply of part (nonperishable asset) to meet demand. Module 1 and Module 2 are both made from Device 12 with cycle time 10 and 4 days, respectively. The demand for Module 1 is 10 units on day 10 and 15 units on day 12. The demand for Module 2 is 8 units on day 5 and 2 units on day 6. The key decision for the CPE is how to best allocate supply from Device 12 to the two modules (represented by the two boxes titled “Supply Amt” in Fig. 14.11). One solution for the situation in Fig. 14.11 might be: (1) immediately allocate 8 of the 10 units of Device 12 on hand to meet demand A (8 units of Module 2 on day 5) 1 day early on day 4 (D 0 C 4); (2) immediately allocate the remaining 2 units of Device 12 on hand to meet demand B (2 units of Module 2 on day 6) 2 days early on day 4; (3) on day 2, allocate 10 units of the projected supply of 30 units of Device 12 to demand C (10 units of Module 1 on day 10) 2 days late on day 12 (D 2 C 10); (4) on day 2, allocate 15 units of the projected supply of 30 units of Device 12 to demand D (15 units of Module 1 on day 12) on time (12 D 2 C 10). The score card for this solution is demand A early, demand B early, demand C late by 2 days, and demand D on time. Table 14.3 summarizes this solution.

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Actual delivery

Demand ID

Type

A B C D

Module Module Module Module

2 2 1 1

Date

Quantity

Date

Quantity

Delta schedule

05 06 10 12

8 2 10 15

04 04 12 12

8 2 10 15

1 2 2 0

Table 14.4 Results of solution 2 Commitment

Actual delivery

Demand ID

Type

A B C D

Module Module Module Module

2 2 1 1

Date

Quantity

Date

Quantity

Delta schedule

05 06 10 12

8 2 10 15

06 06 10 12

8 2 10 15

1 0 0 0

A second option could be: (1) 10 units of Device 12 are allocated to Module 1 on day 0 to cover demand C; (2) 15 units of Device 12 are allocated to Module 1 on day 2; and (3) 10 units of Device 12 are allocated to Module 2 on day 2. The score card for this solution is: demand A is met 1 day late, demand B is met on time, demand C is met on time, and demand D is also met on time. Table 14.4 summarizes this solution. Which is better? If the four module demands are exit demands, the answer depends only on the relative importance of each demand and the business policy on “sharing the pain” if a demand cannot be met on time. If demand A is demand class 1 (the lower the value, the more important the demand) and demands C and D are demand class 3, the first solution is the logical choice. If the demand classes are reversed, the second solution is the logical choice. What if all of the demands have the same demand class? Do we go with solution two since demand A is just one day late? Do we meet part of demands C and A on time and the other part is late (can we split the order)? Or do we meet all of demand C and part of demand A on time? This typically depends on the business policy of the enterprise. If the four module demands are not exit demands, then in addition to the demand class, we need to also assess whether meeting the module demand on time ensures meeting the exit demand on time. For example, if demand A goes into the exit demand for CARD01 for the XYZ customer and the board (modules go on boards to make a card) required is 2 weeks late, there is no point in worrying about meeting module demand A on time. In Sect. 14.1.4, we introduced a slightly more complicated example of allocating Device 12 to the production of Module 1 and Module 2 with additional supply and demand. Here, we identified three possible allocation schemes. The anticipated demand and supply are displayed in Fig. 14.12. One allocation solution is displayed in Fig. 14.13 with an OTD score of 6:50 and an inventory score of 78.

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Anticipated Supply Device_12 Day 00 02 06 10

Supply Cum sup Allocate Remain 20 20 00 20 30 50 00 30 10 60 00 10 20 80 00 20

ID

Part

G A B H

Module_2 Module_2 Module_2 Module_2

W C D N M L total

Module_2 Module_1 Module_1 Module_1 Module_1 Module_2

Demands Cycle Priority time 04 04 04 04 04 10 10 10 10 04

Commit Date

2 3 3 3 2 3 1 3 5 2

05 05 06 06 07 10 12 14 14 15

Allocate supply of Device_12 to demands for Module _1 & Module_2 Amt 15 8 2 2 4 10 15 5 15 4 80

Fig. 14.12 Example of anticipated demand and supply

order ID D G W L A B H C N M

Day 00 02 06 10

Method 1: order dmd by priority, date, quantity, allocate by priority, commit date, do not split a delivery Demands ordered by priority, date, qty allocations , production, delivery Cycle commit alloc stock stock otd Part time Priority date amt date all amt date amt delta score Module 1 Module 2 Module 2 Module 2 Module 2 Module 2 Module 2 Module 1 Module 1 Module 1

Supply 20 30 10 20

10 04 04 04 04 04 04 10 10 10

cum sup 20 50 60 80

1 2 2 2 3 3 3 3 3 5

12 05 07 15 05 06 06 10 14 14

allocation history allocate remain 19 01 29 01 12 −02 20 00

15 15 4 4 8 2 2 10 5 15

cum rem 01 02 00 00

00 02 00 02 02 02 06 06 10 10

15 15 4 4 8 2 2 10 5 15

10 15 02 06 15 −01 04 4 03 06 4 09 06 8 −01 06 2 00 10 2 −04 16 10 −06 20 5 −06 20 15 −06 on time & inv score

0.00 −0.50 0.00 0.00 −0.33 0.00 −1.33 −1.67 −1.67 −1.00 −6.50

inv days 30 0 12 36 0 0 0 0 0 0 78

total cum tot 19 19 29 48 12 60 20 80

Fig. 14.13 Possible allocation solution to the example in Fig. 14.12

The row for demand D for Module 1 can be read as follows: 15 units from the supply of Device 12 available on day 00 are allocated to meet this demand. Fifteen units of Module 1 comes to stock (is completed) on day 10 and is allocated to demand D. Since the commit date for demand D is day 12 and the supply is available on day 10, the supply is 2 days earlier (this is the delta column). The OTD score

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is 0. The algorithm for OTD scoring is: (a) if the demand is met on time or early the OTD score is 0; (b) if the demand is late the smaller of the number of days late and -5 is divided by the demand priority (this caps the “days late” at 5 and weighs it inversely to the demand priority). The inventory score is 30. The algorithm for inventory scoring is: (a) 0 if the supply is just in time or late to meet demand, (b) if the supply is early, then the score is the number of days early times the number units of supply. The bottom table (allocation of supply) of Fig. 14.13 summarizes when each supply is used. For example, the day 00 row tells us 20 units of Device 12 are available on day 00, out of these 20, 19 are immediately allocated to build modules and 1 unit is held for later use. If we examine the “alloc date” (allocation date) column of the top table, there are two rows with 00 in them – rows D and W. If we now examine the column “all amt” (allocation amount), the number of units of Device 12 allocated for demand D and W (with day 00 allocation) are 15 and 4, respectively. This totals 19 matching what is in the bottom table. A second allocation solution is displayed in Fig. 14.14 with an OTD score of 4:67 and an inventory score of 39. This “smarter” solution improves both inventory and OTD and moves the efficiency frontier (Fig. 14.5) out. This improvement comes from: 1. Swapping the allocation of Device 12 between demand D (allocation moves from day 00 to day 02) and demand G (allocation moves from day 02 to day 00) – which improves the OTD of demand G and reduces the inventory days accumulated with demand G 2. Delaying the allocation of Device 12 for demand L from day 02 to day 06 – which maintains OTD, but reduces inventory days 3. Moving the allocation of Device 12 from demand C from day 06 to day 02 – which generates an OTD Method 2: order dmd by priority, date, quantity, allocate by priority, commit date, no split, & tweaks allocation, productions, delivery improvements in allocation Demands

ID D G W L A B H C N M

Part Module 1 Module 2 Module 2 Module 2 Module 2 Module 2 Module 2 Module 1 Module 1 Module 1

Cycle Priorit commit time y date

10 04 04 04 04 04 04 10 10 10

1 2 2 2 3 3 3 3 3 5

12 05 07 15 05 06 06 10 14 14

amt

15 15 4 4 8 2 2 10 5 15

alloc all stk stk date amt date amt delta 00 15 10 15 02 02 15 06 15 −01 00 4 04 4 03 02 4 06 4 09 02 8 06 8 −01 02 2 06 2 00 06 2 10 2 −04 06 10 16 10 −06 10 5 20 5 −06 10 15 20 15 −06 on time delivery score inventory days

otd alloc all stk stk score date amt date amt delta 0.00 02 15 12 15 00 00 −0.50 15 04 15 01 0.00 02 4 06 4 01 06 0.00 4 10 4 05 −0.33 02 8 06 8 −01 0.00 02 2 06 2 00 02 2 06 2 00 −1.33 −1.67 06 10 16 10 −06 −1.67 10 5 20 5 −06 −1.00 10 15 20 15 −06 on time delivery score −6.50 78 inventory days

Anticipated Supply Device 12 Day

Supply

00 02 06 10

20 30 10 20

cum sup

20 50 60 80

alloca te remain

19 29 12 20

01 01 −02 00

cum rem

01 02 00 00

total

19 29 12 20

cum tot

19 48 60 80

cum rem

05 04 00 00

total

cum tot

15 31 14 20

15 46 60 80

Fig. 14.14 Second possible allocation solution to the example in Fig. 14.12

otd score 0.00 0.00 0.00 0.00 −0.33 0.00 0.00 −1.67 −1.67 −1.00 −4.67 39

improv ement

better

better

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Method 3 include Need Date Demands ordered by priority, date, qty allocating by priority, commit date, no splits ID D G W L A B H

Part Module 1 Module 2 Module 2 Module 2 Module 2 Module 2 Module 2

Cycle time 10 04 04 04 04 04 04

C

Module 1

10

3

N M

Module 1 Module 1

10 10

3 5

Day 00 02 06 10

Supply 20 30 10 20

Priorit commit date y 1 12 2 05 2 07 2 15 3 05 3 06 3 06

amt 15 15 4 4 8 2 2

need date 02 01 03 11 01 02 02

10

10

00

14 14

5 15

04 04

allocation history cum alloca cum sup te remain rem total 20 20 00 00 20 50 29 01 01 29 60 02 08 09 2 80 29 -09 00 29

alloc date 02 00 02

10 02 02

06 10 00 10

all amt 15 15 4 4 8 2 2

stock date 12 04 06 14 06 06 10

stk amt 15 15 4 4 8 2 2

10

20

10

5 15

10 20

otd delta score 00 0.00 01 0.00 01 0.00 01 0.00 −01 −0.33 00 0.00 −04 −1.33 −10

5 04 15 −06 on time score

inv days 0 15 4 4 0 0 0

−1.67

0

0.00 −1.00 −4.33

20 0 43

cum tot 20 49 51 80

Fig. 14.15 Third possible allocation solution to the example in Fig. 14.12

A third solution is displayed in Fig. 14.15 with an OTD score of -4.33 and an inventory score of 43. This shifts the firm along the efficiency frontier by trading better OTD for worse inventory, presumably reflecting the firm’s current business strategy. Observe the CPE needs three core functions to handle this type of decision: (1) being able to “peg” or pass down the priority of the exit demand to all intermediate demands, (2) a method to pass in and represent the business policy, and (3) mechanisms to search for options and assess the trade-offs relative to overall efficiency and current business policy.

14.4.2 Simple Binning (or Sorting) with Downgrade Substitution Simple binning with downgrade substitution refers to the classification of a part into one of a set of mutually exclusive and exhaustive categories based on some key performance factor using a series of testing procedures conducted after completing a set of manufacturing processes (see Leachman et al. 1996 for additional details on binning). In semiconductor manufacturing, the most common (but not the only) location for binning is the completion of wafer fabrication, where hundreds of devices are cut from a single finished silicon wafer. Because of the random variation in the wafer fabrication process, working conditions for the devices on the same wafer vary. Therefore, before proceeding any further in the manufacturing process, devices have to be classified into different categories (each having a unique PN) after being cut from the finished wafer (typically today, the actual testing is done on the wafer). Clock speed, for example, is usually among a number of key performance

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Untested Device

# of devices =? 50%

Device_1 Demand = 30 substitution

30%

20%

Device_2 Demand = 40

substitution

substitution


Fig. 14.16 Simple binning with downgrade substitution after wafer fabrication

factors to be tested for each device. Besides electronics, downgrade substitution can also be seen in such industries as consumer goods (bicycles) and building materials (grades of wood). Figure 14.16 illustrates a typical binning scenario after the water fabrication process, where 50% of the time, the device is tested to have a “grade A” or top performance; 30% of the time, it has a “grade B” or medium performance; and 20% of the time, it has a “grade C” or low performance. These percentages are referred as binning percentages and can generally be observed in semiconductor manufacturing as a result of binning testing. Another phenomenon commonly associated with binning is substitution between materials of the same type. Quite often, parts with a higher performance can substitute for parts with a lower performance if necessary. This form of material substitution is generally called downgrade substitution, and typically, it occurs when there is a shortage of the lower performance part accompanied by an overage of the higher performance part. In Fig. 14.16, the dotted arrows indicate that “grade A” devices (Device 1) can substitute for both “grade B” devices (Device 2) and “grade C” devices (Device 3), and “grade B” devices can substitute for “grade C” devices. The binning percentages can and do change over time. The challenge is to make an optimal use of coproducts and substitution to avoid overstating the number of wafer starts required to meet demand. If the demand is 30 for Device 1, 40 for Device 2, and 30 for Device 3 (Fig. 14.16), the challenge is to determine the minimum number of wafers/devices that must be produced to meet all three demands. One simple rule is the maximum of the quantity required for each device divided by its binning percentage. Continuing the same example, we would need 150 devices, which equals maximum (30/0.50, 40/0.30, 30/0.20) D maximum (60, 133, 150). As shown in Fig. 14.17, such a rule will leave an excess inventory of 50 devices, with 45 contributed by Device 1 and 5 contributed by Device 2. If we optimally account for coproducts and substitutions, the minimum number of devices required to meet all three demands is 100 – testing 100 devices creates 50 of Device 1, 30 of Device 2, and 20 of Device 3 (Fig. 14.18). The extra 20 of Device 1 are used to cover the shortfall of 10 of both Device 2 and Device 3. Other factors complicating the determination of the minimum number of starts required to meet demand in simple binning production structures include: demands

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Untested Device

# of devices = 150

75 Device_1 Demand = 30

50%

Mfg Binning Device_1 is .50 x 150 = 75

339

30%

20%

Device_2 is .30 x 150 = 45

45


substitution

excess = 5

substitution

30


Device_3 is .20 x 150 = 30

excess = 45

substitution

excess = 0

Fig. 14.17 Maximum quantity of starts leaves excess inventory of 50 devices

Untested Device

# of devices = 100

Mfg Mfg Binning Binning Device_1 Device_1 is is .50 .50 xx 100 100 == 50 50 Device_2 Device_2 is is .30 .30 xx 100 100 == 30 30 Device_3 Device_3 is is .20 .20 xx 100 100 == 20 20

50 Device_1 Demand = 30

50%

30%

20%

30

20

excess = 0

substitution = 10


substitution = 10

excess = 0

substitution


excess = 0

Fig. 14.18 Optimal number of devices meets all demands and leaves no inventory

for devices that spread throughout the planning horizon, existing inventory, projected WIP completion, and binning percentages and allowable substitutions that change during the planning horizon (date effectivity). In addition, the CPE must locate and isolate these binning situations in a large, complex demand-supply network, as well as maintain full traceability and handle demand priorities (Fig. 14.19).

14.4.3 BSA: Complex Binning, General Substitution, and Alternative BOMs Within the production of modules, an increasingly common manufacturing characteristic is alternative BOM structures, general substitution, and complex binning. In its simplest form, alternative BOM means two or more manufacturing processes are available to produce the same PN. For example, Fig. 14.20 illustrates a scenario in which process P1 and P2 can both be used to produce Module 9. If P1 is selected, the process will consume Device 8A; if P2 is selected, the process will consume Device 8B. Complex binning refers to a situation, where one binning activity immediately invokes another, or substitutions are permitted across binning activities. This is illustrated in Figs. 14.9 and 14.10.

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30%

Device_1A

Untested Device_1 70%

60%

Device_1B

Device_2A

Untested Device_2 40%

Device_2B

Fig. 14.19 Two binning situations that would be solved by separate LPs

untested device WIP = 40 on Day 2

20% 40%

40%

Device_8A inventory = 20

P0

Device_8B inventory = 0

P2

P1

Module_8 demand = 20 priority = 8 Module_9 demand = 20 priority = 1

substitution

Device_8C inventory = 0

P2’ substitution can be viewed as an alternative process P2’

Fig. 14.20 Illustration of alternative BOM structure with substitution

To get a sense for the decision challenges created by alternative BOMs, we will use Fig. 14.20 to look at just the “explosion” question within the CPE. Two processes (P1 and P2) can be used to build Module 9, where P1 consumes Device 8A and P2 consumes Device 8B. In addition, Device 8C can generally be substituted for Device 8B, which means if Device 8B is not available and Device 8C is available, process P2 can use Device 8C to make Module 9. Conceptually, this general substitution can be viewed as the third alternative BOM option (call it P2-prime or P20 /. The explosion engine must determine how to explode demand for Module 9 and Module 8 back to the device level. Should it be half to P1 and half to P2, 2/3 to P1 and 1/3 to P2, or all to P1? Should P20 be considered? The objective is to divide the demand for Module 9 across P1 and P2 (and perhaps P20 / to best use the existing

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341

explosion


8 20% 40%

proj. supply of device (future inventory) Device_8A is .20 x 40 = 8 Device_8B is .40 x 40 = 16 Device_8C is .40 x 40 = 16


P0


P2

P1

16

40%

Module_8 demand = 20 priority = 8 Module_9 demand = 20 priority = 1

substitution 16



Fig. 14.21 Awareness of asset availability affects new starts

inventory and WIP, minimize new starts, and meet other relevant guidelines such as sharing percentages and capacity. Determining the best result requires an extensive search through the entire BOM structure. In Fig. 14.20, assume the priority for demand for Module 9 (which is 1) is higher than that for Module 8, and 20 units are ordered for both modules. A quick search reveals that 20 units of Device 8A are in inventory, which can be used to make Module 9 with process P1 or Module 8 with process P0. There is no current inventory for Device 8B or Device 8C to make Module 9 using process P2 or P20 (the substitution). Most heuristic-based search engines that guide the explosion through alternative BOM structures would explode the 20 units of demand for Module 9 down the P1 process (or leg). This would consume all of the 20 units in inventory for Device 8A and leave the demand for Module 8 totally uncovered; assuming building every unit of module would consume 1 unit of device (Fig. 14.21). To meet the demand for Module 8, a planning engine would require the demand-supply network to produce 100 new untested devices if the engine was not aware of the expected projection of the 40 untested devices (which could be tested into 8 units of Device 8A (D 40 0:20), 16 units of Device 8B (D 40 0:40), and 16 units of Device 8C (D 40 0:40)). On the contrary, if a planning engine is aware of this information, it would only require 60 new untested devices since there is a projection of 8 Device 8A (60 D .20 8/=0:20 D 12=0:20). However, a broader search would uncover all available options at untested device and avoid the conflict for Device 8A between Module 8 and Module 9. There are 20 units of demand for Module 8 and it can only be made from Device 8A. So the question is: are there other options to meet the demand for Module 9? There are 40 units of projected WIP at the untested device and after binning, 8 will become Device 8A, 16 will become Device 8B, and 16 will become Device 8C. Since Device 8C can be substituted for Device 8B, there are actually 32 (16 C 16) future devices that can be used to produce Module 9 (but not Module 8), which is more than enough to meet the 20 units of demand for Module 9. So it is probably not optimal to explode the

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demand for Module 9 down the P1 leg. As illustrated in Fig. 14.22, both demands can be met without using any new untested devices: 1. Assign the 20 units of inventory for Device 8A to be used to produce 20 units of Module 8 (blue dotted arrow). 2. Explode the 20 units of demand for Module 9 into a need for 20 units of Device 8B (red dotted arrow). Net this need against the 16 projected Device 8B, resulting in a need for four new Device 8B. 3. Use 4 out of the 16 projected Device 8C to meet the need for four new Device 8B. Appendix A contains an additional example of the limitations of heuristics to handle alternative BOM paths. Figure 14.23 illustrates how the complexity can grow quickly and Fig. 14.24 illustrates potential opportunities for parallelization through dynamic partitioning (indicated by different colors).


8 20% 40%

proj. supply of device (future inventory) Device_8A is .20 x 40 = 8 Device_8B is .40 x 40 = 16 Device_8C is .40 x 40 = 16

16

P0


Module_8 demand = 20 priority = 8

P1


40%

4

P2

20

Module_9 demand = 20 priority = 1

substitution

16



Fig. 14.22 A smarter option – meeting both demands without using new devices

Device_1

Device_1A Device_1B

# of starts =?

Device_2

Wafer

= Alternative BOM

Raw_ Module_1B

Module_2

Device_3A

Raw_ Module_3A

Device_3B

Raw_ Module_3B

Device_3C

Raw_ Module_3C

Device_3

Fig. 14.23 Complexities across the bill of material structure

Module_1

Raw_ Module_2A

Device_2B

= Binning = BOM = Substitution

Device_2A

Raw_ Module_1A

Module_3

14


Device_1

Device_1A Device_1B

# of starts =?

Device_2

Wafer

= Alternative BOM

Raw_ Module_1A Raw_ Module_1B

Module_2

Device_3A

Raw_ Module_3A

Device_3B

Raw_ Module_3B

Device_3C

Raw_ Module_3C

Device_3

Module_1

Raw_ Module_2A

Device_2B

= Binning = BOM = Substitution

Device_2A

343

Module_3

Fig. 14.24 Opportunities for parallelization through natural partitions

Increasingly, the CPE and the planner have to handle all three complexities: alternative BOM, general substitution, and complex binning. Figure 14.10 illustrates this “bundling.” The following is an example scenario provided by one of the IBM’s most experienced planners working with IBM and customer products. The performance (speed) is specified at the module level when the module process including stress is complete. The module test spec is defined with safety tolerance that guarantees the module will perform to its specification in the application. This performance “spec” is then translated back to device test. The device test spec is also defined with tolerance, “Guard Band” to insure that a “fast” device at device test remains a “fast” device once it has been packaged at the module level. This translation of the module test spec back to device is not 100% perfect. Often times, a different test platform is used for device and module test, further introducing a margin of error. You might ask the question of why do device test at all if it is not accurate. There are two primary reasons for this. First, there can be a lot of variation in the speed distribution from one wafer lot to another. The speed limitations, at a high level, are defined by the design. But depending on where in the process window the lot was processed, you can end up with a “fast” lot, a “medium” lot, or a “slow” lot. Even though there is some margin of error from device test to module test, speed sorting at device allows you to be able to select the right device to package to best meet the demand. If you waited until the module is packaged, it may be too late. Example: Let us say that all of your demand for the next few weeks is on the medium speed module. You would want to release medium devices as opposed to fast devices, thus avoiding down binning of fast modules to medium modules. Also if you did not do device test and you encounter a “slow” lot, you might not end up with enough “medium” modules to meet your demand and miss committed orders.

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Second, the attached BOM assumes one package type. In many cases, devices can be packaged in different module types. These packaging options can serve entirely different markets with different speed requirements. Example: You have two package types with different overall speed requirements as follows: Package 1 Market Speed requirements Gaming Industry

Fast 0%

Medium 80%

Slow 20%

Package 2 Market Speed rqmt’s High End Servers

Fast 40%

Medium 60%

Slow 0%

Obviously, you would never release “fast” devices to Package 1 and have to down bin all the fast module and loose margin as “fast” always commands premium dollars. Also you would never release “slow” devices to Package 2 as there is no market for “slow” modules in this package and you would suffer a lot of excess scrap. The need for reasonably sophisticated decision technology to navigate BSA and find a true picture of a firm’s potential supply is self evident – it can make or break the profitability of a firm. Remember all of the other complexities carry over: demand priority, WIP, inventory, date effectivity, capacity availability, etc.

14.4.4 Lot Sizing Lot sizing refers to a core manufacturing characteristic that the number of units in each lot of activity can have a significant impact on productivity. This preference/ requirement is typically described by minimum lot size, maximum lot size, and multiples. Probably, the easiest and most common example is minimum lot sizing. Within a manufacturing facility, the same tool set is used to process a variety of products. For example, the same testing equipment is used to investigate and sort a wide variety of modules. However, there is a substantial “setup cost” when the tool set switches from one part to another. Therefore, the manufacturing unit wants to get a “return” on this setup cost by processing a minimum quantity of each part. It does not want to do 3 of part A, then 5 of part B, then 2 of Part C, etc.; rather, it wants to do 30,000 of part A, then 5,000 of part B, then 20,000 of part C, etc. This requirement creates a set of challenges for the CPE. First, the engine has to accommodate that in some cases the manufacturing release number can only come in discrete quantities (challenge for LP). Second, the engine needs a mechanism to keep track that the same lot will be accommodating different demand priorities. Third, not all lot sizing requirements are firm: the manufacturing facility may prefer to process 30,000 of part A instead of 3, but it can do just 3 if it is important enough.

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14.4.5 Sourcing If an enterprise has more than one supply location which provides certain parts, it typically wants to maintain some type of balance between the workload on each facility. This “balance” is typically described with a reasonably complex set of business rules. Often, these rules arise from complex contractual obligations to suppliers that vary over time and business conditions.

14.4.6 Fair Share Fair share refers to sharing limited supply or capacity among a set of equally important customers, as opposed to filling their orders in some random fashion. For example, A and B are the firm’s most important customers, and they both have just requested 100,000 units of the same part to be delivered on the same day (200,000 units in total). Suppose the available supply is only 140,000 units, 60,000 short of the combined demand quantity. If no concern is given to the final delivery quantity, the CPE may generate a supply chain plan, which ships 100,000 units to A (i.e., A’s order 100% filled) and 40,000 units to B (i.e., B’s order only 40% filled) despite the fact that they are equally important. With fair share, on the contrary, both customers will have 70% of their order delivered (and share the same degree of “pain”). The importance that all customers with the same level of importance to the company receive the same level of service is quite obvious. This requirement creates an additional challenge for the CPE.

14.4.7 Customer Request Versus Customer Commit Customer request versus customer commit (date, quantity, priority) is emerging as a key complexity directly affecting customer satisfaction. Typically, a customer requests to receive a certain number of part(s) on a certain date, and this is the customer request date and request quantity. Next, the enterprise reviews the request and responds with a date and quantity to which it will commit which becomes the commit date and commit quantity. The customer and supplier may iterate a few times on date and quantity until a firm commit date and quantity are established. The supplier often associates some priority with the firm demand. Historically, this is the only date, quantity, and priority the CPE has for each demand, that is, without using the original request date and quantity information during its solution process. In current practice, at best some postprocessing activity occurs to identify “low hanging” opportunities to meet a few key request dates. To start, let us consider the simple example illustrated in Fig. 14.25, where there is a demand for 100 units of part A with a request date of 06/01 and a commit date of 07/01. Assume WIP exists projected to come to stock on 06/01. A two-pass solution

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• supply for part A • qty = 100 • projected WIP date 06/01

• • • •

demand for part A qty = 100 commit date 07/01 request date 06/01

engine assigns WIP to meet part A demand at what date ?

Information passed to customer

Fig. 14.25 Simple example of handling commit and request dates Anticipated Supply supply ID SUP01 SUP02 SUP03 total

date 2 4 5

amount 30 10 20 60

?

Demands for Part XYZ commit request order id date date amount A 2 none 15 B 3 2 5 C 4 none 10 D 5 none 20 E 6 2 10 total 60

Fig. 14.26 Demand and supply information for Part XYZ

could do the following: in the first pass, WIP is used to cover the 07/01 commit date and then netted out and became unavailable for the second pass. In the second pass, no unnetted asset remains to cover the 06/01 request date. The two-pass process did not pick up the possibility that WIP could cover the request date. A solution that carried both the commit and request date would recognize the WIP could be used to make the request date with the primary search process. A more complicated example is shown in Fig. 14.26. Here, we have five demands for part XYZ and anticipated supplies arriving on days 2, 4, and 5. All demands have a commit date but only two (B and E) have a request date. To keep the example simple, we assume all demands have the same priority. The key business question is what date can we deliver the product to the customer! Can we meet the commit dates? Can we meet any of the request dates? In Fig. 14.27, we show a typical solution focused only on the commit date – all the commit dates but none of the request dates are met. The OTD score for the commit date is 0, but the OTD score for the request date is 5.

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Allocated Supply to meet Demand on time delivery date to for for supply ID date amount customer commit request SUP01 2 15 2 0 NA -1 SUP01 2 5 3 0 SUP01 2 10 4 0 NA SUP03 5 20 5 0 NA -4 SUP02 4 10 6 0 -5 total 60 0

Fig. 14.27 Solution focused only on commit date

Demands for Part XYZ commit request order id date date amount A 2 none 15 B C D E total

3 4 5 6

2 none none 2

5 10 20 10 60

Allocated Supply to meet Demand on time delivery date to for for supply ID date amount customer commit request SUP01 2 15 2 0 NA SUP01 SUP01 SUP03 SUP02 total

2 2 5 4

5 10 20 10 60

2 4 5 6

1 0 0 0 1

0 NA NA -4 -4

Fig. 14.28 Solution obtained with a simple postprocessing routines


Allocated Supply to meet Demand on time delivery date to for for supply ID date amount customer commit request SUP01 2 15 2 0 NA 2 SUP01 2 5 1 0 SUP02 4 10 4 0 NA SUP03 5 20 5 0 NA SUP01 2 10 2 4 0 total 60 5 0

Fig. 14.29 Solution in which both request dates are met

Sometimes, a simple postprocessing routine is executed to identify “easy” opportunities to meet request date. In our example, the supply used to meet demand B has an anticipated supply date of 2, therefore the customer can have the supply on day 2. The results of a postprocessing routine are shown in Fig. 14.28. With this routine, we are able to meet the request date for order B but not the request date for order E. Figure 14.29 shows the results of carrying both commit date and request date within the solution process. Both request dates (demand B and E) are met. This is accomplished by swapping SUP01 and SUP02 between demand C and E. The challenge is to carry both commit and request information (date, quantity, priority, etc.) intrinsically within the CPE and identify opportunities to come closer to customer requests (respecting their relative importance) without sacrificing any commitments which have been made. This requires carrying multiple dates and quantities throughout the explosion component of the CPE, identifying

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opportunities to assign assets to exploded demand to improve the posture of the exit demand, and being robust enough to handle such complexities as binning, substitution, and lot sizing.

14.4.8 Minimum Starts Sometimes, a planner will want the CPE to ensure that a minimum number of starts occur at a certain point in the production process. For example, in Fig. 14.7, the planner may want a minimum of 100 Module 2 per week over the next 20 weeks to smooth the manufacturing flow or to meet contractual agreements (e.g., a vendor may be willing to promise a shorter cycle time if it has a “smoothed” start plan). In these cases, the CPE must explode the exit demand back to the identified minimum start manufacturing activity, compare the minimum start value with the required value, then continue the explosion process. On the return trip (implosion), the CPE must also adapt for the relative priority of the minimum start.

14.4.9 Date Effective Parameters, Substitution Rules, BOM, Capacity, Etc. “Date effective parameters” are a challenge that is often underestimated and underserved – the solution offered often does not do an efficient job at meeting this challenge, leaving a firm with last minute patching and manual intervention. Simply, this challenge is that most “descriptive” elements of the demand-supply network, such as yields, CTs, capacity available, capacity consumed, allowable substitutions, BOM, and so on, have a start date and an end date (date effective), and the CPE must recognize that these elements will change over time. For our discussion, we will use CT. Previously, we used Fig. 14.7 as a simple production flow to explain explosion and implosion. In that example, the CT was fixed over time. Let us make the CT for Module 2 and Card 2 date effective (Fig. 14.30). The CT to produce Module 2 is 8 days in duration from day 1 to day 10 of the planning horizon and 10 days from day 11 to day 25. This means that the production of Module 2 started on day 5 has a cycle time of 8 days and will complete on day 13 (5 C 8). But if the production starts on day 12, the cycle time becomes 10 days and the completion time will be day 22 (12 C 10). The cycle time for Card 2 is 4 days from day 1 to day 14 and then reduces to 2 days from day 15 to day 25. Therefore, the search engine in both explosion and implosion must account for these changing cycle times. Now let us revisit our implosion example. Manufacturing estimates 4 units of Device 2 will be available on day 10. If manufacturing immediately uses these 4 units to produce Module 2, the cycle time will be 8 days and on day 18 (10 C 8) 4 units of Module 2 will be completed. Continuing the projection process, the 4 units

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Fig. 14.30 Flow for production of semiconductor parts with date effective cycle times

349

Wafer_2 cycle time = 60 days; start of BOM chain; one wafer makes 200 devices

Device_2 cycle time = 3 days; requires 1/200 unit of Wafer_2 to build

Module_2 cycle time = 8 days from day 1 to day 10; cycle time = 10 days from day 11 to day 25; requires 1 unit of Device_2 to build

Card_2 cycle time = 4 days from day 1 to day 14; cycle time = 2 days from day 15 to day 25; requires 2 units of Module_2; end of BOM

of Module 2 are immediately used to create 2 units of Card 2 beginning on day 18 (therefore the cycle time is 2 days), which will be available on day 20 (18 C 2). What if capacity was not available on day 10 to start the production of Module 2? If capacity became available on day 11, the CT for Module 2 would increase from 8 days to 10 days. Therefore, a 1-day delay due to a capacity restriction would result in the completion of Card 2 being delayed 3 days to day 23 (11 C 10 C 2). Let us turn our attention to explosion. To meet demand for 1 unit of Card 2 on day 20, the plant must have two completed units of Module 2 available on day 18 (20minustheCTforCard 2 on day18 D 20 2 D 18). This generates an exploded demand of 2 units of Module 2 with a need date or due date of day 18. When does the manufacturing facility need to start the production of 2 units of Module 2? When we had only one cycle time for the module, we simply subtracted that value (8) from the module’s need date. This would drive a Module 2 start on day 10 (18 8). Life is not that simple with date effectivity. As a starting point, we first find the cycle time for Module 2 on day 18, which is 10 days. Remember 10 is the CT for the production of Module 2 that starts on day 18. But we do not want to start production on day 18; rather, we want production to complete (called to stock) on day 18. So we subtract 10 from 18 and get an initial starting date of 8. Does this work? We check the CT for the production of Module 2 on day 8 and find it is 8 days, meaning that if we start the production on day 8 the module will be completed on day 16, 2 days earlier than the time we need them. We might just stop here, since we have found a

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feasible solution. However, another demand may need a start on day 16. We extend our search to determine whether we can start later than day 8. Logically, we might try day 10. The CT for the production of Module 2 on day 10 is still 8 days, so the modules will be completed on day 18 (10 C 8). Just in time! This is not the only complexity associated with date effectivity. Two other common ones are aggregating cycle times and converting to time buckets. In CTs, we might lump the production of Module 2 and Card 2 into one production activity (call it Mod Crd 2). Before introducing date effectivity, this was simple – just add the two CTs and it becomes the cycle time for the new activity (8 C 4 D 12). Now we need to adjust for cycle time changes. In this case, the cycle time for Mod Crd 2 is 12 days from day 1 to day 6, 10 days from day 7 to day 10, and 12 days again from day 11 to day 25. In time buckets, we need to convert the cycle time from daily information into number of buckets. For example, if the time bucket was 3 days in duration, we have 8+ buckets (25/3), and we need to restate each cycle time from days to buckets. Some of the buckets are located on a date effective boundary.

14.4.10 Other Technical Challenges Other challenges include, but are not limited to, demand perishability, squaring sets, soft capacity constraints, alternative capacity, preemptive versus weighted priorities, splitting demand to match partial delays in supply, stability, express lots, delay assembly to test, dispatch lots, foundry contracts (Fig. 14.31: service model; multiple exit demands along the same BOM supply chain), risk-based inventory policy, multisupplier sourcing using inventory, WIP projection, rules governing purchase order change recommendations (allow for date change, quantity change, both or neither).

14.4.11 Challenge of Uncertainty: The Stochastic Nature of the Demand-Supply Network Uncertainty (in parameters, estimated supplies, projected demands, etc.) is no doubt another critical challenge, but we will only briefly touch on this topic in this section. In the near term, uncertainty may force planning applications to take a conservative approach to risk. For example, if (a) the average binning percentage for fast parts is 30%, (b) it has a uniform variability of plus/minus 10%, and (c) there are 100 parts currently being tested, then the actual number of fast parts that we will get from this specific manufacturing activity ranges from 20 to 40. Although on average we will get 30 fast parts from the sorting operation, 50% of the time the actual number of fast parts will be less than 30. Working with the average value is fine over a moderate or long timeframe, but it is impractical if your timeframe is only 1 day. One proposed method called cycle variation looks at taking a more conservative

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IBM receives new contracts for foundry parts (typically finished wafers): Quantities expressed in terms of “daily going rate”(DGR) for finished wafers. DGRs are either shipped directly to external clients (“Exits”) or made available for “Services” at IBM (“Reserves”).

Service Demand

351

typically end products shipped to external clients; consumes DGR Reserves

Service Demand priority applies

Service Demand cannot drive additional DGR starts wafer finish kept for further downstream processing by IBM

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approach with high priority demands than with low priority ones. The downside of this method is that “lower priority demands” had a higher probability of the CPE determining they could be met on time – a small problem! For long range planning, executives would prefer to understand the range of possible outcomes and their likelihood, instead of being given a point estimate. There is a range of work going on (e.g., Kempf 2004) with some involving stochastic optimization while others pulling from DS of wafer fabricators (Burda et al. 2007) and wafer fabrication capacity planning (Zisgen 2005) and still others exploring inventory policy (IBM 2005).

14.4.12 Human Challenge: The Pull Between Simple and Complex Models Little (1992) identified the human challenge as follows. “Since, as humans, we have finite intellectual capacity or “bounded rationality” (Simon), we tend to break complex systems down into small, manageable pieces for analysis, design and control. Once we have decomposed a system into parts, we then have a desire to resynthesize small entities into big ones and work with the large entities as new units.

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“Such hierarchical modeling is a useful approach, but not without pitfalls. Forrester (1961) points out that the parts of the system sometimes interact in unexpected ways and offers system dynamics as an approach for treating this.”

14.5 Core of the CPE: Dynamically Making the Best Use of LP and Advanced Heuristics This section describes the following core aspects of the CPE solution mechanism, with a focus on its capability to dynamically mix and match linear (mathematical) programming (LP) and advanced heuristic resource allocation technology. These core elements are: 1. IMEX (implode–explode) heuristic and its ability to dynamically invoke the Binning LP model to handle simple binning with downgrade substitutions (but not BSA, that is, complex binning, general substitution, and alternative BOM) 2. Supply Chain Optimization Planning Engine (SCOPE), an advanced LP-based engine, which handles the BSA, lot sizing, and preemptive priorities 3. Multidimensional partitioning strategy 4. Structures and algorithms to dynamically mix and match both resource allocation decision technologies (IMEX and SCOPE) to balance scope and scale The focus of this section will be on the two LP formulations, the IMEX heuristic’s ability to dynamically invoke the Binning LP, and the divide-and-conquer strategy, which involves a partitioning process to enable the mixing and matching of IMEX and SCOPE. It is outside the scope of this chapter to describe IMEX in detail, the partition algorithm, fair share, lot sizing, customer request, and date effectivity. It is well known that without the right data structures, all attempts to build successful solutions are doomed to failure. Although we will not spend much time describing the data structures, they are the foundation of IBM’s success. There are about ten key input tables that are tightly coordinated but independent. These tables provide such key information as build options, BOM, yields, cycle times, substitutions, binning, sourcing, receipts, inventory, etc. They also provide a solid base for the solutions to handle the few-to-many relationships and date effectivity with ease (data driven). Figure 14.32 has a list of the input tables and Fig. 14.33 the output tables.

14.5.1 IMEX: Implode/Explode BCD Heuristic IMEX executes three major steps (Fig. 14.34). Step 1 is an MRP type of explosion of demand across the entire BOM supply chain, where hints are established and the binning points are optimized with the “Binning LP” (BINLP, see Figure set 7

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• Cap Priorities • Macro Cap Reqmts Profile for Use at Bottom of BOM • Cap Reqmts Profile • Cycle Time Variability, and Loc Mult • Degrade Option 1, and 2 • Override Option • Forward Allocation rules • Frozen Zone Parameter • Inventory Policy, Qualifier • Lot Size Rules • Penalty Backorder • Penalty Inventory • Reorder Point • Split Indicator • Stability Push • Starts Override Info • Substitution Rules, by Customers, Parents • Priority Option • User Set

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Fig. 14.33 Output tables for the CPE

for an optimized use of downgrade substitutions at a binning point). Hints refer to required manufacturing starts, tentative assignment of WIP or inventory to support a specific start, and demand priority associated with each level of exploded demand (Dangat et al. 1999). The (optional) second step allows the files generated during the first step to be modified to influence the final creation of anticipated supply. For example, the wafer start file could get modified. The third step executes a BOM implosion to create a final BCD solution that meets planning requirements such as demand class, fair share, etc.

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14.5.1.1 Brief Explanation of Low Level Codes and Traditional MRP Explosion To understand IMEX in detail, we must first review the classical MRP explosion and low level codes (LLCs). The reader already familiar with these topics can skip this section. Figure 14.35 shows a simplified part flow from wafer to card. Common Wafer 1 can follow one of the three manufacturing processes to become Device 1, Device 2, or Device 3. The CTs for the manufacturing activities to create Device 1, Device 2 and Device 3 are 15, 10 and 12 days, respectively. Module 1 requires 1 unit of Device 1 and 1 unit of Device 2. Module 2 requires just 1 unit of Device 2. Module 3 requires 1 unit of Device 3. The cycle times required to complete Modules 1, 2 and 3 are 2, 3 and 2 days, respectively. Card 1 requires 1 unit of Module 1 and 1 unit of Module 2, and Card 2 requires 2 units of Module 2 and 1 unit of Module 3. The cycle times required to complete Cards 1 and 2 are 5 and 4 days, respectively. Independent customer (or finished goods) demand can exist for Card 1, Card 2, and Module 3 (marked with an asterisk in the figure). Module 3 is both a finished good and an intermediary part. The first step in an explosion process is to determine the LLC for each manufacturing activity. Generally, activities that are only independent demand or finished

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goods are put into the group of low level code 1 (LLC-1). In Fig. 14.35, parts Card 1 and Card 2 are in LLC-1. As to Module 3, although it has independent demand, it can also have exploded demand placed on it by Card 2. So Module 3 does not qualify to be in LLC-1. Next, LLC-2 contains all parts that have independent demand and/or dependent demand as a result of exploding demand on parts in LLC-1. In our example, parts Module 1, Module 2, and Module 3 are in LLC-2. Following the same logic, all devices are in LLC-3 and the Common Wafer 1 is in LLC-4. The second step is exploding each demand backward through the BOM structure, one LLC at a time, to determine which parts in what quantity on what date have to be available to meet independent or dependent demand. In our example, this step begins by gathering and aggregating all of the demand on cards and then exploding the card demand to the module level. Turn to Fig. 14.35 again but this time, let us focus only on those orange boxes. Wetcountere see whether 10 units of Card 2 are required on day 20, then 20 (2 10) units of Module 2 and 10 units of Module 3 are required on day 16 (20 4). Next is to aggregate all exploded demands on modules with any independent demand and then explode this demand back to devices. To ensure 20 units of Module 2 are available on day 16, 20 units of Device 2 are required on day 13 (16 3), and to ensure 10 units of Module 3 are available on day 16, 10 units of Device 3 must be available on day 14 (16 2). This process continues until the BOM chain is exhausted. Continuing the example, to ensure 20 units of Device 2

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are available on day 13, 20 units of Common Wafer 1 must be available on day 3 (13 10), and to ensure 10 units of Device 3 are available on day 14, 10 units of Common Wafer 1 must be available on day 2 (14 12). Typically, during an explosion, the backwards engine will check whether existing inventory or projected WIP is available to meet the exploded demand. If it finds such assets in the pipeline, it truncates the explosion. Let’s relate this to our example in Fig. 14.36. If 35 units of Device 2 are forecasted to come to stock on day 11 (the black box in the figure), these units will be allocated to meet the exploded demand created by Card 2 (20 units of Device 2 on day 13 to build Module 2, which is needed to build Card 2). As a result, no exploded demand will be sent to Common Wafer 1 to support the build of 20 units of Device 2. 14.5.1.2 IMEX Step 1: Explosion with Binning MRP The first step of the IMEX is sending the demands to a special variation of MRP called “Binning MRP” (BMRP) to create exploded demand for each part on the BOM structure. During the explosion step, the IMEX first examines all parts located at the top level of the BOM (i.e., those with LLC of one, or LLC-1) and separates the binned parts from the nonbinned ones. After the separation, the IMEX creates dependent demand on parts located at the lower levels (i.e., with a larger LLC) by exploding the nonbinned parts with the traditional MRP logic and the binned parts with a “binning LP” (BINLP) algorithm. The information considered in explosion includes, but is

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not limited to, manufacturing starts, receipt due date, and required capacity. The same process repeats itself at every level in the BOM until reaching the lowest one (parts with the largest LLC). This completes the first major step of the IMEX heuristic. The key decision variable in the BINLP algorithm is the required starts in each time period for the binned parts. The objective function ensures that demand is met as much as possible (by imposing a huge penalty on every unit of unsatisfied demand) while lowering the period-end inventory at the same time. The constraints ensure a material balance between inventory, starts, receipts, and demand. Please refer to Appendix B for the detailed formulation. The process flow for explosion is illustrated in Fig. 14.37. The BMRP first uses the traditional MRP logic to explode the demand for Module 1a, Module 1b, Module 2a, and Module 2b to create dependent demand for Device 1a, Device 1b, Device 2a, and Device 2b, respectively. Next, the BMRP explodes those (dependent) demands to create dependent demand for Wafer 1 and Wafer 2. Two small LPs are dynamically created (instantiated) for that purpose: one includes Device 1a, Device 1b, and Wafer 1, and the other includes Device 2a, Device 2b, and Wafer 2. The BMRP would solve both LPs to obtain exploded demand for Wafer 1 and Wafer 2. After that, the BMRP again uses the traditional MRP logic to calculate dependent demand for the Common Wafer. Note that the LPs are generally solved very fast in production runs, often in just a few seconds. In addition to invoking LP runs at simple binning points, please also note these two features exhibited during the explosion process of the IMEX: Initial capacity checking, which means capacity is checked for new starts. And

if capacity is exceeded, the starts will be moved to a different time and/or split based on available capacity.

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Demand data propagation, which means information such as demand class and

lot-size quantity is carried along throughout the explosion process. Other than dependent demand, the IMEX explosion also creates the below information, which is critically important to the heuristic’s implosion step. Required starts associated with such data as part ID, demand class, start quantity, start date, components, etc. (this includes those that have been moved forward or backward in time as a result of capacity checking). Required receipts or units of WIP (each with a need date). Required capacity for meeting all demand. Optimal substitutions as suggested by the BINLP algorithm. Reverse low level codes (RLLCs).

The required starts help the subsequent implosion step in identifying opportunities for earlier starts (in the microelectronics industry, starts often, but not always, mean wafers). Note that only one demand class is associated with each start. RLLCs establish the order in which parts will be processed during implosion – it is essentially the reverse of that followed by the traditional MRP logic. For purpose of further discussion, we define a PN to have an RLLC-1 if it does not have any components.

14.5.1.3 IMEX Step 2: Option to Modify Three Inputs for the Implosion In this step, three types of output created by the explosion can be modified if necessary: required starts, projected receipts, and required capacity. For example, starts created in the past (i.e., those with a negative time) are brought forward to the current time, or a separate model that handles “cascading capacity” (Bermon and Hood 1999; Zisgen 2005) can be invoked to perform a detailed capacity analysis for the requested starts.

14.5.1.4 IMEX Step 3: Implosion The implosion process begins by adjusting the starts located at the top of the reverse BOM (i.e., have an RLLC-1) to be time and capacity feasible. The starts are sorted by demand class, start date, and PN. Further, those with a start date in the past are changed to have a user-specified date or the first day in the planning horizon as the new start date. The stock date (i.e., when manufacturing is expected to complete and parts become available) is also adjusted accordingly. These revised starts are then examined to determine whether capacity is sufficient. If capacity is not enough, it will be allocated on a first-come-first-serve basis. Sorted by demand class and date, starts with a more important demand class have preemptive priorities over those with a less important demand class. If a start date has to be adjusted to accommodate capacity issues, the IMEX heuristic would attempt to move that start earlier in time, or delay it if moving earlier is not possible. When this is all done, the output is an adjusted starts file that is both capacity and time feasible for use in the subsequent implosion steps.

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Next, parts are imploded beginning from those with an RLLC-1. At each reverse LLC, a proper processing order is identified such that substituting PNs (i.e., they substitute for others in short supply) are processed before those being substituted, and shipping locations (i.e., they send out shipments) are processed before those receiving shipments. Supplies such as starts, inventories, purchases, or WIP are collected. Then, the IMEX uses the demands generated by the BMRP run to determine supply allocation. Demands with the same demand class are covered first-come-first-serve. When supply is short, the more important demands may preempt supplies, i.e., taking away supplies that have already been allocated to the less important demands. For lot-sizing, it will be preserved whenever possible during the implosion step, but will be relaxed when supplies are short to better meet customer demand. For example, if a lot of 25 wafers is needed but only nineteen are available on the need date, those nineteen wafers will be taken to continue the implosion process without delay (as opposed to waiting until later in time, six more wafers become available to make up a full lot).

14.5.1.5 IMEX Summary The IMEX heuristic produces a projected supply schedule and commit date estimates satisfying a variety of constraints (temporal, asset-based, and business policy). When capacity is insufficient, starts with a more important demand class are given preference. Similarly, when supplies for a particular component are short, they would be allocated based on the demand class of competing parts. The IMEX uses a binning LP algorithm to handle much of the complexity found in semiconductor-based manufacturing. Generally speaking, it cannot find high-quality solutions when encountering BSA: complex binning, general substitution, or alternative BOM paths (Sect. 14.4.3, Appendix A). For IBM CPE, these complexities are specifically handled by another LP-based solution called SCOPE (Sect. 14.5.2).

14.5.2 Supply Chain Optimization Planning Engine SCOPE is an LP-based supply chain solution developed and deployed by IBM in the 1990s that has been continuously enhanced since. This solution is primarily used to handle the “triple crown of complexity” (i.e., the BSA described earlier sections) for which the IMEX and other heuristic-based solutions are in general unable to identify optimal solutions. The SCOPE solves a supply chain LP model with a cost minimization formulation. So a minimum cost solution is obtained when the model is solved to optimality. As defined in Appendix C, the SCOPE considers such costs as backorder, processing, inventory holding, material substitution, part shipment (ship to customers or ship to other manufacturing locations within the

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enterprise), and a few others. Whenever a feasible solution is identified, these costs will be multiplied by the associated decision variables in the objective function to obtain the (total) cost for the corresponding supply chain plan. In addition to costs, the SCOPE requires the usual input such as customer demand, scheduled vendor shipment, yield, capacity, cycle time, and so on. The input data structure is very much the same as that for the IMEX. Also as in the IMEX, most input data can be provided as time effective that the value would change over time to reflect the real situation. The decision variables in the SCOPE LP model are chosen to correspond to actual supply chain activities: customer shipment (Fmakqj in Appendix C), backorder (Bmkqj ), substitution (Lamnj ), manufacturing start (Pmaej ), interplant shipment (Tmavj ), inventory level (Imaj ), and sourcing (Sauzj and Gauzj ). In any feasible solution, these variables must hold values to satisfy the following five types of constraints: Material balance equations maintain a flow balance between the creation and

consumption (arrival and departure) of any PN at any stocking point, in any period, and at any manufacturing location. These equations handle all the complexities associated with binning, general substitution, and alternative BOM structures. Backorder conservation constraints keep track of unsatisfied demand throughout the planning horizon. That is, in any period they capture the total unsatisfied quantity contributed by all demands with the same PN, customer location, and demand class so it can be backordered and met in a future period. Capacity control equations safeguard capacity utilization, ensuring no resource is overutilized by starts requiring the same resource. Sourcing constraints enable the SCOPE users to control the deviation from sourcing targets, each of which is specified by two numbers, MAXPCT auzj and MINPCT auzj . As defined in Appendix C, they indicate the maximum and minimum percentage of all shipments destined to consumption location(s) u that are preferred to originate from a particular supply location a. If the actual percentage falls above or below the targeted range, penalties will be incurred for the excess or shortage amount and added to the total supply chain cost. Nonnegativity constraints require all decision variables to be greater than or equal to zero.

14.5.2.1 Planning Periods with the SCOPE Like other LP-based supply chain solutions, the planning horizon for the SCOPE is divided into a number of periods (or buckets), which is quite different from the IMEX heuristic. This characteristic is seen clearly in Appendix C, where, for example, the manufacturing start variable Pmaej is defined for every PN m, plant location a, manufacturing process e, and period j . Furthermore, the number of (consecutive) days covered by one period may differ from that covered by other periods. It is obvious that for the same planning horizon, say, 2 years, more periods will increase the

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model size, which means it will take longer for the SCOPE to complete its solution process. On the contrary, if there are not enough periods, or if the period lengths are not set up properly, the SCOPE may not generate supply chain plans with sufficient granularity to meet business requirements. Therefore, it is crucial that the right period configuration (number–length combination) be identified. Different period configurations are generally required for different planning purposes. For an enterprise strategic SCM plan, the planning horizon may span 2 years comprising 28 daily periods, 21 four-day periods, 10 weekly periods, 6 monthly periods, and 1 yearly period. In contrast, the planning horizon for an operational SCM plan may be 1 year, consisting of 30 two-day periods, 8 weekly periods, 2 monthly periods, and 1 six-month period. It takes experience and lots of experiments and fine-tunings to identify the right period configuration. Planning purposes, memory size, CPU speed, and the time window allowed for a supply chain plan to be generated are all factors in the determination of the right period configuration.

14.5.2.2 Cost Setup for the SCOPE SCOPE creates supply chain plans with the lowest cost. Therefore, how costs are set up in the model has a direct impact to what solutions are found. Backorder costs (BOCmkqj ) are probably the most important, which are incurred whenever all or part of a demand cannot be satisfied by the commit date. Normally, a number of customer “tiers” are defined for planning purposes, representing how important a group of customers is to the enterprise relative to the others. Each customer is strategically assigned to a tier, which is associated with a carefully chosen backorder cost. Normally, any enterprise would try to meet the more important demands, that is, those received from the more important customer tiers, on time as much as possible. As such, the backorder cost for those customer tiers would need to be set as much higher than those for the less important customer tiers. Setting up the right backorder costs may be a challenge when the number of tiers is in the dozens, as there is a safe range for the maximum and minimum backorder costs to fall in without risking a numeric runtime error, while the cost difference between some neighboring tiers, e.g., tier one and tier two, also needs to be significant enough in order to meet planning goals.

14.5.2.3 Solution Process The SCOPE begins its solution process by checking, filtering, and processing a variety of input data. In particular, relationships are established between dates and periods in terms of what the dates are for each period, and they serve as the base for transforming raw input data (which is provided in days) to such forms as needed by the subsequent LP processing steps. Data with the same attributes is further aggregated. One such example is customer demand that it is aggregated over the same PN, customer location, demand class, and planning period. Data aggregation reduces the

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size of the model, therefore reducing the time needed by the LP solver. New data such as the relationship between manufacturing input (i.e., starts) and output (i.e., stocks) is also obtained from the BOM, build options, yields, and part attributes. When all data processing steps are finished, various “TAB” files are generated with data displayed in a tabular format ready to be used for model generation. The MPS format is used to represent an LP in the SCOPE solution process. Compared to the input data processing, model generation is quite straightforward that data in the TAB files is written to become one of the following in the MPS file: a decision variable’s subscript, a constraint coefficient, or an objective function coefficient. The data in the TAB files is generated in such a way that it only requires a little more processing for model generation. The SCOPE then calls a subroutine library to solve the LP model. The postprocessing step follows the return of an optimal solution by the optimization solver. A mirrored step of the input data processing, this step processes the LP solution so it can be used by supply chain planners. A comprehensive set of reports are produced directly from the LP solution, including customer shipments, manufacturing release schedules, dependent demand, purchase orders, and vendor shipment schedules. The LP solution can also be disaggregated and linked with input data to produce reports with more granularity.

14.5.2.4 Implementation The SCOPE is implemented in C CC and shell scripts, and can dynamically invoke different optimization solvers including the open-sourced COIN-OR (COmputational INfrastructure for Operations Research, http://www.coin-or.org/) and the IBM Optimization Subroutine Library (OSL, developed at IBM T.J. Watson Research Center). The LPs generated for the module portion of IBM semiconductor manufacturing typically have two to three million variables and one to two million rows.

14.5.3 “Divide & Conquer”: Decomposing the Problem (CPE Partitioning) Despite the constant improvements in hardware performance, determining an optimal central plan for a large enterprise with just the SCOPE (or LP-based solutions in general) along is not even remotely close to being realistic. If the IMEX heuristic is used by itself, the quality of the decisions would most likely be less than desirable for complex product flows. Besides performance and ability to handle complex product flows, there are business reasons to partition and sequence the solution flow. For example, executing an explosion step identifies a prioritized set of required starts/releases needed to satisfy all demand in time. This is essentially a prioritized “wish list.” This information would not be available with a single, monolithic solution process, since the only starts ever calculated are feasible with respect to

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capacity and component availability. The information of prioritized starts is particularly helpful when a planner is attempting to determine what actions to take to meet certain demands that are currently behind schedule or to establish minimum starts. Therefore, developing a fully automated, dynamically partitioning-and-sequencing algorithm which makes the best use of both heuristics and LP technologies was the logical course of action. There are four fundamentals to the “divide-and-conquer” solution strategy: 1. The BOM or production flow can be divided into stages that occur naturally and every manufacturing activity is assigned to one stage. 2. Every part can be classified as complex or simple. 3. There is an explicit explosion action set followed by an explicit implosion action set. 4. Humpty Dumpty can be put back together again. The following sections will explain each of these in more detail.

14.5.3.1 Assigning a Part to a Stage in the Production of a Finished Good Within the production flow of semiconductor-based manufacturing, historically there have been part groupings or manufacturing stages that emerge naturally from the organizational structure. At IBM, there are three major part groupings or stages: Card, Module, and Device/Wafer. The core manufacturing processes at these stages are different and are typically performed at different manufacturing facilities. Therefore, from manufacturing point of view, starts are occurring at each stage and the exact timing and conditions of these starts are handled with some special “local logic.” In addition, a consolidation process often occurs at each stage. To assign a part to a stage, typically, the part is first associated with a specific part type; then, that part type is assigned to a stage. We will use Fig. 14.38 to illustrate this process. First, PNs Module 1, Module 2, Module 3 and Module 5 would be assigned to part type “Exit Module;” then, part type Exit Module would be assigned to stage “Module.” This linkage puts all four modules into the Module stage. Similar structures would be used for other PNs, resulting in all the devices and wafers assigned to the “Device/Wafer” stage, and all the modules and raw modules assigned to the Module stage. There are a variety of checks to ensure a logical consistency across stage assignments. For example, if a user places device Dev 1A1 into the Module stage but wafer Wafer 1A into the Device/Wafer stage, this would be flagged since parts that are linked by binning must be in the same stage. As a second example, assume a planner places Module 3, Raw Mod 3A, and Raw Mod 3B into the Module stage but Raw Mod 3C into the Device/Wafer stage. This would also be flagged since all the component parts for Module 3 must be in the same stage. Figure 14.41 shows the BOM structure in Fig. 14.38 grouped by stage. The algorithms to navigate the hierarchy, traverse linkages among manufacturing activities (resource consumed, part consumed, etc.) make linking associations, and do the consistency checking or flagging are set of network flow “membership”

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algorithms (Sullivan et al. 1991) familiar these days to most computer science students. They were standard practice among the IBM team members who were “living history” (can remember programming before color and spreadsheets). While the Card-Module-Device/Wafer stage decomposition is appropriate for IBM (and other firms that produce similar products), different decompositions may be necessary for different manufacturers. In general, the CPE assumes no specific ways to decompose the supply chain problem.

14.5.3.2 Assigning a Part as Simple or Complex The second dimension of the decomposition is based on “complexity.” The goal of this classification is to assign each part to be processed by either the SCOPE (i.e., a complex part) or the IMEX (a simple part). A part is typically classified as a complex part if one of the following conditions holds: 1. The part can be produced via alternative manufacturing processes at a given location. 2. The part can be produced via alternative general substitutions, and these substitutions are viewed by the business as the equivalent of alternative manufacturing processes at a given location (Figure set 8). 3. The part is involved in a complex binning situation. 4. The user of the CPE prefers the SCOPE to solve the part for some reason. 5. The part is connected to a complex part through product flows or resource sharing.

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The first three conditions are determined directly by the CPE through examining the core input files. The fourth one is a planner decision. For example, if the alternative sourcing possibilities are particularly complex or resource availability is tight, the planner may want the solution to be created by the SCOPE. Any parts that are classified as complex via conditions one through four are called “primary complex parts.” In Fig. 14.38, Module 1 and Module 5 are primary complex parts as both can be made using two different processes P1 and P2. Dev 1A2 and Dev 1B2 are also primary complex as there is a general substitution relationship between them that Dev 1A2 may substitute for Dev 1B2 in any condition and vice versa. A network membership algorithm is used to classify parts as complex via the above fifth condition. Starting from the primary complex parts, this algorithm will conduct an up-and-down search along the BOM supply chain. Any part that is reached by this search is labeled “secondary complex” (it is essentially “guilt by association”). We will first use Fig. 14.20 to illustrate primary and secondary complex parts. Module 9 is a primary complex part as it can be built in three methods: method one uses process P1 and consumes Device 8A; method two uses process P2 and consumes Device 8B; method three still uses process P2 but consumes Device 8C (as a substitution for Device 8B). All other parts in the figure, i.e., Module 8 and untested device, are secondary complex. As another example, Module 1 in Fig. 14.38 is a primary complex part (it can be produced by process P1 or P2), so all of the parts below it (upstream in the BOM) are classified as secondary complex. Figure 14.39 shows all of the parts in Fig. 14.38 classified as simple and complex.

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Fig. 14.39 BOM structure with parts grouped by simple and complex

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Fig. 14.40 BOM structure showing two independent complex groups of parts

Note that the part called Common Wafer is not classified as complex; rather, it is called a “nonbinding part.” A part is classified as nonbinding if the supply is sufficient to meet all demand at any time – that is there is never a need to allocate its supply between competing demands. It is called nonbinding, since it cannot be used to “connect” one part of the BOM structure with another in terms of determining whether a part is complex or not – or in terms of independent paths. In this example, the part Common Wafer is nonbinding. The CPE will assume any exploded demand for this part is met immediately. Without the “nonbinding” part, then all parts would be complex parts and all parts would belong to one single group. Although a simple concept, it is critical to the success of classify parts a complex or simple and being able to identify two mutually independent complex groups of parts shown in Fig. 14.40. It is important to note that the search for secondary complex parts may or may not cross stage boundaries. Figure 14.41 shows how the BOM structure is grouped by stage: LLCs 1 and 2 belong to the Module stage and LLCs 3, 4, and 5 belong to the Device/Wafer stage (for simplicity, the Card stage is not shown). In complex group 1 identified in Fig. 14.40, the complex status of the parts flows from the Module stage into the Device/Wafer stage. Therefore, parts in this group are complex in both stages. This is, however, not the case for complex group 2: the parts are complex in the Module stage but simple in the Device/Wafer stage. Decisions on shared capacity may also drive parts into the complex group, even if they are not connected to primary complex parts through a BOM linkage.

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Fig. 14.41 BOM structure with the module and device/wafer stages

14.5.3.3 Decomposing the Problem across Two Dimensions: How Do We Get a Solution? The methods described in Sects. 14.5.3.1 and 14.5.3.2 form the partitioning algorithm shown in Fig. 14.42. Apply it and we can decompose a semiconductor & package supply chain problem into six subproblems or “partitions” (in the case of IBM): Card-Complex, Card-Simple, Module-Complex, Module-Simple, DeviceComplex, and Device-Simple (Fig. 14.43). Note these partitions are obtained across two dimensions, and the Complex ones will be solved by the SCOPE and the Simple ones will be solved the IMEX heuristic. So now the question becomes: how do we connect all the pieces in the right sequence to create an enterprise-wide, detailed supply chain plan? The key is to mimic the old manual pattern that existed at IBM until the mid 1990s (and is still being done today at many firms). We will adapt and enhance this pattern with the use of a partitioning algorithm (Fig. 14.42), the IMEX heuristic (Sect. 14.5.1), and the SCOPE (Sect. 14.5.2). At a high level, the algorithm for the CPE consists of four major steps: preprocessing, explosion, implosion, and postprocessing (Fig. 14.44). Preprocessing divides the original SCM problem into six partitions, which allows the entire problem to be solved in a controllable, divide-and-conquer manner (one partition at a time). The next step carries out a traditional, MRP-type explosion of the BOM to determine capacity, material, and other requirements. This step is executed in the

368 Fig. 14.42 Steps in establishing partitions

K. Fordyce et al. DETERMINE PARTS STAGE

DETERMINE PARTS WHICH DIRECTLY REQUIRE SCOPE

DETERMINE PARTS WHICH INDIRECTLY REQUIRE SCOPE AT THE SAME STAGE

DETERMINE PARTS WHICH INDIRECTLY REQUIRE SCOPE AT DIFFERENT STAGES

ASSIGN PARTS TO PARTITIONS

COPY DATA INTO PARTITIONS

Fig. 14.43 Partitions of the CPE

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sequence of the Card stage, the Module stage, and the Device/Wafer stage, and both the IMEX and the SCOPE are involved in the explosion within each stage in a cooperative manner. A method called “one level explosion” is executed to connect the explosion of one stage to the next, which will be explained in more detail later. After the completion of the BOM explosion, an implosion step begins to match supplies with the requirements created by the explosion. In essence, this is a mirror step of the previous one: it is executed in the sequence of the Device/Wafer stage, the Module stage, and the Card stage; within each stage, it uses both the IMEX and the SCOPE to consider capacity availability and consumption and create a feasible

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Fig. 14.44 Algorithmic flow of the CPE

supply chain plan; and a method called “one level implosion” is executed to connect the implosion of one stage to the next. Finally, the postprocessing step consolidates outputs from all the partitions to create a single, coherent solution, which is further processed through a series of formatting procedures to produce usable reports for the supply chain planners.

Starting Point for the Solution: Explosion at Cards The CPE begins the explosion for the Card stage by gathering all the pertinent information: BOM, substitutions, inventory, WIP, yields, etc. It then divides this pool of information into two categories: complex (i.e., the Card-Complex partition) and simple (i.e., the Card-Simple partition). For the Card-Complex partition, the SCOPE is used in an MRP instantiation to determine the exploded demand for the Module stage (remember that MRP is concerned with determining the minimum starts needed on the latest possible date to meet demand without any concern for capacity). For the Card-Simple partition, the IMEX which contains the BINLP algorithm (Sect. 14.5.1, Appendix B) is used. Details of each individual demand are maintained, and exploded demands are consolidated into a single picture. It turns out that this is also a good time to handle some aspects of lot sizing and a few other planning requirements. This process is repeated two more times, moving backwards by stages from Card to Module and then to Device/Wafer.

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One Level Explosion Algorithm Between Stages One level explosion is a critical component of the CPE, as decomposing the manufacturing process into stages relies on this algorithm running efficiently to link two consecutive stages. One level explosion passes crucial information such as demand type and demand class from one stage to the next. For instance, when the CPE has finished the explosion of, say, the Card stage, it needs to know how many units of what (dependent) demand to create at the Module stage to support the starts recommended for the Card stage (in turn, these starts would support exit demand). The type of demand created during the solution process is called dependent demand. To allow one level explosion to create dependent demand accurately, an association (or “pegging”) between customer shipments and manufacturing starts is required, where these starts are made for the parts located at the bottom of the BOM for the current stage. This association must have such granularity that a correct portion of each start is pegged to a particular customer shipment, and thus to a particular customer demand (because customer shipment contains the original demand information). Such information is made available by the pegging method.

Continuing the Explosion Process After the one level explosion between cards and modules is complete and pegged dependent demand for the modules is established, the explosion process is repeated at the Module stage creating device demand. The same process is repeated for a third time at the Device/Wafer stage creating wafer demand. These wafer demands are the bottom of the food chain.

Starting Implosion to Create a Supply Statement After the completion of the (last) explosion process for the Device/Wafer stage, the CPE reverses the stage sequence and starts an implosion process for each of the stages. Especially, the implosion step considers the requirements generated during the explosion step and solves the problem within each partition with limited capacity. For the SCOPE solver, typically the capacity allocated to each time period is the maximum of three amounts: 1. The prorated share of the total capacity required for the given time period (determined during the explosion step). 2. The prorated share of the total capacity required for the given time period, the time period before the given one, and the time period after the given one. 3. The prorated share of the total capacity required throughout the planning horizon. The use of the maximum ensures the capacity available to the SCOPE is at least equal to the prorated share of the complex parts during the current, intermediate, and long-term time periods. There are a number of alternatives that are more complex

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and can account for demand priority; these alternatives may also involve an iterative exchange between the two solvers. But generally, the above maximum rules work reasonably well in practice. The establishment of capacity availability allows the SCOPE to solve the complex partition and create a feasible, intelligent solution. The exact capacity consumed by the complex parts can be calculated, and it is subtracted from the available capacity. Then, the IMEX is invoked to solve the simple partition at the same stage, and again, the available capacity is adjusted by subtracting the amount consumed by the simple parts. The remaining capacity will be used later by appropriate parts belonged to a subsequent stage. The implosion step also needs pegging and other necessary methods to link two consecutive stages (e.g., from Module to Card), which is achieved by passing information such as demand class on to the next stage and executing some required (nonlinear) business rules. First Implosion Step The first stage to be considered for implosion is Device/Wafer. Parts are again classified into complex and simple and then solved by the SCOPE and the IMEX, respectively. The goal of the implosion step is to best meet prioritized demand created during the previous step without violating temporal or capacity constraints. The algorithm used to allocate capacity between the complex and the simple partition has been described in the previous section, which can be tweaked by the planner if necessary. It turns out estimating capacity availability/consumption is a quantum more elusive than knowing such things as product flows, inventory, and WIP. These sharing algorithms work well in practice. One Level Implosion Algorithm between the Stages One level implosion is another critical component of the CPE. It runs between two consecutive stage implosion processes, e.g., after the completion of implosion for the Device/Wafer stage and before the start for the Module stage. This algorithm gathers up supplies generated during the implosion process which just finished, merges them with inventory, WIP and other existing supplies, and allocates these supplies to the next implosion process. One level implosion handles all of the “tracking” issues that associate supply with demand priority, some aspects of lot sizing, and capacity allocation as appropriate. In practice, there is a considerable amount of intelligence required to optimally allocate supply to demand at this point of the solution process. The implosion step of the CPE mixes and matches a simple, but powerful and exceptionally fast heuristic (not the IMEX, but one specifically designed for this task) and a scaled-down version of the SCOPE to perform this task. In addition, there is a tremendous opportunity for large grain parallelization – for large problems that require the solution power of the SCOPE; dynamic parallelization is a key to robust performance. This process continues until the last stage is completed.

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Putting Humpty Dumpty Back Together Again Once the implosion for the Card stage is completed, all individual solution pieces need to be put together into a single, consolidated solution. The reconfiguration of the supply chain into stages or partitions is only for the purpose of identifying an intelligent plan – apparently, users of the CPE do not need to know which partition a part was in while the CPE was running, so such information needs to be invisible to the users. The consolidated solution provides the planners with a cohesive, detailed plan or schedule of events in the enterprise with a clockwork precision, allowing them to view the plan from an aggregate level or monitor detailed lot movements and capacity consumption. Again, the pegging/tagging information is used by the appropriate algorithms to achieve this result. The output tables created by the CPE contain early warning reports, coverage analysis, detailed demand pegging (individual lot or customer order level of detail), detailed manufacturing starts, capacity consumption, substitutions, etc. All this information is available to the planners in their “tool of choice.” Additional details about the core algorithms in this CPE can be found in Hedge et al. (2004), Milne et al. (1999), Orzell et al. (2004), Denton and Milne (2006), Denton et al. (2004), and Denton et al. (2005).

Beyond the Holy Grail As the reader has noticed, the goal to create and deploy a centralized, end-to-end enterprise wide supply chain plan with a sufficient level of detail and a reasonable runtime performance has been the “holy grail” since the early 1990s. Today, a substantial portion of that goal has been achieved. As with any science, the accomplishment of one goal brings not only a sense of pride but also a huge dose of reality in what is left to be finished. In SCM, achieving a reasonably strong level of central control can dramatically increase organizational performance – but it also clearly identifies gaps in a timely synchronized response that currently can only be handled with ad hoc manual interventions that operate without a global awareness. A simple example is an order change once the plan has been established; another example is a component part that can be finished earlier than the planned cycle time to meet customer demand but there is no connection between the two activities. In both of these examples, the fundamental question is when to “update the plan.” In the current standard approach, the plan is regenerated based on some amount of elapsed time. As Harpal Singh (2007) observes, this is often not best practice: “We are finding the need of a monitoring mechanism that measures the changes in supply and demand so that the “current” plan can be tested constantly for relevancy. So the issue becomes – when should the plan be re-run? The traditional approach is for a fixed, time based cycle like a week or a month. We are finding that this is not necessarily a good approach because the frequency is in direct proportion to the volatility in the environment.”

The flip side of a gap is an opportunity window and we will look at these opportunities in the next section.

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14.6 Future Direction 1: Extending the Big Bang Over the past 10 years, much of the effort in enterprise-wide central planning has been to move organizations from a decentralized, loosely coupled approach that created a new plan once a month to a reasonably tightly integrated process using more intelligent models to create new plans from once a week to once a day. There is still considerable work to do across business organizations in an enterprise to achieve this level of central planning. Even for firms who have achieved this level, effort is still required to keep it from slipping back into bad habits. We refer to today’s SCM central planning process as the “big bang” approach. An enterprise creates a centralized process and data representation of the firm at some point in time. Then, through some combination of automated models and manual processes, the enterprise creates a “global plan.” Some firms execute a global planning process a few times each day; others do it once a month and take a week to create the plan. In both cases, the new plan arrives after a reasonable amount of effort and replaces the old plan in its entirety. A new “universe” is created via the “big bang!” There are two primary opportunities/challenges for this big bang approach: (1) expanding the use of advanced global planning from leading firms to all firms, and (2) improving the quality of the global plan by capturing more of the complexity in the supply chain solver. The first challenge belongs to business consultants and operations management researchers. The key question to ask for this challenge is this: what are the obstacles at average firms that keep them from bringing the same level of sophistication and supply chain performance as observed at leading firms? Our observation on this topic is that most business consultants and operations management researchers focus mostly on processes and data and fail to understand the importance of the decision models. The second challenge includes reducing solution times with parallel processing and faster algorithms and machines, improving accuracy with smaller time buckets or units of granularity, with more robust representations of the demand-supply network, or with latest techniques such as constraint programming, handling the stochastic nature of information, and automating some aspects of plan review.

14.7 Future Direction 2: Beyond the Big Bang: Supply Chain Physics of the Twentieth Century Meets Its Quantum Revolution At the start of the twentieth century, physicists learned that we do not live in a clockwork universe (Wolfson 2000). The same can be said for supply chain modelers at the start of the twenty-first century. Although centralized processes and models have increased organizational effectiveness, there are clear limits and we are rapidly reaching them.

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Effective centralization refers to the ability to take into consideration all aspects of the decision situation simultaneously and generate an optimal or at least a very good solution. To be effective, a central solution requires a synchronized current view of the entire decision landscape, the ability to handle complex trade-offs, and a reasonably fast runtime performance. Gaps exist and are often created by time lags, summarization, performance, triggers, and formulation. By triggers, we refer to the event that “wakes up” at a specific time during the day and runs the central solution. Once a week, once a day, or once every 3 days, the central solver executes, but the decision to execute is made without any knowledge or monitoring of events since the last execution. Formulation gap refers to the inability to formulate key decision questions in such a manner that lends itself to a central solution as opposed to a sequence of negotiations or collaborations. The reality is, even if we could get the big bang models to finish execution in 5 min, an organization would run it once a day at best to match demand with supply and synchronize the enterprise. As the time between runs decreases (for example, from once a week to once a day), the following limitations become apparent: 1. Understanding and repairing the plan. 1.1 Why does the plan give these results? 1.2 What demands are not being met (alerts) and why? 1.3 What actions can I take to improve the plan? 1.3.1 Identify and book actions to improve the supply posture as it relates to demand. 1.3.2 Less significant items such as inventory picture. 1.4 Monitoring the repair actions as they are being executed. 2. The nature (quality) of the plan – no plan to plan continuity. 2.1 No checks, filters, or alerts (CFAs) on demand information/signals. 2.2 No CFAs on supply information/signals. 2.3 No CFAs on changes in production specification or business policy. 2.4 Each plan is built from scratch. 2.5 No built-in dialog with other key providers of input, such as 2.5.1 Projected supply. 2.5.2 Projected demand. 2.5.3 Capacity estimation. 3. Observe both 1 and 2 require an incremental matching (planning) or net change engine. The following example shows how a “simple” repair action can quickly become complex. In Table 14.5, we have two demands for the same part (P111): D001 with a due date of 4/4/2006, priority of 2, and quantity of 80 and D002 with a due date of 4/5/2006, priority of 1, and quantity of 100. In Table 14.6, we have two anticipated supplies for part P111: S00A and S00B with an anticipated delivery date of 4/4/2006 and 4/6/2006; the quantity is 100 units each.

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Table 14.5 Demands for Part P111

Demand for Part P111

Table 14.6 Supplies for Part P111

Supply for Part P111

ID D001 D002

Date 04/04/06 04/05/06

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Quantity 80 100

ID

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Fig. 14.45 Initial assignment of supply to demand for Part P111

S00A 4/4/06 amt 100

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Fig. 14.46 Simple repair to meet demand D001 on time

Initially, we assign S00A to D002 (Fig. 14.45) since D002 has a higher priority and the other supply S00B would arrive 1 day after the due date for D002. This assignment resulted in D001 being met 2 days late. Figure 14.46 demonstrates a simple repair action to meet demand D001 on time: the analyst requests that the S00B supply be completed 2 days earlier on 4/4/2006. Figure 14.47 shows a smarter repair action. First, the analyst switches the assignment of supply to demand – supply S00A is assigned to D001 and supply S00B is assigned to D002. Then the analyst requests that supply S00B be expedited only 1 day to 4/5/2006 to meet D002 on time. To take organizations to the next leaps in efficiency requires a substantive adjustment in approach. Just as nineteenth century physics had to adjust its equations and

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S00A 4/4/06 amt 100

change date to 4/5, 1 days earlier

S00B 4/6/06

4/5/06 amt 100

D001 0 4/4/06 priority 2 amt 80 D002 0 4/5/06 priority 1 amt 100

Fig. 14.47 Smarter repair action to meet both demands on time

formulations for special relativity, general relativity, and quantization, supply chain solvers must learn to accommodate techniques from intelligent agents and sense and respond and learn to incorporate the concept of collaboration into their solution processes. Collaboration refers to an iterative process that focuses on finding a satisfactory solution. The next search step depends on prior steps and may involve back tracking. Often, the step involves negotiating a temporary change in a subset of the rules governing the game for a limited period of time, and typically contingency occurs to handle uncertainty. Two major areas in which the IBM supply chain team is currently working are understanding and repairing the plan and establishing plan-to-plan continuity with sense, response, and incremental matching (net change engine). Typically when a planner reviews a plan, he or she would visit the following questions: what demands are not being met (alerts) and why; why the solver gave these results in the plan; what the options are to improve the plan; identify and book actions to improve the supply posture as it relates to demand or inventory; and monitoring that the repair actions are being executed. The IBM team is working on a tool called AIIRR (Assess, Identify, Improve, Respond, Repair). Today, each plan is built from scratch, that is, without reference to the prior plans or changes in demand or supply occurred after the execution of the last plan. There are no checks, filters, or alerts (CFAs) on demand information, supply information, and changes in product specification or business policy. There is no built-in dialog with other key providers of input such as projected supply and projected demand. There is no ability to incrementally modify an existing plan. Currently, the IBM team is working on applications to close these gaps.

14.8 Conclusion: New Science Emerges and Extends the Borders of Bounded Rationality When one of the authors, Ken Fordyce, joined IBM in 1977 as a junior programmer with an undergraduate degree in mathematics, a few courses in operations research and statistics, one of the first people he met was Herschel Smith. Herschel was in

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his 60s and Ken was 23. Herschel had built a small LP model to optimize taxes for IBM World Trade. His efforts were recognized and appreciated by IBM, but he was part of an Information Systems organization. At that time, modeling did not have enough traction to be its “own person,” and decision modeling applied to business problems was at most 25 years old. Today, IBM and some of its clients use a combination of LP and some pretty clever “heuristics” to establish a daily plan for their enterprise. These firms cannot think of “life without these decision models.” This same level of success of decision models can be seen in a number of areas. We believe we are witnessing a growing awareness of the importance of decision models and already competing on analytics (Davenport 2006). That said, there is still a long way to go. People are comfortable with their guesses and decision scientists often fail to deliver real value. Politely, life is much cleaner if your modeling work remains an academic exercise. What decision scientists offer is the potential to “Extend the Borders of Bounded Rationality.” Herbert Simon (Nobel Prize Winner in Economics) (1957) observed, “as humans, we have “bounded rationality” and break complex systems into small manageable pieces.” The challenge for organizations is to integrate information and decision technology to push boundaries out and improve performance. Nick Donofrio (IBM Senior Vice President) observed, “access to computational capability will enable us to model things that would never have believed before.” The challenge reaches beyond coding algorithms, linking with data, and turning it on. Each decision-science team must execute its role as “intelligent evolutionists” to ensure the organization adopts complex decision technology in a sustained incremental fashion. This will enable organizations from semiconductor firms to hospitals to be more responsive. “The ability to simultaneously respond to customers’ needs and emerging business opportunities in an intelligent orderly manner is a survival requirement for today’s market place. Our customers continue to tell us that the quality of our responsiveness is as important as the quality of our products. . . The work done by the innovators on our PROFIT team, as well as their colleagues, has enabled us to meet the customer responsiveness challenge. And I’m convinced that the monetary savings associated with the PROFIT work [$80 million in 1999 alone] – though significant in their own right – understate the work’s true value to IBM.” (Nick Donofrio, IBM Senior Vice President in Lyon et al. (2001)

Appendix A: Example of Limitations of a Simple Heuristic to Handle Alternative BOM Figure 14.48 contains a simple alternative flow example with demand and assets. The Red Module can be produced with process P1 or P2. The P1 path consumes Dev01 and the P2 path consumes Dev02. The Blue Module can be produced with two processes also, consuming Dev02 or Dev03. The on-hand inventories, demands,

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Module Green

Which Paths Do I select to meet each Demand ??

Dev 05

Dev 04

DEM 102 blue Module Date = 10 Priority = 5 Amt = 20

DEM 101 Red module Date = 9 Priority = 5 Amt = 15

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P2 Cycle time = 8 yield = 0.4

Cycle time = 8 yield = 1.0

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Inv = 00 Cycle time = 4 yield = 0.9

P1

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P1

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P1 Cycle time = 6 yield = 0.2

Dev 01 Inv = 00


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Fig. 14.48 Alternative paths to meet demands that share common assets

cycle times, and yields can be seen in the figure. The CPE must decide a series of activities to meet the demand for the Red and Blue Module. One possible heuristic to handle alternative BOM might focus on consuming existing inventory first to “optimize” the use of existing assets. The heuristic would work as follows: 1. Process demand in the following order 1.1 Start with the highest (most important) demand class. 1.2 If there is a tie in demand class, start with the demand which has the earliest due date (within that demand class). 2. When alternative BOM paths exist 2.1 Do a “one level explosion” down the first path (e.g., P1) 2.1.1 If inventory exists to meet this demand, consume this inventory. 2.1.2 If there is no or insufficient inventory, explore the next alternative path (e.g., P2). 2.2 Repeat this process of looking to consume existing inventory through each alternative BOM. 2.3 If there is no or insufficient inventory at each alternative BOM, return to the first (P1) process and apply the same heuristic rule to explode the next level

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If we apply the above heuristic to our example in Fig. 14.48, we get the following solution. 1. Process demand Dem101 (Dem101 is processed first since it has the same priority as Dem102 and an earlier due date). 1.1 Explode the P1 path first 1.1.1 Need 15 units of Dev01 on day 3. 1.1.2 Inventory of Dev01 is 0, so try the next path. 1.2 Explode the P2 path 1.2.1 Need 30 units of Dev02 on day 3. 1.2.2 30 units of Dev02 are available, so assign them to Dem101. 2. Process Dem102 2.1 Explode the P1 path. 2.1.1 Need 20 units of Dev02 on day 2. 2.1.2 The inventory of Dev02 is 0 (already allocated all to Dem101), so try the next path. 2.2 Exploded the P2 path 2.2.1 Need 50 units of Dev03 on day 2. 2.2.2 But the inventory of Dev03 is 0. 2.3 Return to the P1 path and explode to the next level 2.3.1 Need 100 units of Wafer B on day -4. 2.3.2 Generate a need of 100 units on Wafer B on day 0 (ignore lot sizing). 3. Results of this heuristic are 3.1 Dem101 is met 3 days early. 3.2 Dem102 is met 4 days late. 3.3 remaining inventory is 0. 3.4 wafer needs are 100 units of Wafer B. With just a little bit of study, we can identify two alternative answers that better meet demand and result in a better inventory position. The first alternative solution is as follows: 1. For Dem102 1.1 Explode the P2 path 1.1.1 Need 56 (D(20/0.4 D 50)/0.9) units of Wafer D on day 2. 2. The results become 2.1 Dem101 is met 3 days early. 2.2 Dem102 is met 2 days late. 2.3 Remaining inventory is 0. 2.4 Wafer needs are 56 units of Wafer D. The following is the second alternative solution. 1. For the 30 units of Dev02 in inventory 1.1 Allocate 20 units to Dem102. 1.2 Allocate the remaining 10 units of Dev02 to Dem101. 1.3 Explode the P1 path for the 10 units of unsatisfied Dem101. 1.3.1 Need 10 units of Wafer C on day.

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2. The results are 2.1 For Dem101 2.1.1 Units are met 3 days early (from 10 units of Dev02 in inventory). 2.1.2 10 units are met on time or potentially 1 day early. 2.2 Dem102 is met 2 days early. 2.3 remaining inventory is 0. 2.4 wafer needs are 10 units of Wafer C.

Appendix B: Detailed Formulation for the Binning LP Bellow is the LP formulation for simple binning and downgrade substitution to obtain the minimum production required for binned part at the next level.

LP Formulation for Simple Binning and Substitution Definition of Constants J K T

Set of all output parts, resulting when the binned part is produced Set of all the parts that have demand, obviously we have K J Number of time periods

Definition of Subscripts j k t

Output PN that results from the binning process, j 2 J PN that has demand, k 2 K Time period, t D 1; 2; : : : ; T

Definition of Coefficients Ij 0 Dkt Rjt Bjt

Inventory of output part j at the beginning of the planning horizon, i.e., t D 0. Demand for part k in period t. Receipts of output part j in period t. Binning percentage of output part j in period t.

Definition of Decision Variables Pt Production of the binned part in period t Sjkt Quantity of output part j that is used to satisfy the demand of output part k during period t. Ijt Inventory of output part j at the end of period t. Zkt Unsatisfied demand of part k during period t.

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Below is the LP formulation to solve the simple binning and substitution production situation. 2 " #3 X X X 4Pt C Minimize 0:00001Sjkt 5 .0:9/t 109 Zjt C 0:0001Ijt C t

j

k

Subject to Ijt D Ij.t 1/ C Bjt Pt C Rjt Dkt D Zkt C

X

X

Sjkt

k

Sjkt

j

Ijt >D 0; Pt >D 0; Zjt >D 0; Sjkt >D 0; 8 all j; t; k: The objective function computes the minimum production required for the binned part. A huge penalty is charged for every piece of unsatisfied demand; naturally, the LP will always try to meet all the demand on time. The first constraint calculates the inventory of the output part j at the end of period t: it is equal to the part’s inventory at the end of the previous period .t 1/ increased by the new stock from the binned part’s production (i.e., Bjt Pt ) and the part’s receipts in period t, then decreased by the total substitution quantity. The second constraint specifies the relationship between the demand and the unsatisfied demand for part k during period t. Finally, the last constraint requires all the decision variables to be nonnegative. This LP formulation can be solved very quickly to optimality.

Appendix C: Detailed Formulation for the Complex LP (SCOPE) LP Formulation for Complex Product Structures. Below is a detailed description of the basic LP model adopted for SCOPE.

Definition of Subscripts j m n z e a v

Time period/bucket Part number. Part being substituted. Group (representing a family of parts). Process (can be a manufacturing or purchase process). Plant location within the enterprise. Receiving plant location.

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Customer location (note that a customer location can never be a plant location). Demand class (indicates relative demand priority). Resource capacity (a resource can be a machine, worker, etc.). Consuming location(s) (which can be a plant within the enterprise or an external demand location).

Definition of Objective Function Coefficients PRCmaej SUBCamnj TC mavj INVC maj DMAXCauzj

DMINCauzj

BOC mkqj

Cost of releasing one piece of part m during period j at plant a using process e. Substitution cost per piece of part n which is being substituted by part m during period j at plant a. Transportation cost per piece of part m leaving plant a for plant v during period j . Inventory cost of holding one piece of part m at plant a at the end of period j . Cost per piece of exceeding the maximum amount of shipments specified for parts in group z from plant a to consuming location u during period j Cost per piece of falling short of the minimum amount of shipments specified for parts in group z from plant a to consuming location u during period j . Backorder cost of one piece of part m at the end of period j for demand class q at customer location k.

Definition of Constants DEMANDmkqj RECEIPT maj CAPACITY waj CAPREQwmaej QTYPERmaenj YIELDmaej SUBQTY amnj

Demand requested during period j for part m at customer location k for demand class q. Quantity of projected WIP and purchase order receipts for part m expected to be received at plant a during period j . Capacity of resource w available at plant a during period j to support manufacturing starts. Capacity of resource w required for part m at plant a for process e during period j needed for manufacturing starts. Quantity of component part m needed per part n during period j at plant a using process e. Output of part m per piece released (or started) at plant a during period j using process e. Quantity of part m required to substitute for one piece of part n at plant a during period j .

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MAXPCT auzj

383

Maximum percentage of total shipments of group z (a collection of parts) leaving supply plant a during period j to support consumption at location(s) u. Minimum percentage of total shipments of group z leaving supply plant a during period j to support consumption at location(s) u. Cycle time (which is the number of periods between the release and completion of parts) for releases of part m using process e at plant a during period j . Transport time for part m from plant a to plant v.

MINPCT auzj CT maej

TT mav

Definition of Decision Variables Inventory at the end of period j for part m at plant a Manufacturing starts of part m during period j at plant a using process e. Quantity of part n which is being substituted by part m during period j at plant a. Internal shipments of part m leaving plant a for plant v during period j . Shipments of part m leaving plant a during period j to satisfy class q demand at customer location k. Backorders of part m at the end of period j for class q demand at customer location k. Total shipments of group z leaving supply locations during period j to support consumption at location(s) u. Amount by which total shipments of parts in group z from plant a to consumption location(s) u during period j exceeds the maximum amount specified as desired in the sourcing rules. Amount by which total shipments of parts in group z from plant a to consumption location(s) u during period j falls short of the minimum amount specified as desired in the sourcing rules.

Imaj Pmaej Lamnj Tmavj Fmakqj Bmkqj Huzj Sauzj Gauzj

LP Model Formulation Minimize XXXX m

C

a

e

C

j

XXXX m

C

PRCmaej Pmaej C

a

v

XXXX a

TCmavj Tmavj C

a

u

z

j

m

k

q

j

XXXX

n

DMAXCauzj Sauzj C

a

INVCmaj Imaj

j

XXXX a

BOCmkqj Bmkqj

SUBCamnj Lamnj

j

XXX m

j

XXXX

m

u

z

j

DMINCauzj Gauzj

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Subject to (material balance constraints) X

X

x3xCCT maex Dj

e

Imaj D Ima.j 1/ C RECEIPT maj C C

X n

X

X

X

x3xCTT mva Dj

v

Lanmj C Tmavj

XX

v

Fmakqj

q

k

Tmvax

X

YIELDmaex Pmaex

SUBQTY amnj Lamnj

n

X

X

n3m is component of n

e

QTYPERmaenj Pnaej

(backorder conservation constraints) Bmkqj D Bmkq.j 1/ C DEMANDmkqj

X

Fmakqj

a

(capacity constraints) XX m

CAPREQwmaej Pmaej CAPACITY waj

e

(sourcing constraints) Huzj D

XX m2z a

X

Tmauj C

X

m2z

q

X

X

Tmauj C

m2z

Tmauj C !

X

! Fmauqj

q

Fmauqj Sauzj MAXPCT auzj Huzj ! Fmauqj C Gauzj MINPCT auzj Huzj

q

(nonnegativity constraints) all decision variables Xi;j;::: 0: The objective function measures a production plan’s total cost (or penalty) over the planning horizon chosen by the planner. The total cost is contributed by activities of manufacturing (Pmaej ), material substitutions (Lamnj ), interplant logistics (Tmavj ), inventory holding (Imaj ), sourcing (Sauzj and Gauzj ), and backorders (Bmkqj ) occurred in all the time buckets. It is a common practice to formulate SCM models based on a number of time buckets with variable durations or lengths. Then, these buckets

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collectively form the planning horizon for the production plan. Different combinations of time buckets are usually used for different planning purposes. For example, planning horizons for an enterprise strategic SCM plan may span 2 years and consist of 28 daily buckets, 21 four-day buckets, 10 weekly buckets, 6 monthly buckets, and 1 yearly bucket. Because of this modeling practice, part of the LP solution process is to aggregate appropriate demands and other items for every time bucket so that the correct quantities are used for model generation. Among the five types of constraints, material balance constraints ensure the inventory balance of material flows at every stocking point in the BOM supply chain as well as through time buckets in the planning horizon. In simple words, they make sure that what goes in equals what comes out. Next, backorder conservation constraints calculate the backorder amount that is needed at the end of every time bucket j , which is the balance between the total shipment made in that bucket and the demand for the same bucket plus the backorder carried forward from bucket j 1. Any (positive) backorders are accumulated over time, and they disappear only when sufficient shipments can be made to cover them and any new demand. Next, capacity constraints safeguard the usage of any resource capacity so that none goes beyond what is available. Obviously, if capacity has an unlimited supply, there will be no need to backorder because manufacturing starts can just be made to satisfy any demand. The sourcing constraints adopted in our model are “soft” in the sense that the ideal sourcing levels MAXPCT auzj and MINPCT auzj can be violated by paying a price, with Sauzj and Gauzj being slack variables to capture the quantity of sourcing overage and underage and then be penalized in the objective function. Finally, nonnegativity constraints ensure that all the decision variables in the model remain either positive or zero.

References Arntzen B, Brown G, Harrison T, Trafton L (1995) Global supply chain management at digital equipment corporation. Interfaces 25(1):69–93 Bermon S, Hood S (1999) Capacity optimization planning system (CAPS). Interfaces 29(5):31–50 Burda R, Degbotse A, Dews B, Milne RJ, Sullivan G (2007) Who would have thought – optimization in fabricator dispatch and artificial intelligence in enterprise wide planning. In: (Working paper) IBM Strategic Systems, 1000 River Road, Essex Junction, VT 05452, USA Dangat GS, Gokhale AR, Li S, Milne RJ, Orzell RA, Reid RL, Tang X, Yen C (1999) Best can do matching of assets with demand in microelectronics manufacturing. U.S. Patent 5,971,585, 26 Oct 1999 Davenport T (2006) Competing on analytics. Harvard Business Review, January 2006, pp 1–9 Denton B, Milne RJ (2006) Method for optimizing material substitutions within a supply chain. U.S. Patent 6,983,190, 3 Jan 2006 Denton B, Hedge S, Orzell RA (2004) Method of calculating low level codes for considering capacities. U.S. Patent 6,584,370, 18 May 2004 Denton B, Forrest J, Milne RJ (2005) A method for considering hierarchical preemptive demand priorities in a supply chain optimization model. U.S. Patent Application: 2005–0171828, also, IBM docket: BUR9–2003–0198US1

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Denton B, Forrest J, Milne RJ (2006) Methods for solving a mixed integer program for semiconductor supply chain optimization at IBM. Interfaces 36(5):386–399 Duchessi P (1987) The conceptual design for a knowledge based system as applied to the production planning process. In: Silverman B (ed) Expert systems for business, pp 163–194 Fogarty D, Hoffman T (1983) Production and inventory management. South-West Publishing, Cincinnati, OH Fordyce K (1998) Matching assets with demand engines for PROFIT and supply chain management. MicroNews (a publication of the IBM Microelectronics Division, 3rd Quarter, 1998) 4(3). www.chips.ibm.com/micronews/vol4 no3/profit.html Fordyce K (2001) New supply chain management applications provide better customer service: serious gets exciting. MicroNews (a publication of the IBM Microelectronics Division, 2nd Quarter, 2001) 6(3). http://www.chips.ibm.com/micronews/vol7 no2/fordyce.html Forrester J (1961) Industrial dynamics. M.I.T. Press, Cambridge, MA Galbraith J (1973) Designing complex organizations. Addison-Wesley, Reading, MA Glover F, Jones G, Karney D, Klingman D, Mote J (1979) An integrated production, distribution, and inventory planning system. Interfaces 9(5):21–35 Goldman S (2004) Science in the twentieth century. Great Courses on CD by the Teaching Company, Chantilly, VA Graves RJ, Konopka JM, Milne RJ (1995) Literature review of material flow control mechanisms. Prod Plan Contr 6(5):395–403 Hackman ST, Leachman RC (1989) A general framework for modeling production. Manag Sci 35(4):478–495 Hegde SR, Milne RJ, Orzell RA, Pati MC, Patil SP (2004) Decomposition system and method for solving a large-scale semiconductor production planning problem. United States Patent No. 6,701,201 B2 IBM white paper G510–6402–00 (2005) DIOS – dynamic inventory optimization, IBM Corporation, 1133 Westchester Avenue, White Plains, NY 10604, USA IBM white paper G299–0906–00 (2006) Collaboration with IBM E&TS Helps ADI Stay ahead of Customer Demand, IBM Corporation, 1133 Westchester Avenue, White Plains, NY 10604, USA Kempf K (1994) Intelligently scheduling wafer fabrication. In: Intelligent scheduling. Morgan Kaufmann, San Francisco, CA, pp 517–544 (Chapter 18) Kempf K (2004) Control-oriented approaches to supply chain management in semiconductor manufacturing. In: Proceedings of the 2004 American control conference, Boston, MA, pp 4563–4576 Leachman R, Benson R, Liu C, Raar D (1996) IMPReSS: an automated production planning and delivery-quotation system at Harris corporation – semiconductor sector. Interfaces 26(1):6–37 Lee HL, Padmanabhan V, Whang S (1997) Information distortion in a supply chain: the bullwhip effect. Manag Sci 43(4) (special issue on frontier research in manufacturing and logistics):546– 558 Lin G, Ettl M, Buckley S, Yao D, Naccarato B, Allan R, Kim K, Koenig L (2000) Extended enterprise supply chain management at IBM personal systems group and other divisions. Interfaces 30(1):7–25 Little J (1992) Tautologies, models and theories: can we find “laws” of manufacturing? IIE Trans 24(3):7–13 Lyon P, Milne RJ, Orzell R, Rice R (2001) Matching assets with demand in supply-chain management at IBM microelectronics. Interfaces 31(1):108–124 Milne RJ, Orzell RA, Yen C (1999) Advanced material requirements planning in microelectronics manufacturing. U.S. Patent 5,943,484, 28 Aug 1999 Norden P (1993) Quantitative techniques in strategic alignment. IBM Syst J 32(1):180–197 Orlicky J (1975) Material requirements planning: the new way of life in production and inventory management. McGraw-Hill, New York Orzell R, Patil S, Wang C (2004) Method for identifying product assets in a supply chain used to satisfy multiple customer demands. U.S. Patent 20050177465A1, 17 Oct 2004

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Promoting O.R.: The Science of Better (2005) Matching assets to supply chain demand at IBM microelectronics. http://www.orchampions.org/prove/success stories/mascdi.htm Simon HA (1957) Administrative behavior, 2nd edn. The Free Press, New York Shobrys D (2003) “History of APS,” Supply Chain Consultants (www.supplychain.com), 460 Fairmont Drive, Wilmington, DE 19808, USA Shobrys D, Fraser J (2003) Planning for the next generation (supply chain planning). Manuf Eng 82(6):10–13 Singh H (2007) Personal communication with Ken Fordyce supply chain consultants (www.supplychain.com), 460 Fairmont Drive, Wilmington, DE 19808, USA Sullivan G (1990) IBM Burlington’s logistics management system (LMS). Interfaces 20(1):43–61 Sullivan G (2005) PROFIT: decision technology for supply chain management at IBM microelectronics division. In: Applications of supply chain management and E-commerce research. Springer, New York, pp 411–452 Sullivan G (2007) Evaluating planning engines in 1994, working paper, chapter in memoirs Sullivan G, Jantzen J, Morreale M (1991) Using Boolean matrices or integer vectors to analyze networks. In: APL91 Proceedings editor Jan Engel, APL Quote Quad, vol 21(4), pp 174–185 Swaminathan J, Smith S (1998) Modeling supply chain dynamics: a multi agent approach. Decis Sci 29(3):607–632 Tayur S, Ganeshan R, Magazine M (1998) Quantitative models for supply chain management. Kluwer Academic, Boston, MA Uzsoy R, Lee C, Martin-Vega LA (1992) A review of production planning and scheduling modules in the semiconductor industry, Part 1: System characteristics, performance evaluation, and production planning. IIE Trans, Scheduling Logistics 24(4):47–60 Uzsoy R, Lee C, Martin-Vega LA (1994) A review of production planning and scheduling modules in the semiconductor industry, Part 2: Shop floor control. IIE Trans, Scheduling Logistics 26(5):44–55 Wolfson R (2000) Einstein’s relativity and the quantum revolution. Great Courses on CD by the Teaching Company, Chantilly, VA Woolsey G (1979) Ten ways to go down with your MRP. Interfaces 9(5):77–80 Zisgen H (2005) EPOS – stochastic capacity planning for wafer fabrication with continuous fluid models. IBM Global Engineering Services, Decision Technology Group Mainz, Germany

Chapter 15

Production Planning in the Plastics Industry Rajesh Tyagi and Srinivas Bollapragada

15.1 Introduction In a highly competitive plastics industry, manufacturing good quality products is not sufficient to maintain a competitive global position. A plastic company must have the ability to deliver customized products to its customers’ doorsteps anywhere in the world – in the right quantity, at the right time, and at reasonable cost. In effect, the supply chain has become part of the product offering. With increasing supply chain complexity, this requirement puts an unprecedented burden on decision systems that allocate global capacities. Major manufacturers supply a variety of plastics to many different industries, including automotive, appliance, computer, and medical equipments. To meet the customer demand, the companies maintain manufacturing plants all over the globe. Until recently, most producers practiced a regional manufacturing philosophy wherein each product was made in the region (e.g., America, Europe, or Pacific) where it was ordered. In recent years, however, as consumer products manufacturing has shifted to the Pacific, the demand for plastics used in those products has also shifted to that region. Even as plastics companies rush to add new manufacturing capacity in the Pacific, most of them still face a geographic imbalance between capacity and demand with excess capacity in the America and insufficient capacity in the Pacific. Consequently, many have been forced to abandon the regional approach in favor of a global approach to their manufacturing operations. Although a better balance between capacity and demand remains the primary motivation, a global supply chain offers additional opportunities for reducing manufacturing costs, including: (1) economies of scale from centralizing production, (2) lower raw material costs, as these can now be sourced globally, and (3) ability to take advantage of tax breaks offered by certain countries to set up and operate plants even if that region has sufficient capacity. The remainder of this chapter is organized as follows. We begin with a brief overview of the plastics production process in the next section, which will be followed by a section on demand planning. We then describe the manufacturing supply R. Tyagi () General Electric Global Research, Niskayuna, NY 12309, USA e-mail: [email protected] K.G. Kempf et al. (eds.), Planning Production and Inventories in the Extended Enterprise, International Series in Operations Research & Management Science 152, c Springer Science+Business Media, LLC 2011 DOI 10.1007/978-1-4419-8191-2 15,

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chain and show how it can be modeled as a mathematical program. After discussing implementation issues and benefits of the optimization model, we conclude with some final remarks.

15.2 Production Process The plastics resin industry is the sixteenth largest industrial sector in USA. The annual demand exceeds 100 billion pounds valued at more than $400 billion. The major producers in USA include The Dow Chemical Company, DuPont, and SABIC Innovative Plastics (formerly GE Advanced Materials). This industry makes different types of plastics to meet a variety of needs. For example, the automobile industry needs dent-resistant body panels; cooking-ware must be extremely heat resistant; aircraft building material should be very light; adhesives must quickly change their form from viscous to solid; and mouth guards must be malleable when heated but then retain their shape. Whatever the final application, the end product is made using a two-stage production process. The primary raw materials of plastics are crude oil and natural gas, referred to as feedstock. A set of feedstock is first converted into resins. By using different sets of feedstock, a number of different resins are produced. Each resin has its own unique characteristics such as hardness, heat resistance, and corrosion. In the second stage, resins are combined to produce the finished product. By combining different resins in different quantities with suitable chemical additives, the finished product can be made to meet customer specifications of its characteristics. This end product is then shipped to the industrial consumers who then convert the finished plastics into a form they can use in their products. Table 15.1 shows the typical bill of materials. There are certain exceptions to this process. On the one hand, some customers will buy only resins and then finish them at their own facilities. On the other hand, plastics manufacturers will sometimes outsource production of certain resins for economic reasons. Some large customers will often insist that the plastics manufacturer transport the resins to the customer site and set up the finishing process there itself; this is done primarily so that the customer has a better control over the finishing process.

Table 15.1 Bill of materials: each plastic is made up of a number of resins, which are combined with certain additives Resins Additives Plastic Plastic 1 Plastic 2 ... ... Plastic p

Resin 1 0.55 0 ... ... 0.4

Resin 2 0 0.6 ... ... 0.4

... ... ... ... ...

Resin r 0.4 0.3 ... ... 0

Additive 1 0 0.05 ... ... 0.2

Additive 2 0 0.05 ... ... 0

... ... ... ... ...

Additive a 0.05 0 ... ... 0

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15.3 Demand Planning The sales/marketing division handles all customer orders. It manages a global network of distribution centers and supporting warehouses to serve customers worldwide. This division is responsible for estimating demand by each region for a few years in advance. The demand is estimated based on a number of factors including conversations between account executives and the customers, industrial trends, and discussions at professional conferences. In addition to estimating demand, this division estimates the selling prices of the end products. Some of the products can be thought of as commodity items for which the market price is well known. However, other products are highly specialized and their future prices must be estimated considering evolving technologies and the competitive landscape. Table 15.2 shows demand and price forecast that the marketing division must make. It is difficult to accurately forecast demand, cost and price data over a long period of time. The forecasts become increasingly unreliable further into the planning period as technological advances and potentially new competitors come into play. Therefore, the sales/marketing division develops a number of potential scenarios and estimates the demand and price data under each of those scenarios. The estimates are deterministic under each scenario. Plastics are normally sold as both made to stock and made to order, and thousands of different grades are produced every year that differ primarily in blend requirements and color. For the purpose of capacity planning, these products are aggregated into product categories based on similarities in manufacturing requirements. During the year, the sales/marketing division aggregates individual customer orders into production orders, and forwards them to manufacturing for production. The manufacturing plants are designed for different capacities, and as a matter of policy, each line is earmarked to produce small, medium, or large lots only. Consequently, the demand for the product categories is broken down by lot sizes as well. Typically, lot sizes vary from few hundreds of pounds to tens of thousands of pounds. Lot sizes of less than 4,000 pounds are considered small, while those above 8,000 pounds are considered large. Small lots are common in specialty plastics manufacturing, whereas large lots are more prevalent in commodity plastics. Once the manufacturing division receives forecasted demand, it must determine how and where to produce to meet it. In the event that it does not possess sufficient manufacturing capacity to meet all forecasted demand, the manufacturing division

Table 15.2 The marketing division is responsible for forecasting the yearly demand and price of the end products. The forecast is done for up to 4 years into the future Demand Price Product Plastic 1 Plastic 2 ... Plastic p

Americas 1,340 4,000 375 623

Europe 330 500 1,000 215

Pacific 203 1,000 258 67

Americas 5,380 2,980 3,654 6,000

Europe 4,056 2,980 3,432 6,000

Pacific 4,801 2,980 3,600 6,000

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must also determine the quantities of each product whose production will yield the maximum profits. Once manufactured, the products are delivered to the appropriate warehouses in the geographic regions.

15.4 Supply Chain Regardless of the characteristics of their end products, most plastics production requires similar manufacturing supply chains consisting of two levels of manufacturing plants (see Fig. 15.1). The first-level plants, called resin plants, convert feedstock into resins, which are then shipped to the second-level plants called finishing plants. At finishing plants, these resins are combined with additives to produce different grades of the end product, which are then shipped to customers. Each plant is generally capable of making a number of the products; however, the production rates will often vary from one product to another. The production capacity must, therefore, be defined in terms of the products the plant can produce. For example, a finishing plant may produce 4 million pounds of product A alone or 6 million pounds of product B alone. Of course, we could divide the manufacturing time between the two products and produce quantities in the corresponding ratios; for example, dividing manufacturing time in the ratio of 40% for product A and 60% for product B will yield 1.6 million pounds (40% of 4 million) of product A

Plastics Manufacturing Feedstocks

Resin Plants

Feedstock 1

Resin Plant 1

Feedstock 2

Resin Plant 2

…

…

…

…

Finishing Plants Finishing Plant 1

Finishing Plant 2

...

... …

…

…

…

…

…

…

…

… …

Feedstock f

Resin Plant m

Market Demand Americas Plastic 1 Plastic 2 … … … … … Plastic p

Europe Plastic 1 Plastic 2 … … … … … Plastic p

Pacific … Finishing Plant n

Plastic 1 Plastic 2 … … … … … Plastic p

Fig. 15.1 Plastics manufacturing supply chain: resins produced at resin plants are combined with additives at the finishing plants to produce the end products

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and 3.6 million pounds (60% of 6 million) of product B. For each plant, our model determines the fraction of time to be allocated to each of the products that it can produce. Table 15.3 shows an example of manufacturing capacity for resin and finishing plants. Table 15.4 shows the conversion costs for both types of plants; the conversion costs include labor, utility, and other plant operating costs. The last column in this table shows plant yield, the amount of inputs needed to produce one unit of output. Another important cost consideration in manufacturing planning is shipping costs from resin plants to finishing plants and from finishing plants to the warehouses in the regions for onward shipments to the customers, as shown in Table 15.5. Table 15.3 The capacity of the manufacturing plants is defined in terms of the quantity (in ‘000 lbs) that the plant can produce of each product Resin 1 Resin 2 ... Resin n Resin plant 1 0 0 ... 8,500 Resin plant 2 13,000 0 ... 8,000 Resin plant 3 14,000 0 ... 7,500 Resin plant 4 13,000 20,000 ... 0 ... ... ... ... ... Resin plant m 0 18,000 ... 0 Finishing plant 1 Finishing plant 2 Finishing plant 3 Finishing plant 4 ... Finishing plant f

Plastic 1 15,000 16,000 17,000 0 ... 0

Plastic 2 0 0 0 32,000 ... 30,000

... ... ... ... ... ... ...

Plastic p 6,000 7,500 8,000 ... 0

Table 15.4 The conversion costs for both types of plants include labor, utility, and other plant operating costs per 000 lbs of output production. The last column indicates the plant yield, the amount of inputs needed to produce one unit of output Resin 1 Resin 2 . . . Resin n Yield Resin plant 1 0 . . . 1,400 1.10 Resin plant 2 1,200 0 . . . 1,000 1.05 Resin plant 3 1,400 0 . . . 1,200 1.10 Resin plant 4 1,100 800 ... 0 1.20 ... ... ... ... ... ... Resin plant m 0 900 ... 0 1.08 Plastic 1 Plastic 2 Finishing plant 1 1,400 0 Finishing plant 2 1,000 0 Finishing plant 3 1,200 0 Finishing plant 4 0 1,200 ... ... ... Finishing plant f 0 1,000

. . . Plastic p ... . . . 700 . . . 1,000 . . . 1,200 ... ... ... 0

Yield 1.08 1.05 1.10 1.20 ... 1.06

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R. Tyagi and S. Bollapragada Table 15.5 The shipping costs (per ‘000 lbs) must include any duties if the source and destination are in different count Finishing Finishing Finishing plant 1 plant 2 ... plant f Resin plant 1 1,000 600 ... 750 Resin plant 2 800 1,000 ... 1,100 Resin plant 3 800 800 ... 800 Resin plant 4 1,200 800 ... 1,200 ... ... ... ... ... Resin plant m 600 1,200 ... 900 Finishing plant 1 Finishing plant 2 Finishing plant 3 Finishing plant 4 ... Finishing plant f

America 1,000 800 800 1,200 ... 600

Europe 600 1,000 800 800 ... 1,200

Pacific 750 1,100 800 1,200 ... 900

Use of sophisticated models in global capacity planning is relatively uncommon. Most manufacturers use a regional manufacturing system to simplify analysis requirements. Any capacity analysis is limited to simple spreadsheet models focusing on marginal changes to the existing plans.

15.5 Problem Statement Each year, the manufacturing division must develop a multiyear production plan. While the corporate goal is to meet all demand, insufficient production capacity may force the manufacturer to settle for product rationalization. In such instances, it must determine the subset of demands it must produce so that the resulting contribution margins are highest. The goal of the production-planning model is to maximize the total contribution margin (i.e., revenues from material produced less total cost of manufacturing and shipping), subject to market demand and price of plastics, plant capacities, production rates, conversion costs, additive costs, and distribution costs. Since the demands and prices vary under different scenarios, the problem is solved for each scenario separately. The model is used for aggregate production planning in the short term (about 1 year) and for capacity planning in the long term (2–5 years). Short-term demand and price estimates do not vary by scenario but can vary significantly in the long term. Management uses the output from different scenarios in making capacity decisions.

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We state the production-planning problem as follows: Maximize Contribution Margin over the planning horizon D Revenues from Plastics Products (Manufacturing Costs C Additive Costs C Distribution Costs) Subject to: Demand constraints. Manufacturing capacity constraints. Material flow constraints.

The decision variables are: (1) the amount of each resin produced at each resin plant intended for use at each finishing plant; and (2) the amount of each plastic product produced at each finishing plant intended to be shipped to each pole. The objective function to be maximized is the total revenue from all plastics produced less the manufacturing, additive, and transportation costs over the entire planning horizon. The demand constraints should be applied to each product in each region to ensure that production does not exceed the demand estimates. An ideal solution would be one where all demands are met and the contribution margin is also maximized. However, capacity limitations and various production and transportation costs may result in some unmet demand while maximizing contribution margin. Manufacturing capacity constraints enforce production capacity, production rates, and yield factors. Recall that for the same plant, production rates vary by product and that the plant’s production capacity is described in terms of the total quantity of each product it can produce if it were to produce that product alone. We therefore determine the quantity of each product produced as the fraction of production capacity devoted to that product multiplied by the production rate for that product. These constraints apply to all resin and finishing plants. Material flow constraints model the flow of material from resin plants to finishing plants. To properly account for transportation cost, we must keep track of quantities shipped from each resin plant to finishing plant and from each finishing plant to each market. We now present a detailed statement of the problem formulation.

15.5.1 Model Formulation Indices: p D Plastics product r D Resin c D Additives m D Market

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f D Finishing plant k D Resin plant t D Planning period (month or quarter for short term production planning; year for longer term capacity planning) M D Arbitrary large number

Decision Variables: xpfmt D Amount (pounds) of product p produced at finishing plant f for market m in year t. yrkft D Amount (pounds) of resin r produced at resin plant k for finishing plant f in year t. zcft D Amount (pounds) of additive c needed at finishing plant f in year t.

Parameters: Dpmt D Demand for product p in market m in year t. Npmt D Selling price of product p in market m in year t. Rpr D Usage of Resin r per pound of product p. Cpc D Usage of additive c per pound of product p. Acft D Cost of additive c per pound at finishing plant f in year t. Sfmt D Transfer C Duty cost from finishing plant f to market m in year t. Tkft D Transfer C Duty cost from Resin plant k to finishing plant f in year t. Ufpt D Variable cost of producing product p at plant f in year t. Vkrt D Variable cost of producing resin r at plant k in year t. Pfpt D Amount of product p that can be produced at finishing plant f in year t. Qkrt D Amount of resin r that can be produced at Resin plant k in year t.

Constraints: 1. Product Demand: For each plastic product, we determine the total production of that plastic by summing up its production at all finishing plants. We then ensure that this total production quantity (lbs) does not exceed the demand forecast for that product. This constraint is applied to the demand for the product in each region in each year of the planning horizon. X

xpfmt Dpmt

for all p; m; t

(15.1)

f

2. Resin Requirements: We apply the bill of materials to the finished plastic to determine the total requirement for each resin at each finishing plant. Since this resin can be produced at many resin plants, we specify that the total production

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quantity of that resin at all resin plants must meet the total requirement for that resin at that finishing plant. We apply this constraint to the requirement for each resin at each finishing plant in each planning year. X

yrkft

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for all r; f; t

(15.2)

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3. Additive Requirements: Additive requirements are calculated in a manner similar to resin requirements. However, unlike resins, the additives are bought from vendors who deliver them to the finishing plants. Therefore, we simply determine the quantity of each additive needed at each finishing plant each year. zcf t

XX p

Cpc xpfmt

for all c; f; t

(15.3)

m

4. Finishing Production Capacity Constraints: Production constraints at finishing plants are a little tricky to implement. Recall that the production rate of a plant depends on the product being produced. Thus, for each plastic produced, we determine the fraction of the plant time that is allocated to a product. We then make sure that the sum of these capacity fractions used by all products produced at the plant does not exceed the plant’s capacity [i.e., 1]. The constraint is applied to each finishing plant in each year. X

1 X xpfmt 1 for all f; t Ppft m

p2fpWPpft ¤0g

(15.4)

5. Resin Production Capacity Constraints: Production capacity constraints at resin plants are implemented in a manner similar to those at the finishing plants. X r2frWQrkt ¤0g

1 X yrkft 1 for all k; t Qrkt

(15.5)

f

Revenue and Cost Calculations: 6. Revenues: Revenue from a particular plastic in a particular market in a particular year is calculated by multiplying the production of that plastic intended for that market in that year by its estimated selling price in that market in that year. The total revenue can be determined by summing this revenue over all plastics, markets, and planning years. XXX t

p

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xpfmt

(15.6)

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7. Finishing Plants Production Costs: The production cost at a finishing plant is computed by multiplying the total production of each plastic by the corresponding production cost. The production costs over the planning horizon are estimated as follows: the finance department knows current costs; future costs are estimated by multiplying the current costs by an average annual multiplier to reflect cost changes into the future. The total production cost at the finishing plants is determined by adding up this cost over all plastics produced in all plants in all planning years. XX X t

Ufpt

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! xpfmt

(15.7)

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8. Resin Plants Production Costs: The total production cost at resin plants is determined in a similar fashion to that for the finishing plants. 0 1 X XX X @ Vkrt yrkft A t

r

k

(15.8)

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9. Additives Costs: The additive costs are computed similarly by summing up the cost for each additive required at each finishing plant in each planning year. XXX t

c

Acft zcft

(15.9)

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10. Transfer Cost for Plastics: The cost of transferring finished plastic products from a finished plant to a warehouse in a market is calculated by multiplying the total shipment from the plant to that market by the applicable shipping rate from that plant to the market. By adding this cost for all finishing plants and all markets in all years, we get the total transfer cost for plastics. Shipping costs are estimated in a manner similar to production costs: future costs are estimated by multiplying the known current costs by an average annual multiplier to reflect cost changes into the future. XXX t

f

Sfmt

m

X

! xpfmt

(15.10)

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11. Transfer Cost for Resins: The cost of transferring resins from resin plants to finishing plants can be similarly calculated as XXX t

k

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X r

! yrkft

(15.11)

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Objective Function: Maximize Total Contribution Margin D Revenues (Production Costs C Additives Cost C Transfer Costs) 0 1 2 ! XXX XX X X X @ A 4 Npmt D xpf mt Ufpt xpfmt t

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! xpfmt C

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XXX t

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!3 yrkft 5 (15.12)

r

The model size will depend on the number of resin and finishing plants and the number of product grades. For example, consider a manufacturing system with 8 resin plants that make 3 resins, and 21 finishing plants that combine these resins to make 24 grades of plastics. The mathematical model for this system for a single period would have about 3,000 variables and 1,000 constraints. Of course, the model is completely linear. As such, the solution time for such a model should be within a few seconds to a few minutes depending on the solver and the computer.

15.6 Implementation and Benefits We implemented the production-planning model at a large US plastics company with global customers and supply chain. We implemented the decision support systems in two stages. In the first stage, we built a prototype and put it to use for one decision cycle. Based on the user feedback, we revised the prototype and put it through a few more decision cycles for further validation. Once the prototype functionality was deemed satisfactory, it served as functional specifications for building and implementing a robust production system. The first set of decisions in building the prototype dealt with selecting appropriate system platforms. The platform should allow for rapid prototyping, often at the expense of execution speed and error checking. For our prototype, we selected Excel as the user interface platform since it was widely used in the company and thus would find easier acceptance among end users. For managing data, a relational database such as Access would normally be the preferred choice; however, we realized that populating data directly into a database is often difficult for the end user, as the data must first be converted into a tabular form appropriate for a relational database. So we decided to keep the data in Excel worksheets. Next, we needed to select a math-programming solver. The primary considerations for the solver

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include licensing cost, execution time, modeling environment, and integration with user interface (Excel in our case). Free open source solvers are also available. In our case, we had a tight deadline and licensing cost was not an important consideration. So we chose LINGO since we were already very familiar with it. LINGO has some useful features: it integrates well with Excel and provides a built-in modeling language. Finally, we chose Excel VBA as the procedural language to implement the GUI, and to integrate it with data and the solver. Now that the system platform had been determined, we set about building the prototype. It took us 3 months to build the initial tool and demonstrate it using dummy data to the global manufacturing leader. It was very well received, and we proceeded with the challenging task of collecting and populating real data. While some data already existed in other systems, others had to be estimated or created specifically for the DSS. The first task was to create product families. Our client sold thousands of different types of plastics. Working with the users, we aggregated products into families on the basis of their resin composition alone, ignoring the finishing process (color etc.), and ended up with 24 product families. Plant manufacturing capacity is an interesting example of something that we had to estimate. Recall that this capacity depends on the product being produced – the same plant will produce different quantities if it were to produce two different products all year. Of course, a plant makes many different products in a year, and since it makes a specific product for only a few weeks in a year, we only knew the weekly capacity for each product. This, however, cannot be directly scaled to annual capacity since numerous cleaning and setup operations would be incurred during normal operations, as the plant would switch production among different products. So we asked the plant managers to estimate the annual capacity by scaling up the weekly production rate and factoring in the setups and maintenance operations they would typically encounter in a year. We then analyzed historical production data at the plant for the previous year to independently compute capacities for verification. Any discrepancies were resolved through discussions with plant managers. Other cost data such as transportation and manufacturing costs were similarly acquired by talking to appropriate personnel. We then populated all the data into the Excel tool. The DSS was now ready for use by the global manufacturing leader of the division. Its first use was to determine how to integrate a newly acquired company that made similar products. The challenge was to decide how to realign production in a way that best utilized the combined production facilities of the two companies. The DSS turned out to be a great vehicle for performing this analysis. The analysis led to a new manufacturing plan and operating budget, which were well received by the senior leadership team. After the first use, some minor changes were made to the DSS to incorporate user feedback and it went through another decision cycle. The leadership team also decided that other divisions should start using similar analysis for their capacity planning needs. Therefore, the company asked its information technology group to institutionalize the tool across the company. With the help from the two outside vendors, a web-based production system was implemented, which fully integrated with the corporate databases so that most of the data needed by the tool is current

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and is automatically obtained from the source systems. During this development, the prototype tool continued to be used for two purposes: (1) long-term (4-year planning horizon) capacity planning, which was the original intent, and (2) shortterm (1 year) aggregate production planning – the manufacturing leader realized that by using monthly data the DSS tool could be used as a production planning tool. Aside from the primary uses of the tool described above, many additional benefits listed below were realized. Return on plant and equipment (P&E) investment: We could estimate the benefit

of adding a potential new plant by including it in the supply chain and analyzing the effect on the total contribution margin and the return on investment. Maximization of total contribution margin: This is the key financial metric for most plastics companies. Facilitation of raw material procurement and manpower planning: The multiyear plan provides visibility to long-term production requirements. Manpower requirements can now be projected for better planning. Similarly, sourcing has a much longer lead-time to acquire materials. Product rationalization: When demand exceeds capacity, the model determines which products contribute the most and thus should be produced.

15.7 Conclusions Over the last decade, the plastics industry has witnessed a major change in how it fulfills its customer demand. As the global manufacturing base has shifted to the pacific, so has the demand for plastics, a common component in many products. Even as new manufacturing facilities are being setup closer to the demand in the pacific, the industry has moved to a global supply chain to fulfill customer needs. This has resulted in increased transportation costs and import duties, and companies must rethink how they allocate the manufacturing capacity to meet specific product needs of their customers more cost effectively. In this chapter, we have presented a production-planning model for the plastics industry that allocates plant capacities to various plastics products or grades that must be produced. Specifically, the model determines what to produce where in order to maximize total contribution margin, an important financial metric for the industry. We described a successful implementation of the model at a large US plastics manufacturer and listed various benefits that were achieved. Of course, aggregate production planning is only one-half of the complete production system. After the plan is developed, detailed production schedules (usually weekly) must be developed to meet individual customer orders. Many considerations go into creating production schedules: the product ordered, order quantity, and due date; weekly production capacity; cleaning and setup times; and inventory policies. While production-scheduling systems are in use at many plastics companies, we have not encountered any significant effort to integrate production planning and production scheduling systems, which stands as an area for future research.

Chapter 16

Model Predictive Control in Semiconductor Supply Chain Operations Karl Kempf, Kirk Smith, Jay Schwartz, and Martin Braun

16.1 Introduction Maintaining agility in a multi-echelon multi-product multi-geography supply chain with long and variable manufacturing lead times, stochastic product yields, and uncertain demand is a difficult goal to achieve. The approach advocated here is based on a practical application of control theory that includes a model of the system being controlled, feedback from previous results, feed-forward based on demand forecasts, and optimization of both the financial results and the control actions applied to achieve them. This Model Predictive Control (MPC) approach has been employed in the continuous-flow process industry for many years, and has been independently suggested for supply chains by a number of academic research teams. This chapter describes a large-scale application of the approach in the semiconductor industry. Moving from raw silicon release to product delivery to customers, Intel Corporation’s supply chain combines a series of fan-out phases with a worldwide network topology. Each manufacturing facility outputs a wider variety of entities than are input, and each transportation operation ships to multiple inventory positions or customers. The raw silicon is converted into hundreds of different products that are shipped to thousands of different customers. MPC is demonstrated on an interfactory problem of managing the fan-out of products from one factory to multiple inventory positions, then on an intrafactory problem of managing the fan-out of products inside one factory. The improved agility compared to previous heuristic approaches is shown to provide Intel with hundreds of millions of dollars per year in savings.

K. Kempf () Decision Technologies Group, Intel Corporation, 5000 W. Chandler Blvd., Chandler, Arizona, USA 85226 e-mail: [email protected]


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16.2 The Basic Business Problem For 40 years, Intel Corporation has developed computer technologies. Today, it supplies chips, boards, and systems for mobile and desktop computers, servers, and networking and communication products. With annual revenues exceeding $43 billion, Intel is the world’s largest chip maker. Although historically known for its leadership in microprocessor architecture and manufacturing technology driven by Moore’s Law (Moore 1965, 1998), Intel today counts supply chain management as one of its core areas of expertise. The supply chain that Intel manages represents an extreme of “fan-out” topology. Starting with raw silicon wafers, it produces a few hundred different products for thousands of customers across the world. There are distinct phases in this process and “fan-out” is a characteristic of each. Phase 1: Silicon wafers are released into the fabrication process containing hundreds of manufacturing steps and requiring weeks of processing. Transistors are built up a layer at a time followed by interconnection with layers of metal to form circuits. A given fabrication factory can produce a number of basic products by altering the number and position of the transistors as well as the pattern of the interconnections. Each factory makes many basic products, each of which is made in multiple factories as a risk management technique. Current process technology can fabricate a microprocessor with several billion transistors in an area the size of a human thumbnail, and each wafer can hold hundreds of such “die”. Fan-out here refers to starting with blank wafers and producing die of many different basic products. Phase 2: Once fabrication is completed, each die is tested and sorted by performance level. This is driven by the stochastic nature of the manufacturing process that produces wafers with some die that are completely nonfunctional and a distribution of maximum operating speeds in the rest. Generally speaking, higher speed die consume more power, generate more heat, and are most suited to application in servers. Lower speed die consume less power, generate less heat, and are ideal for mobile applications. Die that fall in between are useful in desktop computers. Fanout here has two connotations; each basic product flow through fabrication is sorted into a number of performance categories (high, low, and medium speed), and each performance category of each basic product may be shipped to different assembly and test factories. Phase 3: Received from many fabrication/sort factories, die of different basic product and performance categories are initially inventoried until being released into the assembly process containing tens of manufacturing steps and requiring days of processing. Die are joined mechanically, electrically, and thermally to “substrates” that protect the die and allow it to be manipulated as it is being built into computing, networking, and communication systems. These substrates are what come to mind when ones visualizes a “chip”. Fan-out here is due to the fact that a die of any specific speed range of any particular basic product can be placed into a variety of different substrates depending on the target product and application. Note that there are many styles of server substrates, desktop substrates, and so on.

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Phase 4: Once assembly is complete, each die-substrate combination constitutes a semifinished good that is tested to determine its maximum speed. This is relatively fine testing (compared to that done after fabrication) since the product is nearing shipment to the customer. Fan-out here is because there are two- to fivefold more speed-substrate bins in semifinished goods inventory than there are speed bins in the die inventory in front of assembly/test factories. Phase 5: Semifinished goods are released into the finish/pack factory to be configured, marked, packed, and placed into finished goods inventory ready to be shipped to the geographic regions where Intel has customers. The configuration operation provides the opportunity to match current supply and current demand as well as possible. Intel fabricates and assembles die based on a market forecasts, but the manufacturing process is stochastic and results in a speed distribution for each semifinished product. Due to competitive market forces, customers can place orders and subsequently change them throughout the duration of the manufacturing process, making demand unpredictable. The intersection at a point in time of the uncertainty in manufacturing processes and uncertainty in market demand leads to three possible situations that must be managed in the finish/pack factory. (A) If the market orders a lower speed distribution than is available, then faster parts have to be configured to match the demand. This involves an opportunity cost since faster parts have a higher value in the market than slower parts. (B) In the best of cases, supply matches demand in both quantity and speed and no configuration is required. (C) If the market orders a higher speed distribution than is available, then more parts have to be manufactured. This is because higher speed parts can be configured to run more slowly, but lower speed parts cannot be configured to run faster. Manufacturing more material makes additional higher speed parts available to satisfy demand, but also creates additional lower speed parts that are not particularly valuable in this market condition. It is possible in the same market that different products require different configuration actions. The same is true for one product being sold into one market over time. Phase 6: Finished goods are released and transported into inventory positions across the geographies. The fan-out here is the shipment from each finished goods warehouse to many field warehouses. Phase 7: Finished goods are released from the field warehouse and transported to customers. The fan-out is the shipment from each field warehouses to various customers. One useful representation of these phases of supply chain operation is shown in Fig. 16.1. This representation takes advantage of the very high volume of chips that move through Intel’s supply chain. From this perspective, semiconductor manufacturing can be modeled as a continuous flow problem similar to that found in the process industry. The control nodes C1–C6 can be thought of as control valves releasing flow into factories and into transportation links: C1 controlling flow into fabrication/sort factories M1, C2 into transportation links T1, C3 into assembly/test factories M2, C4 into finish/pack factory M3, C5 into transportation links T2, and

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material release (C1) into Fabrication/Sort factory (M1) feeding Fab Die inventory (I1)

C1 M1 C2

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material release (C2) into Transportation (T1) feeding Assembly/Test Die inventory (I2)

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release (C3) into Assembly/Test factory (M2) feeding Semi-Finished inventory (I3)

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release (C4) into Finish/Pack factory (M3) feeding Finished inventory (I4 )

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release (C5) into Transportation (T2) feeding Geography inventory (I5)

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INVENTORY

TRANSPORT

DEMAND

D3

D2

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t

release (C6) feeding Geography Demand by period - D1 in t1 - D2 in t2 - D3 in t3

Fig. 16.1 Semiconductor manufacturing as a flow problem

C6 to satisfy customer demand. The pipes in this flow representation are factories M1, M2, and M3 modeled as very long leaky pipes to represent throughput time (TPT) and yield loss, respectively. Holding tanks model the inventory positions: I1 is the die inventory at the end of fabrication/sort, I2 is the die inventory in front of assembly/test, I3 is the semifinished goods inventory between assembly/test and finish/pack, I4 is the finished goods inventory, and I5 is the inventory in the field. Demand is represented by the moving conveyor belt at the bottom of Fig. 16.1. The goal of agile supply chain execution is to exercise control over the valves C1–C6 and the configuration process in M3 so that the materials in inventory positions I1–I5 are kept between appropriate levels to satisfy demand over time – not too high to incur unnecessary storage costs, not too low to risk costly stock outs. Constraints in the control system represent both the capacity and the rate at which control can be exercised based on the capabilities of the manufacturing and transportation components. The objective is to minimize the cost of missing the starts and inventory targets that have been passed into the controller from more strategic supply chain planning processes (Kempf 2004). The control system must, of course, accomplish this in the face of stochasticity in both the supply and the demand including the manufacturing delay between releasing raw materials at C1 and having material available at I5. Considering that $43 B/year in sales represents over $115 M/day worth of output from the system represented in Fig. 16.1, the difference between good and excellent control can easily be worth hundreds of millions of dollars per year to Intel.

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16.3 The Basic Solution Approach The MPC approach advocated here to provide variability management through inventory control includes both feedback from past performance and feed-forward of both supply and demand forecasts (Kothare and Morari 2000; Qin and Badgwell 2003). The major components of the system are depicted in Fig. 16.2. The Model provides a fast forward projection of the supply system’s dynamic performance using historical and forecasted TPTs, yields, and speed splits as well as historical and planned control actions. The Optimizer takes as input the error between the projected future supply and the forward looking reference trajectory of future demand. It supplies as output a set of future control inputs to the supply system that minimizes the error relative to future demand and the control effort based on the objective function, physical constraints of the system, and strategic constraints supplied from the strategic supply chain planning processes. The variables in this system that we can adjust are called the manipulated variables (MVs) and represent releases out of warehouses and into factories and transportation lanes. The variables in the system that we are trying to influence are called the controlled variables (CVs). They are all related to satisfying demand and can be factory outputs or inventory levels in warehouses. The first step in this algorithm is a state estimation step (see Fig. 16.3). On the one hand, there are measurements of the current state of the system (measured CVs Reference trajectory Past inputs and outputs

Predicted outputs −+

MODEL

Future inputs

Future error

OPTIMIZER

Fig. 16.2 Block diagram of Model Predictive Control

Objective function and constraints

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Fig. 16.3 Current state estimation

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as a result of previous MVs), but these measurements are known to contain noise. On the other hand, the model has predicted the state of the system at this point in time (modeled CVs based on previous MVs), but the model contains errors. This set of signals must be appropriately combined to estimate the true state of the system, and the process for doing this is one way of managing variability in the system. In the second step, control is frozen (MVs are held constant at current values) and the model predicts the future states of the system over time (CVs) including the entire range of dynamics that the model includes as shown in Fig. 16.4. The third step is depicted in Fig. 16.5. The setpoint trajectory is added representing the desired future trajectory of the system based on the demand forecast, inventory position, and factory load. Errors are measured at multiple time points into the future between the CVs predicted by the model and the computed setpoint trajectory. Figure 16.6 depicts the fourth step that is based on a number of inputs. One is the set of CV errors measured in the previous step in the context of hitting the setpoint. Another is the set of constraints on the supply system (e.g., factory capacity,

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maximum allowable rate of change of factory starts, inventory limits) that determine what control MV moves are allowable. A third is the objective function that defines a good answer and quantifies the costs of exercising control of MVs and hitting (or missing) CV setpoints. On the one hand, the optimization minimizes the error between the CVs and the setpoint. On the other hand, it minimizes the effort expended by the system in exercising control MVs. This is a “control-relevant optimization” in that it does not simply minimize error. For example, it might be possible in Fig. 16.4 to match the CVs to the setpoint more closely but only with a more costly set of control MV moves. Or it might be possible to find a lower effort set of control MVs, but only at the cost of larger CV deviations from setpoints. The mapping of this set of control ideas from continuous flow manufacturing (Figs. 16.2–16.6) to the supply chain production, inventory, and demand satisfaction problem (Fig. 16.1) including implementation in Intel’s semiconductor supply chain is the focus of the rest of this chapter.

16.4 Related Work As might be expected, the first exposition of the benefits of applying control theory to production systems was offered by Herbert A. Simon (1952). Using Laplace transform methods, a basic controller was formulated for a deterministic production system involving a single product continuously flowing through one factory into and out of one inventory position. The basic formulation was extended to explore the inclusion of a fixed TPT for the factory, aspects of demand forecasts, and production and inventory cost minimization. Analysis of the formulation provided useful insights into the stability of the system, particularly the impact of the decision rule concerning production rate. This initial work was translated into discrete space

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using z-transforms with particular focus on the impact of forecast techniques for variable demand (Vassian 1955; Bessler and Zehna 1968). Simon’s approach was expanded and presented to a broad audience in the well-known and important book by Holt, Modigliani, Muth, and Simon (HMMS) (Holt et al. 1960) dealing primarily with linear stochastic production systems with quadratic cost criteria. In the two decades that followed there was a modest but steady flow of work on linear deterministic and stochastic systems, the latter including random variations in both supply and demand. These contributions have been reviewed by Axsater (1985) citing more than 50 relevant references. In addition, optimal control was explored (Kleindorfer et al. 1975; Thompson and Sethi 1980), perhaps most practically by considering a production system as a Pontryagin-type problem with the goal of minimizing the sum of the costs of production, storage, and stockouts (Hwang et al. 1967; Sprzeuzkouski 1967). Over the next two decades, as commercial firms began to recognize the importance of production and inventory dynamics to their profitability, research broadened to consider both integrated horizontal systems (i.e., supply chains from suppliers’ suppliers to customers’ customers) and vertical systems (i.e., hierarchical layers inside the firm). Emphasis was placed on reducing inventory variation and demand amplification. Ortega and Lin (2004) have reviewed this work citing a further 50 relevant references covering both deterministic and stochastic problems. Of particular note throughout this period are the contributions of Towill and team (Towill 1982, 1992; Towill and Del Vecchio 1994; Evans et al. 1998; Disney and Towill 2002). Their models include such factors as expected orders trends and target customer service levels from sales and marketing, pipeline lead time estimates from the shop floor, and stock positions from retail outlets. Their master production scheduling decision support system contains (a) an inventory feedback loop to keep finished goods inventory aligned with a goal, (b) an “orders in the pipeline” control loop to dampen oscillations in the system, (c) a lead time loop that is nonlinear and time varying to provide updated estimates of lead times, and (d) a forward loop that estimates average future sales. The literature most relevant to the work described in this chapter lies at the intersection of relatively sophisticated MPC techniques with production, inventory, and supply chain management problems. Although MPC has been widely known in the continuous process industries for many years (Garcia et al. 1989; Lee et al. 1994; Camacho and Bordons 1995), a number of different research groups began to independently explore its application to inventory and supply chain problems over a relatively short period of time. Tzafestas and colleagues (1997) were the first to use an MPC-type approach for inventory control. Their formulation adjusted replenishment to maintain a target inventory level in the face of “disturbances” (using the terminology of the process industry) such as material removed from inventory for demand satisfaction. This work demonstrated robust setpoint tracking for a step change in target inventory level subject to stochastic demand and plant-model mismatch. Bose and Pekny (2000) presented a decomposition approach for the supply chain planning and scheduling problem to compute inventory levels for a distribution

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network handling fast-moving consumer goods. They built an MPC framework that used a forecasting model and a mixed integer programming model. The models were integrated into a simulation that incorporated uncertainty in supply lead time and demand. Centralized, decentralized, and distributed supply chain control structures were analyzed. Perea et al. (2000a) used MPC control on a discrete event simulation of a three-product, four-echelon supply chain to investigate improvements in system performance with centralized control of the nodes. In the same conference session, Rivera and colleagues (Flores et al. 2000) demonstrated the robustness of MPC strategies for lead time uncertainties involving both deterministic and stochastic demand. Perea and team expanded their work to eventually include a fan-out supply chain with three plants, each making three different products, and a distribution network of three plant warehouses, four distribution centers, and ten retailers serving twenty end customers. Their model took advantage of a rolling horizon approach to quickly update the decision-making process of a centralized coordinator whenever changes in demand forecast arose (Perea et al. 2000b, 2001, 2003). Rivera and team focused explicitly on semiconductor supply chains. Their initial work expanded to include the use of process control ideas for manufacturing systems by implementing MPC as a decision policy for reentrant manufacturing lines (Vargas-Villamil et al. 2003). Later, MPC was used as a tactical decision policy for supply chain management (Braun et al. 2003a; Wang et al. 2003, 2004). Braun focused on MPC for partially decentralized control of the last stages of the semiconductor manufacturing supply chain and was primarily concerned with deterministic representations of the manufacturing process (Braun et al. 2003a, b). Wang’s focus was on centralized policies for semiconductor manufacturing problems, issues related to stochasticity, and multi-degree-of-freedom controller formulations (Wang et al. 2003, 2004, 2005, 2007, 2008). Schwartz continued the use of process control ideas by comparing state-of-the-art order-up-to policies against policies based on Internal Model Control and MPC (Schwartz and Rivera 2010). Contributions were made relating to financial optimization of the control-oriented decision policies (Schwartz et al. 2006) and providing control-relevant demand forecasts for enhancing the quality of tactical decisions (Schwartz et al. 2005, 2009a; Schwartz and Rivera 2009b). Schwartz also studied the identification and parameterization of semiconductor manufacturing facilities as coupled nonlinear hyperbolic partial differential equations, allowing for more accurate factory simulations and nonlinear control applications (Schwartz and Rivera 2009c). Seferlis and Giannelos (2004) presented a two-layer optimization-based control approach for multi-node and multi-product supply chains they explored with deterministic and stochastic demand. The approach applied MPC principles to the entire network while maintaining levels of safety stock through feedback controllers on each production and inventory node. The objective of these controllers is to maximize customer service at a minimum cost over a rolling time horizon.

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16.5 The Detailed Mathematical Formulation For more than 50 years, control methodologies have been continuously improved and reduced to reliable practice in a variety of process industries (Garcia et al. 1989; Camacho and Bordons 1995). Process control systems are widely used to adjust flows to maintain product levels and compositions at desired levels. This is analogous to the management goals of high volume supply-demand networks and material flows in these networks can be modeled using a fluid analogy as shown in Fig. 16.1. As a result of using this fluid analogy, one can expect that decision policies based on process control concepts could have a large and beneficial impact on supply chain management. MPC is arguably the most general method currently known of posing the process control problem in the time domain (Ogunnaike and Ray 1994). In MPC, a system model and current and historical measurements of the process are used to predict the system behavior at future time instants. A control-relevant objective function is then optimized to calculate a sequence of future control moves that must satisfy system constraints. The first predicted control move is implemented and at the next sampling time the calculations are repeated using updated system states. This is referred to as a Moving or Receding Horizon strategy and is illustrated in Fig. 16.7. Input variables consist of two types: manipulated variables u, which can be adjusted by the controller to achieve desired operation and disturbance or exogenous variables d. The starts rates for Fabrication/Sort (C1), Assembly/Test (C3), and Finish/Pack (C4) represent MVs for the problem in Fig. 16.1, with suggested targets

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determined by the Central Planning Organization (Kempf 2004). Demand (D1, D2, : : :) in Fig. 16.1 is treated as an exogenous signal. This signal consists of actual demand which is only truly known in the past and for a very short time into the future, and forecasted demand, which is provided by a separate organization such as the Central Sales Organization (Kempf 2004). As noted in Fig. 16.7, the demand forecast is used in the moving horizon calculation to anticipate future system behavior and plays a significant role in the starts decisions made by the MPC controller. Representing quantities of primary importance to the system, y is a vector of output variables. Outputs can be classified in terms of CVs, which must be maintained at some setpoint value or associated variables which may not have a fixed setpoint, but must reside between high and low limits. For the problem in Fig. 16.1, CVs consist of the inventory levels (I1–I5) whose setpoint targets are determined by a separate organization such as Central Planning or Sales. Associated variables include loads on the manufacturing nodes (M1, M2, and M3) in Fig. 16.1 determined on the basis of their WIP and capacity. In MPC control, predictions of y over a horizon are computed on the basis of an internal model arising from mass conservation relationships describing the dynamics of the manufacturing, inventory, and transportation nodes. For the problem in Fig. 16.1, the mass conservation relationship for inventories I can be written as: Ix .k C 1/ D Ix .k/ C Cx YMx .k TPT Mx / CxC1 .k/;

(16.1)

where x can equal 1, 2, 3, 4, or 5. For the manufacturing nodes, an expression for the work in progress WIP is written as: WIPMx .k C 1/ D WIPMx .k/ C Cx .k/ Cx .k TPT Mx /;

(16.2)

where x can equal 1, 2, or 3, and TPT and Y represent the nominal TPT and yield for the manufacturing node, respectively, while C represents the manufacturing starts per time period that constitute inflow for factories and outflow streams for warehouses. These systems of equations can, in general, be organized into a discrete-time state-space model representation amenable to MPC implementation and analysis (Qin and Badgwell 2003; Lee et al. 1994). The goal of the MPC decision policy is to seek a future profile for u, the MVs, which bring the system to some desired conditions consistent with the relevant constraints and the minimization of an objective function. The ability to address constraints explicitly in the controller formulation is part of the appeal of MPC. For the problem in Fig. 16.1, constraints need to be imposed on the magnitudes of factory and shipping starts (16.3), changes in factory and shipping starts (16.4), factory loads (16.5), and warehouse inventory levels (16.6). 0 Cx .k C j jk/ Cxmax x D 1 to 5 j D 1; 2; : : : ; m Cxmin Cx .k C j jk/ Cxmax x D 1 to 5 j D 1; 2; ::; m

(16.3) (16.4)

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max 0 WIPMx .k C i jk/ CAPM x D 1 to 3 i D 1; 2; ::; p x

Ixmin Ix .k C i jk/ Ixmax x D 1 to 5 i D 1; 2; ::; p:

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While there is significant flexibility in the form of the objective function used in MPC, a meaningful formulation for the problem in Fig. 16.1 is: minC.kjk/:::C.kCm1jk/ J D

5 X m X

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The first input target term is meant to maintain the starts close to target values for each time period over the move horizon m based on the targets calculated by Central Planning. The second move suppression term penalizes changes in the starts over the move horizon m. This term serves an important control-theoretic purpose as the primary means for achieving robustness in the controller in the face of uncertainty (Garcia et al. 1989). The third setpoint tracking term is intended to maintain inventory levels at targets specified by Central Sales or Planning over time. These targets need not be constant, and can change over the prediction horizon p. The emphasis given to each one of the sub-objectives is achieved through the choice of weights Q that can potentially vary over the move and prediction horizons. For an MPC system relying on linear discrete-time state-space models to describe the dynamics, with an objective function as described above, and subject to linear inequality constraints, a numerical solution is achieved through a Quadratic Program (QP). Depending on the nature of the objective function, model and constraint equations, other programming approaches (LPs) may also be utilized (Vargas-Villamil et al. 2003). The objective function shown in (16.7) offers considerable flexibility to the supply chain planner. Through judicious selection of the weights Q a planner can directly manage the trade-offs between important supply chain objectives: keeping inventory at targets with QI , minimizing factory thrash with QC , and keeping factory inputs at financially optimal levels with QC . However, in certain supply chain applications, the specific values of the CVs are not as important as whether they lie within certain boundaries. One example is a transportation/logistics problem where there is no steady-state optimal shipment quantity, and there is no cost associated with changes to shipment schedules. In this scenario, a planner can use an alternate predictive control formulation that seeks to drive CVs to targets by using a “funnel”. In this alternate formulation, the optimizer is indifferent to CV values within the funnel, but acts to prevent the CV from exiting the funnel. The setpoint/upper

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boundary of the funnel is the Upper Control Limit (UCL) obtained from more strategic supply chain planning processes. The lower boundary of the funnel is a straight line between the most recent measurement and the setpoint (Maciejowski 2002). At the end of the optimization horizon, the reference trajectory is equal to the setpoint. The slope of the lower boundary is determined by a user-adjustable performance ratio (Qin and Badgwell 2003). The overall effect is to drive the CV to the set point over the course of the optimization horizon. Figure 16.8 depicts this process graphically. The reference trajectory is the staircase pattern, increasing linearly from the latest measurement to the setpoint value across the horizon. The setpoint is defined as the UCL. The area between these two curves is the funnel. During an execution interval, the inventory solver will manipulate shipments across the move horizon to keep inventory levels within the funnel while also considering the effects of forecasted demand. The controller implementation treats the funnel boundaries as soft constraints, where violations are permitted but penalized in the objective function. The alternate predictive control objective function, that could be applied to the network in Fig. 16.1, consists of two soft constraint terms that penalize inventory excursions outside of the funnel. M N X 2 X 2 J D IjC .k C i jk/ high .k C i / C Ij .kCi jk/ low .kCi / : j D1 i D0

(16.8)

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M denotes the move calculation horizon, a tuning value for penalizing excursions outside the funnel, high the upper boundary of the funnel, IjC a slack variable that represents the inventory value when the value exceeds the upper boundary of the

Inventory Level Setpoint/UCL Reference Trajectory Lower Control Limit

Time (Shifts) Fig. 16.8 The control scheme acts to keep the predicted inventory within a funnel

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funnel and is otherwise zero, low the lower boundary of the funnel, Ij a slack variable that represents the inventory value when it drops below the lower funnel boundary and is otherwise zero, and k the current sampling index. The optimization may be subject to the same hard constraints as the standard formulation with bounded inventory levels, MV values, MV changes, and Work-In-Progress levels. In receding horizon fashion, after the inventory solver has determined the solution, the first move (shipment quantity) is implemented and the process is repeated. For a general description of this alternative receding horizon control strategy, the interested reader would be best served by consulting Qin and Badgwell (2003) and Maciejowski (2002).

16.6 An Intel Interfactory Example Intel produces a vast array of products, which share geographically dispersed manufacturing facilities and a sophisticated, centralized MPC-based methodology can be applied to interfactory control within the supply chain network. To manage the multitude of material flows that span multiple continents, Intel utilizes a hierarchical stack of supply chain management decision processes that act to provide demand forecasts and inventory control limits (Kempf 2004). Tactical decision policies based on MPC then take these signals and make decisions concerning how much material to ship between sites. For example, die are held in Fab/Sort Die Inventory before being shipped to Assembly/Test Factories across the globe. At the beginning of each shift, a replenishment policy decides how much material should be shipped to each factory site. Upon arrival, material is held in the factory inventory until it is released for further processing based on a demand signal that for this analysis is considered to be exogenous. Factory inventory levels must be maintained because if inventory drops to zero, there will be no material to process and the factory will be idle. Any inventory replenishment policy must maintain factory inventory levels at targets under these conditions of uncertain demand and supply. Shipment times are stochastic and sitedependent with values falling between 15 and 50 time periods. Demand is stochastic and uncertain with forecast error ranging from 0% up to 300%. This section is concerned with the tactical policy that utilizes information from the strategic systems to make appropriate decisions on how much material to ship from I1 Fabrication/Sort Die Inventory to the I2s Assembly/Test Die Inventories for the supply chain shown in Fig. 16.9. Note the fan-out from I1 to multiple I2s, and that each speed of die in I1 is not shipped to all I2s. In the Intel problem described here, there are 39 different die being shipped to 7 different locations with a maximum of 5 sites receiving any one die type.

16.6.1 The Previous Method Before the implementation of MPC as a tactical decision policy for supply chain management, a typical policy would have been to simply “push” material from I1

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to the I2s according to the demand forecast for each Assembly/Test factory. This policy would consider the nominal shipping time between the two sites, and would ship material so that it would arrive just before being required by the factory. Such a policy performs well when there is little variability in the shipping times and accurate demand forecasts are available. However, in practice these conditions are seldom met. For scenarios where transit times are variable and the demand forecast is uncertain it is necessary to use a more advanced decision policy.

16.6.2 Applying MPC Intel has chosen to apply MPC as a tactical policy that can mitigate the negative effects of uncertainty in the supply and demand signals. For the problem presented in Fig. 16.9, Intel employed the alternative funnel-based predictive control formulation. The objective function consists of two soft constraint terms that penalize inventory excursions outside of the funnel as described by Eq. (16.8). The optimization is subject to hard constraints on shipping capacity. 0 uO j .k C i jk/ uO max j :

(16.9)

To demonstrate the benefit of this formulation, we compare actual Intel supply chain data generated through use of an MPC controller with a simulated version of what would have been realized with the “push-to-forecast” policy. The factory network simulation incorporates the same transportation times and distributions as the actual network.

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16.6.3 Results and Discussion Figure 16.10 shows a summary of policy performance across three different scenarios. Each scenario corresponds to a different product and different geographic locations for the factory sites with different distributions for the shipping times between I1 inventory and the I2 inventories. Numerical values in the table denote a quantitative metric for forecast accuracy, where lower values signify higher accuracy and larger values signify high levels of forecast error. Generally, accurate demand forecasts are available in Scenario #1, whereas the demand forecasts in Scenario #2 have substantial errors. Scenario #3 has accurate forecasts for four factory sites and erroneous forecasts for two sites. Non-shaded boxes denote situations where MPC outperforms (or ties) the push policy. Shaded boxes show where the push policy outperforms MPC. Better performance is defined as fewer time periods with stockouts. If both policies have an identical number of time periods with stockouts, then performance is determined by the policy with lower average inventory levels. The simple “push-to-forecast” policy can perform well if accurate demand forecasts are available. The benefits of the more sophisticated MPC approach are most obvious in Scenario #2, which has the highest levels of forecast error. Figure 16.11 is a more detailed depiction of the results shown in Fig. 16.10. The number of time periods with stockouts and average inventory levels is plotted against increasing amounts of forecast error, where lower values are better for both metrics. The MPC-based policy generally has fewer time periods with stockouts across the forecast error range, and this is especially true under high levels of forecast error. There is a notable exception in Scenario #1 for Factory Buffer #1, where the forecast error metric is 11.6. In this situation, the MPC policy produced 91 time periods with stockouts, whereas the push policy produced only 23.

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A tactical policy may be effective at reducing stockouts, but it may do so at the expense of high inventory levels. Managing the trade-off between inventory holding costs and the cost of not fulfilling an order is a fundamental problem in supply chain management. Note that in the presence of large forecast error values, the MPC policy manages to virtually eliminate stockouts without holding more inventory than the push policy. Figure 16.12 shows the forecast error signal corresponding to Factory #1 in Scenario #1. The 2-norm of the forecast error signal is 35.7, indicating relatively low levels of forecast error. Figure 16.13 shows push policy results for this scenario. The push policy and supply chain network were simulated using MATLAB.

Fig. 16.11 Stockouts (left) and average inventory levels (right) vs. forecast error

Fig. 16.12 Forecast error for Scenario #1, Factory #1. The axes are obscured to protect confidential data. For comparison purposes, the scale is identical to that in Fig. 16.15

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Fig. 16.13 Push policy inventory for Scenario #1, Factory #1. Axes units are omitted to protect confidential data; both Figs. 16.13 and 16.14 are plotted on the same scale

Fig. 16.14 MPC policy inventory for Scenario #1, Factory #1. Axes units are omitted to protect confidential data; both Figs. 16.13 and 16.14 are plotted on the same scale

The policy results in 6 time periods with stockouts and average inventory level of 7 units. Figure 16.14 shows actual data from Intel’s supply chain when an MPC-based policy is used to make tactical decisions. The simplistic push policy outperforms the MPC policy by a small margin, yielding 6 time periods out of 1,000 where stockouts occur. Contrast this with 27 time periods for the MPC policy. This supports the contention that any policy will work well when accurate demand forecasts are available.

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Fig. 16.15 Forecast error for Scenario #2, Factory #1. The y-axis has been obscured to protect confidential data, but the scale is the same as that in Fig. 16.12

Fig. 16.16 Push policy inventory for Scenario #2, Factory #1. Axes units are omitted to protect confidential data; both Figs. 16.16 and 16.17 are plotted on the same scale

Figure 16.15 shows the forecast error signal corresponding to Factory #1 in Scenario #2. The 2-norm of the forecast error signal is 572.6, indicating relatively high levels of forecast error. Figure 16.16 shows simulated push policy results. Figure 16.17 shows actual data from Intel’s supply chain when an MPC-based policy is used to make tactical decisions. The simplistic push policy underperforms, causing stockouts in 118 out of the 1,000 time periods in the time span. The MPC policy manages to eliminate the stockouts.

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Fig. 16.17 MPC policy inventory for Scenario #2, Factory #1. Axes units are omitted to protect confidential data; both Figs. 16.16 and 16.17 are plotted on the same scale

This example, along with all of the results from Scenario #2, demonstrates the superior performance of the MPC-based policy in the presence of highly uncertain future demand. The MPC policy managed to reduce or eliminate stockouts for any scenario where the 2-norm of the forecast error was larger than 100.

16.7 An Intel Intrafactory Example In terms of intrafactory control, MPC was initially conceptualized for augmenting the existing single factory decision and control system for assembling die with substrates, testing their performance, and realizing final configuration and packaging before shipping to customers to satisfy demand. But before any decision and control system is installed at Intel, whether it is augmenting an existing business process or automating a new one, it is verified and validated through simulation (Sarjoughian et al. 2005; Godding et al. 2007; Huang et al. 2006, 2009). Therefore, a discrete event simulation was implemented for the manufacturing process shown in Fig. 16.18 and validated against 60 days of historical data from a particular Intel assembly/test factory. TPT distributions reveal a highly stochastic process, where the mean roughly equals its standard deviation. The implied 100% variation in TPT is typical for assembly-test processing. Die speed distributions are well known for monthly time periods, less well for weekly time periods and are typically not explicitly traceable for day-to-day variability due to the splitting and merging of lots. These variations in TPT and speed represent a large amount of model uncertainty for a real-time control and optimization problem. The simulation accuracy for TPT and speed are shown in Fig. 16.19 and track the real process within the


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tolerances established for validation at Intel. Comparing actual data to simulated data, the simulated TPT mean was C0:4% simulated TPT standard deviation was 10:2%, simulated die speed mean 3:1%, and standard deviation 18:6%.

16.7.1 The Previous Method The previous process for controlling this system utilized a heuristic feedback/feedforward decision and control loop. The information being passed was similar to that used in modern automated control systems, but the process was driven by many layers of human communication. Interactions that occur at this speed are incapable of timely reaction to changes in the market. In addition, given the complexity of managing an ever-increasing number of line-items, there was often a separate process for each product or each product family. In this environment, agile response was infeasible and revenue opportunities were missed.

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16.7.2 Applying MPC Changing these processes for factory execution and tactical demand management could be a source of major improvement opportunities. A solution that would support better agility would have access to historical and current data as well as forecasts of future supply and demand. It would operate in a timeframe consistent with the rate of supply and demand variability. It would seek a global financially optimal execution plan over multiple products and multiple factories. The expectation would be more efficient use of both capacity and inventory to minimize cost and maximize demand satisfaction. We initially performed an experiment to validate that MPC can deliver on these requirements for the internal operation of an assembly/test factory. Experiments were designed to numerically assess how well demand was filled relative to the current methods for a single factory. Data from the same 60-day period from a particular assembly/test factory was utilized. This included (a) incoming die from fabrication facilities, (b) die releases into assembly, (c) output from test with speed data, (d) semifinished goods inventory by speed bin, (e) detailed configuration activity in finish/pack, (f) finished goods inventory by substrate and speed, and (g) finished goods shipped by substrate and speed. In these experiments, it was assumed that the incoming die had to be accepted and the finished goods shipped represented true demand. Success of the MPC system would be measured as (a) use of less material from assembly die inventory, (b) performing less configuration in the Finish/Pack factory thereby (c) enriching the speed bins in semifinished goods inventory, and (d) reduce finished goods inventories. The MPC system for this problem included a formulation with the following size: (a) 18 DVs that consist of initial die arrivals from fabrication factories and final shipments in response to demand, (b) 52 MVs that consist of die releases, semifinished goods inventory releases by substrate and speed, and configuration by substrate and speed, and (c) 29 CVs that consist of all inventories in the assembly/test die bank by speed category, semifinished goods inventory by substrate and speed, and finished goods inventory by substrate and speed. The formulation was designed to minimize the error to the inventory target CVs (semifinished and finished goods), minimize control action for the MVs (die release, release from semifinished goods inventory, and configuration) to meet demand DVs (orders) shipped daily from finished goods inventory. Control performance was measured by the ability of the new method to meet the actual demand over the same 60-day time period.

16.7.3 Results and Discussion The resulting simulations controlled by the MPC system quantify the observed change in production starts (die release and semifinished goods release) and inventory (die bank, semifinished goods, and finished goods) positions. Simulation cases were run for 60 days with no forecast (purely reactive), a 50% uncertain forecast (roughly realistic), and a perfect forecast looking out 21 days. Figure 16.20

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highlights the importance of utilizing demand forecasts, including ones known to contain uncertainties. MPC is shown to perform well even when the demand uncertainty is high. With the same die influx and the same products shipped using a 50% forecast uncertainty, MPC was able to (a) release 60% fewer die from the die bank, (b) require 80% less configuration, (c) position 150% more inventory just before the configuration step, and (d) incur 55% less finished goods inventory. MPC makes the better use of inventory by releasing less material to satisfy the same demand while delaying and therefore dramatically decreasing configuration resulting in a richer speed mix in semifinished goods inventory and less material in finished goods inventory. Overall, this translates to more efficient daily execution of demand fulfillment benefiting Intel by hundreds of millions of dollars per year.

16.8 Conclusions Applying MPC to Intel’s fan-out supply chain has improved agility in spite of long and variable manufacturing lead times, stochastic product yields, and uncertain demand. Whether applied to intrafactory fan-out across production and inventory stages or to interfactory fan-out across transportation and inventory stages, the benefits are substantial. With the ever-increasing complexity of Intel’s multi-echelon multi-product multi-geography supply chain, MPC promises to be a cornerstone for continuously improving Intel’s service to its customers.

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Acknowledgements Three groups need to be acknowledged for collaboration in realizing the work described here. First is Dr. Daniel E. Rivera, Professor of Chemical Engineering and Program Director of the Control Systems Engineering Laboratory at Arizona State University. Professor Rivera was the first to suggest applying Model Predictive Control to supply chain problems at Intel. With the aid of three of his PhD students – Martin Braun, Wenlin Wang, and Jay Schwartz – he constructed the theoretical foundation for our work. Second are Dr. Joseph Lu, Chief Scientist and Senior Fellow at Honeywell International, Honeywell Process Solutions in Phoenix, Arizona and Dr. Duane Morningred, Engineering Fellow at Honeywell International, Honeywell Process Solutions in Moorpark, California. Together they facilitated the implementation of the initial Model Predictive Controller in Intel’s supply chain management system in collaboration with Intel’s Kirk Smith (formerly with Honeywell International). Third are Intel’s Customer Planning and Logistics Group (CPLG) Vice Presidents Keith Reese and Frank Jones. The former supported the early research and the latter the implementation of Model Predictive Control systems in Intel’s supply chain. Their strategic vision was vital to the success of the work described here. The critical contributions of tens of additional members of CPLG are gratefully acknowledged.

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Chapter 17

Models and Methods for Production Scheduling in the Pharmaceutical Industry Dario Pacciarelli, Carlo Meloni, and Marco Pranzo

17.1 Introduction The pharmaceutical marketplace is dominated by large multinational companies, competing worldwide, with a global presence in branded products. Retaining marketshare requires standards of product quality and reliability close to 100%, attained at sustainable cost. Wholesalers and final customers expect reliability and quality from pharmaceutical companies, which face an increasing challenge to achieve such standards. Reliable production plans are critical to this aim. In fact, to achieve 100% availability of final products, it is not sufficient to attain excellence in each phase of the planning process from strategic planning to real-time scheduling, but it is also important to effectively manage the coordination between these different phases. As observed by several authors (e.g., McKay and Wiers 2003a,b), many computerized tools for planning and scheduling have had limited success in practice. Possible reasons include the lack of flexibility of planning and scheduling systems (McKay et al. 1988, 2002), as well as the mismatch between the key aspects of the scheduling problem to be solved and the system implemented in practice, which tends to reflect more the perceived than the real problem. Kempf et al. (2000) present a lucid overview of the many issues involved in assessing the quality of a production schedule. Most of them are also addressed in this chapter, to develop effective models and methods for production scheduling and to investigate the interaction between planning and scheduling functions. The latter topic is very under-researched in the academic literature, but is critical to the successful implementation of planning activities in pharmaceutical industry. To this aim, we use the generic term planning activity to denote any activity from demand management to production planning, except finite capacity scheduling, which is called the scheduling activity. In a commonly adopted classification, the planning function determines the quantities of products and components to be produced in

D. Pacciarelli () Dipartimento di Informatica e Automazione, Università “Roma Tre”, via della Vasca Navale, 79 - 00146 Roma, Italy e-mail: [email protected]


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a given time horizon, while the scheduling function involves the release of work orders to the shop floor, specifying the allocation of shop resources to the different orders over time (Pinedo 2005; Vollmann et al. 1997; Vossand and Woodruff 2003). Following the notation of Kempf et al. (2000), a predictive schedule describes the designed system behavior over the schedule horizon, while a historical schedule describes the operation start times and resource assignment actually executed on the shop floor. The usual way planning and scheduling interact in the pharmaceutical industry, at least in the short-term, is as an open loop control chain, in which the plan determines input data and constraints to be satisfied by the predictive schedule. To a large extent, the customer-facing end of the pharmaceutical supply chain is driven by orders. Demand management is usually based on historical data and statistical considerations, and planning decisions are driven by the need to meet market demand dates. Allocating less production capacity than needed to a product may result in significant lost revenue (see, e.g., Mallik et al. 2002). On the other hand, scheduling decisions are driven by the availability of shop floor resources. Hence, when the quantities required by the plan are excessive relative to the available capacity, the schedule cannot comply with the plan, causing delays in the delivery of final products. However, wholesaler orders are typically the result of a negotiation in which order quantities, due dates, and penalties for late delivery are regulated by contracts (Mockus et al. 2002; Romero et al. 2003b). Hence, the reliability of the whole process is closely related to the ability to negotiate realistic delivery dates for all orders or, alternatively, evaluating the impact of a new order on the delivery dates of those currently in the system. In other words, managing the demand based on historical data cannot guarantee 100% reliability, which requires closing the control loop from production scheduling to the sales department to provide real-time information on the future production capacity which is available for new demand. Effective tools are needed for this purpose that can generate and maintain effective predictive schedules and automatically evaluate alternative schedules as new market scenarios materialize. Producing reliable estimates of future capacity in the pharmaceutical industry is not an easy task. One reason is the complexity of manufacturing processes and the impact of contamination issues on the factory throughput. Another reason is related to the issue of on-time delivery. It is generally recognized that in the vast majority of manufacturing systems, the two main objectives to be pursued in production planning and scheduling are on-time delivery of the final products and maximization of the total value produced by the plant. These two objectives are often conflicting, since the former hampers the organization of production schedules with large lots, increasing the number of setups and idle time and reducing the factory throughput. Relative to other manufacturing processes, the pharmaceutical industry gives higher importance to on-time delivery over throughput maximization, due to the economical and legal implications of late deliveries and stock-outs at the final customers (Huang and Chen 2006; Lin et al. 2002; Mishra et al. 2005; Romero et al. 2003a). The above facts have significant consequences for both real-time operations management and planning activities. On the real-time side, the complexity of production processes and the need for meeting short-term due dates lead to frequent

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under-utilization of shop resources and disregarding of long-term due dates. On the planning side, the lack of reliable information on the actual plant capacity makes estimating reliable delivery dates to negotiate with the wholesalers difficult, thus causing frequent urgent orders, which further increase the short-term pressure on the operations managers (Berning et al. 2004; Bollapragada and Sadeh 2004). The best utilization of production resources and on-time delivery of the final products can be achieved using scheduling support systems both in real time, to organize the short-term production, and in the planning phase, to carefully estimate actual resource capacity and reliable delivery dates for final products (Bauer et al. 1994; Brown and Scherer 1994; Pinedo 2005). This is the subject of this chapter. The sections are organized as follows. In Sect. 17.2, we briefly review the most common aspects of pharmaceutical production processes that are relevant to planning/scheduling activities. We do not deal here with primary pharmaceutical production, in which active ingredients and other basic components of final products are produced, but rather focus on secondary pharmaceutical production, in which the components are dispensed, blended, processed, and packed. Secondary production is more critical from the viewpoint of coordination between planning and scheduling. In Sect. 17.3, we discuss several optimization models that can be used to formulate scheduling problems arising in secondary pharmaceutical production. Section 17.4 deals with solution techniques. To a large extent, scheduling and related issues are still carried out by human schedulers, who develop feasible predictive schedules based on their past experience and intuition. The development and implementation of effective computerized systems for such operational problems require paying attention to a number of aspects that are rarely taken into account simultaneously in scheduling theory or in practice. The complexity and variety of constraints arising from the shop floor characteristics and from organizational issues require powerful scheduling algorithms. The quality of a schedule may typically involve several indices, such as the utilization level of shop floor resources, production costs in terms of personnel and energy consumption, the fulfillment of production targets and others. More importantly, resources and scheduling priorities change over time, which necessitates adapting the scheduling algorithms continuously to the new situation. Finally, no decision maker would unconditionally accept a predictive schedule that they do not consider convincing. Therefore, at least in the transition from manual to computerized scheduling, it is necessary for the computer to generate schedules akin to those produced manually. From the above issues, it follows that effective computerized scheduling algorithms must be modular, easy to implement and to maintain, able to deal with many constraints and objective functions, and able to incorporate observations and suggestions arising from the scheduling practice, i.e., to incorporate user intuition and experience in the scheduling system. In Sect. 17.4, we discuss these issues and describe several algorithmic techniques that can be successfully applied to face such challenge. In Sect. 17.5, we report on a practical experience in three departments of a pharmaceutical company. We chose two small, non-critical departments and a typical complex manufacturing department that allow us to describe in sufficient detail the

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whole process of modeling the scheduling problems, developing solution algorithms and discussing results and feedback from the field. Some conclusions and directions for future work follow in Sect. 17.6.

17.2 Pharmaceutical Manufacturing Systems In this section, we briefly review those characteristics of pharmaceutical production processes that are most relevant to the planning/scheduling decision activities. Typical pharmaceutical supply chains contain at least two manufacturing stages: primary and secondary manufacturing (Cole 1998; Shah 2004). The former is dedicated to the production of active ingredients and other basic components through complex chemical and biochemical processes. Production is typically a push process organized in long campaigns, driven by forecast demand, to reduce the impact of long cleaning and setup times that are necessary to ensure quality and avoid crosscontamination. Primary manufacturing is therefore not very sensitive to short-term demand fluctuation, and the main issue here is careful lot sizing to avoid shortages of active ingredients. Secondary manufacturing is usually a pull process, driven by wholesaler orders, in which active ingredients and other components are dispensed, blended, processed, and packed to produce the final products. Primary and secondary manufacturing are typically decoupled by relatively large stocks of components. Since, in general, secondary production is very sensitive to market demand, scheduling and coordination between planning and scheduling are critical issues to guarantee 100% availability of final products at sustainable costs. In the remainder of this chapter, we focus on secondary manufacturing only, which is the most relevant stage as far as the coordination between planning and scheduling is concerned. Secondary pharmaceutical manufacturing systems consist of a set of multipurpose production facilities that produce a variety of intermediate and finished products through multistage production processes. Facilities are linked by supplier– customer relations, i.e., one facility produces intermediate goods that are processed further by other facilities, reflecting the material flow relationships given by the recipes of the final products. Furthermore, each facility may interact with external (e.g., suppliers) and/or internal (e.g., warehouses) entities. Common production processes are devoted to producing solid, liquid, aerosol, or powdered items according to a family of similar recipes (Cole 1998). For example, one of the main process families is devoted to solid dosage manufacturing (SDM), which includes, among others, the production of tablets. Each common production process consists of a set of self-contained activities. For example, SDM includes: 1. Raw material handling, in which the availability of all materials required by the receipt is checked. 2. Dispensing activities, in which materials are weighed according to the batch recipes and stored in sealed bins.

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3. Binder preparation, where specific agents are prepared to be used to wet powders in the granulation. 4. Granulation, where dry materials from the bins are passed into a granulator and mixed with specific quantities of binder agents to produce granules. This is passed through a wet mill into a fluid bed drier. After drying, the granule is transferred in a handling bin. 5. Blending, where bin contents are blended. This can occur before or after the granulation and may include the addition of new materials. 6. Compression, where granules are fed into tablet presses. 7. Coating, in which a solution is prepared and sprayed over the tablets. 8. Counting, where packages are prepared according to the different orders and market places. 9. Packaging, in which tablets and packages are transferred to the packaging line and processed. It includes individual specific packages and bulk packs when the products are sent to other sites for further processing. 10. Quality assurance and control activities are distributed throughout the whole the manufacturing process. Between coating and packaging, tablets are typically stored in sealed bins to decouple the two activities. A similar separation occurs between dispensing and granulation. Hence, the production is typically organized with three or four main departments, as in Fig. 17.1, which to a certain extent can operate independently of each other. The first two activities are performed in the dispensing department, activities from binder preparation to coating are performed in the manufacturing department, and counting and packaging are performed in the packaging department. When counting activities require a significant amount of work, counting and packaging are performed in different departments (Fig. 17.2). This is a common situation in Europe, where the many different national regulations and languages require the handling of a huge number of different packages in the same plant (Cole 1998; Piachaud 2005; Shah 2004).

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The typical layout of a dispensing department consists of one or more rooms connected to a warehouse, in which raw materials are weighed. Cross-contamination issues require processing of one product at a time in each room and cleaning the room when switching from one product to another. Minor cleaning is sufficient when two consecutive products need the same raw materials, while major cleaning is necessary when the raw materials change. Hence, the production is organized in groups of products requiring the same raw materials to be scheduled consecutively, which are called campaigns. Major cleaning is, however, necessary after a maximum number of products of the same type, called the size of a campaign. Hence, from the scheduling point of view, each room acts as a single machine with sequencedependent setup time and campaigns. The layout of a counting department is very similar to that of a dispensing department, but its production process is characterized by the absence of significant setups and by a larger number of lots per week. The layout of a typical manufacturing department is a complex flow shop with a number of specialized machines (e.g., mixers, reactors, dryers, etc.), in which each product is processed on a subset of machines only. Even though there may be different copies of the same machine, production is often organized by assigning a specific machine to each product type to reduce cross-contamination risks. Hence, routing flexibility is very limited in practice. As in the dispensing department, the production is organized in campaigns to reduce setups. Packaging departments usually contain one or more packaging lines, which can process one product at a time. Hence, this department can be viewed as a set of parallel machines. The number of lots per week processed by the packaging department is usually much larger than for the manufacturing department. In fact, for example, a single lot of tablets can be divided into many different lots of final products, differing from each other only in the package. Planning and scheduling activities are usually loosely integrated in secondary manufacturing industrial practice. In their simplest form, planning and scheduling activities strive to meet delivery dates for all demands subject to constraints on sequencing, resource capacity, and process production (Askin and Standridge 1993; Bauer et al. 1994; Clement et al. 1992). Sequencing constraints specify a partial ordering among the operations for a set of tasks. These constraints are typically dictated by each specific recipe, but sequencing constraints may be required also by the production process, by quality control or by specific management policies. For example, in some cases a task can be started only after some equipment preventive maintenance or calibration task is finished.

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Resource constraints limit the number of resources that can be used for a specific activity at any time. For example, manpower resources are categorized by their respective skills and low skilled operators can perform only a subset of simple tasks. This fact may limit, for example, the number of tool changes that can be performed simultaneously in a shift, when these activities involve complex mechanical operations. Besides such issues, pharmaceutical production processes are characterized by extremely strict requirements of chemical composition and operations execution to guarantee the quality of the final products. Process constraints are dictated by the production process established in the plant. These constraints generally aim to avoid the risk of cross-contamination between different products or between different lots of the same product. Good manufacturing practices prescribe cleaning equipment before, and sometimes after, processing a new material. Therefore, setup times can be large with respect to the processing time of a lot. Some features require incorporation of a number of relevant details into the scheduling models. For example, when switching from a product to another, it may be necessary to clean a machine and to change some tools, which involve mechanical operations. While cleaning operations can be performed by low skilled operators, mechanical operations require specialized operators. Moreover, different mechanical operations can be performed in parallel, and therefore a careful model of the setup time should take into account the number of operators involved and their respective skills, besides the possible need for minor/major cleaning. Constraints may derive from specific plant management policies. For example, a department may prefer to organize production such that the starting/completion time of some particular operation is aligned with the beginning or end of a shift. Other departments may constrain each setup operation to be completely executed within the same shift, although there are no specific technological reasons for such requirement. A good scheduling decision support system (DSS) in this context should provide a feasible predictive schedule which simultaneously ensures that: 1. Enough material at each bill of materials level is produced to satisfy the demands. 2. Early warnings of shortages of intermediates and raw materials are provided. 3. All production constraints, resource availabilities, and business requirements are respected. 4. Good solutions in a practical sense with regard to the multiple conflicting objectives are obtained. Besides these main requirements, the DSS should enable management to: 1. Reduce the amount of work of the human scheduler 2. Permit every department to operate independently to a certain extent, while collaborating with the others for increased transparency and information sharing 3. Reduce response time to changes 4. Monitor and manage conflicts among departments

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17.3 Optimization Models The construction of optimization models for planning and scheduling purposes is necessarily the result of an iterative process, during which all different aspects of the problem are taken into account and formalized progressively with the agreement of all stakeholders. It is possible to distinguish at least two different methods for formally defining a problem. Descriptive models provide a formal way to list the relevant aspects of the problem, while formulations explicitly define all the variables of the problem as well as their logical and mathematical relations. While descriptive models are more suitable for defining the problem in a preliminary phase, when interacting with the stakeholders, formulations enable the construction of solution algorithms. In other words, descriptive models are necessary to define all the aspects of the problem to be solved, and formulations are necessary to design solution methods. In the next section, we give a brief description of the most frequent aspects arising in pharmaceutical industries using a descriptive model. In the subsequent section, we introduce some formulations that have been shown to be particularly effective in solving practical production scheduling problems.

17.3.1 Descriptive Models A common classification scheme for scheduling problems focuses on the characteristics of the storage facilities. From this point of view, at least four relevant settings can be distinguished (see, e.g., Schwindt and Trautmann 2002): unlimited intermediate storage (UIS), finite intermediate storage (FIS), no intermediate storage (NIS), and zero wait (ZW). Hall and Sriskandarajah (1996) model the absence of intermediate buffers (NIS) as a blocking constraint. In this case, a job, having completed processing on a machine, remains on it until the next machine becomes available for processing. The FIS case can be viewed as NIS problem by modeling each position of an intermediate buffer as a dummy machine with zero processing time (McCormick et al. 1989). In ZW settings, a job, having completed processing on a machine, must immediately start processing on the subsequent machine without waiting. Other relevant settings are related to scheduling problems with setups (see, e.g., the extensive review of Allahverdi et al. 1999) and to the management of perishable items. A commodity is said to be perishable if some of its characteristics are subject to deterioration over time with respect to customer/producer requirements. The perishability issue is approached in various ways in the scheduling literature. A common approach when scheduling perishable goods with high decay rate consists of introducing tight ZW constraints to avoid product degradation (see, e.g., Grabowski et al. 1997; Hall and Sriskandarajah 1996; Pinedo 1995). A more detailed descriptive model for scheduling problems is the well-known ˛=ˇ=” classification scheme of Graham et al. (1979). With this notation, ˛ indicates

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the scheduling environment, ˇ describes the job characteristics or restrictive requirements, and ” defines the objective function to be minimized.

17.3.2 Problems Formulations In general, most production scheduling problems can be formulated as mixed integer linear programs by defining suitable variables and constraints. However, as observed by Jain and Meeran (1999) and Pinedo (1995), mathematical programming techniques have so far been of limited practical use, at least for job shop scheduling problems of practical size. Their observation is particularly valid for complex production scheduling problems in which a solution has to be found within strict time limits, or when the models can change due to reconfigurations of the production environment, rendering the mathematical properties that can be exploited to speed up solution algorithms less and less effective (Alle et al. 2004; Burkard and Hatzl 2006). A different approach to the formulation of scheduling problems is based on the observation that the basic variables to be considered are the starting times of the operations to be scheduled, while the constraints can be viewed as time relations among pairs of variables. Within this stream of research, graph models have been demonstrated to be particularly effective for modeling and solving very general scheduling problems. The first successful model in this context is the disjunctive graph formulation for the job shop scheduling problem proposed by Roy and Sussman (1964). The job shop scheduling problem is the problem of allocating machines to competing jobs over time, subject to the constraint that each machine can handle at most one job at a time. With this first formulation, a job consists of a sequence of operations, each to be processed without interruption on a given machine. This version of the job shop scheduling problem is clearly a simplification of many relevant problems that can be encountered in the industrial practice. The disjunctive graph does not take into account a number of additional constraints arising from scheduling practice. However, several authors have observed that it can be easily adapted to deal with many practical issues (see, e.g., White and Rogers 1990, and Schutten 1998) such as assembly and disassembly sequences, setups, due dates, release times, maintenance operations, material handling delays, and other operational side constraints. Different graph formulations have been proposed in the last two decades to overcome the limitations that remain in these models. McCormick et al. (1989) introduce the so-called precedence constraints graph to study a flow shop scheduling problem in an assembly line with finite capacity buffers between machines (FIS). They model the positions of the intermediate buffers as machines with zero processing time and show that once a sequence for the jobs has been found, the starting time of the jobs on all machines can be easily computed on a precedence constraints graph. Sanmarti et al. (1998) introduce the schedule graph representation to study a production scheduling problem arising in a

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multipurpose batch plant. The schedule graph generalizes the precedence constraints graph to represent general job shop problems with both UIS and NIS situations. Romero et al. (2004) combine the schedule graph representation, or S-graph, with a feasibility check based on linear programming to deal with different intermediate storage policies, including ZW, FIS, and common intermediate storage (CIS). Mascis and Pacciarelli (2002) introduce the alternative graph model, which generalizes the previous formulations. An advantage of the alternative graph with respect to the schedule graph is that ZW and perishability constraints can be represented directly within the graph, without the need for additional feasibility checks. This compact representation enables the development of more effective optimization procedures, and it has been successfully applied to the solution of complex production scheduling problems (Meloni et al. 2004; Pacciarelli 2002; Pacciarelli and Pranzo 2004; Pranzo et al. 2003). A different stream of research aims at formulating integrated routing and scheduling problems. The most common production environment in this context is the parallel machines case, when the problem is to decide when to start an operation and on which machine. Examples in this context can be found, for example, in Ovacik and Uzsoy (1997). This problem has also been extensively studied in the form of a constraint satisfaction problem by several researchers from the field of artificial intelligence (Sadeh et al. 1995). Finally, the field of resource constrained project scheduling investigates the case occurring when the operations may require a general amount of some production resources, which are available in a given quantity (Brucker et al. 1999).

17.4 Solution Methods In this section, we describe a general algorithmic approach to design scheduling algorithms that are easily adaptable, modular, and suitable for incorporating human experience in the computerized methods. In these methods, the search process is performed by several heuristic procedures guided by a general optimization strategy. These features are particularly useful when facing practical problems, where models suffer from a lack of strong mathematical properties that can be used to design effective solution algorithms. Moreover, simple heuristics are more easily accepted and trusted by the human schedulers, who can understand their principles and suggest modifications to improve their performance over time. In this section, we describe the algorithmic building blocks: constructive algorithms, mainly used to obtain initial feasible predictive schedules, and iterative algorithms, which can be used to improve the initial solution. Section 17.4.1 focuses on greedy methods, which can be used to find a feasible solution for a large variety of practical scheduling problems. Section 17.4.2 describes simple and general local search techniques. Section 17.4.3 deals with strategies to improve the performance of a heuristic or to combine different algorithms. Finally, Sect. 17.4.4 illustrates methods for improving the effectiveness of simple local search strategies.

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17.4.1 Constructive Algorithms Among simple constructive techniques for scheduling problems, we focus on greedy algorithms such as list scheduling and insertion algorithms. List scheduling algorithms typically sort the operations to be performed according to a given criterion and then, starting from an empty schedule, build a complete predictive schedule by adding one operation at a time to the end of the partial schedule, in the order induced by the sorting criterion. When routing and scheduling decisions have to be made simultaneously, as in the parallel machines environment, the added operation is assigned to a machine according to a given dispatching rule. The sorting criterion and the dispatching rule specify the list scheduling algorithm. Several studies of greedy algorithms can be found in Bhaskaran and Pinedo (1992), Blackstone et al. (1982), Chang et al. (1996), Geiger et al. (2006), Haupt (1989), Morton and Pentico (1993), and Panwalkar and Iskander (1977), among others. Sorting criteria can be broadly classified according to different dimensions, such as static or dynamic, deterministic or randomized criteria, and more. A static criterion sorts the operations according to information available in the initialization phase of the algorithm. A dynamic criterion sorts the operations according to information that changes during the run of the algorithm, thus requiring the recalculation of the criterion for each unscheduled operation every time an operation is added to the partial schedule. Deterministic criteria always select the next operation according to a certain sorting criterion, whereas stochastic criteria may choose randomly from a subset of candidate operations. More sophisticated constructive algorithms are the so-called insertion algorithms. An insertion algorithm repeatedly selects a promising operation and then chooses its position in the partial schedule, according to a given policy. A common choice consists of tentatively inserting the selected operation in all the possible feasible positions in the partial schedule, and choosing the locally best position. Insertion algorithms are among the best performing simple constructive algorithms for scheduling problems (Nawaz et al. 1983; Nowicki and Smutnicki 1996), but typically require larger computation times than list scheduling algorithms.

17.4.2 Iterative Algorithms Iterative algorithms are based on the concepts of move and neighborhood. A move is a rule for transforming a solution s into a different solution s 0 . A solution s 0 that can be reached from s using a single move is called a neighbor of s. The set of all the neighbors of s is called the neighborhood of s. A neighborhood structure N associated with a given move defines, for each solution s, its neighborhood N .s/. Given an initial solution s, there are three basic elements that compose an iterative algorithm: (a) the neighborhood structure N , (b) a procedure for selecting a new solution in N .s/, called the selection criterion, (c) a method for terminating the procedure, called the stopping criterion.

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Common neighborhood structures for scheduling problems are based on three kind of moves: swap, exchange, and insert (Brucker et al. 1996, 1997; Werner and Winkler 1995). The swap move consists of exchanging the position of two adjacent operations in s, while leaving the rest of the schedule unchanged. The exchange move considers two operations i and j in s, and produces a new sequence s 0 in which i replaces j and vice versa. The swap move is therefore a particular type of exchange move. The insert move consists of choosing an operation i and producing a new schedule in which i is removed from its position and inserted before or after another operation j. Figure 17.3 provides an illustration of the three moves. The selection of a new solution within the neighborhood is typically performed by evaluating each solution s 0 2 N .s/ with a scoring function f .s 0 /. The scoring function depends on the problem under study and can be an approximation or a surrogate of the value of the objective function to be minimized in s 0 . Common approaches to choosing the new solution s 0 2 N .s/ consist in selecting a solution s 0 2 N .s/ such that f .s 0 / < f .s/. Algorithms based on this approach are called local search algorithms. The most common choices are to choose s 0 such that f .s 0 / D minff .x/ W x 2 N .s/g, called the steepest descent strategy, or the first solution s 0 2 N .s/ such that f .s 0 / < f .s/, called the first improvement strategy. The iterative algorithm consists of replacing s with s 0 and continuing until the stopping criterion is satisfied. Common stopping criteria consist of stopping the algorithm when a time limit is reached or if there are no improving moves in the neighborhood of the current solution, i.e., when f .s/ minff .x/ W x 2 N .s/g. If the latter case holds, s is called a local minimum with respect to f .x/ and N .x/.

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Fig. 17.3 An illustration of the three moves

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17.4.3 Constructive Metaheuristics As observed, e.g., by Lawrence (1984), greedy heuristics may exhibit erratic behavior, possibly because (a) locally promising decisions may not lead to good global solutions, and (b) poor choices cannot be changed once made. In order to overcome these limitations, general strategies have been developed to improve the reliability and efficiency of greedy heuristics, incorporating them into more sophisticated constructive metaheuristics. Common strategies are based on the following ideas: Maintain a set of partial promising solutions. This is the basic idea of the beam

search technique (Morton and Pentico 1993), which consists of a limited breadth first visit of the branching tree. At each step of the search process only the most promising ˇ (beam width) candidates are maintained, and all the other partial solutions are discarded. The beam search procedure has been applied by Kanakamedala et al. (1994) to a rescheduling problem arising in a multipurpose batch chemical plant, and to job shop problems by Sabuncuoglu and Bayiz (1999) using dispatching rules, by Werner and Winkler (1995) using an insertion heuristic and a local search procedure, and by McMullen and Tarasewich (2005) for mixed-model scheduling with setups. Restart and randomize the construction. The mechanism of the greedy randomized adaptive search procedure (GRASP) (Feo and Resende 1995) consists of repeatedly constructing a solution using a greedy algorithm with a randomized criterion, and then improving the solution with a local search. If the new solution improves the current best solution, the new solution is stored as the current best. Binato et al. (2001) develop a GRASP algorithm for the job shop scheduling problem. Backtrack mechanism. An effective strategy to guide the exploration of the search tree is limited discrepancy search (Harvey and Ginsberg 1995), with its variants and extensions (Korf 1996; Walsh 1997). In limited discrepancy search, a discrepancy is defined when the child chosen is not the best one according to the heuristic criterion. The underlying idea is that solutions with small discrepancy are more promising and worth visiting early in the search. Reiterated construction. The iterated greedy approach iterates a destruction– construction cycle to improve the performance of a greedy heuristic. The algorithm consists of three different phases, destruction (some components of the current solution are removed), construction (the greedy algorithm is applied to the partial solution to build a new complete solution), and the acceptance criterion (the new solution is evaluated). The iterated greedy approach has been demonstrated to be very successful for solving the flow shop scheduling problem (Ruiz and Stutzle 2006). Look-ahead strategy. This is the main idea of the Rollout algorithm (Bertsekas et al. 1996, 1997) or Pilot method (Duin and Voss 1999; Voss et al. 2005) to overcome the myopic effect greedy heuristics. A master program enlarges a partial solution using one or more greedy algorithms as a look-ahead strategy. Meloni et al. (2004) applied this approach to solve several versions of the job shop problem.

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Fig. 17.4 Pseudocode of pilot/rollout algorithm

We next illustrate the rollout/pilot method. The basic idea of this metaheuristic is to consider a solution s to an optimization problem as a collection of m components, s D .s1 ; s2 ; : : : ; sm1 ; sm /, e.g., a sequence of jobs to be defined. The problem is then solved sequentially by fixing the components s1 ; s2 ; : : : ; sm1 ; sm one at a time. A partial solution Sk in which the value of k components is fixed is called a k-state. For example, Sk D .s1 ; s2 ; : : : ; sk1 ; sk / may correspond to the first k jobs in a schedule. Starting from k D 0 with no fixed components, the kth iteration consists in evaluating a scoring function p.e/ for each seemingly feasible component e which can be added to the current state Sk1 and choosing the most promising component, i.e., the one having the smallest scoring function. The iteration is repeated for k D 1 to m, when a complete solution is found. The scoring function p./ is a lookahead strategy, which in the rollout method is guided by one or more sub-heuristics, called pilot heuristics. More precisely, consider a state Sk1 and let e be a possible value for the sk th component. The i th pilot heuristic H i ./ is a constructive algorithm that, starting from the partial solution Sk D .Sk1 ; sk D e/, produces a complete solution with objective function value H i .Sk1 ; e/. The value of the scoring function p.e/ for .Sk1 ; sk D e/ is then based on the values H i .Sk1 ; e/ of all the pilot heuristics, e.g., the minimum value for all pilot heuristics. Then, if p.e0 / D minfp.e/g, then sk D e0 . Figure 17.4 shows a sketch of the rollout algorithm applied to the problem of sequencing a set M of jobs.

17.4.4 Iterative Improving Metaheuristics Iterative improving heuristics typically converge to local minima. To overcome this limitation, a variety of strategies have been proposed to escape from local minima.

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These can be broadly classified as iterative improving metaheuristics. Among the metaheuristic schemes that appear more suitable for applications in industrial planning and scheduling are iterated local search (ILS), tabu search (TS), variable neighborhood search (VNS), and simulated annealing (SA). Several applications of these techniques to planning and scheduling problems are reported in Brucker et al. (1996), Brucker et al. (1997), Brucker and Hurink (2000), Glover and Kochenberger (2003), and Morton and Pentico (1993). In this section, we give a brief description of these methods, and then discuss in more detail the VNS scheme. TS metaheuristic (Glover 1990; Glover and Laguna 1997) is an iterative algo-

rithm with steepest descent criterion, which accepts non-improving moves and uses a tabu list to restrict the neighborhood at each step, based on the past moves. Non-improving moves allow the algorithm to escape from local minima, while the tabu list prevents the search returning to solutions already selected in recent iterations. More precisely, at each step, the inverse of the last applied move is added to the tabu list. The moves in the tabu list are then forbidden in the next steps. A move remains on the tabu list for a limited number of iterations, called the length of the tabu list, which can be fixed or variable. This mechanism can be overruled when a solution associated with a tabu move satisfies an aspiration criterion by leading to a solution with a good objective function value. In a basic TS algorithm, the only parameter to set is the length of the tabu list. More sophisticated TS schemes have been proposed in the literature, and this method is certainly among the most successful heuristics for a large number of planning and scheduling problems (e.g. Franca et al. 1996; Nowicki and Smutnicki 2005). SA is an optimization technique based on the analogy with cooling solids (e.g., metals) (Aarts and Korst 1989; Aarts and VanLaarhoven 1988). The basic idea is that when solids cool, the particles behave in a way that can be viewed as a local search for the minimum energy configuration. The literature on the SA is quite rich in successful applications of the technique to planning and scheduling problems (VanLaarhoven et al. 1992). SA escapes from local minima using a nondeterministic move selection. At each iteration of a SA algorithm, a candidate move is randomly selected and the corresponding change in the objective function value f is evaluated. The candidate move is always accepted if it leads f

to an improvement, and is accepted with probability e T if the move is not improving (i.e., f is positive). In SA terminology, the parameter T is referred to as the temperature. When T assumes high values, the probability of accepting a non-improving move becomes close to one for all moves and the algorithm acts as a random search procedure. On the other hand, for very low values of T , only improving moves will be accepted. The SA algorithm starts with a high value of T which slowly decreases over a large number of iterations. The detailed specification of this mechanism is called the cooling schedule. ILS performs a local search in a neighborhood structure until a local minimum is found. Then a perturbation mechanism allows the procedure to escape from local minima by generating a new starting point and restarting the local search phase. Finally, an acceptance criterion decides whether to accept the new local

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Fig. 17.5 Algorithmic scheme of variable neighborhood search (VNS)

minimum or not (Lorenço et al. 2001; Lourenço et al. 2002). A perturbation can be implemented with a number of randomly selected moves, called kick moves, or with a problem-specific strategy, e.g., as a short search in a secondary neighborhood, pursuing a secondary evaluation function. There are several variants of this metaheuristic, and several applications to planning and scheduling in production and supply chain systems are presented in the literature (den Besten et al. 2001; Detti et al. 2007; Stuetzle 1998). VNS is a local search technique focused on systematic neighborhood changes (Hansen and Mladenovic 2001; Mladenivić and Hansen 1997). The main idea of this algorithm is to exploit several different neighborhood structures. It is based on the observation that a local minimum for a given neighborhood is not necessarily a local minimum for different neighborhoods. The sketch of the algorithm is given in Fig. 17.5. Let fN1 : : : Nk g be a set of neighborhood structures. The VNS starts the search process from an initial solution in the first neighborhood i D 1, i.e., N1 . In the generic iteration, the algorithm scans the i -neighborhood looking for an improving move; once an improving move is detected, it is performed and the VNS starts searching in the first neighborhood (N1 ). If no such improving move exists, the algorithm starts searching in the .i C 1/-neighborhood. The search terminates when no improving moves are available in any of the considered neighborhoods. Since the VNS accepts only improving moves, the final solution of a VNS algorithm is a local minimum for all the considered neighborhoods fN1 : : : Nk g.

17.5 Case Study In this section, we report on our experience with a practical implementation of a DSS for production scheduling at a pharmaceutical production plant located in Italy. The plant supplies different European countries and the production flow is organized into

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the four main phases of Fig. 17.2. However, while there are only one dispensing and one counting department in the plant, manufacturing and packaging activities are organized with several departments. The plantwide planning organization follows a 4-week rolling horizon strategy in which departments are called to solve their specific scheduling problems. The plant’s ERP system defines a set of due dates for the final products to be delivered in weeks 3 and 4. These become due dates for the packaging department, which are propagated backward to the counting and manufacturing departments, by assuming approximately 1 week of lead time for each product, and to the dispensing department, by assuming approximately 2 weeks of lead time for each product. The impact of a late delivery can be minor, as when the delay is absorbed by the wholesaler inventory system, or major, when it may cause a stock-out at final customers. In the latter case, a hard deadline is associated with the product besides the due date. A deadline can be viewed as a due date whose violation causes an infinite penalty, and therefore it should not be violated by the scheduler. At the beginning of week 1, the dispensing department schedules the production orders for weeks 1 and 2 and implements the predictive schedule for week 1. If some product is delivered late with respect to the due date defined by the plan, its scheduled delivery time is used as a release time for the subsequent department. At the beginning of week 2, the manufacturing and counting departments schedule the production orders for weeks 2 and 3 and implement the schedule for week 2. Similarly, at the beginning of week 3, the packaging department schedules the production orders for weeks 3 and 4 and implements the schedule for week 3. The whole process is repeated every week, i.e., every week each department schedules the production for the next 2 weeks, to recover possible differences between the historical schedule and the predictive schedule. At the end of each week, there may be production orders in some department that have not been delivered on schedule, or the production of one or more urgent orders is required by the planner, e.g., in case of stock-outs for some products. In such cases, it may be necessary to reschedule the production in different departments, since late deliveries at a department may cause lack of materials at the subsequent department, while urgent orders cause extra requirements at the corresponding department. Late and urgent orders are managed as orders with strict deadlines to be processed as soon as possible. Clearly, deadlines make it difficult to organize long campaigns and thus reduce the actual capacity of the departments. This reduction may cause, in turn, late deliveries at the end of the week and such negative effects may propagate over several weeks. All these problems motivate the need for more coordination among the departments and with the planner. In this section, we consider the production scheduling of three departments: dispensing, counting, and manufacturing. The first two are the simplest and least critical departments, which can be described quite easily. The third department is more complex and represents the typical complexity encountered in secondary manufacturing. A description of the considered departments is given in the next sections, after which we describe the models of the scheduling problems and the solution algorithms. Computational experience and feedback from the field conclude this section.

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17.5.1 Dispensing Department The dispensing department is in charge of checking the availability of all raw materials required by each recipe and preparing sealed bins containing the needed amount of raw material according to the batch recipe. The sealed bins are then put in a buffer waiting for processing in the manufacturing department. The weighing operation is carried out in two independent rooms, where the raw materials picked up from a warehouse are weighed and put in a bin. The remainders are sent back to the warehouse, unless the following job requires the same components. Cross-contamination issues require the department to process one product at a time and to clean the room when switching from one product to another. Minor cleaning is sufficient when two consecutive products need the same raw materials, while major cleaning is required when the raw materials change or when the maximum number of consecutive products of the same type, called the size of a campaign, is reached. From the scheduling point of view, each room acts as a single machine with sequence-dependent setup times and campaigns. There may be planned temporary room unavailability, mainly due to lack of personnel, which must be taken into account when scheduling production. Note that a job can start processing before an unavailability and complete after the interruption, since the weighed materials can be stored in sealed bins which are reopened after the interruption. Hence, according to Lee (1996), the unavailability constraint classifies as resumable. For each production order there may be a release time and a due date or a deadline, when there is a risk of stock-out. The primary objective in evaluating the quality of a predictive schedule is to respect all the deadlines and, as far as possible, all the due dates. Secondary objectives are makespan minimization, minimization of the number of late jobs and maximization of the total value produced in the scheduling horizon. In the first implementation of the system, the primary objective was the minimization of the maximum lateness, i.e., the maximum over all products of the difference between the product completion time and its due date. As secondary objectives, in lexicographic order, we chose makespan minimization and the minimization of the number of late jobs. The discussion of the results with plant managers led us to some changes. First, the primary objective was modified to the minimization of the maximum tardiness, i.e., the maximum between zero and the maximum lateness. In fact, since negative values of the lateness correspond to zero tardiness, there can be many schedules with different (negative) lateness, which are all optimal in terms of the tardiness. Among this larger number of solutions, we are more likely to find schedules with smaller makespan values, i.e., there is more margin for the minimization of this secondary objective. The second change was the introduction of priorities among products to reflect product value and company preferences. Priorities are used if the total amount of time necessary to complete all production orders exceeds some resource capacity. In such cases, several production orders have to be postponed to the subsequent weeks, and this task is carried out by taking into account the priorities among products.

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17.5.2 Counting Department The counting department is where materials required for packaging are prepared. Packages, labels, and patient information leaflets are taken from a warehouse, counted and prepared for the subsequent packaging operations. Although each counting operation is relatively simple, the counting department serves several areas of the plant. Moreover, due to the number of different packages that must be used in different countries, this department typically deals with a much larger number of lots than the dispensing department. The counting department is composed of three independent rooms, although some room may be temporarily unavailable, and there is no significant setup between two consecutive operations. In the counting department, there are also release times (when the packages becomes available), deadlines, and due dates (propagated backward by the packaging department). The objectives are, as in the dispensing department, minimizing tardiness, makespan, and number of late jobs. In this department, there are also priorities among products when the demand exceeds the capacity.

17.5.3 Manufacturing Department In the plant there are several manufacturing departments, each devoted to a particular line of products, in which raw materials are processed to form finished drugs. We focus on a department with 24 different machines, where the presence of several constraints makes the scheduling task particularly difficult. Some machines may be unavailable due to preventive maintenance or, if available, can be shared with other manufacturing departments. In the latter case, the production is organized with reserved time intervals for each department. In our model, a machine is considered unavailable when busy with maintenance or when reserved to other departments. The sequences of machines traversed by different production orders are quite different, and range from very simple (a single operation) to complex, including job recirculation and assembly of different subproducts. Operations cannot be preempted, i.e., no machine can stop processing an operation to execute a different one. However, some machines can interrupt processing an operation for a while to resume its execution later, for example, during unattended time intervals between two shifts. Processing times range from a fraction of hour to several shifts. Buffer space between machines is not an issue; therefore, it is considered of infinite capacity in our model. Each job has a release time (the completion time at dispensing department) and a due date or a deadline. Cross-contamination issues require cleaning the machine when switching from one product to another. Minor cleaning is required when consecutive products belong to the same campaign, major cleaning is required otherwise. Machine tooling issues imply the presence of sequence-dependent setup

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times, since changing different tools requires different amounts of time. These constraints can be summarized as sequence-dependent setup times with maximum campaign sizes. A further constraint concerns personnel availability. Workers are needed to supervise machines during processing and setup operations, and their number may vary from shift to shift, the night shift being less supervised. The number of workers needed to supervise a machine varies from zero (for completely automated machines) to two (for complex setup operations), and typically their number is not sufficient to allow all the machines to operate simultaneously. In addition, workers have different skills, and each one is able to supervise a limited set of machines. As in the counting and dispensing departments, the objective functions are minimization of maximum tardiness, makespan, and number of late jobs.

17.5.4 Descriptive Models In this section, we formally define the two scheduling problems with the .˛=ˇ=”/ classification scheme of Graham et al. (1979) and then describe our scheduling algorithms. We use the following notation:

Pk , identical parallel machines production environment with k machines Jk , job shop production environment with k machines ri , release times di , due dates Di , deadlines sij , sequence-dependent setup times M CS , maximum campaign size RA, resumable availability constraints on the machines RC , resource availability constraints (personnel, tools, machines, . . . ) Tmax , minimization of the maximum tardiness Cmax , minimization of the makespan U , minimization of the number of tardy jobs

With this notation, the Dispensing department can be classified as P2 jri ; di ; Di ; sij ; M CS; RAjTmax; Cmax ; U; the Counting department as P3 jri ; di ; Di ; RAjTmax ; Cmax ; U; and the Manufacturing department as J24 jri ; di ; Di ; sij ; M CS; RC jTmax; Cmax ; U: We notice that although the dispensing and counting departments are considered simple departments from the practical point of view, since they are not critical and

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by far less complicated than the manufacturing department, the resulting scheduling problems are quite difficult NP-hard problems (Garey and Johnson 1979; Pinedo 1995). Moreover, some constraints like the maximum size of a campaign and resumable machine availability are not frequently addressed in the scheduling literature.

17.5.5 Algorithms for Dispensing and Counting In this section, we describe the algorithms developed for the first two, noncritical departments. Following the discussion of Sect. 17.4, we focus on solution algorithms that are modular, easy to implement and modify, and suitable for incorporating human experience in the computerized methods. In a first step of the algorithm development phase, we implemented several simple greedy procedures. Some algorithms were simple list schedules such as the Earliest Due Date, Shortest Processing Time, and the like. Other greedy heuristics were insertion heuristics or more sophisticated algorithms. The purpose of the first phase was to verify the correctness of the models with the users, and to seek useful properties of the problem instances to exploit in the algorithms. Discussion with the users allowed us to validate the models described in the previous section and to a description of the behavior of human scheduler when building a feasible schedule. The human schedulers in the plant did not follow any formal procedure to schedule production orders and did not even use any formal definition of the quality of a predictive schedule. The schedules were simply the result of schedulers intuition and past experience. However, the schedules produced by hand in the counting department were quite similar to those from the MJS algorithm illustrated in Fig. 17.6, a modified version of the Jackson (1955). In the Dispensing department, the schedulers strove to obtain large campaigns in addition to respecting the due dates. Algorithm Delta in Fig. 17.7 is a surrogate of their behavior. In a preprocessing step, the processing horizon is divided into intervals of length , and the due dates in the interval Œk; .k C 1/ are replaced with k. Then, when scheduling jobs according to the MJS algorithm, the production orders of the same type having the same due dates are scheduled to form campaigns. Hence, a larger value of favors the formation of larger campaigns over the tardiness minimization objective. Once an initial schedule was built, small changes were made to improve the overall quality of the schedule. We implemented the heuristics of Figs. 17.6 and 17.7, among others, and the results were considered feasible, although not particularly successful, by the schedulers. Hence, we used these simple procedures as a good starting point for further algorithmic development. In a second step of the algorithm development phase, we used the rollout method of Fig. 17.4 to improve the performance of the basic heuristics. Finally, the VNS approach of Fig. 17.5 was applied to further improve the solutions of the rollout algorithm. We considered five neighborhood structures associated with the following moves. The first three moves disregard the parallel machine environment and aim at improving the single machine schedules independently of each other. The latter two moves involve two machines.

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Fig. 17.6 Pseudocode of the modified Jackson schedule

Fig. 17.7 Pseudocode of the algorithm delta ()


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SWAP: Two adjacent jobs assigned to the same machine Mx are swapped. INSERT: A job is removed from its current position in the schedule of Mx and

is inserted on the same machine, m positions forward or backward. EXCHANGE: Two (non-adjacent) jobs assigned to the same machine Mx are

swapped. M-MOVE: A job assigned to machine Mx is removed from Mx and inserted in

the sequence assigned to machine My . M-SWAP: A job assigned to machine Mx and a job assigned to machine My are

swapped.

17.5.6 Algorithms for Manufacturing In this section, we describe the algorithm implemented for the manufacturing department. Due to the complexity of the production process, the common practice of this department was to schedule production orders with a time horizon of a few days. Therefore, no practical experience was available to design efficient algorithms for this department. We first developed a constructive heuristic able to generate feasible solutions for the problem. We then embedded the algorithm in a rollout scheme followed by a local search phase, based on the ILS framework. The constructive algorithm for the manufacturing department is composed of two phases as shown in Fig. 17.8. In the first phase, called backward, a provisional schedule is built backward in time, by taking into account only some constraints. In the second phase, called forward, a detailed schedule † is obtained starting from and assigning a feasible starting time to all operations. The aim of the latter phase is to correctly evaluate setups and resource availability constraints, which are taken into account only approximately in the backward phase. Specifically, in the backward phase, workers are all viewed as having the same skill and campaigns are not considered, i.e., the cleaning operations between consecutive operations of the same type are always minor. The forward procedure assigns starting times to operations and skilled workers to machines, updating the worker availability for the shifts in which the operation is executed. This operation is made more difficult due to the availability of workers with different skills. In a second step of the algorithms development phase, we use the rollout method to improve the performance of the constructive heuristic. The algorithmic scheme is detailed in Fig. 17.9. Finally, the ILS approach shown in Fig. 17.10 is applied to further improve the solution from the rollout algorithm. The ILS applies two moves in tandem: SWAP: Two adjacent jobs in are swapped. SWAP-M: A job ja assigned to machine Mx is swapped with job jb , where jb

is the next operation in assigned to machine Mx . These two moves are defined on the provisional schedule . Therefore, to calculate the corresponding solution, the forward phase of the algorithm must be executed to obtain a new schedule †. Note that a move may also lead to infeasible schedules.

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Fig. 17.8 Pseudocode of the constructive algorithm for the manufacture department

Fig. 17.9 Pseudocode of the Rollout procedure for the manufacturing department

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Fig. 17.10 Algorithmic scheme of the ILS procedure in the manufacturing department

17.5.7 Results and Feedback from the Field The algorithms described in the previous section have been tested on a set of instances for each department, each representing from 2 to 3 weeks of planned production. However, we notice that the evaluation of its performance is limited to a short time horizon and is related to the transition from the manual to the automated scheduling system. More quantitative and accurate conclusions can only be drawn after an extensive observation of the system, carried on in a more stable situation. Moreover, the lack of detailed information on the performance of the previous scheduling process does not allow a direct and complete comparison with the new system. Nevertheless, from these early results several conclusions can be drawn. As far as the dispensing and the counting departments are concerned, the assessment of the new system was based on the feedback from the department managers. The solutions provided by the stand-alone greedy algorithms were not considered satisfactory, since it was simple for the users to improve the solutions with local changes in the schedules. On the other hand, the schedules obtained after the rollout and VNS phases were rarely improved by users. The increase in capacity for the dispensing department has been estimated as up to 1 h/day by the department manager, which approximately corresponds to a 2% productivity increase. Similar results were obtained for the counting department. In the manufacturing department, a more systematic comparison was carried out by asking the human schedulers to produce the predictive schedules by hand for several weeks. These schedules were then compared to those produced automatically. In all cases the computerized schedules outperformed the manual schedules, also when taking into account key performance indicators (KPIs) that differed from those explicitly included in the objective functions and listed in Sect. 17.5.4. Figure 17.11 reports the average improvement attained by computerized schedules over manual ones, as far as several KPI are concerned. Specifically, under Tmax , Cmax , and U , we report the improvement of the maximum tardiness, the makespan, and the number of tardy jobs, respectively. Under Setup Time, we report the decrease of the cumulative time required by the machines to switch from a product type to another. Ttot denotes the total tardiness improvement, i.e., the reduction of the sum of the tardiness of all jobs. Finally, under Lead Time we indicate the decrease in

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MANUFACTURING DEPARTMENT: PRELIMINARY RESULTS 54%

12,

14%

11%

11,

12% 10% 8% 6%

2%

1%

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1%

2,7

4%

7%

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2% 0%

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C max

U Setup Time KPI IMPROVEMENTS [%]

T tot

Lead Time

Fig. 17.11 Preliminary results

the average time elapsed from the jobs’ release to their completion. We notice that the automated system is able to outperform the manual one from the viewpoint of conflicting objective functions, i.e., it is able to improve the products on-time delivery while increasing department productivity. Moreover, while a computerized schedule for the next 2 or 3 weeks is produced within a few minutes, human schedulers need several hours of work to produce a satisfactory schedule. The common practice at the manufacturing department was actually to schedule production orders every day, just for the subsequent few days, without constructing a complete 2-week schedule. The qualitative feedback from the field is highly positive. The system has been approved by the production managers and accepted by the schedulers. Key success factors of the project included the following facts: (a) the solutions provided by the system were considered convincing by the human schedulers, (b) the algorithms used by the system were understood by the schedulers, who participated actively in their tuning, (c) the system allowed the actual planning horizon for the manufacturing area to be extended from a few days to 3 weeks, (d) the system allows a significant reduction of the time needed to generate new schedules, and (e) it allows for negotiation of deadlines for urgent orders with the planner, based on the quantitative impact of urgent orders on department productivity. Other strengths include the improved traceability allowed by the computerized system, which in turn enables better production activity control, communication, and information sharing among departments.

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Nevertheless, there are aspects of the systems that need further development. Optimizing the production at the four departments led other problems to emerge that were not perceived as relevant with the manual scheduling system. The transportation system can become a bottleneck, in particular when the predictive schedules at dispensing and manufacturing are very different, which requires storing and handling a large number of bins of dispensed materials. Hence, the management expressed the need to coordinate the schedules at dispensing and manufacturing by taking into account the material handling system. A further need deals with human resource management. In the current version of the scheduling system, each unit of personnel must be preliminarily assigned to a certain shift in each working day, while the department managers prefer to manage the assignment of persons to shifts and machines with the largest flexibility, to better cope with specific needs of each worker, besides the production process. In fact, the computerized scheduler is currently used by taking into account only the amount of workers needed in each shift, while the actual assignment to shifts and resources is performed manually. Finally, there is a need to measure the difference between predictive and historical schedules, to evaluate the reliability of the schedules. This ongoing process will be important to detect systematic discrepancies and their causes, and consequently to update models, algorithms, and production data.

17.6 Discussion and Conclusions In this chapter, we described several algorithmic techniques that can be effectively used to rapidly automate the production scheduling process in a pharmaceutical manufacturing plant and presented a practical implementation of these techniques. Most of the discussion also applies to a large number of manufacturing systems. Clearly, when changing the context, key variables and attributes of the problem will change, as well as KPIs. However, for the techniques to be successfully applied, there are some characteristics that must be retained by the production process. In fact, the appropriateness of any technique depends on the structure of the problem and on the manner in which the technique is used. This holds in particular for the content of this chapter. To the success of the techniques described in this chapter, the reliability and stability of the data set, at least within the scheduling horizon, are important. In the pharmaceutical industry, a large amount of production data is deterministic, being subject to national or international regulations, and only weakly exposed to daily changes. In different production environments, when the data set is very uncertain, or subject to frequent changes, then it may not make sense to implement this kind of techniques to produce schedules that would be in any case unreliable or suboptimal. If the weekly bucket of work changes everyday, scheduling on a weekly horizon is unlikely to meet company expectations. Similarly, since computerized systems work with numbers, if the processing times loaded in the computer differ

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from the actual processing times in the shop floor, the performance promised by the predictive schedule will not be met by the historical one. In such cases, more suitable techniques can be used to deal with uncertainties, such as stochastic programming, fuzzy systems, and simulation-based methods. In any case, important features of a good computerized scheduling system include the modularity of the algorithms, the ease of implementation and maintenance operations, the possibility to easily incorporate in the scheduling algorithms observations and suggestions arising from the scheduling practice. All the methods described in this chapter exhibit these features. Automated scheduling systems can increase plant productivity, but they must be accepted by the human schedulers, who must understand the principles of the algorithms to manage possible unexpected events. However, there are always problems that are better faced by humans than by computers, such as disruption management, personnel management, and more in general the management of undefined or ambiguous issues. A number of problems remain that need further research. The coordination between planning and scheduling, as well as the interaction between human and automated scheduling systems, are critical to the success of planning and scheduling activities. Despite this fact, they have not received enough attention in the academic literature, and need further development. Automated planning and scheduling tools offer new opportunities to improve productivity, but the current interaction mechanisms between planning and scheduling do not seem adequate to fully exploit their potential. New concepts are necessary in this field. A number of issues also remain to be addressed to improve the effectiveness and reliability of models and algorithms. In recent years, there is a clear trend in the academic literature toward richer scheduling models, but practical scheduling problems are still significantly more difficult than the typical academic ones. This is a research direction that needs further attention in the coming years. On the other hand, there are obvious limits to the performance attainable by simple metaheuristic techniques. Simple and general algorithms are useful when prototyping a scheduling system, but when dealing with well-established problems, ad hoc algorithms can obtain better performance. Deeper analysis of heuristic algorithms and new mathematical properties for practical problems will be useful from an academic as well as from an industrial point of view.

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Chapter 18

Developing a Computerized Scheduling System for the Steelmaking–Continuous Casting Process Hubert Missbauer, Wolfgang Hauber, and Werner Stadler

18.1 Introduction This paper reports on the development, structure and implementation of a computerized, optimization-based scheduling system for the steelmaking–continuous casting (SM-CC) process of a steel plant in Austria. In accordance with the main objectives of this volume, we concentrate on the scientific problems that arise in the design and implementation of the system and try to identify research topics that are still unsolved in the scheduling literature. The purpose of the paper is threefold: (1) to describe the planning problem and the solution that is actually implemented, (2) to make explicit some of the expert knowledge that is necessary for structuring this practical problem and (3) to identify shortcomings in scheduling research. The paper is organized as follows: Sect. 18.2 describes the SM-CC process in the steel plant under consideration. Section 18.3 describes the role of SM-CC scheduling in the overall planning system of the plant and the resulting definition of the scheduling problem. Section 18.4 presents the planning concept and some considerations leading to this concept (namely the relative merits of monolithic versus hierarchical planning systems). Related literature is presented in Sect. 18.5. Section 18.6 describes the architecture of the scheduling system. Section 18.7 describes some experiences gained throughout the implementation, especially those of integrating a scheduling system into human problem solving and the organization of the plant. Section 18.8 presents conclusions and summarizes the suggested research directions.

H. Missbauer () Department of Information Systems, Production and Logistics Management, University of Innsbruck, Universitätsstrasse 15, 6020 Innsbruck, Austria e-mail: [email protected]


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18.2 Description of the SM-CC Process The manufacturing process in the iron and steel industry, which is one of the major industries in the world economy, transforms iron ore, scrap and some other input factors to steel products such as plates, tubes. The manufacturing process can roughly be divided into three stages: (1) Ironmaking: Production of molten iron (termed hot metal) mainly from iron ore, coke and a fluxing agent. (2) SM-CC: Processing of the hot metal into steel with a well-defined chemical composition, and solidifying the steel to cuboids, called slabs. (3) Production of the finished products by hot rolling, cold rolling, hot-dip galvanizing, etc. In this article, we primarily consider the SM-CC phase, which is often – as in our case – a bottleneck in the manufacturing process (see Tang et al. 2002a, p. 55). This makes scheduling of the SM-CC process crucial to maximizing the throughput of the plant. Furthermore, the SM-CC process has a number of characteristics or technological constraints1 that make predictable start and finish dates of the operations and early information about the consequences of disturbances highly desirable. This has motivated a steel plant in Austria to develop and implement a computerized scheduling system for performing this scheduling task. In the following, we describe the SM-CC process and also outline the essential properties of the hot rolling stage downstream of SM-CC, because hot rolling sets essential constraints for SM-CC. The downstream manufacturing stages (cold rolling, galvanizing, etc.) are less important to our problem and are not described here. We also do not deal with ironmaking, since the blast furnace produces hot metal at a rather constant rate with short-term fluctuations. The SM-CC process (depicted in Fig. 18.1) consists of four stages: Hot metal preparation, steelmaking (LD-process), refining (secondary metallurgy) and continuous casting. The hot metal from the blast furnace is poured into a transfer ladle and enters the hot metal preparation where it is de-slagged and de-sulfurized. After this operation, it can be conveyed directly to the converter or stored in a hot metal mixer, which serves as a buffer. After pouring the hot metal into the converter, the transfer ladle is idle and can be used again. After the converter is filled with scrap and hot metal, the LD-process starts for this charge of steel (typically 160 tons). After finishing the LD-process, the converter is tapped, that is, the crude steel is poured into a ladle, which must be present at this time. Refining, which transforms the crude steel into the steel grade required for the customer order, is performed by one conditioning stand, two ladle furnaces and two vacuum degassing units. Refining can require alloying, de-sulphurization, de-phosphorization, de-gassing, etc. 1

Specific technological constraints that heavily influence production planning are frequently observed in process industries; for case studies, see (Günther and van Beek 2003).

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Hot metal preparation and buffer stock

LD-Converters Conditioning Ladle Stand Furnaces

Vacuum Degassing Units (RH)

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CC 6 Conv 7

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Desulfur ization CC 3

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Fig. 18.1 The steelmaking – continuous casting process under consideration

In the converter and refining stage, many operations can be allocated to alternative facilities working in parallel (e.g., to ladle furnace 1 or 2 or to a ladle furnace or a degassing unit). Some refining operations can be split up into two operations that are performed consecutively on different facilities. Hence, the actual routings and even the number of operations are determined at the scheduling level. There are preferences for routing alternatives due to technological reasons, transportation, etc., that must be regarded as soft constraints. The allocation of explicit monetary costs to these routing alternatives is difficult. The continuous casters (abbreviated CC; one double-strand and three singlestrand) solidify the steel into slabs (typical: 215 1; 650 mm, up to 31 tons) in a continuous process. The steel is poured from the ladle into a tundish and solidifies in a crystallizer. The steel is moved downward at 1.00 – 1.80 m/min and is cooled until it can be cut to the appropriate length. If the casting process is interrupted (cast break), a restart lasts about 1 hour. We define as a cast a sequence of charges that have to be cast consecutively on the same continuous caster. In contrast to the converters and refining facilities, the continuous casters require significant sequence-dependent changeover activities between the charges. This can be due to changes in the steel grade, which can require separation of the material, tundish change or even a setup (casts break), and/or due to changes in the slab format. This requires scheduling of the continuous casters following these criteria. The operation times at the continuous casters can be varied within a certain range; e.g., the cast can be slowed down in the case of limited material availability. After the continuous casting process is finished, the ladle must be reworked at the tilting device, after which it is available for the next charge. All transport operations are performed by cranes.

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The throughput of the plant is 4.4 million tons of crude steel per year, and the number of charges is up to about 100 a day. After continuous casting, the next stage in the material flow of the slabs is the hot strip mill (HSM) and the heavy plate mill (HPM) where the slabs are rolled to coils and to heavy plates, respectively. In the following, we concentrate on the description of the HSM. Before rolling, the slabs are reheated in a reheating furnace and then enter the line that consists of a roughing mill that must be passed several times in both directions and a finishing mill. Scheduling this facility is a difficult task mainly due to the tight sequencing constraints: The cost of roller replacement is very high, so a high endurance is an important goal. Roller wear and hence their endurance depends on the rolling sequence, which requires cyclic scheduling. The criteria for the sequence optimization are widths, gauge and hardness of the orders (for a detailed description, see Tang et al. 2002b). A number of operations research models for this problem, often Travelling Salesman type models, have been described in the literature (overview in Tang et al. 2001). In practice, decision rules often are established that result from the technical constraints. For instance, rolling should be performed from wide to narrow, from thick to thin, etc. Note that these cycles do not conform to the cycles at the continuous casters.

18.3 Definition of the Scheduling Problem The requirements for SM-CC are the slabs to be produced and the required due dates. These requirements are determined by a centralized planning system that encompasses the entire manufacturing process; it coordinates customer demand (actual customer orders and forecasts) and the manufacturing system by means of a rough-cut plan over a two-month horizon for the entire plant. The crucial part of this rough-cut plan is a coordinated schedule for continuous casters and HSM. In the following, we outline this coordination; then we describe the interface to SMCC scheduling. Production planning for SM-CC and hot rolling is difficult because cyclic production is required at each stage, but the criteria that are relevant for the cycles are different. The complexity is increased by the modes that link SM-CC and hot rolling. After continuous casting, the slabs can be stored in a slab yard and reheated before rolling. This leads to high energy consumption. Therefore, linkage modes between SM-CC and hot rolling have been developed that convey the slabs to the HSM at high temperatures (hot charging). This can be done in different ways (see Tang et al. 2001 for a description of these modes and their technical properties). Hot charging requires close coordination between the schedules for SM-CC and HSM. Even if the slabs are stored in the slab yard, close coordination between SM-CC and HSM is desirable in order to reduce inventory holding costs and space requirements. Moreover, the material handling task in the slab yard where the slabs are piled can be rather complex (see Tang et al. 2002b for a specific problem of slab stack shuffling). For the slabs that enter the HSM, the discharging interval between two successive slabs from the reheating furnace must be determined (mill pacing), which

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determines the exact timing of the rolling operations and hence the throughput of the mill. The discharging intervals should be as short as possible to maximize throughput, but without triggering collisions of the slabs. (For a description of this problem for which published material is rare, see Baosteel (n.d.)) This requires consideration of all relevant bottlenecks in the line. Since the reheating furnaces can also pose a bottleneck, coordination with the schedule of the reheating furnaces is a relevant topic (see Petersen et al. 1992). These detailed scheduling decisions are less relevant for our problem. Designing a coordinated schedule of SM-CC and HSM can be considered as a two-stage scheduling problem with cyclic production at each stage according to different criteria. While the single-machine scheduling problem with sequencedependent setups has received considerable attention in the literature and a modest amount of research has addressed permutation schedules in flow shops with sequence-dependent setups (reviews in Zhu and Wilhelm 2006, Allahverdi et al. 2008), research on non-permutation schedules for this type of flow shops is rare. Due to the constraints of the specific problem, general approaches of two-stage cyclic scheduling are hardly applicable, and only a small number of specialized methods have been developed (for a review of the literature, see Ouelhadj 2003). Cowling and Rezig (2000) develop an integrated planning method for continuous casters and HSM taking advantage of the fact that coils can be manufactured from slabs with different dimensions and steel grades. Ouelhadj (2003) develops a multi-agent system for this integrated scheduling; Cowling et al. (2004) present a case on the use of multi-agents for integrated dynamic scheduling of steel milling and casting. In the steel plant under consideration, two scheduling systems (specialized software packages) are used for scheduling HSM and continuous casters. (For a description of software packages of this type, see A. I. Systems 1995, 1997.) The systems use proprietary logic and consider the sequencing constraints at the respective facilities, customer demand and hot charging (the hot charging ratio is on the order of 50%). The coordination of the schedules for continuous casters and HSM is largely based on the capabilities of experienced planners and, together with the scheduling system for the continuous casters, yields a preliminary schedule for the continuous casters. This preliminary schedule is defined in terms of (1) casts (charges and their sequence) and planned cast breaks, (2) allocation of casts to the continuous casters and (3) sequence of the casts on the continuous casters. The scheduling system for the continuous casters considers sequencing constraints that result from different steel grades and slab formats of the charges. It also takes into account the limited capacities of upstream facilities and limited availability of hot metal, but only at a fairly aggregate level. The preliminary schedule for the continuous casters is the starting point for the SM-CC scheduling system. Given this interface, the SM-CC scheduling system has to schedule all casts and all charges on all resources (the facilities, cranes and ladles). This decision problem can be structured into four sub-problems as follows: Scheduling the continuous casters: determining the casting speed and the start

and finish dates of each cast, and for each charge within each cast. Allocation of charges among alternative facilities working in parallel.

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Sequencing of the charges on each facility. Timing: determining the start and finish dates of each charge on each facility.

The objective of SM-CC scheduling encompasses the following components: Maximize output per period or, equivalently, minimize the makespan for the

given set of casts. This is the dominant objective. Minimize flow times. Minimize the duration of cast breaks. Minimize the frequency and extent of changes in the casting speed.

It is difficult to define a single objective function or, more generally, a criterion that determines the optimal compromise between these objectives. Since the scheduling algorithm is rule-based (see below), this is not necessary in the scheduling context. For the LP model that solves one sub-problem (see Sect. 18.6.4), the objective function is the weighted sum of these components, but the impact of the weights on the final solution is limited. This problem structuring does not fully determine the decision problem because how SM-CC scheduling is to be integrated into the existing planning architecture remains to be specified. Since separate scheduling systems for SM-CC and CC-HSM decompose the overall problem into subproblems that are closely related, a coupling mechanism must be designed that minimizes the optimality loss due to this decomposition. Following the scheduling literature and the literature on the decomposition of the planning problems in manufacturing planning and control systems, there are two obvious possibilities to define the interface between SM-CC scheduling and the integrated planning system for continuous casting and HSM: 1. The CC schedule can be taken as fixed, which determines the required finish dates of the last refining operation of each charge. This means a leading role for CC-HSM scheduling with no feedback from SM-CC scheduling. However, since the integrated planning system for continuous casting and HSM does not consider resource constraints at a sufficient level of detail (insufficient anticipation function; see Schneeweiss 2003, p. 33 ff.), this would require a revision of this integrated planning system, which would be a highly complex problem. 2. The importance of output maximization or, equivalently, minimization of the makespan for the static scheduling problem under study suggests that the timing decisions of the CC schedule should be ignored and only the casts and their allocation to continuous casters should be used. In this case, the start and finish dates would be determined entirely by SM-CC scheduling. This procedure would ignore a lot of information from the upper planning level and would lead to a complex scheduling problem for which applicable solutions (in contrast to basic research) hardly exist (see Sect. 18.5). Thus, neither of these two approaches seems desirable. We are not aware of any approach in the literature that lies in between these extreme solutions. Therefore, the following intermediate solution has been developed: 3. We define a cast topology as the relative positioning of the start/finish dates of casts/charges on different continuous casters (e.g., a cast on CC5 starts

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after a certain cast on CC4 is finished). The cast topology is derived from the preliminary CC schedule and, together with the casts and their allocation to the continuous casters, defines the production target for SM-CC scheduling. The cast topology is treated as a constraint of the scheduling problem and binds the SM-CC scheduling system to certain rules for the allocation of scarce hot metal and capacity.

18.4 Definition of the Planning Concept: Monolithic Vs. Hierarchical Planning One of the major issues in the design of a manufacturing planning and control system is the question of whether the planning and control system should follow a strictly centralized approach or a hierarchical approach. Designing a set of rules specifying which approach should be chosen under which condition is an important goal of research on manufacturing planning and control systems, and our knowledge on this question is very limited. Therefore, we highlight this decision problem and the theoretical underpinning of the design decision for our case. In the centralized approach, a detailed scheduling system determines the exact times the orders are processed on the facilities simultaneously for all orders and facilities. Ideally, this is performed by a model that considers all decision variables and their interdependencies (“monolithic” model). The alternative is a hierarchical approach, which in our case requires structuring the SM-CC scheduling system into two levels: A top level that determines the release or start dates of the charges based on the estimated flow times that consider routings, capacity availability, etc.; and a base level that performs short-term sequencing and timing of the charges on the specific resources. This represents a hierarchical decomposition of the scheduling problem. (For conceptual issues on hierarchical planning, see Schneeweiss 2003.) The problem we encounter here is a special case of a much more general problem often analyzed in the literature. Especially during the 1980s and 1990s, a number of alternative manufacturing planning and control concepts have been developed that can replace, or at least complement, the MRP/MRP II concept, such as workload control, Kanban, CONWIP or Drum-Buffer-Rope – OPT (for an overview of these alternative concepts, see Zäpfel/Missbauer 1993). A body of literature has emerged that compares the performance of these concepts in different environments. Since most of these manufacturing planning and control concepts are hierarchical (MRP II, workload control, Kanban) and some are more centralized (especially Drum-Buffer-Rope supported by OPT or a similar scheduling software), especially the comparison between MRP/MRP II and Drum-Buffer-Rope (see Steele et al. 2005) compares hierarchical and more centralized concepts. Scientific guidance for whether a centralized or a hierarchical planning system should be preferred is difficult to provide based on the current state of our knowledge. Since the centralized approach is ideally based on a monolithic model

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and hierarchical structuring is a decomposition, a planning hierarchy cannot lead to a better solution than a monolithic model provided that the planning problem can be modelled precisely and the model can be solved sufficiently close to optimality. However, the reasons for hierarchical structuring of a planning system are more related to uncertainty, complexity of the decision process and “bounded rationality” of the decision makers (i.e., their limited ability to understand a complex situation and decision process). These issues are difficult to integrate into the simulation models that are often used to compare production planning methods. Thus, the decision on whether a centralized or a hierarchical approach should be chosen must consider a number of factors that are usually not considered in decision models or simulations (for a case study of a tire manufacturer, see Meal 1984). Based on both theoretical insights and practical experience, some guidelines for this decision can be derived. Based on these guidelines, a centralized solution seems preferable in our case. The relevant factors are the following: The requirements for the precision of a schedule are very high. Early comple-

tion of charges at the refining stage leads to a fall in temperature, while late completion can cause cast breaks, which must be avoided. The buffer stock at the continuous casters must be low, so the timing of the charges must be controlled very strictly. Multiple resource constraints (facilities, ladles, hot metal), together with the precision required, lead to high complexity of detailed scheduling. Scheduling decisions at a work centre severely affect the entire plant. The number of charges in the system is low (about 15), which facilitates detailed scheduling, but makes it difficult to determine and maintain planned flow times based on aggregate data, e.g., the current workload in the shop. Note that Input/Output Control (Belt 1976, Wight 1974) or clearing function models (see Chap. 14 of this volume) that aim at controlling work in process and lead times by balancing order release and output are similar to fluid relaxations in scheduling theory or fluid approximation in queuing theory. Fluid relaxations work best when the number of orders in the shop is large (for the asymptotic optimality of fluid relaxations, see Bertsimas and Sethuraman 2002, Bertsimas et al. 2003). “Generally, one arrives at the fluid approximation by increasing the number of particles n to infinity, while simultaneously accelerating time and aggregating space both by a factor of n” (Chen and Mandelbaum 1994, p. 48). Computerized scheduling systems are apparently very difficult to implement in job shop environments; the gap between scheduling theory, largely based on OR methods, and practical application seems to be very wide here (see McKay et al. 1988). However, the characteristics of process industries facilitate the applicability of optimization-based scheduling systems (see McKay and Buzacott 2000). In our case, the casts and charges, their possible routes and processing times are known, which eliminates a lot of uncertainty. Thus, an optimized schedule should greatly support the scheduling task. The literature on SM-CC scheduling is dominated by the detailed scheduling approach (see Sect. 18.5), supporting the correctness of the decision to pursue this approach (the conceptual phase of this project was in 1996/1997, and much of the relevant literature is more recent).

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Thus, a detailed scheduling system is to be developed that precisely determines the start and finish dates of all operations.2 The system is similar to a “Leitstand” (Adelsberger and Kanet 1991; Pinedo 1995, p. 304 ff.), which is “a computer-aided graphical decision support system for short-term interactive production scheduling and control”. A Leitstand has five major components: a graphics component, usually (as in our case) a Gantt chart; a schedule editor for manually generating schedules; a database management system; an evaluation component; and an automation component for automatic schedule generation (see Adelsberger and Kanet 1991, p. 43). These functions can be performed by specialized software or by modules of manufacturing planning and control software. The next design decision concerns the level of decision support provided by the system. Planning software can support the decision maker in different ways, where we can roughly distinguish (1) providing relevant data, (2) performing simple analyses (such as regressions), (3) evaluation of decision alternatives (“what-if” simulation) and (4) calculating a preferred alternative. (For a taxonomy of decision support systems, see Alter 1980, cited in Arnott and Pervan 2005.) Again, this is an important design decision where theoretical support is very limited. This decision determines the division of labour between the information system and the human decision maker(s). The optimal level of decision support provided by the scheduling software depends on factors that are difficult to quantify, such as the ability to explicitly represent the relevant knowledge, the unambgiuity of the objective function, and the speed and quality of human problem solving. Supporting the schedulers by automatically generating a schedule essentially assumes that the algorithm can generate a reasonable starting point for the scheduler (McKay and Buzacott 2000, p. 87). In our case, this assumption seems reasonable, and the scheduling system optimizes the schedule rather than merely presenting the relevant information that enables the planner to resolve resource conflicts (e.g., a target schedule for each charge and the resulting resource conflicts that have to be resolved manually). A scheduling system of the latter type has been successfully implemented in a similar steel plant3 , which might indicate that different levels of decision support can be successful in similar environments.

18.5 Related Literature Scheduling problems in the steel industry have been studied extensively in the literature, often concentrating on the sequencing problems at the continuous casters and at rolling mills and on integration of continuous casting and hot strip mill (for reviews, see Tang et al. 2000a,b; 2001). Research on SM-CC scheduling as formulated 2

Missbauer et al. (1998) give related arguments on why a hierarchical solution has been proposed for an automotive supplier where the situation was entirely different. 3 The plant has been visited by two of the authors. No published material on this scheduling system is known to the authors.

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in our case (with pre-determined casts) has been limited, and most of the literature is quite recent. The problem can be formulated as an MILP model, as presented in Tang et al. (2002a), Bellabdaoui and Teghem (2006). Missbauer et al. (2009) present an MILP formulation that includes the limited availability of ladles (ladle loop) and alternative routes with the number of operations depending on the route. This model closely matches the situation in the steel plant under consideration. In scheduling theory, SM-CC scheduling can be classified as a special case of hybrid (or flexible) flow shop scheduling, but the specific situation (alternative routes with “soft” preferences, continuity constraints and variable processing times at the continuous casters, ladle loops) preclude straightforward application of algorithms for hybrid flow shop scheduling (for an overview, see Linn and Zhang 1999, Pinedo 1995, p. 112 ff.). Thus, algorithms specifically suited for SM-CC scheduling must be developed. The problem structure and the basic ideas for solution methods can be described as follows: If the start date of a charge at the continuous caster and maximum waiting times at all facilities are known, then a target schedule for this charge can easily be determined for the standard route by backward scheduling If there is only one continuous caster, a target schedule for a cast can be developed by trying to concatenate the target schedules for the charges within the cast (Fig. 18.2). Resource conflicts upstream of the continuous casters must be resolved by shifting operations backward, assigning an operation to an alternative facility or by slowing down casting operations. Harjunkoski and Grossmann (2001) develop an MILP formulation for this task. The entire schedule can be obtained by concatenating the schedules of the casts as depicted in Fig. 18.3. The sub-schedules in the upper part of Fig. 18.3 are the cast

Conflict! Converter 1

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Fig. 18.3 Aggregating the schedules for a number of casts (example). The modules depicted in the upper and middle part of the figure are the cast schedules from Fig. 18.2. Source: Harjunkoski and Grossmann (2001, p. 1653)

schedules of Fig. 18.2. These schedules for the casts are considered as units (middle part of Fig. 18.3) and then concatenated in a sequence that is to be determined (lower part of Fig. 18.3). Harjunkoski and Grossmann (2001) describe a model of this type that determines the processing sequence of the casts. For the entire planning system, consisting of the sub-problems of grouping of charges into casts, scheduling and LP improvement of the schedule (see below) they report a makespan that is between 1% and 3% from the optimum for an industrial-scale example. If several continuous casters are working in parallel, this substantially complicates the problem, because in the case of resource conflicts the start dates of all casts that have not yet been started (note the continuity constraint) can be delayed, that is, the relative positioning of the casts on the timeline can be changed. For instance, if hot metal availability is the relevant bottleneck, it must be decided whether all continuous casters work at low speed or one continuous caster is interrupted for several hours and the speed of the other continuous casters increased. Thus, the methods for solving problems with a single continuous caster cannot be applied to our plant. Tang et al. (2002a) formulate the SM-CC scheduling problem with several continuous casters as a mixed-integer programming model and develop a solution methodology based on Lagrangean Relaxation. Two sets of coupling constraints are relaxed, which yields one sub-problem for each charge. The sub-problems are solved by dynamic programming, a heuristic is used to obtain a feasible solution to the original problem. Huegler and Vasko (2007) develop a heuristic that calculates an SM-CC schedule for pre-determined casting times and start dates of the

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charges on the continuous caster. A fitness value for the schedule is calculated, and an evolutionary programming algorithm is used to optimize the casting times (and hence the schedule obtained by the heuristic). The authors state that they abandoned the attempt to formulate an adequate integer programming formulation due to the increasing complexity of the problem. Bellabdaoui et al. (2005) describe a construction heuristic that starts with the availability date of the continuous casters and with the shortest casting times. The heuristic first schedules the continuous casting and refining stage and then the converters. The casting operations are slowed and the start dates of casts are delayed if necessary. No practical implementation was reported in these papers. If we look at the subproblems of SM-CC scheduling (scheduling the continuous casters, allocation, sequencing, timing; see Sect. 18.3), only the allocation and sequencing decisions are binary and lead to high complexity. If these decisions are already made (e.g., manually or by rules), the remaining problem is to determine the optimal start and finish dates of the operations and the optimal casting speed. This can be formulated as a linear programming model (Tang et al. 2000a) and solved efficiently within less than 1 minute (our experience) using standard methods. From this literature review, we can conclude the following: (1) While the SMCC scheduling problem can be handled quite well in the case of a single continuous caster, problems with several continuous casters are substantially more complex. (2) Optimization methods for solving these complex problems have only recently been described in the literature, and even today most available methods have not been tested extensively in a practical setting. (3) The sub-problem of determining timing and casting speed simultaneously can be solved routinely. (4) Thus, it is necessary to develop and extensively test a planning method suited for this practical case.

18.6 Description of the Scheduling System 18.6.1 Structure of the System The following considerations are most relevant for determining the structure of the scheduling system: Extensive time-consuming research efforts have not been possible during the

project. Robust, reliable results are preferred. Furthermore, the computer run time must not exceed a few minutes. This limits the applicability of combinatorial optimization methods and also of metaheuristics, because the practical applicability of metaheuristics for the problem under study has not yet been tested sufficiently. The “soft” allocation and routing constraints and the importance of scheduling rules in the manual planning system suggest a rule-based scheduling procedure. This leads to the implementation of a sequential scheduling heuristic, similar to the Giffler/Thompson (1960) method, as the frame for the application of the

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rules. Due to the continuity constraints at the continuous casters, this scheduling task must be performed backward in time, based on the planned start dates of the casting operations. A scheduling heuristic of the Giffler/Thompson type treats each operation as a single entity and thus ignores the interdependencies of the operations/charges due to the casts. If, for a certain CC schedule, a feasible schedule for the plant cannot be found, the scheduling heuristic is unable to identify the necessary changes in the CC schedule (see Bellabdaoui et al. 2005, whose approach is different in this respect). Hence, two additional planning levels are necessary: (1) Determining a refined CC schedule that adjusts the initial CC schedule (from integrated CC-HSM scheduling) to the resource constraints and yields an (approximately) feasible CC schedule. (2) If small infeasibilities of the CC schedules cannot be resolved by the scheduling heuristic, then a simultaneous determination of all start/finish dates and casting speeds by means of an LP model (Tang et al. 2000a; Harjunkoski and Grossmann 2001; see Sect. 18.5) is performed. This leads to the following three-level planning hierarchy: Level 1: Revising the preliminary CC schedule. Level 2: Scheduling all operations at all facilities, including cranes and ladles. Level 3: Adjusting start/finish dates and casting speeds by means of an LP model.

The structure of the scheduling system is depicted in Fig. 18.4. In the following, these levels are described, concentrating on level 2 that is the core of the system.

18.6.2 Revising the CC Schedule If no resource constraints were relevant, all casts would be cast as early as possible at the maximum allowable speed. If limited availability of hot metal and/or upstream facilities makes this infeasible, the optimal combination of delayed start dates and reduced speed of the casts must be determined with respect to the objective described above (especially throughput maximization). The starting point is the preliminary CC schedule that results from the integrated schedule of continuous casters and hot strip mill and frozen decisions (e.g., allocation of charges to converters). The task is to determine a CC schedule that is balanced with upstream capacities and allows a feasible schedule for the facilities upstream of the continuous casters. The planning process is performed for casts and determines the start and finish dates and the casting speed for each cast (and thus for each charge within the cast). The facilities upstream of the continuous casters are aggregated into capacity. The planning process is carried out in the following steps (see Missbauer et al. 2009 for details): Step 1: Set all start dates of the casts to the target value (from the cast topology);

set the casting speed to the maximum value. If this is feasible, this is the final schedule. Else go to Step 2.

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CC scheduling

Coordination

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Coordinated schedule for continuous casters and HSM CC schedule: casts, allocation to CCs, cast topology

Revising the CC schedule

Start dates and casting speed for all charges on CCs

Scheduling of all facilities upstream of the continuous casters

Allocation and sequencing decisions; preliminary start and finish dates

LP improvement

Fig. 18.4 Structure of the scheduling system. The dashed line marks the interface to the upper planning level

Step 2: For given start dates of the casts, reduce the casting speed of all casts as

much as necessary. This is performed by a forward-pass heuristic. For this step, the planning horizon is divided into short periods that are considered sequentially. Step 3: Set Casting speed D Max [speed from Step 2; minimum allowable speed]. Set the start dates of the casts to the earliest possible value. The start sequences are determined from the cast topology. The feasibility of a CC schedule in Steps 2 and 3 is checked by comparing the cumulative demand for resources (calculated from the CC schedule and standard lead times) and the cumulative supply (projected hot metal supply from the blast

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Fig. 18.5 Feasibility check for a CC schedule. For given casting speed (that determines the resource requirements over time), first the cast on CC4, then the cast on CC3 are started as early as possible

furnace, available capacity of the facilities). A CC schedule is feasible with respect to a certain resource if the cumulative consumption of the resource is less or equal to the cumulative supply of the resource at each time. Hence, the earliest start date of a cast for a given casting speed is the earliest start date for which this condition holds for all relevant resources (see Fig. 18.5). For the calculation of the cumulative available capacity, it is assumed that the capacity is only available for a limited time, which is a rather straightforward heuristic. Steps 2 and 3 are repeated until convergence. Step 4: If necessary, the casting speed is smoothed over time. Note that frequent

and/or high variations of the casting speed should be avoided. The decomposition of the problem into the subproblems of (1) determining the casting speed for given start dates and (2) determining the start dates for given casting speed is a heuristic that lacks any guarantee of optimality, but in our experience it works well in practice. Simultaneous optimization of speed and start dates is an important topic for future research (see Krappinger 2004 for an approach using Lagrangean techniques and dynamic programming).

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18.6.3 Scheduling of All Facilities Upstream of the Continuous Casters The starting point is the planned start date of each charge at the continuous casters (from level 1). The scheduling level (1) allocates the charges to facilities, (2) sequences the charges on the facilities and (3) determines preliminary start and finish dates for each operation. It considers the following complicating factors: (1) parallel facilities (including cranes), (2) routing preferences, (3) multiple resource constraints (limited availability of circulating ladles and hot metal) and (4) “frozen” decisions from the previous schedule. Due to the complex nature of the problem and the importance of decision rules for allocation and sequencing, a sequential scheduling heuristic is applied that schedules one operation in each iteration. If conflicts occur (i.e., if operations would overlap if they were assigned to their preferred facility at the preferred time), these conflicts are resolved either by assigning operations to an alternative facility or by preferring one operation and by delaying other operations. This conflict resolution is governed by a set of rules. In the following, we first describe the scheduling procedure; then we outline the essential logic of the rules.

18.6.3.1 The Scheduling Algorithm The scheduling procedure is based on the Giffler/Thompson algorithm (Giffler and Thompson 1960), which is a construction heuristic for job shop scheduling. It yields a schedule that is active and hence efficient with respect to regular objectives. The method can also be used as a basis for metaheuristics (Storer et al. 1992). Since it schedules operations successively and without considering the continuity constraints at the continuous casters, the procedure is applied iteratively backward and forward in time. The algorithm for backward scheduling is depicted in Fig. 18.6. For forward scheduling, it uses basically the same logic with reversed time. Scheduling is performed in the following steps: Step 1: Backward scheduling of all charges/operations based on the target CC

start dates. If this yields a feasible schedule, then stop, else (e.g., if some operations are in conflict with frozen operations or are scheduled in the past) go to Step 2 (see Fig. 18.7). Step 2: Forward scheduling based on feasible start dates of the first operations (Fig. 18.8). If cast breaks occur (i.e., the finish dates of the last refining operations do not allow the continuity constraints to be met), forward scheduling can be repeated several times with adjusted casting speeds. Backward and (repeated) forward scheduling is performed iteratively until the schedule is stable, that is, it remains (nearly) unchanged.

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Parameters: Number of ladles=P Target schedule for CC's Initialize: pool of ladles = {all ladles} (P ladles) Initialize: Set of available Operations = the P casting operations with latest target due date

Schedule the P casting operations with latest target due date at their desired times; allocate the ladles from the due date backward pool of ladles = {} Set of available operations = {all predecessors of scheduled casting operations}

Set of available operations = {}?

yes Stop

No Determine a preferred facility for each operation (according to current situation and scheduling rules)

For all facilities which are preferred facility: Select the operation with longest "backward waiting time" (target due date minus latest due date). Result: One operation for each faciltiy

Select the faciltiy with latest due date of the selected operation. Schedule this operation on the selected facility. Remove this operation from the set of the available operations.

Add predecessor to set of available operations.

no

Scheduled operation on Conv., CC or ladle rework?

yes Add predecessor to set of available operations. Update ladle pool.

Fig. 18.6 Scheduling algorithm for backward scheduling

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Conv 7 Conv 8 Conv 9 infeasible CS Standard route occupied

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Fig. 18.7 Backward scheduling

Conv 7 Conv 8 Conv 9 CS

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Fig. 18.8 Forward scheduling

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18.6.3.2 The Rule Base of the Scheduling Algorithm Allocation Rules Deciding on the allocation of an operation to alternative facilities in many cases is not a local decision for this specific operation, but it determines or indicates the choice of the routing of the charge – whether the standard route, an alternative route or even an emergency route is selected. For each charge, one routing has to be selected from the alternative routings. Hence, the allocation rules must consider the possible routes and the preferences of the dispatchers. The preferred routing for a specific situation is determined by technical preferences, but it also depends on situational factors such as the load situation at the facilities, downtimes of production facilities or cranes, and has been selected by “experience” using extensive knowledge and scheduling rules resulting from this largely undocumented knowledge. Automated scheduling requires the representation of at least the most important part of this knowledge. Knowledge acquisition and representation is a complex topic, which cannot be discussed here. In this project, knowledge acquisition was performed by means of interviews and extensive cooperation with planners and steelmaking experts and by analyzing prototype schedules. For the representation of the knowledge on route selection, sophisticated techniques such as fuzzy logic have been avoided (for fuzzy approaches in scheduling, see Schultmann et al. 2006), instead a rule-based approach has been adopted. For each steel grade (which is known for each charge), route preferences are defined as master data that determine the standard route and alternative routes (Fig. 18.9). This definition is (1) backward (if operation n is allocated to a certain facility, then the preferences for the allocation of predecessor operation n 1 to alternative facilities are defined), (2) without “backward-memory”, that is, the above definitions LD Converters Conditioning Ladle Stand Furnaces

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Fig. 18.9 Route preferences for a steel grade (example)

Continuous Casters

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are independent of the allocation of the successor operation n C 1. Since a crisp (in contrast to a fuzzy) representation of the preferences is required, the preferences are represented as time limits: for instance, an operation should be allocated to facility m, but it is allocated to alternative facility i if this reduces the waiting time by more than x minutes. The route preferences and the time limit determine the preferred facility in the scheduling algorithm (Fig. 18.6) for the specific situation. In the context of the sequential scheduling heuristic applied in our case (see Sect. 18.6.3.1), the route preferences yield priorities that determine the predecessor facility provided that the successor facility is known. Hence, these priorities can be used for backward scheduling and are termed backward priorities. Since we also need priorities for forward scheduling that determine the successor facility provided that the predecessor facility is known, the backward priorities have to be transformed into forward priorities (we use the analogous term) in a consistent manner (i.e., in a way that yields identical schedules from backward and forward scheduling unless this is prevented by constraints). This problem is very specific, and we have been unsuccessful in finding a solution in the literature. Thus, the following procedure is used for each steel grade (we omit the index denoting the steel grade): The proportion of the charges assigned to a facility m for which a certain preb decessor facility i is chosen (termed backward proportion pim ) is calculated as a function of the time limit that models the preference (facility i is chosen instead of the standard facility i ’ if the delay at i ’ would be longer than the time limit). This function must be non-increasing and is estimated from empirical data. This b that a charge that is bound for facility m is yields the backward proportion pim allocated to predecessor facility i (see Fig. 18.10). f Calculation of the forward proportion pim that a charge that is processed by facility i is next processed by facility m. First, the relative throughput rates

Backward proportion

b pim

Time limit Time limit

Fig. 18.10 Calculation of the proportion of the charges for which a certain predecessor facility is chosen

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of the facilities (i for all i ) are calculated using the following set of linear equations: X b m pim 8i i D m2Ni

where Ni denotes the set of possible successor facilities of facility i . The output of facility i bound for facility m must be equal to the input to m arriving from i : f

b i pim D m pim

8i I 8m 2 Ni

which yields: f pim D

b m pim i

8i I 8m 2 Ni :

Calculation of the time limit that models the forward priority using the inverse

of the functional relationship between the time limit and the proportion of the charges for which a certain route is chosen (Fig. 18.10). This requires some adjustments of the results that are not described here. Note that this inversion – and hence the procedure as a whole – is not based on a sound theory, but is heuristic in nature.

Sequencing Rules Once the allocation decision is made, sequencing of the operations on the facilities is less complex due to the flow shop structure. In many cases, the sequence of charges that are cast on the same continuous caster is the same as their sequence on the continuous caster. The complexity results from two factors: First, conflicts between charges from different continuous casters can occur. Second, different operation times at the degassing units and/or two refining operations on certain charges can require a different sequence on these operations compared to the CC sequence. We apply the FCFS sequencing rule: In forward scheduling, the operation with the longest waiting time (earliest possible start date minus target start date) is preferred. Analogously, in backward scheduling, the operation with the longest backward waiting time (target due date minus latest possible due date) is preferred.

18.6.4 LP Improvement: Adjusting of Timing and Casting Speed Up to this point, a preliminary schedule is determined. But the scheduling level 2 only partly considers the interdependencies of the charges due to the casts, and the LP improvement that simultaneously considers operations of the charges and

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their interdependencies (due to the casts) corrects this. The starting point for this procedure is the charge allocations and sequences from the scheduling level 2. The LP improvement level decides on (minor) adjustments of timing and casting speed in order to (1) improve the schedule, (2) eliminate infeasibilities (short cast breaks) that can result from level 2. Due to the complex material flow structure and the multiple resource constraints, the details of the LP formulation are rather cumbersome (see Missbauer et al. (2009) for the model; the formulation is similar to Tang et al. 2000a). The basic structure can be depicted as follows: The essential variables are the start dates of the operations and the casting times: startj;m : : : : : : : : : :: start date of charge j at facility m castTimej : : : : : : :: duration of the casting operation of charge j The essential sets of constraints are the following: Start dates of the operations at the facilities (production sequence, seen by the

facility): startj;m C OpTimej;m startS.j;m/;m

8j I 8m performing operations on j;

where OpTimej;m denotes the operation time of charge j at facility mI S.j;m/;m the index of the charge following charge j at facility m. Routing of the charges startj;m C OpTimej;m C wj;F .j;m/ D startj;F .j;m/ 8j I 8m performing operations on j; where wj;F .j;m/ denotes the waiting time of charge j at facility F .j; m/ that performs the next operation. Figure 18.11 illustrates the structure of the LP model with the help of a Gantt chart. A number of additional constraints are necessary to model the specific technology of the continuous casters (cast breaks, setups, tundish change, constraints on casting speed) and the flows of scrap, hot metal and ladles. The objective function is the weighted sum of finish dates of the orders, flow times (of hot metal, scrap and charges), duration of cast breaks and a suitable measure of variations of the casting speed. The LP encompasses about 10,000 variables and 8,000 constraints depending on the actual planning horizon and is solved by Simplex or – beyond about 1,800 constraints – by an interior point method. Note that the computer runtime is critical here, because scheduling is a real-time activity and the LP model consumes about 60% of the total computer runtime of the three planning levels (which is about 35 s as of May 2006). Figure 18.12 depicts a schedule for the steel plant as it is displayed to the planner. Comprehensive information can be displayed on the screen when needed.

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Fig. 18.11 Illustration of the LP structure (transportation ignored for simplicity)

Fig. 18.12 Schedule displayed to the user

The planning horizon is usually 72 h for all planning levels, scheduling all resources (including cranes; this turned out to be necessary) in detail. The frozen horizon (with respect to allocation and sequencing; timing is not relevant) is about 2 h, mainly due to material already available at the facilities. Schedule updates are performed in certain intervals (a few minutes) or manually, if necessary.

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18.7 Implementation Issues When a scheduling system is implemented in a plant with the characteristics described, it is of utmost importance to recognize that manual scheduling is based on extensive expert knowledge and has been working for many years. The implementation of an automated scheduling system is an intervention into a planning system that works properly, and failures in the planning method/rules or in the implementation can make things worse. Great care must be taken to prevent such failures and to integrate the scheduling system into the plant organization and into the decisionmaking process of the planners. In the following, these two issues are highlighted.

18.7.1 Integration Into the Plant Organization The concept of the scheduling system was developed in 1996/1997, and detailed system design and programming was carried out in 2002–2004. The final implementation phase, including extensive support of the planners by the development team, lasted about 3 months in Spring 2004. The long duration indicates that the transition from manual to computer-assisted scheduling in such a complex environment is a major change. It affects numerous processes in the plant and inevitably leads to doubts and concerns that must be taken seriously and discussed with care. In the manual scheduling system, there was not just one planner (“the” scheduler): The dispatchers at the continuous casters established the CC schedules. This was the basis for determining the start dates at the converters, which was done by the schedulers there. The CC schedule and the converter start dates largely determine the entire schedule (the jobs of dispatchers and the schedulers also encompass a number of technical tasks that are not described here). In contrast to this “distributed decision making” (Schneeweiss 2003), a “Leitstand” like the one developed here essentially assumes a single “planner”. To our knowledge the theoretical questions related to this change are largely unsolved (e.g., centralization of the planning process vs. coordination of several planners; who should have a dominant position; where to provide slack in the schedule that allows adjustments at the facility level). Since these changes are difficult to design and even to predict, the experiences from the implementation are extremely valuable. It turned out that the dispatchers and the schedulers design and maintain the schedule collaboratively. The collaboration, as is often the case, is largely informal. At the moment, it is too early to say which division of labour will emerge in the end.

18.7.2 Integration with Human Problem Solving Decision making as an organizational process, that is, which person or organizational unit makes which decision, what is the source of the relevant data, to which

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person or organizational unit are the results transmitted, etc., can be explored using the well-known techniques of information systems analysis. In contrast, it is frequently observed that the decision-making algorithms or rules actually applied by an individual or a group can be explored only with great difficulty, if at all. Nevertheless, the scheduling system must be linked with the decision-making process of the planners (instead of simply replacing it) for the following reasons: (1) The rules applied by the planners contain information about the planning problem that cannot be obtained in another way. Note that this argument stresses the evolutionary aspect of manufacturing planning and control systems that can be regarded as an adaptation to the complex details of the planning and control problem. (2) Disturbances in the manufacturing process require immediate action. For instance, if a charge is delayed due to a technical problem, the cast can be slowed down to bridge the delay and avoid a cast break. This can only be done by the planners (the system is too slow and its knowledge base presumably is insufficient), which means that the planners must understand the schedule produced by the system, including the interdependencies of the operations and their timing, at each moment. The importance of this aspect is confirmed by the observation that in the manual scheduling system decision rules have been established that may have led to a deviation from optimality, but made the schedules more transparent. For instance, the following structure of the material flow had been established: Converter a feeds CC x, converter b feeds CC y. Since both “virtual flow lines” were unbalanced (CC x is faster, CC y is slower than the converters), the converter b produced a charge for CC x at certain intervals. The start dates of the charges at the converters had been adjusted to this policy. This indicates that optimality and transparency of a schedule may be conflicting goals, and the limited ability of the planners to understand all relationships in a schedule (“bounded rationality” according to H. Simon) can impose restrictions on the optimality (defined in the usual sense) of a schedule. We are not aware of any decision support on this aspect from the scheduling literature; it seems to be a topic largely neglected by research. These topics have been considered in the following ways: The decision rules concerning assignment of charges to facilities and frozen hori-

zons (which decisions should be frozen for which horizon) have been explored carefully and cross-checked with other experts. This has also been a part of the implementation process. The goal is a set of rules that require manual changes of the schedules only in special cases. Due to the complex production technology, it has not always been easy to distinguish whether an established rule is necessary due to technical reasons or it is more of a routine. Informal procedures such as changes in order pegging are supported. The system supports manual scheduling and manual changes/adjustments to the schedule. The extent and frequency of manual interventions is an important indication of the quality and acceptance of the scheduling algorithm (in the extreme case, the system is just an “electronic Gantt chart” for manual planning). After more than 5 years of experience, it can be concluded that the schedules generated by the system are well accepted, although some manual modifications can be necessary.

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The system supports integration of specific constraints to the material flow (e.g.,

Converter a feeds CC x) into the rule base of the scheduling algorithm. This can also compensate for shortcomings in the scheduling rules. Estimating the effects of the scheduling system on the performance, especially on the output of the plant, is important for the economic justification of the system and of the respective changes in the organization. Methods available for ex-ante estimation of the effects are often insufficient (see Vollmann et al. 1997, p. 394 ff. for this topic). Even a comparison of optimization results with real-life data can be (and in our case is) invalid because it is hardly possible to consider the effects of disturbances and rescheduling in the calculation. In our case, an ex-post estimation of the effects on the performance indicators is difficult because of capacity expansions that were installed at the same time. Taking into account comparisons of optimization results with real-life data and experiences with a scheduling system in a similar steel plant, an increase in output of about 1% is estimated. A second positive effect is the improved transparency of the material flow especially during the change of shifts.

18.8 Conclusions This article described a successful implementation of a computerized scheduling system for a SM-CC process. Detailed schedules are generated by an algorithm that combines a rule-based and an optimization-based approach. After more than 5 years in use, it can be concluded that the system works well and is accepted by the planners. Applying scheduling theory in this practical case leads to a number of interesting insights that should stimulate future research. Most important are the following topics: Structuring complex practical problems and deciding which concept of decision

support (e.g., organizational planning hierarchy, automatic scheduling yes/no) is a difficult process, and much of the scheduling theory is based on a pre-defined structure. Perhaps, the so-called “theory-practice-gap” in fact is a lack of theory on problem structuring and model building. Despite the extensive literature on scheduling in manufacturing systems there is considerable need for scheduling methods in special environments such as the steel industry. Especially when practical applicability (which also means: maturity) of the research is required, the problems are far from being solved. Scheduling algorithms – and Operations Research models in general – aim at obtaining optimal solutions. This can lead to complex solutions and, as in our case, to solutions that are difficult to understand for the decision makers. Handling this conflict is at the interface of mathematical optimization and human problem solving and is a difficult research topic. The scheduling task in practice encompasses much more than optimizing a schedule at a certain time. Integrating the optimization into the scheduling task, both at the level of the plant organization and at the level of the individual

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decision maker, is extremely important. In our experience, this topic largely determines success or failure of such a project – probably to a larger extent than the algorithm, whose development is, although difficult, a well-defined problem.

References Adelsberger H, Kanet JJ (1991) “The Leitstand – a new tool for computer-integrated manufacturing.” Prod Inventory Contr J 31(1):43–48 Allahverdi A, Ng CT, Cheng TCE, Kovalyov MY (2008) “A survey of scheduling problems with setup times or costs.” Eur J Oper Res 187:985–1032 Alter SL (1980) Decision support systems: Current practice and continuing challenges. Addison Wesley, Reading, MA Arnott D, Pervan G (2005) “A critical analysis of decision support systems research.” J Inform Technol 20:67–87 Baosteel (n. d.) “Software Design for Mill Pacing. Baosteel Technical Report”. Published on http:// www.baosteel.com/english n/e07technical n/tech020802.htm. Retrieved on 28.04.2007 Bellabdaoui A, Teghem J (2006) “A mixed-integer linear programming model for the continuous casting planning.” Int J Prod Econ 104(2):260–270 Bellabdaoui A, Fiordaliso A, Teghem J (2005) “A heuristic algorithm for scheduling the steelmaking continuous casting process.” Pacific J Optim 1(3):447–464 Belt B (1976) “Integrating capacity planning and capacity control.” Prod Inventory Manag 17(1):9–25 Bertsimas D, Gamarnik D, Sethuraman J (2003) “From fluid relaxations to practical algorithms for high-multiplicity job shop scheduling: the holding cost objective.” Oper Res 51(5):798–813 Bertsimas D, Sethuraman J (2002) “From fluid relaxations to practical algorithms for job shop scheduling: the makespan objective.” Math Program Ser A 92:61–102 Chen H, Mandelbaum A (1994) Hierarchical modelling of stochastic networks, part I: Fluid models. Stochastic Model Anal Manuf Syst Yao, David D. Springer, New York et al Cowling PI, Ouelhadj D, Petrovic S (2004) “Dynamic scheduling of steel casting and milling using multi-agents.” Prod Plann Contr 15(2):178–188 Cowling PI, Rezig W (2000) “Integration of continuous caster and hot strip mill planning for steel production.” J Sched 3:185–208 Giffler BL, Thompson GL (1960) “Algorithms for solving production scheduling problems.” Oper Res 8:487–503 Günther H-O, van Beek P (2003) Advanced planning and scheduling solutions in process industry. Springer, Berlin Harjunkoski I, Grossmann IE (2001) “A decomposition approach for the scheduling of a steel plant production.” Comput Chem Eng 25:1647–1660 Huegler PA, Vasko FJ (2007) “Metaheuristics for meltshop scheduling in the steel industry.” J Oper Res Soc 58(6):791–796 Krappinger R (2004) Maschinenbelegungsplanung bei der Stahlerzeugung – Ein hierarchischer Ansatz auf Basis der Lagrange-Relexation und der dynamischen Optimierung. Thesis, University of Innsbruck, Austria (in German) Linn R, Zhang W (1999) “Hybrid flow shop scheduling: a survey.” Comput Ind Eng 37:57–61 McKay KN, Buzacott JA (2000) “The application of computerized production control systems in job shop environments.” Comput Ind 42:79–97 McKay KN, Saffayeni FR, Buzacott JA (1988) “Job-shop scheduling theory: what is relevant?” Interfaces 18(4):84–90 Meal HC (1984) “Putting production decisions where they belong.” Harv Bus Rev 102–111

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Missbauer H, Hauber W, Stadler W (2009) “A scheduling system for the steelmaking - continuous casting process - A case study from the steel-making industry.” Int J Prod Res 47(15):4147– 4172 Missbauer H, Zäpfel G, Hauber W (1998) “A progressive figure based production planning system for a component manufacturer.” Int J Prod Econ 56–57:463–481 Ouelhadj D (2003) A multi-agent system for the integrated dynamic scheduling of steel production. Ph.D. Thesis, University of Nottingham Petersen CM, Sørensen KL, Vidal RVV (1992) “Inter-process synchronization in steel production.” Int J Prod Res 30(6):1415–1425 Pinedo M (1995) Scheduling. Theory, algorithms, and systems. Prentice Hall, Englewood Cliffs, New Jersey Schneeweiss Ch (2003) Distributed decision making. 2nd edn. Springer, Berlin Schultmann F, Fröhning M, Rentz O (2006) “Fuzzy approach for production planning and detailed scheduling in paints manufacturing.” Int J Prod Res 44(8):1589–1612 Steele DC, Philipoom PR, Malhotra MK, Fry TD (2005) “Comparisons between drum-buffer-rope and material requirements planning: a case study.” Int J Prod Res 43(15):3181–3208 Storer EH, Wu SD, Vaccari R (1992) “New search spaces for sequencing problems with application to job shop scheduling.” Manag Sci 38(10):1495–1509 Systems AI (1995) BetaPlanner product specification. A. I. Systems, J. Wybran Ave., Brussels, Belgium Systems AI (1997) AlphaPlanner product specification. A. I. Systems, J. Wybran Ave., Brussels, Belgium Tang LX, Liu JY, Rong AY, Yang ZH (2000a) “A mathematical programming model for scheduling steelmaking-continuous casting production.” Eur J Oper Res 120:423–435 Tang LX, Liu JY, Rong AY, Yang ZH (2000b) “A multiple traveling salesman problem model for hot rolling scheduling in Shanghai Baoshan Iron & Steel Complex.” Eur J Oper Res 124:267–282 Tang LX, Liu JY, Rong AY, Yang ZH (2001) “A review of planning and scheduling systems and methods for integrated steel production.” Eur J Oper Res 133:1–20 Tang LX, Luh PB, Liu JY, Fang L (2002a) “Steel-making process scheduling using Lagrangian relaxation.” Int J Prod Res 40(1):55–70 Tang LX, Liu JY, Rong AY, Yang ZH (2002b) “Modelling and a genetic algorithm solution for the slab stack shuffling problem when implementing steel rolling schedules.” Int J Prod Res 40(7):1583–1595 Vollmann TE, Berry WL, Whybark DC (1997) Manufacturing planning and control systems, 4th edn. McGraw Hill, New York Wight OW (1974) Production and inventory management in the computer age. Boston Zäpfel G, Missbauer H (1993) “New concepts for production planning and control.” Eur J Oper Res 67:297–320 Zhu X, Wilhelm WE (2006) “Scheduling and lot sizing with sequence-dependent setup: A literature review.” IIE Trans 38:987–1007

Chapter 19

A Multi-Model Approach for Production Planning and Scheduling in an Industrial Environment Abdelhakim Artiba, Valerie Dhaevers, David Duvivier, and Salah E. Elmaghraby

19.1 Introduction There are many domains across industry and commerce that are of such complex nature that current decision-support approaches and tools are inadequate due to the (often drastic) simplifications required to enable analytical treatment. This can only get worse as emerging technologies and practices make the problem domains even more complex. Examples include supply chains and logistics, production planning and scheduling, business process re-engineering, and multi-modal transportation. It is our thesis that decision making problems must be approached with several tools. Too often, these tools are concerned with either a methodology (such as mathematical programming) or solution tools (such as a particular software), with only few conceptual aspects (language for representation, integration, etc.) that tie them together. This has created a ‘Tower of Babel’ situation where the users are confronted with a variety of incompatible tools, each with its own language, which is understood only by the high priests of the discipline. The lack of appropriate and integrated mechanisms has led many researchers to attempt to develop the so-called ‘multi-model systems’, combining, in one integrated application, not only different tools in the areas of analysis and optimization, but also different approaches. This chapter is another attempt in that direction. In this chapter, we propose a unifying approach for decision making, be it in the planning phase or in the real-time operational phase of the process. We label this approach the ‘multi-model approach’. It combines three main modes for the resolution of complex decision making problems in industrial environments: Analytical Models, heuristics/meta-heuristics, and Simulation. This is a nascent research topic in manufacturing, logistics, enterprise modeling, and supply chains. The ultimate objective is to develop a unifying schema, together with the language that implements it, that integrates different concepts relating to different solution methodologies and

A. Artiba () Professeur des Universités, Vice-Président Recherche et Valorisation, Université de Valenciennes et du Hainaut Cambrésis, UVHC – LAMIH, F-59313 Valenciennes Cedex 9, France e-mail: [email protected]


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different modes of analysis in a coherent information-decision-verification system. In this approach, the different steps of analytical modeling, optimization, approximation, and simulation use the same integrated formalism and environment. An important feature of this approach is the originality of the decision-making mechanism over a hierarchy of complex discrete systems that model the real world problem. This chapter is based on an application of this scheme of multi-model systems to a plant in Belgium which produces high quality steel wires and cables. Production is organized as a ‘hybrid flowshop’ composed of seventeen machines organized into seven stages with multiple machines at several stages (hence the ‘hybrid’ designation). The process is composed of discrete and continuous operations. Several constraints make this discrete-continuous planning and scheduling problem hard to tackle. For instance, preemption is not allowed; the need for different auxiliary resources to support production; sequence-dependent and sequence-independent setups; the machines in the same stage are not identical, with some machine(s) able to process some, but not all, products, etc. The product is made-to-order; whence the process has to satisfy, at the lowest possible cost, irregular demand of many customers at high quality requirements, taking into account the availability of raw materials and auxiliary services (delivery, maintenance, etc.). Careful study of the prevailing culture at the production facility convinced us that the control of production must be conducted in three interleaving phases. The first is a loading phase which selects, from a long list of customer demands, the appropriate products to produce over the specified planning horizon. The second is a scheduling phase in which the selected load is sequenced over various stages of the facility. The third is a verification/correction phase which takes into account all the real life constraints and requirements (such as allowance for workers personal times, shift changes, materials handling constraints, etc.) that were assumed away in the previous two phases. The loading model is proved to be a straightforward linear program (LP) of reasonable size which takes into account the value of each order placed on the plant, its due date, its processing requirements, and the available shop capacity at various stages of production. The LP can be solved rather efficiently and quickly by commercially available software. On the surface, the scheduling problem appears to be a straightforward application of standard theory on hybrid flowshops (most of the stages containing several machines in parallel). However, the real life situation confronted us with two unique characteristics, which make it stand apart from other common scheduling procedures. First, the machines available at any stage were not equally able to process all the jobs that require processing at the stage – in fact, some machines are specialized and can process only a subset of the products. Second, the product (steel wire/cable) is conveyed from one phase of production to another wrapped on ‘bobbins’. These are large drums with a high rim, which vary in size and capacity; each bobbin is capable of accommodating a rather narrow range of the material. All bobbins share two important characteristics: they are bulky and they are expensive (to purchase, handle, maintain, and store). Therefore, to conserve space and minimize cost, it was

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mandatory that any proposed scheme for scheduling the production activities over various stages must attempt to minimize the number of these bobbins, apart from their proper use through careful scheduling of their loads to achieve maximal output. The scheduling scheme should also take into account the fact that any bobbin is loaded at some stage and is returned empty to its starting stage at the end of processing its load at a lower stage along the production line. The construction of the mathematical model was a challenge, as different jobs require different size bobbins to carry them between the production stages, with the possibility of the cable on several bobbins at the start of a stage to be combined (braided) into a single cable that is carried on one bobbin at the end of the stage. The picture is further complicated by the fact that a bobbin has finite capacity. This limits the size of the ‘job’ it carries, which, in turn, causes an order for a particular product to be split among a number of bobbins, each representing a job, with the added requirement that the set of jobs belonging to the same customer order should be processed as close to each other as possible throughout the process to facilitate subsequent inspection and shipping operations. The different models used at the different levels of decision are described below, as well as the interaction mechanisms allowing the collaboration process over these methods. The multi-model system was designed to take capacities into account at each level of the production process to avoid generating infeasibility between decision levels due to the finiteness of available capacities. These levels have also been designed so as to avoid incompatibilities between the decision levels, taking into account unequal granularities in terms of time horizons. In other words, we tend to reduce the gap between the aggregate level in the long term and the detailed schedules in the short term. The scheme that we propose exemplifies our philosophy: It combines different methods that are constructed in such a way that they lead to the selection of the proper tool (analytical, heuristic, meta-heuristics, simulation), at the right place (depending on the required level of detail), at the right time (depending on the available amount of time for decision-making), to perform the required task so as to integrate the best performance/efficiency of each selected tool (see Fig. 19.1). It then proceeds to validate/correct the analytical models to take into account all the real life factors that were ignored or simplified in the analytical models. The simulation model we have constructed was also required to cope with random disturbances to the smooth flow of production in the shop, such as ruptures of wire and other operational malfunctions in the machines or in personnel. Moreover, when considering the flexibility of the lines of production, the scheduling strategy should keep the requisite inventory of empty bobbins at the start of each flow line so as to cope with the highly uneven demand. Our end result, which was implemented in the plant, is a system that exhibits greatly improved performance combined with robustness under stochastic occurrences compared to the man-made plans that were in place. Figure 19.1 gives a global view of the structure of the decision-support system and identifies its various constituents.

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19.1.1 Review of Literature Production of the immense variety of products of iron and steel is among the most important foundations of the modern industrial age; little wonder that there is a vast number of books, articles, and other forms of publication on the subject. We limit ourselves in these brief remarks to the most recent publications on topics that are intimately related to the main thrust of this chapter. The interested reader may consult the references cited in each of these papers for more in-depth discussions of particular topics. Serial production in which material flows linearly over various stages of the production process (with single or multiple facilities at each stage) is quite common; see for instance the paper by Park et al. (2002) which investigates and suggests an efficient solution to the problem of scheduling the steel making line in a steel mill consisting of three major processes: molten steel making, continuous slab casting, and hot charged rolling; and the paper by Li et al. (2006) which treats a production scheduling problem for hot rolling seamless steel tube in China. Considering the complexity of the problem and the acceptable time for solving it, a bottleneck-based heuristic scheduling method was devised that considers simultaneously all production processes in three production units and ‘optimizes’ them as a whole; see also the other contributions of Li and his co-workers (Li et al. 2005a,b) which are related to steel tube production and the references cited therein, and the paper by Zhu and Yu (2002). That complexity motivated Denton et al. (2003) to attempt to improve responsiveness by shifting from a pure make-to-order system toward a mixture of hybrid make-to-stock cum make-to-order system. Choosing which semi-finished products to make to stock and how to manage their inventory are difficult problems in their own right, which were resolved via the implementation of a decision-support tool for choosing the designs of made-for-stock slabs. Kalagnanam et al. (2007) developed a system that provides at 30 min intervals an operational level production design and schedule that is cognizant of the plant level facility and capacity constraints for a rolling horizon of two days.

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The concept of combining mathematical optimization with Monte Carlo sampling and simulation of the plant operation for purposes of improvement and validation, which constitutes a pivotal element of our approach, can be found in other recent approaches; see for instance the paper by Appelqvist and Lehtonen (2005) who provide for production scheduling in a steel mill in Finland an approach where only the main constraints are included in the scheduling algorithm but the schedule is subsequently validated using a discrete-event simulation model that includes additional detail. Their conclusion that “The optimization can be kept simpler as validation with a simulation model increases the credibility and accuracy of the resulting schedule” coincides with our experience. As will become clearer below in our detailed description of the production process, a central concern to us is the issue of the bobbins which are re-circulated to their respective origin stations after serving as carrier of the cable from one stage to another. This problem is not unique to cable manufacturing; it is quite common in all processes in which the mode of conveyance of the product is committed for one unit of the product for a short period of time after which it is released and is re-used for other products. The paper by Elmaghraby and Elimam (1984) describes the approach to the mathematical modeling and optimization of such a process in the context of the production of asbestos-cement pipes in which the mandrels play the same role as the bobbins in our application. By the very fact that our approach utilizes several approaches to resolve the production problem of concern, we direct the reader’s attention to the appropriate reference to the adopted technology/methodology at the point of its introduction.

19.1.2 System Architecture Ultimately, our system should be linked to a Manufacturing Execution System (MES) – the popular name of the execution tools for gathering information in real time. To the best of our knowledge, existing MES applications do not address all of the manufacturing execution processes required to replenish the supply chain while dynamically responding to unpredictable changes. Adaptive manufacturing strategies also require defining the process management layer between Enterprise Resource Planning (ERP) and process control, and translating demand-driven supply chain requirements into a set of capabilities, systems, and workflow integration investments. Therefore, adaptive manufacturing organizations need an execution platform that connects manufacturing processes with enterprise and supply chain processes, a platform that enables closed-loop mechanisms and provides decisionsupport to production personnel so that they can deliver on their performance goals. Up till now, the scientific literature has mostly addressed this issue by proposing different integration models between ERP and MES systems. While these ‘patch-up’ models may resolve some inter-operability problems, they fall short of adding decision-making tools capabilities to either system. We present here a different approach by proposing a generic manufacturing execution platform (dubbed

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‘PlanOrdo’1, described in detail below) where real time discrete event simulation models complement the traditional ERP/MES model. Intuitively, good decisions can improve one or more key performance indicators. In a complex system, it is usually difficult to calculate the impact of a decision on the overall system. In this context, using on-line simulation is attractive but requires that the simulation model be integrated into the enterprise information system landscape and the communication network. It must also combine multiple optimization techniques in order to support different decision-making scenarios. Our proposed architecture is presented in Fig. 19.2. The main objective of the architecture is to provide real time decision-support in response to unforeseen perturbations to the manufacturing environment. Such perturbations may occur at the top level (a new order) or at the bottom level (machine breakdown). The implementation of the real-time control modules of this architecture, while taking perturbations into account, is not discussed in this chapter. However, hereafter, we explain the logic and functional aspects of this function. The different software components are spread over several servers linked by local area networks and/or a global network. The connection to the real manufacturing processes is supported by OPC (OLE for P rocess C ontrol) clients connected to different OPC servers collecting data directly from the process. These clients take charge of raw data acquisition, preliminary storing, and processing. Depending on the storing/processing rules in an OPC, a client can store processed data in an SQL database, exchange the data via XML clients, and generate messages to other components if significant events so 1

For the French ‘planification’, meaning ‘planning’, and ‘ordonnancement’ meaning ‘scheduling’.

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require. SQL/XML clients or servers assure data exchange with external software, such as ERP, Web servers, Supervisors, etc. They can also generate messages to other components of the system on the basis of their configurations. The nucleus of our decision-making system is a System Dynamic Builder (SDBuilder) (Artiba et al. 1997); essentially a hybrid of discrete-event simulator, a set of rule-based expert systems, and a state-graph search-based optimization engine. In this application, we do not use the state-graph engine because we developed the LP and ILP models with LINGO (Lindo Systems Inc. 1999) whereas the heuristics do not need the use of this engine. The ideas lying at the base of the SD-Builder are partly inspired by the RAO method and simulator (Artiba et al. 1998). The system dynamics in the SD-Builder is based on an object oriented representation of the elements of the complete discrete system (CDS), and on a description of the state changes using production rules. The SD-Builder runs models designed for different parts of the controlled system and responds to incoming messages by initiating decision-making logic depending on the disturbing event and the current state of the system. At the beginning of the decision making process, the SD-Builder is initialized with data from the ERP (such as production orders, production sequence, etc.) and from the MES (such as order status, machine status, etc.). The SD-Builder can run simulation models, trigger an expert system inference process, initiate an optimization program using state graph search techniques, or just execute an appropriate rule from decision tables, among others. Simulation can also be used like a meta-heuristic support to evaluate the key performance indicators under a set of constraints or production rules. In its turn, it sends messages representing decisions made to other actors of the architecture including OPC clients to act on the controlled process. Other modules or systems can be integrated into this architecture for reporting, monitoring, mailing, or other purposes. Finally, a user interface is required to configure the system composition, address the modules, specify the data to be acquired, the message types and formats, and to monitor events. The user interface is also used to develop the calculation models, to define the key performance indicators, and to model the decision making logic itself. We consider the interpretability between the planning and execution levels, i.e., the reactive planning assuming accurate data relative to the current state of the production system. To achieve this goal, we develop hybrid optimization methods combining simulation and exact and approximate optimization algorithms capable of dealing with complex decision problems. At the execution level, we develop an optimization scheme capable of dealing with new real time technologies and ‘smart tag’ technologies such as Radio Frequency Identification (RFID). Other modules or systems can be integrated into this architecture for reporting, monitoring, mailing, or other purposes. We recognize that the joint optimization of production and maintenance operations planning is a complex undertaking that is still an open research field in industrial engineering. But there is ample evidence that planning the operations of production and maintenance as two separate entities is far from optimal and leads to wasteful efforts. Hence, our attempt to integrate the two functions.

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An incremental approach is used to master the fundamentals of optimization methods for the solution of appropriate classes of generic problems. Some of these production/maintenance problems are amenable to formal modeling and analysis, while others can only be defined jointly with industrial partners and approached via heuristic methods. The flexibility and robustness of the developed optimization methods and algorithms are primordial prerequisites for flexibility in application. Generic simulation models are then developed in parallel, as well as different integration schemes between solution methodologies and specific classes of problems. Advanced programming techniques to improve/optimize production planning and scheduling are herein used to achieve pre-defined objectives. The MES works at a more detailed level in order to provide efficient production tracking and data acquisition, reflecting the actual status of material flows and resources. This allows making small adjustments in the scheduling and planning decisions in order to react to deviations from the planned production and to unexpected or unplanned production orders. As the data acquisition application of the MES can be integrated with the planning, scheduling, and maintenance functions, decision-making methods will be improved in order to optimize, at the same time, the production scheduling and the maintenance planning, taking into consideration not only the real time stop/start events of machines but also the pattern of breakdowns based on historical data.

19.1.3 Chapter Organization The discussion proceeds as follows. We first describe in Sect. 19.2 the real life factory that motivated this study. Section 19.3 describes the loading model and Sect. 19.3.1 the scheduling models. Section 19.4 details the experimental studies that were made prior to the actual implementation of the multi-model system. The implementation proper is described in Sect. 19.5; and the chapter concludes with Sect. 19.6 which summarizes the lessons learned from this experience and the directions in which our current research/development efforts are directed.

19.2 The Plant Description The plant that is the object of this study produces high quality steel wires and strands that are used in pre-stressed concrete and major civil engineering projects (such as oil rigs, bridges, airports, stadiums, and hotels) all over the world. The plant is organized as a hybrid flowshop composed of seven stages (sets of machines), see Fig. 19.3: 1. 2. 3. 4.

Cleaning, one machine Stretching, five machines Stranding, three machines Stabilization, five machines

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Fig. 19.3 The plant flow layout

5. Packaging, five machines 6. Sheathing, one machine, and finally 7. Packing, two machines Continuous and discrete operations are linked thanks to the intervening soldering and cutting operations in order to store semi-finished products on bobbins. Thus, stretching, stranding, and stabilization are continuous operations while the remaining operations are discrete in nature. Production is divided into four main families according to the external diameter of the wire: (1) Monofil, or single wires, (2) PT2 , or small size strands with external diameter less than 9.30 mm, (3) MT3 , or medium size strands with external diameter between 9.30 and 11.30 mm, and (4) GT4 , or large size strands with external diameter 11:30 mm. These products are obtained from bare wires and galvanized wires. Each product is composed of a sequence of operations that must be scheduled with respect to resources and precedence constraints. The resource constraints include input/output limitations, setups, and non-preemption (an operation cannot be stopped to start another one). The precedence constraints simply require that no operation can begin before the end of all the preceding operations. The duration of an operation depends on both the product and the chosen machine. The production of the shop is not cyclic, and demand variation from one month to another is significant. In this make-to-order process, several kinds of setup are present: sequence-dependent, sequence-independent, and weight-proportional setups are encountered. Additional considerations add to the complexity of the production function; they include: the use of unique support facilities; maximum/minimum input/output diameter and maximum input/output load; not all products are compatible with all machines; not all lines are able to make all products or perform packaging; some orders cannot be late (due to severe external transportation constraints, for instance), etc.

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The plant works on a three-shift basis except for weekends. One of the important aspects of our work is that we take into account auxiliary resources because of the crucial role they play in the total production effort. In this plant, the most important auxiliary resource is that of bobbins that convey the wire/cable from one operation to another: lack of bobbins of the requisite variety at any stage stops production of at least one family of products in the whole line; hence a scheduling strategy should provide bobbins when and where they are required without oversupply (due to their cost and volume). Figure 19.4 gives a simplified view of the flow of bobbins between various stages of production. The objectives, as expressed originally by the plant management are several: 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

Maximize the added value of goods produced Maximize throughput in terms of tons/period Respect the different levels of priority of customers’ demand Fulfil due dates on time with no penalty for earliness or tardiness Minimize the lead time between order receipt and order shipment Maximize machine utilization (load balancing of lines, the ratio of setup to production times) Maximize the flexibility of the production lines. As much as possible keep empty bobbins so as to accommodate uneven demand Minimize the use of temporary workers Minimize the quantity and value of in-process inventory Minimize setup time

Objective 3 is more fittingly considered as a constraint, and we shall consider it as such. Objective 4 is reflected in the criterion function in which both earliness

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and tardiness are penalized. Objective 5 and 6 are realized when throughput is maximized; hence the latter may be considered as proxy for these two. Objective 8 is dominated by objective 6 since optimizing the load balance among the lines leads to a decrease in the use of temporary workers; hence the maximization of throughput can also be considered as a surrogate for it. The size of the in-process inventory is limited by the number of available bobbins in the plant, and therefore, objective 9 may be absorbed in the proper planning of the bobbins used. The correct scheduling of the bobbins now replaces it as an objective. This discussion whittles down the criteria to the following five concerns:

Throughput (tons/period) Capacity utilization (setup/production times) Value of goods produced Manufacturing interval Quantity and value of in-process inventory

It is clear that purely analytical models are simply incapable of satisfying all five criteria simultaneously. This partially explains our resort to a multi-model approach in which sections of the total process are optimized either through formal mathematical programming models or via heuristic and meta-heuristics procedures as detailed below, while the interaction among the various components of the plant under more realistic (such as random) conditions taking all objectives into account is taken care of using simulation.

19.3 The Loading/Scheduling Model In the following model, we shall use throughput as the sole measure of performance. We present the final version that was in fact implemented in the plant; the reasons for discarding prior versions are explained in Sect. 19.5 on implementation. Careful study of the flow of material in the plant revealed that the stranding stage may be considered as the bottleneck stage in all flow lines. Reality is a bit more complicated than this. True, in most cases, the bottleneck is effectively the stranding stage; however, in some cases, depending on the list of outstanding orders, the stretching stage is the bottleneck. In this study, we shall consider the stranding stage to be the only bottleneck stage, and the analytical models we propose concentrate on this stage only. But the mathematical model(s) presented are able to optimize the stretching stage in a similar way. As a consequence, loading and scheduling the other six stages are adjusted to comply with the needs of the stranding stage. There is also a possibility (not described here, since we focus on the target plant) to group similar orders before starting our models. Then, the starting dates (and also due dates) are adjusted via the mathematical models. And finally a priority (or a ranking) is computed by the dispatching rules. The simulation models use this priority vector to select the orders to be performed on the machines. The cleaning

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and stretching operations shall be referred to as the ‘upstream’ operations, and the stabilization, packaging, sheathing, and packing operations as the ‘downstream’ operations. In order to estimate the overall production time, it is necessary to build an upstream model that computes the effective starting dates, adjusted from the list of orders, that takes into account the durations of the upstream stages (cleaning and stretching). For similar reasons, it is necessary to build a downstream model to determine the delivery dates. In the current application, we assume for simplicity that these two models (upstream and downstream models) are simple linear approximations obtained by simple linear regressions on the concerned stages. However, more accurate/sophisticated models are under construction and will be used instead in future versions of our framework. The Loading model is a planning model under finite capacity; it is used to check the correspondence between the demand and the (maximum) load of the shop. The stranding model is used to perform detailed planning on the stranding stage. It might be used at two levels: it is used to compute bounds over several criteria so as to refine the stopping rule of our meta-heuristics; and it is also used to refine starting/ending dates to avoid overloading the shop. We are using here the model with the V -matrix (to be explained presently) corresponding to a planning horizon of T periods of HPTP (hours per time period); in this application HPTP D 8 h. The granularity of this model, defined by the value of HPTP, may be reduced so as to obtain detailed schedules for shorter periods, or increased to obtain plans for longer periods. Processing job j on machine m is of duration p.j; m/. Let V .j; m; t/ be a vector of 0s and 1s in which there are exactly p.j; m/ 1’s starting at t, denoting that job j is initiated on machine m in period t. 0

1 p.j;m/ ‚ …„ ƒ B C V .j; m; t/ D @0; : : : ; 0; 1; : : : ; 1; 0; : : : ; 0A : " t

(19.1)

Let V .j; m; t; k/ denote the kth component of the vector V .j; m; t/. Then, V .j; m; t; k/ D 1 for k D t; : : : ; t C p.j; m/ 1, otherwise V .j; m; t; k/ D 0. Using this time-indexed formulation, the successive rows of the V -matrix are in fact the successive possible schedules (starting times) of the job j on machine m starting in period t and ending in period t C p.j; m/ 1 (it is also assumed that machine m is always available at this point when building the V -matrix). The successive rows are the successive possible schedules of job j on machine m. This matrix is automatically generated by PlanOrdo before launching Lingo in quest for a solution of the mathematical model. It is possible to include periodical machine unavailability (week-ends, holidays, maintenance) using a calendar defined via the Graphical User Interface (GUI) of PlanOrdo. The detailed description of the Stranding Model that was finally implemented in the plant shall be postponed till Sect. 19.5.1 in the discussion of Implementation in order to avoid going through the history of various models that were developed then discarded for one reason or another.

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19.3.1 The Stranding Stage Recall that the stranding stage contains three machines. Denote the set of all three machines by M and label the machines as M1 ; M2 ; M3 . Assume a time horizon of length T ; it is important to have T as small as possible to limit the number of variables in the mathematical models. The nature of the products and the specialized nature of machine M1 suggested that we should segregate the products into two classes: those that require processing on machine M1 , labeled ‘the set J1 ’, and those that may be processed on either of the other two machines M2 and M3 , labeled ‘the set J2 ’. Knowledge of the identity of a job immediately identifies the class to which it belongs. Thus each job, where a job is a portion of a given order, has associated with in its processing time, the designation of the machine m 2 M on which it is to be processed in this stage, if any, and a due date. (Absence of the need for processing in the standing stage is reflected in zero processing time.) The job also has a penalty for earliness and for tardiness, which are usually unequal, assumed linear in their argument. If the due date of a job is a deadline – in the sense that it must be met – the penalty for tardiness is put prohibitively large to prevent its occurrence.

19.3.2 The Set J1 On Machine M1 This is the problem of scheduling jobs on a single machine. Each job has processing time denoted by p.j; 1/, and a due date denoted by d.j; 1/, in which the “1” refers to set J1 . The objective is to maximize throughput with as many jobs as possible on time. For the sake of simplicity of implementation, we interpret this latter objective to mean to ‘minimize the number of jobs late,’ rather than implement a more elaborate dynamic programming (DP) model that guarantees the minimal-cost justin-time realization. For the sake of economy of space, we forfeit the discussion of the exact DP model here. A complicating factor in this application is the realization that the various jobs are not ready at the same time since the stranding stage is the third stage in the flowline (see Fig. 19.3): depending on their times of processing in the previous two stages, jobs shall arrive at the stranding stage at different times. We circumvent this complication by adjusting the start time of the jobs in the prior stages to suit the schedule imposed by the stranding stage. This is accomplished by absorbing the processing times in the preceding cleaning and stretching stages in the processing times in the stranding stage. It is well known that sequencing jobs in DD-order (due date order) achieves the desired optimum. This is a simple rule to implement, and is of complexity O.n1 log n1 /, where n1 D jJ1 j, the number of jobs in the set J1 . Since the definition of a job was derived from the size of a customer’s demand order, it is possible that many jobs shall have the same due date. In this case, we order the jobs in SPT-order (shortest processing time order) to gain the advantage of recuperating the bobbins early.

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19.3.3 The Set J2 on M2 [ M3 We shall present two integer linear programming (ILP) models. The first, presented next, requires the off-line prior determination of the set of occupancy vectorsfVj gj 2J2 ; the second, which is presented in Sect. 19.5.1, also needs such a vector. As it turned out, the first model, while more elegant and concise in its statement, is rather demanding in its computing requirements; the second model does not exhibit such undesirable behavior despite its larger size. The output of this step, among other things, is a schedule of the bobbins’ requirements over time, and the residual bobbins after subtracting the requirements of the generated schedule from the totality of available bobbins. Note that this step of the solution does not determine the availability of the bobbins but assumes that these jobs have a higher priority than all others and therefore they have first call on the bobbins available. 19.3.3.1 A Model With the Occupancy Vectors fV.:/g The vector V .j; m; t/ was defined above (see page 12) with 1’s in positions t; t C 1; : : : ; t Cp.j; m/1 and 0’s everywhere else. This vector represents the occupancy of machine m by job j if it starts processing in period t. Let wj denote the weight (in tons) of job j , which defines its processing time. This latter is denoted by p.j; m/, where m is either M2 or M3 . Let X.j; m; t/ for j 2 J2 be a 0,1 variable defined by: 1 1; if job j 2 J2 starts processing X .j; m; t/ D @ on machine m 2 M2 \ M3 at time t; A 0; otherwise 0

There are five sets of constraints. 1. Require that each job must start at some time in the planning horizon: X

X

X.j; m; t/ D 1; for all j 2 J2 :

(19.2)

t 2Fj m2M2 [M3

2. Constrain the use of machines to 2 at any time (machines M2 and M3 ), X X

X

X .j; m; t/ V .j; m; t/ 2e:

(19.3)

t 2Fj j 2J2 m2M2 [M3

where e is a vector of 1’s of dimensionality T . The symbol Fj in the first summation represents the feasible start times of job j 2 J2 . This is determined a priori based on the job’s processing times in prior stages.

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3. Determine the requisite number of bobbins and respect their availability. With abuse of language, let B1 denote the class of bobbins used on the input side of the stranding operation as well as their number; and let B2 be similarly defined for the output side. Let B1 .t/ denote the number of input bobbins to the stage at time t, and B2 .t/ denote the number of output bobbins, for 1 t T . We know that some jobs in class J2 require 7 bobbins of class B1 on the input side, and some jobs require only 3 bobbins on the input side. All jobs require one bobbin of class B2 on the output side. In the following (19.4), bj is either 7 or 3, depending on the job j , which is known when the identity of the job is known. In (19.5), all coefficients fbj g are equal to 1 since each job requires one bobbin on the output side: X

X

bj X .j; m; t/ V .j; m; t/ B1 .t/ 0; for each t 2 T; (19.4)

j 2J2 m2M2 [M3

X

X

X .j; m; t/ V .j; m; t/ B2 .t/ 0; for each t 2 T; (19.5)

j 2J2 m2M2 [M3

4. Constrain the maximal number of bobbins used at any time; B1 .t/ B1 ; 8 t D 1; : : : ; T

(19.6)

B2 .t/ B2 ; 8 t D 1; : : : ; T

(19.7)

5. Define the completion time Cj of each job: Cj D

X

X

.t C p .j; m/ 1/ X .j; m; t/ ; for all j:

(19.8)

t 2Fj m2M2 [M3

Recall that Fj denotes the earliest feasible start time of job j . 6. Define the earliness/tardiness of each job, assuming that job j has due date dj : Cj dj D uj j ; for all j:

(19.9)

Here, uj measures the tardiness, and vj measures the earliness, of job j . 7. The non-negativity constraints on the u’s and v’s (the X ’s are already nonnegative by definition): uj ; uj 0; for all j:

(19.10)

The objective is to maximize the total profit. This is composed of four components: the value of the product as indicated by its selling price or value after accounting for the production cost, denoted by sj ; the earliness penalty, valued at

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aj per unit time early; the tardiness penalty, valued at bj per unit time tardy, and the cost of acquisition of the two types of bobbins priced at h1 and h2 ; respectively, max z D

X sj aj vj C bj uj h1 B1 h2 B2 :

(19.11)

j 2J2

19.3.3.2 Model Complexity With n2 D jJ2 j, this model has 3n2 C 5T equations in 2 C 3n2 C 2n2 T variables (the X ’s, u’s, v’s, C ’s, and the two B’s). If n D 30 jobs and T D 90 periods (each representing a half shift, over 15 days with three shifts each day), we would have only 540 constraints in 5,492 original variables. This size problem, despite its large number of integer variables, does not overwhelm commercially available software. The handicap of this model is the need to construct the vectors V , and there are 2n2 T of them (some 5,400 in this instance!).

19.3.4 The Problem of The Bobbins As explained in the Introduction, the most significant support facility in the plant is the bobbins which convey the wire/cable from one station to another. Simply stated, it is desired to have, at the right time, the requisite number of bobbins of the correct size at the input and output sides of the each stage in order for the operation at that stage to take place. Since we wish to be frugal with the total number of bobbins in the plant, empty bobbins should be returned to their proper station as soon as possible. For the sake of concreteness of discussion, we shall take a specific example that involves three operations linked as shown in Fig. 19.5. It is assumed that the output of Operation 1 (denoted by O1 ) at station 2 is split between Operations 2 and 3 (denoted by O2 and O3 ; respectively). There are two end products (PRODUCT 1 .P1/ and PRODUCT 2 .P 2/) that are produced in stations 3 and 4. We assume the following parameters. Operation k consumes pk of processing time. The return time of a bobbin to its initial position in the operation for re-use is rk . The daily demand for the two end products is known as D2 and D3 (corresponding, respectively, to P1 , which is the output of O2 , and P2 , which is the output of O3 ). It follows that the daily production of O1 is the sum of the two demands. Let D1 denote the “demand” on O1 , D1 D D2 C D3 : As we are assuming “flowshop operation” on a continuous basis, suppose there are H minutes per day. If it is an 8 h operation, then H D 480 .allowance for meal breaks and personal time/. Let sk denote the interval (spacing) between any two units of product k; k D 1; 2; 3. Then,

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Station 2

PRODUCT 1

n2 tur Re

Station 1

Return 1

n2 io t a er Op

3

Op er ati on

rn tu Re

Operation 1

3

PRODUCT 2 Station 4

Fig. 19.5 Example of bobbin movement

sk D

H ; k D 1; 2; 3: Dk

Let the total cycle time of a bobbin in operation Ok be denoted by ck . Then, ck is the sum of the processing time and the return time, ck D pk C rk I j D 1; 2; 3: It immediately follows that the number of bobbins required, denoted by nk , is given by H I k D 1; 2; 3: nk D ck To illustrate, suppose we have the following additional data: Demand for product P1 is D2 D 75 units/day, and the demand for product P2 is

D3 D 50 units/day.

Processing times and return times are as follows: Operation

Processing

Return

Demand

1 2 3

10 14 18

2 3 2

125 75 50

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Suppose that the allowance for meal breaks and personal time is 80 min/day. From the above data, we may derive the following values: Operation 1 2 3

Cycle Time

Interval (min)

12 17 20

400 125 400 75 400 50

D 3:2 D 5:33 D 8:00

Bobbins ˙ 12 D4 ˙ 3:2 17 D4 5:33 ˙ 20 D3 8

The operation of this line of production will be as follows. O1 will produce one unit every 3.2 min. The two other operations shall be also synchronized to the same period: the spacing in O2 will be the same, 3.2 min, and the spacing in O3 will be twice as long, 6.4 minutes. 60% of the output will be diverted to O2 , and the remaining 40% will be diverted to O3 as shown in Fig. 19.5. Of every three units produced in O1 , two are processed in O2 and one in O3 . To minimize delays, the first two are diverted to O2 and the third unit to O3 . Therefore, there shall be 2 units of P1 produced every 9.6 min. To achieve the desired quantity of 75 units, we need 10C 75 9:6 D 370 min. During this interval, there would have 2 ˘ been 360 D 37 units of P 2 produced. The remaining 13 units can be produced 9:6 by diverting all the production of operation O1 to O3 That is, starting at time 370 all production of O1 is allocated to O3 at spacing 3.2 min. This will complete the production of the 13 units in 13 3:2 C 18 D 59:6 min. Therefore, the total time elapsed to complete the demand is 370 C 59:6 D 399:6 min, which is close enough to the allocated time of 400 minutes. The total number of bobbins used in this operation is 11, allocated as follows: 4 bobbins to both O1 and O2 , and 3 bobbins to O3 .

19.4 The Pre-Implementation Studies Before we implemented our decision-support system in the plant, we conducted extensive experimentation in order to (1) tailor-fit the general software available to us to the requirements of this particular plant; and (2) adjust the various parameters of the heuristics used to the operational patterns of the plant. This section details this ‘fitting and calibration’ phase of our study. We label our framework ‘PlanOrdo,’ which aims to provide the decision-makers with a generic framework for optimization and simulation of discrete-continuous planning and scheduling problems. Using a divide-and-conquer approach, our framework is based on a model composed of several cooperating sub-models; at least one of them is dedicated to simulation and another deals with the optimization aspects of the problem. We believe that PlanOrdo is applicable to generalized flowshop planning and scheduling problems. The basic aim of our framework is to provide an environment to be used as a testbed for comparing the performance of different planning and scheduling strategies. Each scheduling strategy is composed of a dispatching rule – providing an initial solution/schedule – enhanced by an optimization method.

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The simulation model is based on the RAO simulator (Artiba et al. 1998). Generally speaking, our RAO simulation model is composed of a generic set of patterns defining the functioning of common/generic resources, plus some specific patterns dedicated to the specific application’s problem. The GUI of PlanOrdo controls several units, which are briefly described below. The scheduling unit comprises a set of widely used dispatching rules, such as Shortest Processing Time (SPT)-rule, enhanced by several meta-heuristics. These simple dispatching rules are not only fast and easy to implement, to control and to run, but also easy to explain to the responsible personnel in the shop. Another advantage is that these rules apply to a wide range of scheduling problems. Major drawbacks are that they do not always produce really good results, and they are not stable when some parameters, such as the load of the simulated process, varies (Smith and Seidmann 1983; Carlier and Chrétienne 1988; Avinash et al. 1995; Hansmann and Hoeck 1997). To tackle this problem and increase the robustness of dispatching-rulegenerated solutions, the optimization unit is composed of several meta-heuristics. As indicated above (see Sect. 19.3.1), it is also possible to focus on bottleneck(s) using mathematical models. Several units of PlanOrdo can be assembled to implement a wide range of hybrid methods based on various models. According to Talbi (2002), a hybrid model is the assembly of different units, at different levels of combinations. More precisely, in this chapter, our hybrids are composed of a set of dispatching rules, a simulator, a multi-criteria method and several optimization tools. In this particular application, the performance of three hybrid models is evaluated: the first is very fast but too ‘fragile’ in face of heavily loaded periods; the second requires more computing time to give good results via a meta-heuristic; and the last is more effective than the second since it is based on a mathematical model used to accomplish the planning aspects at a detailed level of granularity. These hybrids are described in the following sections. All the hybrids presented here have been implemented to be used either directly in our framework, or as “stand alone” optimizers. Note that the mathematical models are used before starting any of the hybrids. Table 19.1 gives a bird’s eye view of the mixture of procedures and hybrids used and their properties.

Table 19.1 Summary of procedures and hybrids used Name Loading Stranding Meta Hybrid 1 Hybrid 2 ? Hybrid 3 ? S C Hybrid 1 ? S C Hybrid 2 ? ? S C Hybrid 3 ? ? L C S C Hybrid 1 ? ? L C S C Hybrid 2 ? ? ? L C S C Hybrid 3 ? ? ?

DR

Simulator

? ? ? ? ? ? ? ? ?

? ? ? ? ? ? ? ? ?

Lex-Sort

Promethee

? ? ? ? ? ?

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In this table

“Loading”: Loading model “Stranding”: Stranding model “Meta”: meta-heuristic; e.g., hill-climber used in the hybrids “DR”: dispatching rules “Simulator”: simulation model “Lex-Sort”: guide the meta-heuristic via the lexicographic sort “Promethee”: guide the meta-heuristic via the Promethee multi-criteria method

These hybrids are described next.

19.4.1 Description of the Hybrid Models For the sake of clarity, we do not represent the loops between the different decisionmaking levels (corresponding to the various models). To be as close as possible to the shop capacity, we start with an estimated shop capacity (based on historical data) at the loading level. We verify the feasibility of the calculated load at the evaluation level, where the detailed constraints are taken into account by the simulation model. Then, we validate/correct the shop load and return to the loading model if needed. The number of iterations depend on the precision we impose. To recap, our hybrid models are composed of two mathematical models, a set of dispatching rules and a simulation model. The first mathematical model is the Loading Model. It is used to ensure that the list of products over the specified planning horizon is compatible with the (maximum) shop capacity before proceeding any further. The second mathematical model is either the Stranding Model or the Stretching Model, depending on the bottleneck, which is easily detected thanks to the Loading model. The result is used to determine the starting and ending dates of the aggregated orders. The dispatching rules compute the initial solutions to be sent to the simulation model. Provided with an initial solution, the simulation model then generates a final solution composed of a list of scheduled orders and a set of related performance measures (see Fig. 19.6). The functioning of our hybrids is the following: 1. Retrieve the list of orders (over typically several months) and compute the average estimated load of the shop via the loading model, and estimate the potential bottleneck(s). 2. If the load is compatible with the capacity of the shop, refine the estimation of the load of the shop using the detailed planning model applied to bottleneck(s) stage(s), for instance, apply the Stranding model to the Stranding stage. 3. Use the result (in terms of estimated/ideal values for various criteria) of the previous stage as bounds to compute the gap between the (real) schedules to be computed in this step via Hybrid1. If the gap between the values of the criteria associated with the solutions of Hybrid1 and the “ideal” values given by the previous stage is less than a predefined threshold, then stop.

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Fig. 19.6 The Hybrid(x) model

4. If optimization of the previously obtained schedule(s) is needed, then start Hybrid2 (simple to parameterize but strict ranking of criteria is needed) or Hybrid3 (a bit more complicated to parameterize but with no limitation on the relative importance of criteria and it is easy to consider a lot of additional criteria). 5. Present the results visually using either a classical lexicographical sort or Promethee II and the GAIA plane (Brans and Mareschal 1994). In this hybrid model, the impact of different dispatching rules on the criteria under consideration (see Sect. 19.2) was investigated. Starting from simple dispatching rules, such as SPT and LPT, these rules have been extended and combined to obtain a set of accurate experimental dispatching rules. These dispatching rules are greedy algorithms. They work by sorting orders/products according to predefined sorting keys, for example, based on the processing time of orders. The dispatching rules are used to compute initial solutions. In our implementation, an initial solution is a permutation of integers representing the ranking of each shop-order to be produced by our simulation model, and by the actual plant if the proposed solution is accepted by the human planner (HP). Five representative dispatching rules have been selected from the set of available rules. They are (the notation Key1 C Key2 C Key3 means that the dispatching rule

510 Fig. 19.7 Schema of the Hybrid1 model

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E.R.P. / Planning Part-Solutions

Dispatching rules Initial-Solutions

Simulation model Final-Solutions

under consideration sorts orders first according to Key1, then Key2, and finally Key3 (lexicographical sorting algorithm)):

Rule 0: D C F C HP Rule 1: D C F C SPT Rule 2: D C F C LPT Rule 3: SPT C F C D Rule 4: LPT C F C D

where D is the starting date, and F is the due-date. In the remainder of this section, these five rules are compared. The first rule D C F C HP rule is a special one, since it uses information provided by the human planner. Our first hybrid model, Hybrid1, is composed of a set of dispatching rules and a simulation model. The dispatching rules compute the initial solutions to be sent to the simulation model. Provided with an initial solution, the simulation model then generates a final solution composed of a list of scheduled orders and a set of related performance measures (see Fig. 19.7). Hybrid1 runs fast and is easy to understand and tune, but it is not robust enough to systematically generate good, or even satisfactory, solutions during periods of heavy load mainly due to the number of orders in the shop and/or the overheads resulting from malfunctions. A second hybrid model (Hybrid2) was developed to improve on these results. It is based on a general local iterative search method, named “hill-climber”. This iterative improvement algorithm belongs to the widely used Stochastic Local Search methods (Hoos and Stützle 2005). To speed up the local search, the hill-climber is based on the first improvement strategy as explained below. This is very easy to design and does not require huge computational resources. However, it can easily become trapped in a local optimum. Basically, when applying the hill-climber to a maximization problem, the process is the following: starting from an initial solution, the hill-climber tries to move uphill thanks to an “operator”. This operator is applied to the current solution to generate a new one, performing an elementary movement of the hill-climber in the search space. If this new solution, according to the selection function, is better than the current solution, then the new solution replaces the current solution. The operator

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Start Compute Initial-Solution Evaluate Initial-Solution via simulation Current-Solution

Initial-Solution

Generate a New-Solution Evaluate New-Solution via simulation

No

Improvement ? Yes

Current-Solution

New-Solution

Stop ?

No

Yes

End Fig. 19.8 Schema of Hybrid2 model

is applied to the new current solution in order to generate another solution, and this cycle is continued until a predefined stopping criterion is met. The cycle is called “an iteration.” To be more precise, it performs the algorithm displayed in Fig. 19.8. In this hill-climber, there are two main parameters: the operator and the selection function. In our implementation, the operator swaps the ranks associated with two randomly selected orders. It is responsible for moving the hill-climber from one solution to a neighboring solution in the search space. Using this simple operator, we can focus on the selection function itself. In this hybrid, the selection function is based on a lexicographic sort, which allows us to rank solutions according to the criteria computed by the simulation model. So, in the particular case of our hybridization, the evaluation of each solution requires one run of the simulation model. Hybrid2 is based on lexicographic sort, and therefore, requires us to formulate the problem so that it is possible to strictly rank the criteria to be optimized. The relative importance of these criteria, defined by the decision maker, determines this ranking. However, this is neither easy nor realistic when considering some real life

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problems. Moreover, when comparing two scheduling strategies, a decision-maker often accepts a solution that is worse from the point of view of the main criterion if this solution leads to significant improvements in some other criterion (or criteria). The choice of the best strategy taking into account several criteria is not evident. The best way to achieve this goal is to incorporate a multi-criteria method into the hill-climber itself, which is done in our third hybrid model. We will not go into more detail about our Hybrid2 model because that would require a thorough presentation of its components, which is beyond the scope of this chapter; the interested reader may consult the following references which, collectively, contain a thorough and complete documentation of this hybrid: Duvivier et al. (2002); Duvivier et al. (2003, 2005, 2007, 2006b) and Roux et al. (2005). Our third hybrid, Hybrid3, is composed of a set of dispatching rules, a simulation model, and the Promethee II multi-criteria method (Preference Ranking Organization METHod for Enrichment Evaluations). Promethee II was defined by Brans et al. (1986), it is based on the outranking concept, which is built upon pairwise comparisons of the solutions (additional informations might be found in Brans and Mareschal (1994) and Pomerol and Barba-Romero (2000)). Promethee II provides a complete ranking based on pair-wise comparisons of solutions. This method is used to compare and rank solutions according to selected criteria. This ranking method was chosen due to its efficiency and to the fact that it is easy to understand and to tune by decision makers. Practically, Promethee II calculates positive and negative preference flows for each solution. Based on the difference between these flows, the net flow (denoted by ˆ) is obtained and is used to rank the solutions. It is not the purpose of this chapter to explain the Promethee II methodology in detail; the interested reader may consult the book by Pomerol and Barba-Romero (2000). There is also a good URL dealing with Promethee (http://www.promethee-gaia.com). In our framework, the net flow ˆ is computed according to Promethee II and used to compare the newly generated solution to the best current solution of the run, after each iteration of the hill-climber; see Fig. 19.9. It should be noted that, compared to Hybrid2, the major advantage of Hybrid3 is its ability to incorporate “gradation,” “tinge,” and “fuzziness” in the judgment of the decision-makers while comparing scheduling strategies. In Hybrid3, the lexicographic sort is replaced by Promethee II, which acts as the selection function. Promethee II also permits escape from local optima thanks to a tolerance threshold on each criterion, and/or accept compromises among criteria when trapped in a local optimum. We remark that the integration of Promethee II into a meta-heuristic has been attempted before (De Lit et al. 2001; Rekiek 2001), but we believe that this is the first time that a hybrid including the simulation of a highly constrained discretecontinuous problem and Promethee II has been done. Several aspects of our work are such that the previously proposed methods are not applicable: most of these methods require a lot of evaluations to find good solutions since they are mostly based on a population of solutions. In our case, an evaluation requires 1 to 3 seconds on a Pentium IV 3 GHz, so we need to use a meta-heuristic which is not based on

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Fig. 19.9 Schema for Hybrid3 model

Start Current-Solution

Dispatching_Rule (Nr)

Evaluate Current-Solution via simulation Best-Solution New-Solution

Current-Solution

Move_From (Current-Solution)

Evaluate New-Solution via simulation Φ

Promethee (New-Solution, Best-Solution) Deterioration (Φ < 0)?

Yes

No

Current-Solution

New-Solution

Plateau (Φ = 0 )?

Yes

No

Best-Solution No

New-Solution

Stop ? Yes

End

a population of solutions (which is too demanding to evaluate). One of the aims of the present research was to study the combination of the simulation model and the multi-criteria hill-climber illustrated by Hybrid3. Continuation of this research on hybrid models should perhaps use more efficient/sophisticated meta-heuristics such as Simulated Annealing, Tabu Search or Ant Colony Optimization, etc.

19.4.2 The Optimized Criteria Our simulation model includes four criteria. Table 19.2 gives the notation and a short description of each optimized criterion (1)–(4). Criteria (2)–(4) are measured in hours (H). Additional criteria, such as lack of auxiliary resources, are indirectly optimized while choosing among the set of scheduling strategies generated by PlanOrdo.

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A. Artiba et al. Table 19.2 Notation and optimized criteria Notation Description n Number of jobs Due date of job j dj Cj Completion time of job j Earliness of job j Ej D maxf0; dj Cj g: Criterion (1) NT P Tj (2) T D n1 (3) Cmax D maxfCj g P Ej (4) E D n1

Description Number of tardy jobs Average tardiness Completion time of jobs (makespan) Average earliness

In this chapter, the objectives are to minimize criteria (1)–(3), and maximize criterion (4). Criteria (1)–(3) are the classical ones, but it might seem strange to maximize the average earliness (criterion (4)). In fact, this criterion is maximized to cope with the possible malfunction aspects of the process in order to increase the probability of meeting the jobs’ due dates. In passing, we remark that these criteria were chosen according to the decision-makers of this plant, but PlanOrdo is not limited to them.

19.4.3 Experimental Results (In accordance with the confidentiality agreements for this project, no direct or detailed comparisons between the simulations and reality are presented). The objective of our hybrid models is to schedule one month of production in the plant. Our simulation takes 1–3 s to model this period on a Pentium IV 3 GHz; the exact time depends on the number of orders in the period and the kind of statistics computed during the simulation. The following paragraphs present an example of the use of the three hybrid models implemented in PlanOrdo when 134 orders are simulated in a period of one month. Each hybrid is evaluated on the four criteria discussed in Sect.4.2, from the most important to the least important. The ranking is NT T Cmax E (cf. the criteria in Table 19.2). This configuration is also used to rank the scheduling strategies. The experimental results are based on the five selected dispatching rules. Each scheduling strategy is composed of a dispatching rule – providing an initial solution/schedule – enhanced by an optimization method (“hill climbing,” or HC for short). The results of five scheduling strategies are compared to the results of dispatching rules used without optimization. When optimizing the solution of the j th dispatching rule via a hill-climber (HC), 20 runs of 4,000 iterations/evaluations were launched, and average results are computed to take into account the stochastic aspects of this hybrid. An iteration requires one run of the simulation model.

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Table 19.3 Hybrid1, comparison of several dispatching rules NT s T (H) Cmax (H) E (H) 0 0 753.94 271.53 0 D+F+HP 0 0 757.71 277.40 1 D+F+SPT 2 D+F+LPT 0 0 760.17 256.84 3 SPT+F+D 2 17.65 749.77 287.42 4 LPT+F+D 6 10.55 756.68 266.58 All average 1.60 5.64 755.66 271.95

19.4.3.1 Results of Hybrid1 Based on the dispatching rules defined above, the results of our first Hybrid1 are compared with the real-life situation. In most cases, Hybrid1 is able to obtain results identical to those generated by the human planner (in terms of delays, makespan, etc.). However, in busy periods (due to the number of orders in the shop and/or stoppages resulting from malfunctions), this first hybrid is not robust enough to systematically generate good, or even satisfactory, solutions. Each strategy, identified by a number ‘s’ and by the name of the associated dispatching rule, provides one solution in the Hybrid1 results. Table 19.3 compares them for the selected period. The best one, according to the four predefined criteria, is highlighted. The last row (denoted by “All”) gives the average values. For the selected period and configuration, the best dispatching rule is DCFCHP. This shows that the best strategy consists of respecting starting dates computed by the human planner, an undesirable feature from our point of view. A caveat here is in order: this ranking cannot necessarily be generalized to another period due to the influence of the parameters on the performance of dispatching rules. But then, one does not expect the parameters to be re-evaluated at each run of the decision-support system! 19.4.3.2 Results for Hybrid2 To improve on the above results, the second Hybrid2 was constructed. It includes a non-deterministic hill-climber (HC). The results obtained after 4,000 iterations are summarized in Table 19.4. Starting from the results of each dispatching rule, 20 runs of 4,000 iterations/evaluations were launched and average results were computed to take into account the stochastic aspects of this hybrid. For each strategy, this table gives its identification number as well as the average value and standard deviation of each criterion. For a given criterion, the average gain is the absolute difference between its initial value (obtained via the related dispatching rule) and its average final value (given by the final solutions). The sign is positive whenever an improvement is obtained. The best strategy, according to the four predefined criteria, is highlighted. The bottom row (denoted by ‘All’) gives the average values and standard deviations for all the strategies. In this hybrid, a lexicographic sort is applied to determine the best strategy.

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A. Artiba et al. Table 19.4 Results of Hybrid2 after 4,000 iterations Final strategy/solution Gain after 4000 evaluations Hybrid2

NT

s 0 1 2 3

All

4

average std.dev. average std.dev. average std.dev. average std.dev. average std.dev. average std.dev.

0 0 0 0 0 0 0 0 0 0 0 0

T (H) 0 0 0 0 0 0 0 0 0 0 0 0

Cmax (H) 744.95 1.34 742.88 2.79 744.09 1.36 741.07 2.51 744.65 2.29 743.53 2.53

E (H) 278.57 1.34 280.56 1.37 277.06 2.38 279.84 2.05 275.47 2.57 278.30 2.69

NT 0

T (H) 0

Cmax (H) 8.99

E (H) 7.03

0

0

14.84

3.16

0

0

16.09

20.23

2

17.65

8.71

-7.58

6

10.55 12.04

8.89

5.64

6.35

1.60

12.13

In Table 19.4, the fact that both the average and the standard deviation are 0 implies that the hybrid has succeeded in finding a solution with no tardy jobs. Hybrid2 obtains better results than the simple dispatching rules on all criteria. A major drawback of this hybrid is that each run requires from 1 to 3 h on a Pentium IV 3 GHz-based PC (varying according to the number of orders in the selected period and the kind of statistics computed during the simulation). However, Table 19.4 shows two important results: first, that, at the end of each run, there are no late orders (columns NT and T are filled with zeros); and second, that Hybrid2 reduces the makespan .Cmax / by an average of at least eight hours. This latter result means that a minimum of eight hours is gained in the working time of a team of eleven workers during the simulated month. Similar results were obtained when running Hybrid2 over other time-periods. The exception was for very heavily loaded periods. In such cases, the results obtained by Hybrid2 are close to those obtained by the human planner, but still (much) better than Hybrid1.

19.4.3.3 Results of Hybrid3 When a lexicographic sort is used a small gain on a criterion may induce significant losses on another criterion of a smaller weight. In real-life problems, when comparing two scheduling strategies, a decision maker often accepts a solution that is slightly worse from the point of view of the main criterion if this solution leads to significant improvements in some criterion (or criteria). This issue results from the near impossibility to introduce the concept of ‘gradation’ and ‘fuzziness’ in each criterion. Similarly, the simulator computes the schedules with great accuracy, whereas process times can vary by a significant amount when considering

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the production process (mainly due to workers’ interaction) and still be considered close. Consequently, it is more realistic to add a threshold to the measures that are expressed in hours. As previously explained, in order to improve Hybrid2 so that it is possible to introduce equally ranked criteria and/or gradation, Hybrid3 utilizes the Promethee II multi-objective method. This method allows the introduction of indifference .q/ and/or preference .p/ thresholds for each criterion. In this research, three sets of parameters have been tested according to the decision makers. Table 19.5 summarizes the used parameters and the results obtained. In the sets of parameters related to Hybrid3.1 and Hybrid3.2 the criterion NT is considered to be the ‘usual’ one (i.e., the Promethee parameter “type/shape” of this criterion is set to (1)) whereas in Hybrid3.3 it is defined as a linear criterion (i.e., the Promethee II parameter “type/shape” of this criterion is set to (5)). For Hybrid3.1, the other criteria are V -shape type whereas they are linear in the two other hybrids. The values of the parameters are detailed in Table 19.5. For instance, consider Hybrid3.2. If two solutions differ by at least four hours .p D 4/ in the makespan .Cmax /, one solution will be preferred over the other on the basis of this criterion. On the other hand, if the difference is less than a half hour .q D 0:5/, this difference is not significant and the two solutions are considered as equivalent for this criterion. Hybrid3.1 is used to demonstrate (experimentally) that our multi-criteria hillclimber is able to provide similar solutions to those generated by Hybrid2 without the inherent restrictions of the lexicographic sort, which requires a strict ranking of the criteria. Hybrid3.2 and Hybrid3.3 illustrate the possibility of incrementally adjusting the relative importance of criteria according to the decision-maker’s judgement. Using Promethee II, all this is possible by simply adjusting the weights of the criteria (theoretically, there is no restriction on the values of these weights). For instance, provided that the type/shape is set to the same value, if the weights C1 and C2 are identical, then C1 C2 . This means that these two criteria are considered equally important. And it is possible to gradually adjust their relative importance by simply modifying the associated weights. Table 19.4 shows that there is no dominant strategy for the considered sets of parameters. So, to determine the best strategy (highlighted in Table 19.4) for the three sets of parameters, two methods were used: a lexicographic sort and the Promethee II multi-objective method. Similar results were obtained except for Hybrid3.3. For this hybrid, the best solution according to the lexicographic sort is strategy 4, whereas using the Promethee II multi-objective method the best compromise corresponds to strategy 1. This shows that the Promethee II multi-objective method allows the discounting of the Cmax criterion (of 0.91 H) while improving the E-criterion (of 6.09 H). This perfectly matches a situation where, when comparing two scheduling strategies, a decision-maker accepts a solution that is worse than another on one criterion if this solution leads to significant improvements in some (or all) of the other criteria. Moreover, a sensitivity analysis study shows that these results are robust with respect to the weights.

4

3

2

1

0

s

4

3

2

1

0

s

NT

Cmax (H) 744.23 2.32 742.91 2.69 744.45 1.63 741.82 2.37 743.15 1.56 743.31 2.31

E (H) 278.56 1.58 281.41 0.97 275.98 3.80 280.34 2.46 275.90 2.22 278.44 3.24

NT

T (H) 0 0 0 0 0 0 11.04 8.07 0 0 2.21 5.64

Cmax (H) 744.44 2.11 744.77 3.17 744.19 2.40 747.03 3.55 743.86 2.85 744.86 3.01

E (H) 276.11 1.92 277.81 1.67 272.68 2.88 284.20 4.45 271.72 3.70 276.50 5.37

Final strategy/solution

average 0 std.dev. 0 average 0 std.dev. 0 average 0 std.dev. 0 average 1.55 std.dev. 0.69 average 0 std.dev. 0 average 0.31 std.dev. 0.69

Hybrid3.3

T (H) 0 0 0 0 0 0 0.01 0.06 0 0 0.00 0.03

Final strategy/solution

average 0 std.dev. 0 average 0 std.dev. 0 average 0 std.dev. 0 average 0.05 std.dev. 0.22 average 0 std.dev. 0 average 0.01 std.dev. 0.10

Hybrid 3.1

0 0 17.64 10.55 5.64

0 0 2 6 1.59

12.34

13.53

7.96

15.72

14.81

Cmax (H) 9.71

6.48

9.32

-7.08

19.14

4.02

E (H) 7.02

0 0 6.62 10.55 3.43

0 0 0 6 1.29

0

T (H) 0

NT

10.80

12.82

2.75

15.98

12.95

Cmax (H) 9.50

4.55

5.15

-3.22

15.84

0.41

E (H) 4.57

Gain after 4000 evaluations

0

T (H) 0

NT


4

3

2

1

0

s

0 3

0 0

q p

Min

4

0

3

Cmax

0.5 3 Min

1 5 Min

p Goal

5

5 Type q

T

NT

Criterion

Min

4

0.5

5

Cmax

Hybrid3.3

Min

3

1

Min

T

NT

Goal

Cmax (H) 744.14 2.09 741.81 2.69 744.18 1.51 741.44 2.84 743.95 2.53 743.10 2.62

Hybrid3.1

T (H) 0 0 0 0 0 0 0 0 0 0 0 0

Type

0 0 0 0 0 0 0 0 0 0 0 0

NT

Max

4

0.5

5

E

Max

4

0

3

E

E (H) 277.48 1.57 279.46 1.34 273.90 2.43 278.80 1.57 275.06 2.20 276.94 2.81

Finalv strategy/solution

Criterion

average std.dev. average std.dev. average std.dev. average std.dev. average std.dev. average std.dev.

Hybrid3. 2

All

Table 19.5 Results of Hybrid3 after 4,000 iterations

All

All

0 17.65 10.55 5.64

0 2 6 1.60

Min

0

0

1

NT

12.56

12.74

Min

4

0.5

5

Cmax

Max

4

0.5

5

E

4.99

8.48

8.34 -8.62

Parameters of Promethee II

Min

3

0.5

5

T

2.06

E (H) 5.95

15.99 17.06

15.91

Cmax (H) 9.80

Hybrid3.2

0

0

0

T (H) 0

NT


518 A. Artiba et al.

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19.4.3.4 Comparison of Results The average results from various hybrids were then compared. These results show that Hybrid2 and Hybrid3 clearly outperform Hybrid1, which is (only) used when fast results are needed. Moreover, the average results from Hybrid3 (based on a multi-objective hill-climber) are similar to those generated by Hybrid2 (based on a lexicographic sort) according to the configuration NT T Cmax E. Table 19.5 shows that Hybrid3.1 provides us with similar solutions to those generated by Hybrid2, so that it is possible to parameterize Hybrid3 such that its internal multiobjective function matches the lexicographic sort of Hybrid2. In this particular case, Hybrid3 obtains exactly the same results as Hybrid2, provided that the pseudorandom number generators of both hybrids are initialized with the same seed. These results are not presented to reduce the size of this section; we have used the results of Hybrid3.1 as the closest to Hybrid2, even though Hybrid3.1 tends to favor E to the detriment of Cmax . By this, it is meant that Hybrid2 is based on a lexicographic sort. It is possible to reproduce exactly its functioning by setting all type/shape of the criteria to (1) and compute the weights so as to force strict ranking of the criteria. We do not present this type of result here since the only conclusion would be that Hybrid3 is able to act exactly as Hybrid2, which we already know. We have decided to use criteria of type/shape (3) (others may also be used) so as to allow several thresholds. Using these parameters, Hybrid3 tends to favor E to the detriment of Cmax , as stated above – it is a kind of compensation. In a similar way, it is possible to parameterize Hybrid3 to favor solutions with at most (or exactly) one late order. The parameters must reflect the scheduling strategies applied by the human planner to the underlying industrial problem. So the sets of parameters related to Hybrid3.1 through Hybrid3.3 have been chosen as a good compromise between the possibilities of Hybrid3 and those of the underlying process. At this point, two main results may be highlighted: first, Hybrid3 performs at least as well as Hybrid2 which experimentally demonstrates the efficiency of our multi-objective hill-climber; second, thanks to the embedded multi-objective method, Hybrid3 is able to incorporate gradation in the judgment of decision-makers (illustrated by the three sets of parameters associated with Hybrid3) which experimentally demonstrates the versatility of our multi-objective hill-climber. To pursue our comparisons, we can compare average values/tendencies related to the strategies given in Table 19.5, while paying close attention to the standard deviations. These results show that there is a tendency for the average gains in all the criteria to decrease as we move from Hybrid3.1 to Hybrid3.3. However, there is a problem with this conclusion. Direct comparison of average values of a given criterion between hybrids is only possible if the standard deviations of the solutions are close to each other, which, unfortunately, is not the case here (see Table 19.4). One way to overcome this difficulty is to use additional statistical methods to characterize these dispersions (Duvivier et al. 2007). Another one is to compare the set of best solutions using a lexicographic sorting algorithm. This might lead to another pitfall due to the stochastic aspects of our hybrid models: in the context of

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the assessment of the usability of stochastic algorithms, it is quite useless to give the best solution ever found if we do not know how often it is found! 19.4.3.5 Lessons Learned from Experimentation In our experimentation, we have applied a set of very simple heuristics (dispatching rules) enhanced by a multi-objective hill-climber to optimize the scheduling part of a real-life problem. The results obtained are similar to, or better than, those generated by the human planner, even when considering heavily loaded periods. The results are consistent and show the robustness of the generic multi-criteria optimizationcum-simulation model that was developed. Besides the automation of the tedious scheduling task, our framework provides decision-makers with a set of solutions, resulting from a panel of scheduling strategies. Our experiments demonstrated to our satisfaction the efficiency and the versatility of the hybrid models formulated. It appears that the particular context of time-limited decision-making, which implies that whichever method is implemented it must run in a short amount of time, a limited number of evaluations leads to the conclusion that more efficient meta-heuristics (embedding diversification mechanisms) do not have enough time to converge toward good solutions. Consequently, we have chosen to implement fast meta-heuristics with no diversification mechanism (viz., hill-climbers), hybridized with dispatching rules which are chosen so as to start hill-climbers from good sparse solutions in the search space. We have studied the behavior of our hybrids over several periods; the results remain consistent with those generated by the human planner (in terms of delays and makespan). We have also implemented several variants of our hill-climber including more efficient meta-heuristics (such as Tabu Search). As explained in the introduction to this chapter, we have focused on another aspect of our study, the introduction of multi-criteria aspects into our schedule builder. In this context, it is preferable to consider a simple hill-climber (with fewer parameters to tune) before generalizing this approach to more efficient meta-heuristics. In order to be able to assess the effect of the hybridization of our hill-climber with the Promethee II multi-objective method without unduly expanding the number of parameters, we have chosen a very simple algorithm: a non-deterministic hill-climber. However, when considering efficiency aspects, it is clear that this simple hill-climber must be replaced by more efficient meta-heuristics if the available time span for decision making permits such sophistication.

19.5 The Implementation The PlanOrdo framework has been installed in the target plant for a few years and has been iteratively enhanced while interacting with the decision makers. In its current version, the framework is installed as a package composed of a main program

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comprising the Graphical User Interface and several peripheral tools. The internal code is composed of more than 22,000 lines of Delphi code, the graphical interface is composed of more than 122,000 lines. There are also several thousands of lines encoding the simulation models and the external meta-heuristics. More than 500 pages of reports have been written and transmitted to the target plant as well as more than 13 published papers (several papers are submitted and/or to be submitted) mainly dealing of specific aspects of the framework. (These reports may be secured by writing to any of the first three coauthors). However, this chapter is the first attempt to present an overall view of most of the accomplished work in PlanOrdo. The code is divided into several units. The implementation details of all these units are out of the scope of this chapter. We shall detail only a few interesting functioning aspects of the implementation. The implementation is based on the interaction of several tools emanating from different fields. Each tool is packaged in the corresponding unit or accessible via a link to an external application, etc. The performance unit is used by every tool at different levels so as to compare and to rank the solutions according to several predefined criteria. It is composed of several tools such as a lexicographical sort, a multi-criteria method, statistical tests, and several visualization interfaces. The simulation unit is used for precisely scheduling the production. It captures all the details of the functioning of real-life problems and is based on the RAO simulator (Artiba et al. 1998). The optimization unit is used at different levels in the hybrid methods. It is composed of dedicated heuristics (such as dispatching rules), meta-heuristics (such as hill-climbers, tabu search, etc.) and exact methods (such as ILP (integer linear programming) models solved by (Lindo Systems Inc. 1999), or, more precisely, it is used in the loading model to solve the mathematical model via Lingo). Optimization is used when computing detailed planning via the Stranding Model. It is also used when starting several dispatching rules so as to quickly compute a bunch of schedules. And finally, it is responsible for launching the internal or external metaheuristics when (iteratively) optimizing the schedules. When considering interacting models/tools, the problem is not only to parameterize each model/tool independently, but also to make them work together to achieve a common global objective. However, each model/tool has its own granularity, its own view of the search space, its own functioning (diversification and intensification mechanisms while walking through the search space, for instance), etc. Moreover, due to the fact that several tools might consider the problem at different scales, they do not necessarily have the same information/criteria to incorporate in their cost functions. On top of this, the different tools must also be able to communicate service messages (mainly for synchronization aspects) as well as information messages (data, part solutions; etc.). In our implementation, the synchronization is ensured by a classical master– worker model whereas the communications are ensured by a database and some specific files playing the role of a blackboard.

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The objective is to make all the hybridized models/methods/tools to collaborate to a common objective (the efficiency of the hybrid) in such a way that each model/method/tool has the opportunity to make the right contribution at the right place and at the right time. In the context of limited time decision-making, efficiency is defined in terms of obtaining the best solution – or at least an acceptable solution – in a given/fixed amount of time. In our implementation, several kinds of hybridizations are used. Using the terminology introduced by Talbi (2002) the loading model, the stranding model, the Hybrid1 eventually followed by a meta-heuristic are based on a High-level Relay Hybridization scheme where self-contained tools are executed in sequence. Similarly, the Promethee II multi-criteria method and the simulation model are hybridized following a Low-level Relay Hybridization scheme where a given tool is embedded into a single-solution meta-heuristic. Dedicated tools, which are really efficient on a narrow spectrum of problems, fail as soon as the problem faced varies slightly from the initial problem for which they have been designed. It is possible to use several of these tools to try to cover the spectrum defined by the target problems/instances (such as the dispatching rules in PlanOrdo) but it is assumed that the problems are quite varied such that it will be necessary to define/implement too many of these dedicated tools to efficiently cover the whole spectrum. The meta-heuristics are non-dedicated tools that are relatively efficient; but without specializing some parts of these methods, the results are not equally good. The exact non-stochastic approaches are theoretically able to perform systematically at 100% efficiency. However, when introducing the duration of the quest in the definition of the ‘efficiency,’ then this is not the case – the procedure is typically aborted well before reaching the optimal solution. PlanOrdo is based on the following concept: the loading model is able to quickly stop the search if there is no solution. The stranding model is able to refine the starting windows of the orders so as to reduce the search space of the following tools (relay). It is also able to give bounds to improve the stopping condition of our meta-heuristic. The dispatching rules are chosen so as to scatter/spread/disseminate several hill-climbers over the search space. In fact, the set of dispatching rules is the only diversification mechanism of our hybrids 1 to 3, since the hill-climber is only able to focus the search. The set of dispatching rules adds a kind of elementary diversification mechanism to the hill-climber.

19.5.1 The Stranding Model As mentioned in Sect.19.3.3, the initial attempts at implementing the original Stranding model resulted in discovering serious discrepancies between the predictions of the model and its simulated performance. The question then was: is it possible to modify the Stranding model so as to reduce the gap between the results of the simulation and the model?

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We tried to identify the causes of this gap, and discovered the fact that the bobbins are kept during a certain amount of time on the subsequent stage (Stabilization) may be a factor contributing to the discrepancy. Therefore, as a correction, in addition to the V matrix related to the stranding stage, we need to introduce a W matrix related to the Stabilization stage (and ideally it is also possible to introduce a U matrix to take the previous stage (Stretching) into account, which was not done in this application). Week-ends might also lead to increase this gap, and using a 7/5 ratio is not sufficient to adapt the results of the original Stranding model. The difficulty in the actual implementation lies in modifying the Vj matrix so as to introduce unavailability of the machines due not only to week-ends, but also to maintenance and holidays. Different scenarios might occur depending on the location of the time units corresponding to the processing time of job j , and on the location of the time units corresponding to the unavailability of the machines. As all unavailability instances are supposed to be known in advance (week-ends, holidays, preventive maintenance), it is also feasible to modify the V matrix so as to take these gaps in availability into account. At this point, it seemed that we will need to compute and introduce a lot of new variables so as to transform p.j; m/ into p.j; m; t/. However, close study of the mathematical model revealed that we are doing the same thing twice: the modifications are done once into the V vector and again into the p.j; m; t/ vector. This, in turn, raised the question: is it possible to formulate the whole problem without using p.j; m/ or p.j; m; t/? The answer is yes, and the final model contained the following: – Vector V .j; m; t/ remains defined as before; viz., it is a vector of length T with 1’s in positions t; t C 1; : : : ; t C p.j; m/ 1 and 0’s everywhere else for job j . This vector represents the occupancy of machine m by job j if it starts processing in period t. – Define the variable X.j; m; t/ for j 2 J2 as a 0,1 variable: 1 1; if job j 2 J2 starts processing X .j; m; t/ D @ on machine m 2 M2 [ M3 at time t A : 0; otherwise 0

For the sake of economy in computing, we restrict the index t in the definition of X .j; m; t/ toh rangei over the “feasible start times” of job j , which would be limited to Fj , ts0 ; tf0 . These values are determined a priori based on the job’s processing times in prior stages and are embedded in the V and W vectors so as to take machine unavailability instances into account. – Considering the processing time p.j; m0 / of job j on machine m0 (belonging to the stabilization stage), define a vector W .j; m; t/ of length T with 1’s in positions t C p.j; m/; t C p.j; m/ C 1; : : : ; t C p.j; m/ C p 0 .j; m0 / 1 and 0’s everywhere else. This vector represents the occupancy of machine m0 by job j if it starts processing in period t on machine m. It is also used to take into

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A. Artiba et al. Table 19.6 Notation Nomenclature m The identification of the machine #m in the stranding stage, m D 1; : : : ; jM jI M D fM1 ; M2 M3 g Class of jobs to be processed only on machine M1 ; n1 D jJ1 j J1 Class of jobs to be processed only on machines M2 or M3 ; n2 D jJ2 j J2 j Identity of a product or a family of products to be scheduled, j D 1; : : : ; n where the number of jobs n D jJ j; J D J1 [ J2 ; J1 \ J2 D ; T Length of the planning horizon (number of time units) t The elapsed time, t D 1; : : : ; T the ‘time unit’ defines T (a shift, half a shift, or one hour) p.j; m/ The processing time of product i on machine m dj The due date of job j The completion time of job j Cj The tardiness of job j uj vj The earliness of job j Marginal reward for earliness of job j aj Marginal penalty for earliness of job j bj (Maximum) number of available bobbins b in the plant Bb wj Tonnage of job j l.b/ Maximum load of bobbin b (in tons) Number of bobbins (3 or 7) needed for job j (input side) ˇj

account the fact that the bobbin is kept in the next stage during p.j; m0 / units of time; – Introduce pf .j; m/ and pbf .j; m/ to replace p.j; m/ in the bounds of the sums in the constraints of the model; these values are obtained while computing the V (and W ) matrices in the PlanOrdo framework (outside the mathematical model). Before continuing the presentation, Table 19.6 summarizes our notation. This leads to the following model that has been implemented effectively in PlanOrdo. 1. Each job must start at some time in the planning horizon: X X .j; 1; t/ D 1; for all j 2 J1

(19.1a)

t 2Fj 3 X X

X .j; m; t/ D 1; for all j 2 J2

(19.1b)

t 2Fj mD2

2. At any time, there can be at most one job in process on machine M1 and two jobs in process on machines M2 and M3 : X X X .i; 1; t/ V .j; 1; t/ e; for m D M1 : (19.3a) j 2J1 t 2Fj

X X

j 2J2 t 2Fj

X .j; m; t/ V .j; m; t/ e; for m 2 fM2 ; M3 g:

(19.3b)

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3. The following constraints are used to ensure that there are enough bobbins on the stranding stage (thanks to the V matrix), and during the time they are kept on next stage (via the W matrix): X X 7w.j / X .j; 1; t/ V .j; 1; t/ B2 e; l.2/

(19.12)

j 2J1 t 2Fj

3 X X X w.j / ˇ.j / X .j; m; t/ V .j; m; t/ B1 e; l.1/ mD2

j 2J2 t 2Fj

(19.13) X X w.j / X .j; 1; t/ V .j; 1; t/ l.5/

j 2J1 t 2Fj

C

X X w.j / X .j; 1; t/ W .j; 1; t/ B5 e; l.5/ t

(19.14)

j 2J1

3 X X X w.j / X .j; m; t/ V .j; m; t/ l.4/ mD2

j 2J2 t 2Fj

C

X X

X

j 2J2 t 2Fj m2M1[M 2

w.j / X .j; m; t/ W .j; m; t/ B4 e; l.4/ (19.15)

Note that in the actual implemented model constraints (12) and (13) are not implemented since there is no lack of bobbins B1 and B2 in the considered scenario (i.e., the stranding stage is the bottleneck). 4. Define the completion time Cj of each job: Cj D

X

.t C .V .j; 1; t/ e/ 1/ X .j; 1; t/ ; for all j 2 J1 (19.16)

t 2Fj

Cj D

X

.t C .V .j; m; t/ e/ 1/ X .j; m; t/ ;

(19.17)

t 2Fj

for all j 2 J2 with m 2 fM2 M3 g 5. Define the earliness/tardiness of each job, assuming that job j has due date dj : Cj dj D uj j ; for all j:

(19.18)

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where K is a large positive value to ensure that z is positive, and Cmax is defined by: Cmax Cj ; for all j: (19.19) 6. The non-negativity constraints on the u’s and v’s (the X ’s are already nonnegative by definition): uj ; j 0; for all j: (19.20) 7. The objective is to minimize the total cost. This is composed of two components: the earliness gain, valued at aj per unit time early; the tardiness penalty, valued at bj per unit time tardy min z D K C

X

T bj uj C bj Cmax aj j

(19.21)

j

in which K is a large positive value. Using this objective function, tardiness is penalized with the larger weight, makespan is then considered to be minimized and earliness to be maximized (to prepare for unanticipated breakdowns and malfunctioning occurrences). It is also easy to modify this function so as to work in JIT mode by penalizing both earliness and tardiness. In our implementation, aj D 1 and bj D nT. With n2 D jJ2 j this model has 3n2 C 5T equations in 2 C 3n2 C 2n2 T variables (the X ’s, u’s, v’s, C ’s and the two B’s). If n D 30 jobs and T D 90 periods (each representing a half shift, over 15 days with 3 shifts each day), we would have only 540 constraints in 5,492 original variables. This size problem, despite its large number of integer variables, does not overwhelm commercially available software. The major drawback of this model is the need to construct the vectors V , and there are 2n2 T of them (some 5,400 in this instance)!

19.6 Discussion and Lessons Learned from the Project One of the objectives of our platform was to assist the human planner by automating the tedious tasks of planning and scheduling and providing her/him with various scenarios (what we call scheduling strategies) to be applied to the actual shop. Does the plant have better control on its bobbins inventory after the implementation of the PlanOrdo system? It may be a bit too strong to state a blanket ‘yes,’ but significant success was achieved in this regard. We had detected a problem concerning the number of bobbins when running our first models. Due to the limited number of bobbins, the production manager could not plan certain mix of products to circumvent the lack of bobbins. He had to keep some bobbins free to have flexibility in facing changes in the planning at the execution levels – at the expense of delaying the delivery of some customer orders. The production manager had an (approximate) idea about the lack of bobbins when certain products should be planned at the same period, but could not justify the number and the kind of bobbins in

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shortage. The bobbins are expensive to buy, and they represent considerable investment in money and space. With our integrated optimization and simulation models, we were able to determine the minimum and maximum number of bobbins, by category, necessary for smooth and efficient production. These results were extremely important to the enterprise for solving this complex problem and were very useful to the plant in buying the exact number of bobbins to meet the enterprise objectives in terms of flexibility and cost reduction. The loading model has proved to be useful in the context of limited-time computer aided decision since it gives its answer in a very short time. The simulation model is able to check this answer and to refine the diagnostic operation, but it needs several minutes to several hours to run. This may not be very useful if the decision maker needs a quick (but not necessarily exact) answer. Even when all the stages in the plant are considered, the scheduling model has a set of local views: when a machine is available, the launch of the job that corresponds to the highest priority is accomplished quickly, albeit (sometimes) weakly due to the dispatching rules. The stranding model makes its decision based on the whole planning horizon: it is able to decide to delay a group of orders if this leads to global improvement. The meta-heuristics are somewhere between the stranding model acting globally and optimally but focusing on the bottleneck stage, and the scheduling model taking all machines into account but (only) doing low-level local optimization, taking into account all low level details (workers, tools, etc.). An objective that was internal to the research team was to accurately replay the actual shop and to obtain similar values for average measures (tardiness, or makespan, or lack of bobbins), that is used to determine whether the list of orders might be performed in a given time horizon (in the context of time-limited computer aided decision). This objective was fully achieved. We can in fact faithfully duplicate the shop performance over a specified planning horizon and evaluate its various measures of performance. ERP systems are designed to integrate all internal business processes of an enterprise. However, ERP systems still show a number of limitations for the management of production. First, ERP inherits two major shortcomings associated with its central MRP planning function: the assumption of unlimited capacity of resources and its non-stochastic nature. Also, ERP systems need additional external systems to monitor and collect real-time data. Finally, ERP systems are designed to collect trails of events but are not meant to help the decision-making process. Consequently, the ERP system’s inability to handle uncertainties and unexpected events limits its use for supporting decision making processes in dynamic production environment. MES provides a common user interface and data management system. MES’ functionality is typically broad and can support the management of production operations from the point of order release to manufacturing to the point of product delivery to finished goods. Using current and accurate data, MES can guide, trigger and report on plant activities as events occur. However, MES applications do not address all of the manufacturing execution processes required to replenish the supply chain while dynamically responding to unpredictable changes. Adap-

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tive manufacturing strategies also require defining the process management layer between ERP and process control, and translating demand driven supply chain requirements into a set of capabilities, systems, and workflow integration investments. Therefore, adaptive manufacturing organizations need an execution platform that connect manufacturing processes with enterprise and supply chain processes, that enables closed-loop mechanisms, and that provides decision-support to production personnel so they can deliver on their performance goals. The complementary research work between academics and industrialists led us to propose innovative concepts and methodology for designing and implementing generic solutions for real-time decision-making problems. Most of the literature on scheduling assumes that machines, or, in general, productive facilities, are available at all times. However, due to maintenance activities, machines cannot operate continuously without periods of unavailability. All this work has been done in collaboration with a target plant so that it was continuously possible to compare the results of the models to the real production process. Of course, there is still a gap (due to unpredictable workers interactions, among other factors), but the results given by our framework are close to the real results. It is possible to precisely compute the load of the shop on periods ranging from one week to several months if needed. Our future work will concern the evaluation of the real time control modules of this architecture (considering perturbations). This challenge deals with academic research and their industrial implementation. As the implementation of our planning and scheduling platform was successful, the application managers wanted to go farther in the decision process. The first direction concerns the integration of Enterprise Resource Planning (ERP) system and the real-time information, while the second direction should deal with the integration of production and maintenance decisions. We are still refining the models so as to include financial aspects and to help the plant personnel to closely determine the effective cost of each product. Acknowledgment This chapter reports on two projects supported by the Walloon Region of Belgium and the European Social Fund, which the authors acknowledge with gratitude. The authors thank the staff at Fontainunion for their invaluable cooperation and contribution to this work.

References Allahverdi A, Gupta JND, Aldowaisan T (1999) A review of scheduling research involving setup considerations. Omega 27(2):219–239 Appelqvist P, Lehtonen JM (2005) Combining optimisation and simulation for steel production scheduling. J Manufact Tech Manag 16(2):197–210 Artiba A, Levecq P, Riane F, Botta V, Guinet A (1997) A unified framework and a classification scheme to model production systems. In: Proceedings of the First International Conference on Operations and Quantitative Management, Jaipur, 467–474. ICOQM, January Artiba A, Emelyanov V, Iassinovski S (1998) Introduction to intelligent simulation: The RAO language. Kluwer

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Li J, Tang L, Pang H, Wu H (2005) Decision support system for steel tube production planning and scheduling. Proceedings of 2005 Chinese Control and Decision Conference, Haerbing, 1891–1894 Li J, Li L, Tang L, Wu H (2006) A case of rule-based heuristics for scheduling hot rolling seamless steel tube production. Expert Syst 23(3):145–158 Lindo Systems Inc. (1999) LINGO, The Modeling Language and Optimizer. LINDO Systems Inc., 1415 North Dayton Street, Chicago, Illinois 60622, USA, 1999. De Lit P, Latinne P, Rekiek B, Delchambre A (2001) An ordering genetic algorithm for assembly planning. Int J Prod Res 39(16):3623–3640 Mareschal B (1988) Weight stability intervals in multicriteria decision aid. Eur J Oper Res 33:54–64 Nowicki E, Smutnicki C (1996) A fast taboo search algorithm for the job shop problem. Manag Sci 42(6):797–813 Nowicki E, Smutnicki C (1998) The flow shop with parallel machines: A Tabu search approach. Eur J Oper Res 106:226–253 Osman IH, Kelly JP (1995) Meta-heuristics: An overview. In: Proceedings of Meta-Heuristics’95 (MIC’95). Kluwer, July Osman IH, Laporte G (1996) Metaheuristics: A bibliography. Ann Oper Res 63:513–628 Papadimitriou CH, Steiglitz K (1982) Combinatorial optimization: algorithms and complexity. Prentice-Hall, Englewood Cliffs, NJ, USA. ISBN: 0-1315-2462-3 Park H, Hong Y, Chang SY (2002) An efficient scheduling algorithm for the hot coil making in the steel mini-mill. Prod Plan Contr 13(3):298–306 Pinedo M (2002) Scheduling – Theory, Algorithms, and Systems, 2nd edn. Prentice Hall. ISBN: 0-13-028138-7 Pinedo M, Chao X (1999) Operations scheduling with applications in manufacturing and services. Irwin/McGraw-Hil. ISBN: 0-07-289779-1 Pomerol J-C, Barba-Romero S (2000) Multicriterion decision making in management: principles and practice. Kluwer, New York Rayward-Smith VJ, Osman IH, Reeves CR, Smith GD (eds.) (1996) Modern heuristic search methods. Wiley. ISBN: 0-471-96280-5 Reeves CR (ed.) (1995) Modern heuristic techniques for combinatorial problems, Advanced topics in computer science. McGraw-Hill. ISBN: 0-07-709239-2 Rekiek B (2001) Assembly line design multiple objective grouping genetic algorithm and the balancing of mixed-model hybrid assembly line. PhD thesis, Université Libre de Bruxelles, Bruxelles, Belgium Riane F, Artiba A, Iassinovski S (2001) An integrated production planning and scheduling system for hybrid flowshop organizations. Int J Prod Econ 74:33–48 Roux O, Duvivier D, Dhaevers V, Meskens N, Artiba A (2005) A multicriteria approach to rank scheduling strategies. International Conference on Industrial Engineering and Systems Management (IESM’05), Marrakech, Morocco, 673–682, May Smith ML, Seidmann A (1983) Due date selection procedures for jobshop simulation. Comput Ind Eng 7(3):199–207 Talbi E-G (2002) A taxonomy of hybrid metaheuristics. J Heuristics 8(2):541–564 Vignier A, Billaut JC, Proust C (1999) Les flowshop hybrides: e´ tat de l’art. RAIRO/RO 33(2):117–183 Avinash MW, Sarker BR, Lal AM (1995) A comparative study of some priority dispatching rules under different shop loads. Prod Plann Contr 6(4):301–310 Zhu B, Yu H (2002) A mathematical programming model for scheduling steel making – continuous casting – hot rolling production. Control and Automation, The 2002 International Conference on Control and Automation, June 16–19, 107. ISBN: 0-7803-7412-6

Chapter 20

Fuzzy Logic-Based Production Scheduling and Rescheduling in the Presence of Uncertainty Sanja Petrovic, Dobrila Petrovic, and Edmund Burke

“In theory, there is no difference between theory and practice. But, in practice, there is.” Jan van de Snepscheut, California Institute of Technology

20.1 Introduction Production scheduling represents a major administrative and management issue in modern production planning and control. Ever since the first results of modern scheduling theory appeared some 50 years ago, scheduling research has attracted a lot of attention from both academia and industry. The diversity of scheduling problems, the large-scale dimension and dynamic nature of many modern problemsolving environments make this a very complex and difficult research area. A typical production scheduling problem concerns the allocation of resources (e.g. equipment, materials, labour) to tasks over time under some constraints (Pinedo 2002). A variety of machine configurations (e.g. single machine, parallel machines, flow shops, job shops) and different embedded constraints related to release due dates, preemptions of jobs, available resources, etc., have led to a wealth of different scheduling models. However, in the majority of these models, scheduling is treated as a single criterion optimisation problem. Criteria such as makespan (i.e. the completion time of the last job), the number of tardy (early) jobs, the total flowtime and so on have been considered individually. However, to bring mathematical models nearer to many real-life scheduling problems, multiple criteria have to be considered simultaneously. The need to introduce multiple criteria decision-making methods to scheduling has been recognised in the last decade or so (T’kindt and Billaut 2002). Deterministic scheduling theory has been providing a large number of production scheduling models and algorithms for a number of years. Such models assume that all parameters are well known and precisely defined (Parker 1995).

S. Petrovic () Automated Scheduling, Optimisation and Planning Research Group, School of Computer Science and IT, University of Nottingham, United Kingdom e-mail: [email protected] K.G. Kempf et al. (eds.), Planning Production and Inventories in the Extended Enterprise, International Series in Operations Research & Management Science 152, c Springer Science+Business Media, LLC 2011 DOI 10.1007/978-1-4419-8191-2 20,

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However, underlying almost every real production/manufacturing process are activities fraught with uncertainties, and this prevents the results of deterministic scheduling theory from being widely applied to practice. Consequently, the deterministic scheduling models and algorithms have been extended to the stochastic case, mainly to models that assume that processing times are random variables with specified probability distributions (Weiss 1995). However, probabilistic characteristics of scheduling parameters are often lacking in practice. Information about production/manufacturing processes can be both imprecise and incomplete, or sometimes, it does not even exist. For example, the duration of tasks involving humans is seldom known precisely; due-dates are not always as rigid as they are supposed to be in the mathematical models or data concerning the production of a new product may not exist and so on. In these cases, the application of the standard methods of probability theory becomes difficult and often even inappropriate. In general, uncertainty has been addressed from different perspectives, among which fuzzy set theory has attracted considerable attention and has been successfully applied to various application domains, particularly where intuition and judgment play an important role (Zimmermann 1996). A fuzzy set is a very general concept that extends the notion of a standard set to accommodate gradual membership through various degrees. Fuzzy scheduling models have been developed in the last decade mainly, using fuzzy sets in a number of different ways (see, e.g., Slowinski and Hapke 2000). Imprecise scheduling parameters have been represented as fuzzy numbers, where operations on them have involved fuzzy arithmetic. The fuzzy parameters, which have received significant research attention, have been processing times and due dates (e.g. Kuroda and Wang 1996; Hong and Chuang 1999; Sakawa and Kubota 2000). However, there are models that have included some other fuzzy parameters such as fuzzy precedence relations to reflect the satisfaction level with respect to the precedence between two jobs (Ishii and Tada 1995), machine breakdowns (Li et al. 1994), and customer demand (Petrovic et al. 1999). Fuzzy dispatching rules have been developed, where linguistic qualifiers, modelled by fuzzy sets, have been used to express the concepts of similarity or closeness between two values; for example, ‘fuzzy versions’ of well-known dispatching rules such as the shortest processing time and the slack time rule have involved fuzzy qualifiers such as shorter processing times, much longer slack times, etc. (Grabot and Geneste 1994). In addition, fuzzy sets have been used to represent flexible constraints whose violation has to be minimised. Most often, the models included flexible temporal constraints, where the best schedule requested the least relaxation of release dates or due-date constraints (Fargier 1997). Various aspects of fuzzy sets theory have been used to consider multiple criteria simultaneously. One approach has been to define satisfaction grades for each criterion which reflect the decision maker’s preferences for the achieved values (see, e.g., Ishibuchi and Murata 2002). Satisfaction grades take values from the interval [0, 1], where 0 represents full dissatisfaction, and 1 represents full satisfaction with the achieved value of a criterion. Various aggregation operators have been used to combine the satisfaction grades achieved for all the criteria into a single value (Pedrycz and Gomide 1998). In Petrovic et al. (2007), a job scheduling problem is treated as

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Scheduling and Rescheduling in the Presence of Uncertainty

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a multi-criteria problem, where the decision function is defined as a linguistically quantified statement, such as ‘the schedule satisfies many criteria’, ‘the schedule satisfies most criteria’, etc. The linguistic qualifiers ‘many’, ‘most’, etc., are modelled using fuzzy sets. In most real-life environments, scheduling is an ongoing process where circumstances in both external business environments and internal production environment may change dynamically. To ensure high customer service level and profitability, there is a need to respond to disturbances such as machine breakdowns, delays in the arrival of materials, the arrival of rush orders, changes in orders, etc. As a consequence, the existing schedule has to be updated/modified either to restore its feasibility or to produce a new schedule that will have a higher level of quality in the changed environment. The importance of rescheduling has been recognised in the last decade, in particular. Vieira et al. (2003) describe a framework of rescheduling strategies, policies and methods. The authors state that production scheduling has had limited impact on practice, mainly because scheduling results do not consider fully the dynamic aspects of the scheduling environments. Aytug et al. (2005) reviewed different approaches to scheduling/rescheduling under uncertainties classifying them into three groups: (1) completely reactive approaches (also called dynamic scheduling and online scheduling by Vieira et al. (2003), (2) robust scheduling approaches, and (3) predictive–reactive approaches. Completely reactive approaches dispatch jobs in real time (when necessary) taking information that is available at the moment (e.g. Hall and Potts 2004; Subramaniam et al. 2005). Robust scheduling approaches generate schedules in such a way as to minimise the differences between the schedules that anticipate disruptions and realised schedules (Herroelen and Leus 2004). Finally, a predictive–reactive approach is defined as a two-step procedure. In the first step, a predictive schedule that optimises shop floor performance is generated. In the second step, the predictive schedule is modified into a reactive schedule after a disruption occurs (Mehta and Uzsoy 1999). An important issue is the frequency of rescheduling, where three main approaches have been developed (Church and Uzsoy 1992): (1) continuous rescheduling, (2) periodic rescheduling and (3) event-driven rescheduling. In this chapter, two decision support systems developed to support scheduling and rescheduling decision-making in real-life manufacturing companies are presented: 1. Fuzzy Multi-objective Genetic Algorithm for Scheduling (FMGAS) developed through collaboration with a printing company, Sherwood Press, in Nottingham, UK. 2. Large Holloware Scheduling (LHS) developed through collaboration with a pottery company Denby, Denby, UK. Although the manufacturing processes in the two companies are quite different, both decision support systems have embedded fuzzy multi-criteria scheduling and rescheduling methods. Scheduling parameters, such as the processing times of printing operations and the due dates of printing jobs are specified imprecisely. Similarly, the delivery times of raw materials required for product glazing and the occurrences

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of raw material shortages are vaguely known. All these parameters are modelled using fuzzy sets. In addition, in both manufacturing environments, there is a need to consider multiple objectives simultaneously. In the printing environment, the objectives of interest are the minimisation of average jobs’ tardiness, the number of tardy jobs, the setup times, the idle times of the machines and the maximisation of jobs’ throughput times. In the pottery environment, where disruptions such as shortage of raw materials occur, rescheduling has to be carried out considering both the efficiency and the stability of the new schedule. Schedule efficiency is measured as makespan, whilst schedule stability concerns changes in the new jobs’ allocation among the machines and their sequences. Finally, in both companies, scheduling and rescheduling rules are employed based on experience, intuition and the knowledge of the schedulers. These are formally presented in FMGAS and LHS using fuzzy IF-THEN rules, and the corresponding reasoning is based on fuzzy inference mechanisms. Fuzzy rules are developed in FMGAS to determine lot sizes into which each job should be split, taking into consideration the current workload on the shop floor, the size of the job and its urgency. In LHS, two sets of fuzzy rules are defined: (1) to determine, in the case of disruption, whether it is necessary to do rescheduling or not, and (2) to select an appropriate rescheduling method to apply. These decisions are made considering the time within the schedule horizon at which the raw material shortage occurred, the time necessary to obtain the requested raw material, the importance of the efficiency of the new schedule and the importance of its stability. Various tests have been carried out using real-life data or simulated data to analyse the quality of schedules generated using the fuzzy multi criteria methods. The chapter is organised as follows. Section 20.2 introduces a real-life scheduling example from the printing industry. The problem statement is given in Sect. 20.2.1. The satisfaction grades that are used to evaluate the performance of the generated schedules are presented in Sect. 20.2.2, followed by the description of the fuzzy multi-criteria genetic decision support system, FMGAS, that is developed to solve the given scheduling problem in Sect. 20.2.3. Experiments performed on real-life data are presented in Sect. 20.2.4. The sensitivity analysis, which is essential in evaluating the quality of the obtained schedule, is discussed in Sect. 20.2.5. A reallife scheduling problem considered in a pottery company is presented in Sect. 20.3. The statement of the problem is given in Sect. 20.3.1. A fuzzy multi-criteria predictive/reactive scheduling methodology implemented in the system LHS is described in Sect. 20.3.2. Conclusions and some directions for future work are outlined in Sect. 20.4.

20.2 Fuzzy Multi-criteria Scheduling in the Printing Industry 20.2.1 Problem Statement In this section, a complex real-life scheduling problem faced by a printing company Sherwood Press Ltd, Nottingham UK is presented. The notation used in the problem statement and throughout the section is outlined as follows:

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N J M Mi (i, j) pQij rj dQj CQ j TQj wj F Ff a T t Q1 Q2 Q3 Q4 Q5 Sk ˚

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Total number of jobs job, j D 1; : : : ; N Total number of machines Machine, I D 1; : : : ; M Operation of job Jj processed on machine Mi ; I D 1; : : : ; M; j D 1; : : : ; N Fuzzy processing time of operation (i,j) Release date of job Jj ; j D 1; : : : ; N, which denotes time when the job can start its processing Fuzzy due date of job Jj ; j D 1; : : : ; N Fuzzy completion date of job Jj ; j D 1; : : : ; N Fuzzy tardiness of job Jj ; j D 1; : : : ; N, Importance of job Jj ; j D 1; : : : ; N Total number of families of jobs Family of jobs, f D 1; : : : ; F Time needed for setup of a machine when different families of jobs are processed one after the other Planning horizon Total number of discretized time periods of the planning horizon ‚ Period in the planning horizon, t D 1; : : : ; T Average weighted tardiness Number of tardy jobs Total setup time Total idle time of machines Total flow time Satisfaction grades of the objective k; k D 1; ::; 5 Overall quality of the schedule measured as an aggregation of the satisfaction grades of the objectives

There are 18 machines on the shop floor, which are grouped into 7 work centres for printing, cutting, folding, card-inserting, embossing and debossing, gathering, finishing (includes stitching, trimming and packaging). Each work centre has a number of machines, some of them being identical. Printing orders with the same printing requirements and the same due date form a job. To each job, the production manager assigns a pre-determined order of processing on machines in some or in all of the work centres. A machine can process one job at a time. Jobs are grouped into families on the basis of their colour requirements. Printing machines require reconfiguration and/or cleaning when jobs of different families, which require different colours, are scheduled for printing one after the other. Therefore, a change of the family of jobs to be processed on the printing machines incurs a setup time (cost). This scheduling problem is modelled as a job shop problem in which N jobs J1 ; : : : ; Jj ; : : : ; JN have to be scheduled on M machines M1 ; : : : ; Mi ; : : : ; MM : The processing of job Jj on machine Mi is referred to as an operation, represented

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by an ordered pair .i; j /, with the processing time denoted by pij . The production manager estimates the processing time of each operation, taking into account the nature of the machine in use. Some machines are automated and can be operated at different speeds, while others are staff-operated and, therefore, the processing times are staff dependent. The processing times of jobs is imprecise due to either incomplete knowledge (e.g. there is lack of evidence, lack of certainty of evidence, or simply the data about the production line is imprecise) or uncertainty in the printing environment. The uncertain processing time of operation .i; j / is modelled bya 1 2 fuzzy set pQij with a triangular membership function given by a triplet pij ; pij ; pij3 , where pij2 identifies a typical value of the processing time, whereas pij1 and pij3 are the lower and upper bounds of the processing time. The latter two numbers reflect the spread of the possible processing times that is specified imprecisely as “about pij2 ”. In this model, they are set to be equal to 0:9 pij2 , and 1:1pij2 , respectively. A membership function for the uncertain processing time “about pij2 ” is shown in Fig. 20.1. To each job Jj , a release date rj , and a due date dj are assigned. These denote the date when the job can start its processing, and the date when the job is due to finish its processing, respectively. Following the scheduling practice in Sherwood Press, we classify jobs into three groups on the basis of their priorities and the corresponding tolerance of their tardiness. Jobs of high priority (Priority 1) are expected to be processed in time, with no delay, and therefore, there are crisp due dates defined for them. Jobs of medium priority (Priority 2) have due dates that can be extended by up to two additional days, whereas jobs of low priority (Priority 3) are given a delay tolerance of up to 1 week on top of the originally set due dates. The imprecise due dates dQj ; j D 1; : : : ; N , are represented by trapezoidal membership functions, as illustrated in Fig. 20.2, where membership degrees represent satisfaction grades associated with jobs’ completion times with respect to their due dates. Based on its priority group, a weighting coefficient wj ; j D 1; : : : ; N , is assigned to each job Jj . Weighting coefficients 0.7, 0.4, and 0.3 are assigned to jobs of Priority 1, Priority 2 and Priority 3, respectively.

µ ~pij

~ pij 1.0

Fig. 20.1 Membership function of the fuzzy processing time “about pij2 ”

0

p1ij

pij2

pij3

t

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µ d~ j 1.0

~ dj

Priority 1

Priority 3 Priority 2

0

dj

dj +2

dj +7

t (in days)

Fig. 20.2 Membership functions of three fuzzy sets Due date of job Jj if the job has Priority 1, Priority 2 and Priority 3

If it is not possible to meet the due dates for all the jobs, to deliver at least parts of the job orders to the customers, the production manager can decide to split the processing of some jobs into lots. Usually, two lots are introduced. Therefore, the decision has to be made whether to split a job into lots, and, if so, the sizes of the lots (i.e. the number of items to be processed in each lot) have to be determined. Also, the production manager has to allocate the machines for lots processing. The usual practice is to load one machine in a work centre heavily, but this practice is not followed strictly. The production manager has to generate a non-preemptive sequence of operations of N jobs on M machines with the aim of maximising the performance of the schedule. The performance of the schedule is measured using the following five objectives: Q1 : the average weighted tardiness of schedule s Q1 .s/ D

N 1 X wj TQj ; N

(20.1)

j D1

where TQj is the fuzzy tardiness of job Jj , which takes into account the difference between the estimated completion and due date of the job. Due to the fuzzy processing times of job operations, the completion time and tardiness of the job are also fuzzy. The standard fuzzy addition based on the extension principle, which preserves the triangular membership function of the result, is used to calculate the job’s completion time (Dubois and Prade 1978). In a production environment with uncertain processing times of jobs, an interesting question arises regarding the measurement of jobs’ tardiness, and that is how to compare a fuzzy completion time with a fuzzy due date, represented by a triangular and a trapezoidal membership function, respectively. This issue will be discussed in Sect. 20.2.2.

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Q2 : the number of tardy jobs in schedule s Q2 .s/ D

N X

uj ;

where uj D

j D1

1 if job Jj is tardy : 0 otherwise

(20.2)

In direct contrast to a deterministic environment where the condition of job’s tardiness is easy to check, in a fuzzy environment this issue has to be considered with care. This will be explained in detail in Sect. 20.2.2. Q3 : the total setup time in schedule s Q3 .s/ D

T F X X

aXf t ;

(20.3)

t D1 f D1

where

Xf t D

1 if there is a need for a setupfor familyf of jobs at period t 0 otherwise

and parameter a denotes the setup time needed to make a machine ready for a changeover of the job families. Q4 : the total idle time of machines in schedule s, where the idle time is defined as the time within the planning horizon during which the machines are used neither for processing of jobs nor for the setup of the machines: Q4 .s/ D

M X

0 @.CQ ia sQib /

i D1

N X j D1

1 pQ ij A

T X F X

aXf t ;

(20.4)

t D1 f D1

where CQ ia and sib are the fuzzy completion time of the last operation and the start time of the first operation on machine Mi within the planning horizon , resN P pQij is the total fuzzy processing times of all the operations on pectively, and j D1

machine Mi . Q5 : the total flow time defined as the total time that jobs spend on the shop floor in schedule s N X Q5 .s/ D .CQ j rj /: (20.5) j D1

20.2.2 Satisfaction Grades Obviously, the stated objectives are measured in different units i.e. they are incommensurable. For example, the weighted average tardiness of jobs is given in time units (in minutes), while the number of tardy jobs takes an integer value from the interval Œ0; N . To enable the handling of multiple objectives simultaneously, and

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also to enable the production manager to express his/her preferences with respect to the achieved values of the objectives, the values of the objectives are mapped into satisfaction grades. The satisfaction grades take values from the interval [0, 1], where 0 and 1 represent full dissatisfaction and full satisfaction with the achieved objective value, respectively. The satisfaction degree of the objective linearly increases from unsatisfactory to satisfactory values of the objectives. The calculation of the satisfaction grade Sk ; k D 1; : : : ; 5, of each measure of performance Qk is briefly explained here, while more details can be found in Fayad and Petrovic (2005); Petrovic et al. (2008b). S1 : For each job, the satisfaction grade of its tardiness is calculated taking into account the intersection of the job’s fuzzy completion time and its fuzzy due date. Two approaches are investigated: 1. The possibility measure CQj .dQj / introduced by Dubois et al. (1988), which evaluates the degree of possibility that fuzzy completion time CQ j occurs within the fuzzy due date dQj (Itoh and Ishii 1999): n o CQj .dQj / D sup min CQ .t/; dQ .t/ ; j

j

j D 1; : : : ; N

(20.6)

where CQj .t/ and dQ .t/ are the membership functions of fuzzy sets CQ j and j dQj , respectively, as illustrated in Fig. 20.3. The area of intersection measures the part of CQ j that is completed by due date Q dj (Sakawa and Kubota 2000). It is illustrated in Fig. 20.4. The satisfaction grade of the tardiness of job Jj is defined in the following way: area.CQ j \ dQj / = area.CQ j /: µ(t) 1 Fig. 20.3 The satisfaction grade of the tardiness of job Jj based on the possibility measure

~

d

j

(20.7)

~

Cj

0

t

µ(t) ~

1 Fig. 20.4 Satisfaction grade of the tardiness of job Jj based on the area of intersection

0

dj

~

Cj

t

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The satisfaction grade of the job’s tardiness based on the possibility measure reflects a more optimistic evaluation of the jobs’ tardiness compared to the one based on the area of intersection because the former measure considers the point of intersection of the two fuzzy sets with the highest membership degree only, while the latter one considers the part of the fuzzy completion time that falls within the fuzzy due date. The satisfaction grade of the average weighted tardiness is calculated as the average of the satisfaction grades of the tardiness of all the jobs. S2 : Due to the fact that a job’s tardiness is fuzzy, a threshold of the satisfaction grade of the job tardiness is introduced to determine whether a job is tardy or not. A linear function is defined to map the number of tardy jobs in a schedule into a satisfaction grade of the number of tardy jobs. S3 ; S4 and S5 : Fuzzy values of these measures of performance are defuzzified first, using the centre of gravity, and after that they are mapped into satisfaction grades using linear functions. The parameters of the linear functions used in the calculation of satisfaction grades reflect the scheduling policy of the company. For example, when assessing Q2 the production manager is fully dissatisfied with a schedule with over 15% of the tardy jobs, while fully satisfied with a schedule with no tardy jobs. Consequently, a linear function is defined to determine the satisfaction grade, where the satisfaction grade linearly decreases from 1 (when there is no tardy job, i.e. the percentage of tardy jobs is 0) to 0, when the percentage of tardy jobs is greater or equal to 15.

20.2.3 Fuzzy Multi-objective Scheduling The FMGAS consisting of three modules was developed for the scheduling problem described above. The three modules are: 1. A fuzzy rule-based system for lot sizing. 2. A genetic algorithm for machine allocation. 3. A fuzzy multi-objective genetic algorithm for sequencing of jobs’ operations on the machines. Each of the three modules is explained in the remaining part of the section.

20.2.3.1 A Fuzzy Rule-Based System for Lot Sizing A fuzzy IF-THEN rule-based system was developed, which decides whether to split jobs and determines the size of each lot (Petrovic et al. 2008b). The fuzzy rules employ four linguistic variables in their premises (IF part of the rules): 1. Size of the job: takes into consideration the number of items to be produced, and it is described by the linguistic terms: Small, Medium and Large.

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2. Slack of the job: represents the available time for the job processing so that the job meets its due date, and is described by three linguistic terms: Small, Medium and Large. 3. Workload on the shop floor: determined by the total number of items to be processed on the shop floor, and is described by two linguistic terms: Not Large and Large. 4. Priority of the job indicates its importance, and is expressed by three crisp numbers: 1, 2 or 3. The membership functions for these linguistic terms were defined in collaboration with Sherwood Press following their policy for lot-sizing. As an example, membership functions of the three linguistic terms defined for the variable size of job are presented in Fig. 20.5a.

a

µ

1.0

Small

5

25 30

Medium

80

60

Large

100

120

number of items (in thousands)

Linguistic terms for the premise variable size of job

µ

b

Medium Small Negative Large Negative

−20

Medium Positive

1.0

Large Positive

0

20

percentage of the size of the job

Linguistic terms for the conclusion variable change of lot

Fig. 20.5 Membership functions for linguistic terms used in the premise and conclusion of the rules

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IF size of job is Small THEN change of lot is Large Positive IF size of job is Medium THEN change of lot is Small IF size of job is Large THEN change of lot is Large Negative Fig. 20.6 Fuzzy rules for lot-sizing with size of job in the premise

Fuzzy IF-THEN rules are defined to mimic the way that the production manager makes a decision regarding the lot sizing, assuming that each job is split into a maximum of two lots. The rules derive the size of each lot based on imprecise values of the identified variables (apart from the priority of the job, which has a crisp value). However, to cover all the domain of the premise variables, 54 .D 3 3 2 3/ fuzzy rules would need to be defined. This would be a difficult, if not impossible, task. To overcome this difficulty, we define fuzzy rules for each of the four fuzzy variables independently, and we introduce a conclusion variable change of lot, which indicates a necessary change (increment/decrement) of the current size of the first lot. The variable change of lot is described by the linguistic terms Large Negative, Medium Negative, Small, Medium Positive, Large Positive. The membership functions of the fuzzy sets are given in Fig. 20.5b. The initial lot-sizing decision is to split each job into two lots of equal size, i.e. each lot will be 50% of the job size. As an illustration, three fuzzy rules with the size of the job in their premises are given in Fig. 20.6. For example, if the size of job is Small, then such a job should be processed as much as possible in the first lot, i.e. the change of lot is Large Positive. The defined four groups of fuzzy rules are applied in sequence, one by one. The change of lot sizes is based on the truth value of the premise of each of the rules. This approach also enables the implementation of the complex policy of the company. The size of the job and its slack can be decisive for the lot-sizing decision, namely, if the size of job is Small enough and its slack is Small enough, then there is no need to split the job into lots, independently of the workload of the shop floor and the priority of the job. Therefore, only the fuzzy rules from the first two groups can be considered.

20.2.3.2 A Genetic Algorithm for Machine Allocation A machine has to be allocated to each operation of a lot. A genetic algorithm was developed, which aims to balance the load of the identical machines in each work centre. Generally, a genetic algorithm is an iterative stochastic search algorithm, which maintains a population of solutions called chromosomes (Sastry et al. 2005). Each chromosome consists of genes. In the developed genetic algorithm, each chromosome consists of two layers. In the first layer, each gene represents a job whose value shows the number of lots the job is split into. These numbers determine the number of genes in the associated second layer. The genes in the second layer present the machine allocated for each lot for its processing. Chromosomes are iteratively modified by means of a mutation operator, applied with a certain probability,

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which swaps the machines allocated to two lots. If the job is not split into lots, then the mutation operator replaces the allocated machine by a machine randomly selected from a given set. The selection phase employs a roulette wheel to determine which chromosomes will survive for the next iteration. The roulette wheel calculates the probability of selection of each chromosome based on its fitness. In addition, the elitist strategy is employed which means that the best chromosome in each iteration is saved and transferred to the next iteration. The objective function (i.e. fitness function) measures the dispersion of the machines’ load from the average load of all machines in the work centre. The rationale behind such a defined objective function is that if all the machines have a load which is equal or close to the average load then the machines in the work-centre are well balanced. The detailed description of the algorithm can be found in Petrovic and Fayad (2005). Experiments with the following different approaches to machine allocation were conducted: 1. An allocation that focuses mostly on a single machine in a work-centre. 2. Using the developed genetic algorithm that performs load balancing. 3. Random allocation of lots to appropriate machines. It can be concluded that the load balancing of the machines does improve the overall quality of the schedule, especially the objectives related to tardiness, because jobs do not wait to be processed on a single machine, but use identical machines instead. The benefit of a good load balancing is also evident in the shorter makespan of the schedules generated by the load balancing approach and also in the similar completion times of operations on all machines in a work centre.

20.2.3.3 A Fuzzy Multi-objective Genetic Algorithm for the Sequencing of Jobs’ Operations on the Machines A fuzzy multi-objective genetic algorithm was developed for the sequencing of jobs’ operations on the machines (Fayad and Petrovic 2005). The genetic algorithm evolves the decision on which dispatching rules to be used for sequencing the operations of jobs (lots) on each machine taking into consideration both the imprecise processing times and the due dates of jobs, and the determined sizes of lots. The dispatching rules considered are: Early Due Date (EDD), Shortest Processing Time (SPT), Longest Processing Time (LPT), Most Work Remaining (MWR), Same Family Together (SFT) and Highest Priority (HP). The following characteristics of the jobs are used in the ranking of their operations: EDD uses the due date of the jobs, SPT and LPT employ the required processing times of the jobs, MWR calculates the required processing time for all the remaining operations of the jobs, SFT takes into account the family of jobs that the operation belongs to, i.e. priority is given to operations of the same family as the previous operation, and HP is based on the priority of the jobs. These rules operate in the following way: if there is only one operation ready for processing, it will be processed immediately; if there are more than one operations available, the dispatching rule associated with the machine will

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determine the order of their processing. The dispatching rule associated with each machine determines a starting and completion time of each operation, i.e. it determines a schedule. Each chromosome in the developed genetic algorithm contains 2 subchromosomes of length M (the number of machines): the first one represents the machines, while the second one represents the dispatching rules associated with the corresponding machine. A one-point crossover operator is defined, which combines two sub-chromosomes to create an offspring. In the case of the machine sub-chromosome, a repair mechanism ensures that machines are not duplicated. The mutation operator swaps two genes in both sub-chromosomes. These two operators are applied with certain probabilities. A roulette wheel selection is used in the selection phase together with the elitist strategy. Typically, meta-heuristics represent schedules in job shop problems as permutations of the operations to be scheduled on the machines. Although such a representation creates a larger search space than the representation by means of dispatching rules associated with the machines (the representation used in our genetic algorithm), it was shown experimentally that solutions generated by utilising the latter representation are of higher quality and can be produced in far less time (Montgomery et al. 2006). A satisfaction grade is calculated for each of the objectives in the generated schedule. Different aggregation operators can be used to combine the obtained satisfaction grades of the objectives. The two most often in use are the average satisfaction grade and the minimum of all the satisfaction grades. These aggregation operators enable the production manager to express his/her preferences. The average aggregation operator allows compensation for a bad value of one objective, namely a higher satisfaction grade of one objective can compensate, to a certain extent, for a lower satisfaction grade of another objective. The minimum operator is noncompensatory, which means that the solution with a bad performance with respect to one objective will not be highly evaluated no matter how good is its performance with respect to the other objectives.

20.2.4 Experiments on Real-Life Data Data sets obtained from Sherwood Press are used to carry out a series of experiments. The input data for each job includes: the identification number of the job, the order of machines on which the job requires processing, the fuzzy processing time of the job operations on each of the machines, the required quantity of the items, the family of the job, the release date, the due date and the priority of the job. The planning horizon is 4 weeks, discretized in minute steps. A month that is considered to be rather busy with an initial number of 158 jobs and a workload of 6388763 required items is chosen to illustrate the developed system. The system was developed using Visual CCC and the testing was performed in the Windows XP environment on a PC Pentium with 2 GHz and 512 MB of RAM.

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20.2.4.1 Experiments on Lot-Sizing Four different approaches to lot-sizing were investigated: 1. A simple heuristic that is sometimes used by the company, which splits a job whose processing time is longer than one working day (15 h) into two lots. 2. The fuzzy rule-based system which splits jobs into two lots. 3. The fuzzy rule-based system which splits jobs into three lots, first making the decision on lot-sizing for two lots, and then applying the same set of rules for splitting up the second lot. 4. No lot sizing. Each approach leads to a different number of lots as given in Table 20.1. The overall satisfaction grade of the schedules, calculated using the average operator, and the satisfaction grade of each objective are given in Table 20.2. Average values obtained after 10 runs of the algorithms are presented together with the best results achieved (the overall best results are shown in bold). The fuzzy lot-sizing approaches yield overall better results than the simple heuristic, and no lots approach. It is interesting to note that the satisfaction grade of the setup time S3 is higher in the fuzzy approaches to lot-sizing, when there are more operations to be scheduled (splitting jobs into lots increases the total number of operations). However, the higher number of operations gives better opportunities for batching, i.e. the grouping of jobs of the same family, which leads to a smaller

Table 20.1 Number of lots in different approaches to lot sizing

Approach to lot-sizing Simple heuristic Two lots Three lots No lots

Table 20.2 Average and best values of the overall satisfaction grades and satisfaction grades of the objectives

Approach to lot-sizing Simple heuristic Average Best

Number of lots 207 266 299 158

˚

S1

S2

S3

S4

S5

0.46 0.49

0.42 0.43

0.64 0.73

0.26 0.30

0.37 0.38

0.61 0.61

Two lots Average Best

0.51 0.51

0.43 0.44

0.71 0.72

0.43 0.43

0.37 0.38

0.61 0.60

Three lots Average Best

0.49 0.51

0.42 0.42

0.66 0.69

0.44 0.49

0.37 0.38

0.59 0.60

No lots Average Best

0.42 0.43

0.41 0.42

0.57 0.62

0.11 0.11

0.36 0.34

0.62 0.65

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Table 20.3 Tardy first lots obtained using different approaches Approach to lot-sizing Number of Number of Percentage of (Total number of jobs) tardy jobs tardy first lots tardy jobs

Percentage of tardy first lots

Simple heuristic (207) Average Best

14.4 11

11.6 10

7 5

81 91

Two lots (266) Average Best

15.4 14

8.5 8

5 5

55 57

Three lots (299) Average Best

20.4 22

11.8 11

7 7

58 50

setup time. It is recorded that the 2-lot and 3-lot fuzzy approaches to lot-sizing generate 172 batches, while the simple heuristic and no lots approach lead to 169 and 161 batches, respectively. The data in Table 20.2 related to the due-dateoriented objectives (average tardiness and the number of tardy jobs) in the schedules with lots need to be interpreted carefully. The presented satisfaction grades take into consideration the completion of the last lot of jobs. It can be observed that the average satisfaction grades of the averageweighted tardiness S1 , achieved in all the approaches, with and without lots, are similar, while the average satisfaction grade of the number of tardy jobs S2 is highest in the schedules with 2 lots. To further investigate the effect of lot-sizing, we treat each lot as a separate job, with the same parameters (release time, due date, etc.) as the job it originated from. Table 20.3 presents the average and best values of the number of tardy jobs (i.e. lots), the number of tardy first lots, the percentage of tardy jobs (where the total number of jobs is considered to be equal to the total number of lots), and the percentage of tardy first lots among the all tardy jobs. The number of jobs in each approach to lot sizing is shown in the parenthesis. The 2-lot approach gives the lowest number of tardy first lots. The completion time of the last lot determines the ultimate quality of the schedule, but the smaller number of tardy first lots will increase the satisfaction grade of the customer. The conducted experiments justify the splitting of jobs into lots and also indicate that it is worthwhile to consider more sophisticated rules for lot-sizing, which take more parameters into account, instead of using only the available capacity on one day (i.e. a working shift) on the shop floor.

20.2.5 Sensitivity Analysis The aim of the sensitivity analysis is to investigate the consequences of changes of the basic model parameters, such as processing times of jobs, on the measures of performance of a schedule. A series of numerical experiments are designed to perform sensitivity analysis of the scheduling problem under consideration (Petrovic et al. 2008a). Numerical experiments are used due to the fact that any analytical

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approach appears to be too complex, and it is practically impossible to derive analytical formulae which would describe the effects of changing jobs processing time on the measures of performance of a schedule. In addition, while in the majority of other studies of sensitivity analysis, a single objective has been considered (see, e.g., Sotskov 1991; James and Buchanan 1998; Hall and Posner 2004), we take into account multiple objectives simultaneously, which makes the sensitivity analysis even more complex. The change of a job processing time in each experiment is done by expanding the support of the corresponding fuzzy set, where the support includes all processing times with membership degrees greater than zero. The support of the fuzzy processing time is expanded in such a way that the lower bound remains fixed, while the upper bound increases gradually, in discrete steps, with the increment of 2 5% of its modal value pij up to 100%. Increasing the upper bound of the processing time means that the value of the processing time has become increasingly pessimistic. An interesting question to be discussed when the pessimistic processing time of a job is increased is what changes of the initial schedule are useful to examine. A number of characteristics are important for a schedule, such as the start time of the processing of each operation, the operation sequence on each machine, a dispatching rule associated with each machine, which determines the operation sequence on that machine, and the measures of performance of the schedule. Every increase of the processing time of an operation leads inevitably to the corresponding increase of the start time of the later operations. Therefore, the start time of the operations will not be used as the indicator of the change of the initial schedule. It is more meaningful to evaluate the change of measures of performance, and consequently, the change of the overall quality of the new schedule. Also, it is decided to keep the dispatching rules associated with the machines in the initial schedule fixed, which may cause the change of the operation sequence on each machine. Due to the change of operation processing times on a machine, the associated dispatching rule can change the operations sequence on that machine. It can also affect the sequence on some other machines to which the dispatching rule Most Work Remaining is assigned. A number of numerical experiments are designed to help the production manager to evaluate the quality of a schedule with respect to all the measures of performance from different aspects. The results of some of them are presented in the remainder of the section. The first experiment is designed to evaluate the effects of the change of the upper bound of an operation processing time on a single objective, on the corresponding satisfaction grade, and on the overall satisfaction grade (the aggregation operator that is used is the average of the satisfaction grades). The experiments show that changes of operation processing times produce some irregular effects on the performance measure of the initial schedule. The obtained values of the objectives are partially conflicting: some among them are improved, while some of them deteriorated. Such phenomena are expected due to the high complexity of the scheduling problem, and also, due to the fact that the scheduling problem is stated and solved by applying discrete mathematical structures. Irregularities are complete in the sense that all possible situations occur: (1) starting from a certain change of the operation processing time there is either improvement or deterioration of the objective values,

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Table 20.4 The range of changes of the performance values as a consequence of changing the upper bound of the processing time of each operation, one by one Objective values in Performance the initial schedule Qk Sk ˚ Q1 789 [minutes] [272, 372] [0:005, 0.01] Q2 9 [jobs] [1, 3] [0:095, 0.031] 722 [minutes] [17, 16] [0:014, 0.016] Q3 189370 [minutes] [2486, 23001] [0:029, 0.002] Q4 460602 [minutes] [7290, 160692] [0:056, 0.002] Q5 ˆ 0.547 [0:022, 0.007]

(2) under some changes of the operation processing time some objectives improve, while under other they deteriorate. It could be very useful for the production manager to record the range of changes of the objective values, of the corresponding satisfaction grades, and of the overall satisfaction grade of the schedule due to the prolongation of operation processing times. As an illustrative example, Table 20.4 presents the objective values of the initial schedule generated by FMGAS, and Qk ; Sk ; k D 1; : : :; 5, and ˚ which denote the difference between the objective values, satisfaction grades, and the overall satisfaction grade, respectively, in the new and in the initial schedule. Note that the positive (negative) values in the Qk column mean deterioration (improvement), respectively, while in the Sk and ˚ columns, positive (negative) values denote improvement (deterioration) of the satisfaction grades and the overall satisfaction grade, respectively. Such an experiment can help the production manager to identify the critical operations, i.e. the operations whose prolongation of the upper bound of the processing time, affects most the objective values and the overall quality of the schedule. The sensitivity analysis can also help the production manager in evaluating the effect of prolonging all the operation processing times on a single machine. That can happen if the machine is running at a lower speed, to decrease the cost, or if new or less experienced staff is allocated to operate the machine. Experiments, similar to the previous ones are performed, but this time the processing times of all the operations sequenced on a machine are increased. The recorded changes in the objective values and the overall quality of the schedule can help in identifying the machines that are critical with respect to the performance of a schedule.

20.3 Scheduling/Rescheduling in the Pottery Company 20.3.1 Problem Statement In this section, a prototype of a decision support system, LHS, developed and implemented in collaboration with Denby Pottery Ltd., UK, is presented. A fuzzy multicriteria scheduling/rescheduling method underpins the LHS system. Denby Pottery Ltd. manufactures a wide range of ceramic tableware products. The manufacturing

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process includes two sequential phases. In the first phase, different shapes/forms are made of clay and fired into semiproducts (called biscuits). After that, the biscuits are glazed and fired in kilns to become the end-products. The problem under consideration is scheduling in the glazing section and large holloware glazing, in particular. There are three flow lines, where on each of them, eight operations such as rinsing, dipping, resist dipping, two types of whiffing, banding and trimming are carried out in sequence. The flow lines have the same speed and, therefore, we can assume that we have three identical parallel machines. However, there are some technological requirements that specify the machine upon which each product can be glazed. At present, the glazing section leader builds the schedule by hand, dealing with approximately 30 large holloware jobs on a weekly basis. The number of jobs can vary according to the production plan. Jobs are independent and non-preemptive, i.e. once a job begins processing it has to be completed without interruptions. Set-up time is not considered. A description of a job includes the number of items of a specific product to be produced. Table 20.5 presents a sample of 10 jobs, where, for example, some jobs can only be processed on machines M1 and M3 . The pattern refers to the glaze that is needed to process the job, where each job, typically, requires one glaze only. The item refers to the shape of the product to be processed. The standard time refers to the processing time of 100 items, while the processing time refers to the time required for processing the job taking into consideration the batch size of the job. However, there are uncertain disruptions on the shop floor that have an adverse effect on the job completion times and consequently on the schedule. One of the disruptions with the greatest impact occurs when a machine runs out of glaze, or a glaze has a low quality and, therefore, cannot be applied. Formally, the glazing scheduling problem can be stated as a non-identical parallel machine scheduling problem with N jobs to be scheduled on M D 3 machines, where each job Jj ; j D 1; : : :; N , can be processed on a subset Mj of the three machines. The job processing times are denoted by pj ; j D 1; : : :; N . The objective is to generate an allocation of the jobs to the machines and sequences of the jobs

Table 20.5 Scheduling data

Pattern Regency green Greenwich Marrakesh Regency green Boston Boston Boston Spirit Fire Blue Jetty

Item Large Jug Large Jug Sauce Boat Teapot Base Small Jug Sauce Boat Tea Pot Base Large Teapot Base Large Jug Sauce Jug

Standard time/100 items [in hours] 4 4 6 4 1 1 2 6 5 4

Processing time [in hours] M1 10 X 2 X 5 X 7 X 1 X 2 X 1 X 4 X 1 X 6 X

M2 X X X X X

M3 X X X X X X X X X X

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on the allocated machines in such a way as to minimise the makespan Cmax , i.e. the time that is taken to complete all the jobs taking into consideration uncertain glaze shortages. The makespan is defined as Cmax D fCj jj D 1; : : :; N g, where Cj is the completion time of job Jj . In the following sections, the application of the developed decision support system for solving a typical scheduling/rescheduling problem in the glazing section is demonstrated.

20.3.2 Fuzzy Multi-criteria Predictive–Reactive Scheduling The developed fuzzy predictive–reactive scheduling method developed and implemented in LHS is presented in Fig. 20.7. The predictive schedule is generated in LHS in such a way as to be able to absorb the uncertain disruptions caused by glaze shortages. One way to achieve this is to extend initially specified job processing times with certain, so-called, ‘idle times’ (see, e.g., Mehta and Uzsoy 1999). In LHS, uncertain disruptions are modelled using fuzzy sets and corresponding idle times are calculated using fuzzy arithmetic. LHS can handle two types of disruption, which can have a low or a high impact. A low impact disruption can be recovered in an assumed time, and, therefore, can be absorbed by the predictive schedule. On the contrary, a high impact disruption cannot be recovered in a reasonable time relative to the duration of the generated schedule. In this case, to respond to the disruption, rescheduling is carried out, i.e. the predictive schedule is modified into a reactive schedule. LHS generates two decisions: (1) whether it is necessary to reschedule and (2) which rescheduling method to apply. Two sets of fuzzy IF-THEN rules are developed and implemented in LHS

Fig. 20.7 Predictive reactive scheduling (adapted from Mehta and Uzsoy 1999)

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to support rescheduling. One set of the fuzzy rules determines when to reschedule, whilst the other one determines which rescheduling method to use. In the following sections, the developed method is described including the modelling of uncertain disruptions using fuzzy sets, predictive scheduling, rescheduling methods and the fuzzy rules for rescheduling decision-making.

20.3.2.1 Modelling of Uncertain Disruptions Using Fuzzy Sets Traditionally, uncertain disruptions have been modelled using probability distributions that concern the occurrences of well-defined events. Corresponding probabilities are assessed or estimated taking into consideration the repetition of events. For example, an uncertain disruption that has most often been treated in the literature is machine breakdowns, typically described by two parameters, such as the mean time between failures and the mean time for repair (Aytug et al. 2005). These two parameters may be determined based on maintenance records. However, historic data may not be available for other sources of disruptions as is the case with glaze shortage events. The company does keep records of neither the numbers of glaze shortages that happened in the past nor the times that were required for the glazes to be supplied. However, the glazing section leader has imprecise knowledge and accumulated experience related to glaze shortage occurrences and may estimate the corresponding data using linguistic terms, such as ‘the number of glaze shortage occurrences is much higher than a certain number’ or ‘the glaze is usually delivered in about a certain period’. This data can be represented using fuzzy sets. Figure 20.8 shows the two linguistic terms mentioned above, which are modelled by fuzzy sets. Fuzzy set Og , which represents the number of glaze shortage occurrences per K P standard time period, is defined using a discrete fuzzy set Og D Og nocgk = kD1

nocgk , where g; g D 1; : : :; G represents a glaze, nocgk ; k D 1; : : :; K, represents

a

µOg (noc g i )

b

1

µ Rg (trg )

1

nocg1 nocg2 nocg3 nocg4 Fuzzy set Og that represents number of glaze shortage

nocgi

trg1

trg2

trg3

trg4

trg

Fuzzy set Rg that represents glaze delivery

Fig. 20.8 Fuzzy sets that represent uncertain disruptions caused by glaze shortage

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the possible numbers of glaze shortages and Og nocgk represents subjectively determined possibilities of having nocgk glaze shortages (see Fig. 20.8a). The delivery time of glaze g is represented by a fuzzy set Rg with a continuous membership function Rg trg . The trapezoidal membership function Rg models the delivery time that is highly likely between trg2 and trg3 , but is definitely not shorter than trg1 and is not longer than trg4 ; it is specified by four parameters tr g1 ; trg2 ; trg3 ; trg4 (see Fig. 20.8b). The idle time idj ; j D 1; : : :; N to be added to the processing time pj of job Jj is equal to the estimated total glaze shortage time during the processing of job Jj . It is calculated as the product of the fuzzy number of glaze shortage occurrences per unit time period, Og, and the fuzzy glaze delivery time, Rg, using the principles of fuzzy arithmetic and the concept of level 2 fuzzy sets (Zadeh 1971). The fuzzy set obtained is defuzzified, i.e. represented by a scalar that most appropriately represents the fuzzy set. The extended processing time pdj for job Jj is calculated as pdj D pj C idj ; j D 1; ; N . More details of the procedures are presented in Duenas and Petrovic (2008).

20.3.2.2 Predictive Schedule Once the idle times are added to the initial processing times, the predictive schedule PS is constructed using dispatching rules. The dispatching rules are applied for the scheduling problem under consideration because they have low computational complexity and are easy to implement in practice. However, they do not provide an optimal schedule. Two dispatching rules that perform well when applied to parallel machine scheduling problems with the objective of minimising makespan without preemptions (Pinedo 2002) are selected, including Least Flexible Job (LFJ) and Longest Processing Time (LPT). In the method developed they are combined as follows. To deal with the technical restrictions, where not all jobs can be glazed on any of the three available machines, the LFJ rule is applied first. This rule allocates the job that can be processed on the smallest number of machines, to the machine that is freed. In the case when there is more than one job that can be allocated to the machine, another dispatching rule, LPT, is applied to select a job. This rule assigns the job with the longest processing time to the free machine. In this case, the extended processing times pdj ; j D 1; ::; N are considered. It is worth noting that the planned job completion times determined by the predictive schedule are used to plan the subsequent manufacturing activity, i.e. firing of the glazed products in the kiln. When the predictive schedule is generated, it is released to the glazing shop floor. The actual schedule realised on the shop floor will keep the same job allocations and sequences on all the machines as the predictive schedule. However, the job completion times in the predictive and realised schedule may be different. In the predictive schedule, the job completion times are determined taking into account idle times that are added to the processing times of all the jobs. The actual, i.e. realised job

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Table 20.6 Predictive schedule PS Machine Jobs’ sequence M1 27 18 11 8 12 21 10 4 M2 M3 16 1 17 23

22 6 24

5 15 2

26 7 13

3 25 20

19

30

28

14

29

9

Machine M1 M2 M3

19.9 30.6 20.1

21.3 33.0 21.1

26.1 35.4 28.1

29.7 37.2 31.2

32.3

34.7

37.0

33.7

36.1

37.3

Completion times 6.2 11.4 15.5 8.2 15.9 23.4 7.8 12.8 15.8

18.1 27.0 17.9

Cmax D 37:3

completion times, will depend on the actual disruption, i.e. a specific glaze shortage occurrence. The processing times of the affected jobs will be extended by the glaze shortage duration; on the contrary, the processing times of the jobs that are not affected remain the same as specified initially. A simulation tool is implemented in LHS to analyse the performance of the predictive schedule compared with the realised schedule (Duenas et al. 2005). For example, a typical schedule in the glaze department considers 30 jobs. The predictive schedule PS generated by LHS is given in Table 20.6.

20.3.2.3 Multi-criteria Rescheduling (Reactive Scheduling) Two rescheduling methods are developed and implemented in LHS to be applied in situations when the impact of the disruption is too high to be absorbed by the predictive schedule: (1) Left-shift rescheduling and (2) Build a new schedule. In Left-shift rescheduling, the jobs affected by the disruption are cancelled and the start times of remaining jobs, i.e. those jobs that have not been processed, are left-shifted to the time when the glaze shortage occurs, keeping the same sequence of the jobs as in the initial predictive schedule. In current practice, the glazing section leader always carries out left-shift rescheduling, whenever a glaze shortage occurs. In the Build a new schedule method, when a glaze shortage occurs, a new schedule is generated from scratch considering the jobs that have not been processed yet. This new schedule is built using the same scheduling algorithm, i.e. we employ the dispatching rules proposed for generating an initial predictive schedule, but obviously in this case, with a smaller number of jobs. To measure the rescheduling methods performance, i.e. the quality of the new reactive schedule, two performance measures are considered: (1) reactive schedule efficiency, defined as makespan Cmax of the reactive schedule RS and (2) stability/instability of RS. The instability IST of the reactive schedule is typically measured as the starting time deviations between the predictive and the reactive schedules. However, in the context of the pottery company, where the subsequent process, i.e. the kiln firing follows the glazing process, it is of importance to consider

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job completion time deviations rather than the starting time deviations. Therefore, the stability/instability of RS is calculated as follows: IST .RS/ D

N X ˇ ˇ ˇCj .PS/ Cj .RS/ˇ: j D1

The difference between the two rescheduling methods can be illustrated by an example. Let us assume that the glazing section leader detects a bad level of quality in Boston glaze at the moment of producing the first job that involves this glaze. However, in this scenario the glaze production section is not able to deliver the glaze in an assumed time, and therefore it is necessary to carry out rescheduling. The jobs that use Boston glaze are J5 ; J6 and J7 and they start processing at 19.9, 27.0 and 33.0 hours, respectively, (shown in grey in Table 20.6). These jobs are deleted from the schedule. When Left-shift rescheduling is applied, after the jobs are deleted, all the jobs sequenced on the machines where they are allocated for processing (M1 and M2 ) are shifted forward. The schedule obtained after applying the Left-shift rescheduling method is presented in Table 20.7. As one can see, in this case the makespan is not affected since it is determined by machine M3 , and this machine is not affected by the disruption. Therefore, the efficiency objective has not been affected. The instability of the schedule is measured using the formula given above: 30 ˇ ˇ P ˇCj .PS/ Cj .RS/ˇ D 105:9. Since jobs J5 ; J6 and J7 are canIST .RS/ D j D1

celled in the reactive schedule RS, to calculate IST(RS), their completion times are considered to be equal to zero, i.e. C5 .RS/ D C6 .RS/ D C7 .RS/ D 0. When the Build a new schedule method is applied, a new schedule is generated involving all the jobs that have not been processed yet, considering the moment of disruption to be the earliest starting time of all the affected jobs. In the predictive schedule PS (Table 20.6), the first job affected by the disruption is job J5 , which starts processing at 19.9 h, on machine M1 . Jobs J10 and J24 have started being processed on machines M2 and M3 , respectively, and have a completion time of 23.4 and 20.1 h. Therefore, all the jobs after jobs J22 ; J10 and J24 on machines M1 ,

Table 20.7 Reactive schedule RS obtained using Left-shift rescheduling Machine Jobs’ sequence 27 18 11 8 22 26 3 19 30 M1 M2 12 21 10 4 15 25 16 1 17 23 24 2 13 20 14 M3 Machine Completion times M1 6.2 11.4 15.5 8.2 15.9 23.4 M2 7.8 12.8 15.8 M3 Cmax D

37.3

18.1 27.0 17.9

IST D 105:9

19.9 28.4 20.1

24.70 30.2 21.1

28 29

28.3

30.9

33.3

35.7

28.1

31.2

33.7

36.1

9

37.3

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Table 20.8 Reactive schedule RS obtained using Build a new schedule Machine Jobs’ sequence 27 18 11 8 22 2 26 3 14 M1 M2 12 21 10 4 20 30 25 16 1 17 23 24 13 19 15 28 M3 Machine M1 M2 M3

Completion times 6.2 11.4 15.5 8.2 15.9 23.4 7.8 12.8 15.8

Cmax D

35.5

18.1 27.0 17.9

19.9 30.1 20.1

20.9 32.5 27.1

25.7 34.3 30.0

29.3

31.8

32.2

34.6

29

34.2

9

35.5

IST D 106:6

M2 and M3 , respectively, are rescheduled. Table 20.8 shows the schedule obtained after applying the Build a new schedule method. Obviously, the job allocations in the new schedule may be different when compared to the predictive schedule, as is the case in the example considered here. In this case, the makespan, Cmax D 35:5, is smaller than that found using the Left-shift rescheduling method, where Cmax D 37:4. This occurs because the Build a new schedule method applies the rules LFJ and LPT, which minimise the makespan, while the Left-shift rescheduling method does not consider the makespan. In addition, to determine which method generates a more stable schedule, the instability of the new schedule is calculated as follows: IST .RS/ D

30 X ˇ ˇ ˇCj .PS/ Cj .RS/ˇ D 106:6: j D1

In this case, the rescheduling method that performs better with respect to the efficiency objective (makespan) is the Build a new schedule method, whilst the Leftshift method performs better for the stability/instability objective. It may be concluded that the selection of the rescheduling method to be applied depends on which of the objectives considered is more important: the efficiency or stability/instability.

20.3.2.4 Fuzzy Rules for Rescheduling Decision Making LHS includes two sets of fuzzy IF-THEN rules. The first set of rules is defined to determine whether it is necessary to reschedule or not, whilst the second set of rules is used to select a rescheduling method to be applied. The rules are of Sugeno type, where the IF part (antecedent) involves linguistic variables and the THEN part (consequent) is a crisp action. In LHS, the standard Sugeno type of rule is modified by associating weights with the crisp actions. The linguistic variables considered in the first set of rules are: (1) TD-the glaze delivery time, defined as the time needed to obtain the required glaze, (2) TO-the time of the glaze shortage occurrence. The linguistic variable TD can take

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b µ

TD

short

medium

1

2 3 4

early

long

6

C max = 37.3

TO

td

middle

6

8 15 24

late

30

to

Fig. 20.9 Fuzzy sets that represent (a) the glaze delivery time and (b) the time of the glaze shortage occurrence

three imprecise values: short, medium and long. The linguistic variable TO can also have three values: early, middle and late, defined with respect to the makespan Cmax of the predictive schedule, as illustrated in Fig. 20.9. These membership functions are context sensitive, and they have to be determined for each scheduling problem separately, usually in collaboration with the section leader. For example, in the scheduling problem under consideration, the membership functions of the glaze delivery time TD is specified considering the average of the added idle times idj ; j D 1; : : :; 30. For instance, the average of the idle times is 1.92 h and can be interpreted as the time during which each flow line may be idle waiting for the glaze to be delivered. Therefore, the imprecise value short glaze delivery time varies from 0 to 2 h (see Fig. 20.9a). The value medium glaze delivery time is modelled by a triangular function starting from 1 to 6 h. As it was decided that 4 h is a long time for the flow line to be idle, since it represents half of the production day, the value high glaze shortage duration is considered to be 4 h and longer. The membership functions of the time of the glaze shortage occurrence TO are determined considering the predicted makespan Cmax D 37:3 h, as follows. If a glaze shortage occurs in the first day of production it is considered to be early. Therefore, the imprecise value early varies from 0 to 8 h (see Fig. 20.9b). If a glaze shortage occurs at the beginning of the fourth production day, it is considered to have occurred late. Hence, the value late varies from 24 to 37.3 h. If a glaze shortage occurs between 6 and 30 h, it is considered to be in the middle of the schedule horizon. The corresponding rules, which determine whether to reschedule or not, take into consideration TD and TO, as presented in Table 20.9. For example, if the time needed to provide the required glaze TD is short, it is assumed that the glaze shortage time can be absorbed by the predictive schedule regardless of the time of the glaze shortage occurrence TO. The same decision is recommended when the TD is medium and the TO is early, since it is assumed that the glaze will be delivered soon enough to continue with the predictive schedule that will finish with the predicted Cmax . When the TD is medium and the disruption occurs in the middle or late in the schedule horizon, the decision obtained is to reschedule, because it is assumed that the time needed to provide the required glaze is too long to be absorbed by the

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Table 20.9 Fuzzy rules for determining whether to reschedule or not Rules IF AND THEN

1 short early DNR

TD TO

2 short mid DNR

3 short late DNR

4 med early DNR

5 med mid R

6 med late R

7 long early R

8 long mid R

9 long late R

DNR: Do not reschedule; R: Reschedule

a

b

µ EF low

µ ST

high

50 100 ef Efficiency performance priority

low

high

st 50 100 Stability performance priority

Fig. 20.10 Fuzzy sets that represent performance priorities

predictive schedule. In the case when the TD is long, it is assumed that the glaze shortage time cannot be absorbed by the predictive schedule regardless of the time of the glaze shortage occurrence TO. Therefore, the decision recommended is to reschedule. To determine which action to take, DNR – Do not reschedule or R – Reschedule, the fuzzy inference is performed. The rules are evaluated first by determining truth values of their IF parts (premises). Then, the activation rates ’DNR and ’R are calculated as the sums of the truth values of the rules that have corresponding DNR and R actions, respectively. The action with the higher activation rate is recommended. More details can be found in Petrovic and Duenas (2006). In the case when the decision is to reschedule, an appropriate rescheduling method has to be selected and applied. The linguistic variables involved are: (1) TO-the time of the glaze shortage occurrence, defined in the same way as in the previous set of rules, (2) EF-the priority of efficiency performance, which represents the section leader’s preference with respect to the efficiency, i.e. the makespan of the reactive schedule, and (3) ST-the priority of stability performance, which represents the preference with respect to the stability, i.e. the minimum number of changes of job completion times in the reactive schedule compared to the initial predictive schedule. The linguistic variables EF and ST can take two imprecise values, low, and high, which are defined on a hypothetical scale from 0 to 100 (see Fig. 20.10). Table 20.10 describes the fuzzy rules defined. The same fuzzy inference is applied as on the previous set of fuzzy rules. However, this time it includes the weights ar and br associated with the two actions, R1 – apply Left-shift rescheduling and R2 – apply Build a new schedule method, respectively. The weights satisfy the

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Table 20.10 Fuzzy rules for selecting a rescheduling method Rules IF AND AND

1 2 TO early early EF low low ST low high

3 early high low

4 early high high

5 mid low low

6 mid Low High

7 mid high low

8 9 mid late high low high low

11 late high low

12 late high high

a1 R1 a2 R1 a3 R1 a4 R1 a5 R1 a6 R1 a7 R1 a8 R1 a9 R1 a10 R1 a11 R1 a12 R1 b1 R2 b2 R2 b3 R2 b4 R2 b5 R2 b6 R2 b7 R2 b8 R2 b9 R2 b10 R2 b11 R2 b12 R2

THEN

ar 0.63 0.72 0.59 0.33 0.13 0.93 0.09 br 0.37 0.28 0.41 0.67 0.87 0.07 0.91 R1 : Left-shift rescheduling; R2 : Build a new schedule

0.39 0.61

Table 20.11 Decisions obtained using the fuzzy rules Inputs Activation rates

R1

’DNR 1.000 1.000 0.417 0.208 0.833 0.833 0.125 0.000 0.000 0.000 0.000 0.000 0.000 0.125 0.000 0.000

Cmax – – – – – – 31.33 36.10 37.35 37.23 37.23 37.35 37.35 31.33 36.10 37.23

Test 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

10 late low high

to 5 5 7 7.5 4 5 7.7 17 28 17 28 17 28 7.7 17 28

td 1 3 3.5 3.5 3.5 3.5 3.5 3.5 3.5 3.5 5 3.5 3.5 3.5 3.5 3.5

ef 80 80 80 80 55 55 80 80 80 55 55 55 55 80 80 80

st 45 45 45 45 45 80 45 45 45 45 45 80 80 80 80 80

’R 0.000 0.000 0.093 0.139 0.000 0.000 0.157 0.722 0.736 0.722 0.736 0.722 0.736 0.157 0.722 0.736

˛R1 – – – – – – 0.111 0.092 0.706 0.109 0.663 0.499 0.704 0.157 0.352 0.649

˛R2 – – – – – – 0.228 0.775 0.178 0.757 0.220 0.367 0.180 0.182 0.515 0.234

Decision DNR DNR DNR DNR DNR DNR R2 R2 R2 R2 R2 R1 R1 R2 R2 R1

0.85 0.15

0.84 0.16

0.97 0.03

0.62 0.38

R2 IST – – – – – – 228.81 134.67 78.31 63.49 56.39 48.43 40.50 228.81 134.67 37.35

Cmax – – – – – – 30.58 33.98 35.81 36.48 35.29 36.29 36.34 30.58 33.98 37.23

IST – – – – – 227.20 131.34 79.75 63.49 58.68 49.88 42.56 227.20 131.34 38.70

Best value

following constraint, ar C br D 1. The higher the weight, the more confidence is given to the corresponding rescheduling method. A Genetic Algorithm is designed and implemented to determine the appropriate weights, based on a set of ‘learning cases’ (Petrovic and Duenas 2006). The developed fuzzy rules are tested using various inputs including: to – the time of glaze shortage occurrence, td – the glaze delivery time, ef – the priority of the efficiency performance, and st – the priority of the stability performance. They are specified as follows: td is determined at the glaze production section, to is recorded at the shop floor as the time when the disruption occurs, and ef and st are determined subjectively by the glazing section leader. Some of the results obtained using the developed fuzzy IF-THEN rules and real-world data are presented in Table 20.11.

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When the glaze delivery time is short (Test 1), the system yields the decision not to reschedule, since this kind of disruption does not have a high impact on the original schedule performance and, therefore, can be absorbed by the predictive schedule. When the time of the glaze shortage occurrence, to, is early (less than 7.7 h, Tests 2–6), the decision obtained, again, is not to reschedule. It is assumed that since the glaze shortage occurrence is early in the schedule and the glaze shortage duration is medium .td D 3:5/, the schedule is able to absorb the disruption impact. In Tests 7, 8, 10, 14 and 15, the decision obtained is to reschedule applying R2 – the Build a new schedule method. Comparing the efficiency .Cmax / and the instability (IST) obtained using both rescheduling methods, it is shown that in all these five tests R2 yields better performance of both measures. When the glaze shortage occurrence time, to, is increased to 28, (Tests 9 and 11) the decision obtained is to build a new schedule. Indeed, the R2 method generates a more efficient schedule than the R1 method, although a slightly better stability is obtained using the latter method. However, this is not of importance because the priority of efficiency performance (ef D 80 and ef D 55, in the two tests, respectively) is higher than the priority of stability .st D 45/. On the contrary, in Tests 12 and 13, the decision obtained is to apply R1 – Left-shift rescheduling. When applied, the R2 method generates a more efficient schedule than the R1 method, although a better stability is obtained using the latter method. However, this is acceptable, because the stability priority .st D 80/ is much higher than the priority of efficiency performance .ef D 55/. Therefore, considering the section leader preferences, applying the R1 method is more appropriate than applying the R2 method. In Test 16, the decision obtained is to apply Left-shift rescheduling. In this case, it is shown that the rescheduling method that yields the better performance for both, efficiency .Cmax / and instability (IST) is R1 . Finally, analysing the tests results, it is concluded that in the case when both performances, efficiency and stability, have the same priority, the linguistic variable that affects the decision on the rescheduling method to be used is the glaze shortage occurrence time. For example, when the disruption is considered to occur late (to D 28 in Test 16) the rescheduling method to apply is R1 – Left-shift rescheduling, whilst if it happens in the middle of the schedule horizon (Test 15), it is better to apply R2 – Build a new schedule.

20.4 Conclusion The main standpoint of the research presented in this chapter is based on two ideas. First, real-life scheduling is carried out in the presence of various sources of uncertainty such as: (a) uncertainty in scheduling and rescheduling parameters, (b) the imprecise preferences of scheduling experts, specified as aspiration values, i.e. values that are acceptable for schedule performance measures or relative importance of the performance measures, (c) subjective knowledge and experience that the scheduling experts use in practice. Representing and treating these uncertainties

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in a form which is acceptable to practitioners may bring scheduling models nearer to real-life scheduling problems. Fuzzy theory provides a suitable framework for modelling various types of uncertainty, and problems where subjective judgement and intuition play an important role. Second, real-life scheduling takes into consideration multiple, usually conflicting and incommensurable objectives. They describe different performance measures to be considered simultaneously, taking into consideration the preferences of the scheduling experts. Consequently, scheduling methods should provide an important and active role for scheduling experts. Two prototypes of the scheduling and rescheduling systems, FMGAS and LHS, have been developed that incorporate fuzzy multi-objective scheduling methods. Multiple objectives have been identified for both scheduling in the printing environment and rescheduling on the pottery shop floor. The results obtained support the application of fuzzy sets in modelling uncertain scheduling parameters (e.g. due dates, processing times) and rescheduling parameters (e.g. the number of disruption occurrences, disruption duration). Fuzzy rules for lot sizing have proved to be superior to the simple heuristic approach to lot sizing based solely on the capacity of the shop floor. Also, fuzzy inference applied for rescheduling decision making has shown good results. Further work is being undertaken including: An analysis of possible improvements in scheduling that might be gained at the

printing and the pottery company by using the systems developed. This includes evaluation and comparison of the schedules generated at the shop floors and the schedules constructed by the systems. The development of rescheduling methods to be implemented in FMGAS to respond to disturbances in the printing environment, such as the arrival of new orders, machine breakdowns, delays in the arrival of raw materials, changes in current orders. An extension of LHS to include other rescheduling methods. An investigation of different techniques to measure the level of disruption uncertainty to quantify its impact on the schedule performance. Acknowledgements The authors thank the Engineering and Physics Science Research Council (EPSRC), UK, for supporting this research (Grant No. GR/R95319/01 and GR/R95326/01). The authors also acknowledge the support of our industrial collaborators Sherwood Press Ltd., Nottingham, and Denby Pottery Company Ltd., UK.

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Petrovic D, Roy R, Petrovic R (1999) Production supply chain modelling using fuzzy sets. Int J Prod Econ 59(1–3):443–453 Pinedo M (2002) Scheduling: theory, algorithms, and systems. Prentice Hall Sakawa M, Kubota R (2000) Fuzzy programming for multiobjective job shop scheduling with fuzzy processing time and fuzzy duedate through genetic algorithms. Eur J Oper Res 120(2):393–407 Sastry K, Goldbreg D, Kendall, G (2005) Genetic algorithms. In: Burke E, Kendall G (eds) Search methodologies: introductory tutorials in optimisation and decision support techniques. Springer, pp. 97–125 Slowinski R, Hapke M (eds) (2000) Scheduling under fuzziness. Physica-Verlag, Heidelberg Sotskov YN (1991) Stability of an optimal schedule. Eur J Oper Res 55(1):91–102 Subramaniam V, Raheja AS, Reddy KRB (2005) Reactive repair tool for job shop schedules. Int J Prod Res 43:1–23 T’kindt V, Billaut J-C (2002) Multicriteria scheduling: theory, models and algorithms. Springer Vieira GE, Herrmann JW, Lin E (2003) Rescheduling manufacturing systems: A framework of strategies, policies, and methods. J Scheduling 6:39–62 Weiss G (1995) A tutorial in stochastic scheduling. In: Chretienne P, Coffman EG, Lenstra Jr JK, Liu Z (eds) Scheduling theory and its applications. Wiley, pp 33–64 Zadeh LA (1971) Quantitative fuzzy semantics. Infor Sci 3:159–176 Zimmerman HJ (1996) Fuzzy set theory and its applications. Kluwer Academic Publishers, Boston

Chapter 21

The Summing-Up Karl Kempf, Pınar Keskinocak, and Reha Uzsoy

At the end of this volume, several years in the making, it is worth reflecting upon the objectives the editorial team had in mind when we started this project. We were motivated by the disturbing observation that although academia seemed to view the problems of planning production and inventories as a largely solved problem, with basic formulations established and agreed upon, there was broad consensus among our industrial colleagues that the available models and solution techniques were a long way from representing the full richness and complexity of the task as they encountered it in their daily operations. Our objective with these volumes was to address this situation by bringing together a leading group of researchers and practitioners to delineate the broader boundaries of the problem and present the state of the art in various related areas. A representative sample of leading-edge industrial contributions illustrates the state of the art in industrial practice, while a selection of research contributions with a more academic bent explores specific aspects of new areas. We believe that the volumes have largely achieved these objectives. The level of interest from the community, as demonstrated by the number of submitted contributions, far exceeded our original expectations. The willingness of both researchers and industrial practitioners to contribute was extremely encouraging, and the diversity of the contributions supports our thesis that a broader vision of the problem domain is needed. The number of contributions involving ideas from what have been hitherto considered quite distinct areas, such as finance (e.g., Heching and King 2010; Cheng et al. 2011; Ettl et al. 2011), demand management (Ettl et al. 2011), traffic and kinetic theory (Lefeber and Armbruster 2010; Ouyang and Daganzo 2010), and game theory (Garg et al. 2011) suggests that problems of planning production and inventories are now beginning to be studied within a much broader context, and are attracting the attention of a broader set of researchers than before, which bodes well for the future of the field.

R. Uzsoy () Edward P. Fitts Department of Industrial and Systems Engineering, 300 Daniels Hall, Campus Box 7906, North Carolina State University, Raleigh, NC 27695-7906, USA e-mail: [email protected]


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Another important aim of the volume was to provide a new generation of researchers and practitioners a perspective on the historical evolution of the field. The contributions by McKay and Ovacik provide an excellent overview of the evolution of industrial practice since the late nineteenth century. The chapter by Fordyce and his colleagues is also a fascinating chronicle of the evolution of production and supply chain planning at IBM, a company whose influence on the field has been profound and whose researchers have made many important contributions over the years. Given the diverse research streams that bear on the problems of planning production and inventories in the extended enterprise, the volume also offers state of the art reviews of major related areas: the use of discrete-event simulation (Ankenman et al. 2010), inventory models (Ozer 2010), perishable inventories (Karaesmen et al. 2010), optimization models (Missbauer and Uzsoy 2010), pricing and coordination issues (Maddah et al. 2010, Aydinliyim and Variraktarakis 2010), and continuous modeling techniques based on differential equation models (Lefeber and Armbruster 2010). The research contributions also span, and integrate, concepts from a number of these usually distinct areas, suggesting again that the field continues to attract the interest of a broad spectrum of researchers from different disciplines and perspectives. A major benefit of the handbook format is that it provides a forum for broadranging essays that allow thought leaders to review their concept of an area in a more philosophical vein than is possible within the narrow confines of a scholarly journal oriented to rigorous, quantitative research contributions. These essays bring together insights from a wide range of research efforts and observations of industrial practice into a tight, well-reasoned exposition of a point of view. The contributions of Elmaghraby (2010) on the nature of manufacturing capacity; Dionne and Kempf (2010) on the centrality of data issues to effective production planning; Graves on how the pervasive uncertainties in demand, supply and production can be addressed; and Carlson and Rafinejad (2010) on the increasingly recognized importance of sustainability in the supply chain provide thoughtful perspectives on how problems can be approached, and how the future of the field may evolve. Finally, the contributions to the handbook suggest that industrial practitioners and researchers are very active in the field, both in developing and in implementing systems to address the planning needs of their firms or industries, and in influencing academic research through collaboration with academic researchers. The industry contributions represent the automotive, semiconductor, chemical, pharmaceutical, computer, retail, and ceramic industries, spanning a wide range of production processes, organizational environments, and planning needs. Industrial contributors are represented both in research contributions (e.g., Chandrashekar and Narahari 2011; Ettl et al. 2011; Cheng et al. 2011; Garg et al. 2011) and in applications; academic researchers are also involved in contributions focused on industrial applications (Artiba et al. 2011; Missbauer et al. 2011; Katircioglu and Gallego 2011; Ormeci Matoglu and Vande Vate 2011). The contribution of McKay (2010) addresses the methodology of research into how planning and scheduling is actually executed and how analytical tools might be integrated into the ongoing business processes.

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On the bright side, then: there is an extensive activity in both industry and academia, and an influx of new ideas from areas such as game theory, control theory, and physics that, while always present to some degree, have never really been part of the mainstream in this field. There is also an increasing integration between what have hitherto been quite separate research domains, such as mathematical programming, stochastic optimization, inventory theory, and queuing. However, upon reading the contributions in the volume, a number of areas of concern still linger; there still remains great potential for further improvements in both our understanding of the problems in this domain, and our ability to deliver practical, useful solution tools to industry. Alas, it appears that our contention in the introductory chapter, that our research efforts to date have provided considerable insight into particular aspects of the industrial problem but no integrated solution, remains as valid today as it was 55 years ago, when it was raised by Simon and Holt (1954). Although the contributions in these volumes indicate that the study of planning across the extended enterprise is indeed leading to a broadened, enriched definition of the problem, we are still far from an integrated, end-to-end solution that supports the entire range of business processes involved in coordinating the flow of material through the extended enterprise in an optimal manner. As Ovacik (2010) points out, although major ERP software vendors have provided planning solutions with their systems, the influence of academic research on these approaches appears to have been quite limited. While a number of academics have launched companies providing solutions to various problems in this area, their numbers are relatively small, and there is still clearly a long way to go to fully incorporate academic results into industrial practice. The divide between research streams, such as mathematical programming, inventory models, queuing and simulation, appears to be still alive and well, although there are some encouraging notes. The renewed interest in models with load-dependent lead times discussed by Missbauer and Uzsoy (2010) brings together concepts from queuing and mathematical programming to provide what appears to be a promising new direction that is yet in its infancy, but attracting growing attention. Several chapters, such as Orcun and Uzsoy (2011), Lefeber and Armbruster (2010) and Ouyang and Daganzo (2010) employ a system dynamics perspective to study the effects of different planning approaches in the supply chain; and the chapter by Fordyce et al. indicates an interesting evolution of optimization, data management, and artificial intelligence approaches in addressing these complex problems. As Graves (2010) points out in his chapter, the management of the many dimensions of uncertainty that are a major component of most industrial planning problems is currently handled by rather simplistic methods. The most complete mathematical formulations for such problems are related to stochastic dynamic programming. Various methods have been developed for special cases such as multistage stochastic programming with recourse (Birge and Louveaux 1997), approximate dynamic programming (Powell 2007), and Markov decision processes (Puterman 2005), but these approaches face a number of challenges: the curse of dimensionality due to the exponential growth in state spaces, the difficulty of

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explaining these involved and rather subtle procedure to managers, and obtaining the necessary data to parameterize these models. While inventory models such as those discussed by Ozer (2010), Karaesmen et al. (2010) and Katircioglu and Gallego (2011) address the uncertainty of demand, the treatment of uncertain supply from capacitated production systems remains an area with many interesting research problems, although recent work such as that by Levi et al. (2008) shows interesting progress. Although many aspects of uncertainty appear to have been addressed individually under relatively idealized conditions, there is a strong need for a taxonomy of the different types of uncertainty and the most appropriate ways to address them and their interactions in different industries with different economics. It is not clear that we understand the real-world uncertainties, their drivers, their behavior, and how the uncertainties impact production planning. The whole topic of uncertainty remains a major challenge to both academic research and industrial practice, although major strides have been made in specific areas. A particularly interesting issue in planning in the extended enterprise is the pervasive need to integrate models of systems that operate on different time scales. While factories and distribution systems operate in essentially continuous time, it is common for planning systems to operate on a periodic basis, say weekly or monthly. How to represent the behavior of a production-inventory system operating in continuous time to an aggregate planning system is an ongoing challenge to which there is currently no completely satisfactory solution. Continuous flow models such as those of Lefeber and Armbruster (2010) and Ouyang and Daganzo (2010) approximate the discrete event nature of the factory floor as a continuous flow; the clearing functions discussed by Missbauer and Uzsoy (2010) can be viewed as an aggregate representation of the behavior of the shop floor over a planning period. In contrast, the planning and scheduling models developed for continuous processes such as those encountered in chemical processing (Floudas and Lin 2005) use discrete time periods that are shorter than the basic task processing times, leading to highly detailed scheduling models that fulfill some of the functions of the more aggregate planning models, but whose practicality in the discrete manufacturing environment is not entirely clear. This discussion raises the question of the relationship between the business processes used by the company and the planning models and systems the company deploys. As Ovacik (2010) has cogently argued, a monolithic integrated optimization model of an entire supply chain, or even the segment of the supply chain owned by a single firm, is technically extremely challenging, and usually does not match the structure of the underlying organization. One conjectures that planning models and tools tend to be commissioned by owners of particular segments of the organization responsible for particular segments of the supply chain, such as inventories or production. This “silo” approach may have contributed to the current state of research: models address one aspect of the problem in considerable depth while either neglecting or grossly simplifying its interactions with others. The industrial ERP systems hold out the possibility of eventually addressing this, since they provide a centralized data structure that, at least in theory, spans the business processes of the entire organization. However in many cases, these systems are not designed

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with planning in mind; in some cases, planning functionality appears to be almost an afterthought, added on as an enhancement that will leverage the considerable investment in transactional data. Any industrial practitioner, or any academic trying to implement and test a planning model in an industrial context, will immediately agree that the major difficulty confronting the success of the project is the availability of data. While the unified, company-wide data models of the ERP systems hold out the possibility of easily available, high-quality data, as discussed by Dionne and Kempf (2010), the reality is far different. Many ERP systems are designed with specific business processes in mind, leaving little room to accommodate planning approaches that do not fit within these; and the costs of interfacing third-party planning software to ERP systems and maintaining these interfaces over time have become a significant barrier to entry for smaller firms specializing in novel planning systems. It is interesting to note that while the planning problems of the extended enterprise are certainly being taken up in a broader context, the basic treatment of data is no different from what it was 50 years ago. Parameters are treated as either a known probability distribution, or as a deterministic constant, with little discussion of how these would be estimated and maintained in a rapidly changing industrial environment. A systematic basis for deciding which data items are more crucial than others would allow us to focus data collection and cleaning efforts on the parameters that matter most, those to which the solutions are most sensitive. Upon sober reflection, there is actually very little guidance available on basic issues such as how long planning periods should be, or how the effects induced by using a fixed, finite planning horizon should be addressed. The only reference we are aware of on the latter issue is Grinold (1983), and this is an issue encountered by every finite-horizon planning model, and hence probably most of those used in practice. Some discussion of the choice of planning periods and different assumptions as to when activities may start in a period is given by Dessouky and Leachman (1997), but this is still an area that has not been systematically explored as far as we can see, but is quite fundamental to models based on discrete time periods. Finally, an important consequence of the diversity of problem formulations and mathematical approaches is the difficulty of comparing alternative approaches in a meaningful manner. This is particularly true when one tries to identify real benefits in an industrial situation, where many ongoing environmental events cloud any discussion of causality and claims of contribution to improved efficiency or effectiveness – either between approaches or when a method is used. A controlled experiment in an industrial context is rarely possible, so a great deal of comparison must be carried out using simulation. However, as Fischbein and Yellig (2011) point out, building and validating large-scale simulation models is a major project in its own right, limiting the degree to which fully realistic simulation models of large extended enterprises can be used to test planning systems. An added complication is that the planning model is generally part of a larger business process, and a truly realistic test would then require simulation of not only the physical operations of the extended enterprise but also the operations of its extensive, and often unstructured, decision processes.

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Setting aside this difficulty, experimental issues emerge even in an academic environment. For example, consider the problem of multiple time scales alluded to above. Suppose we wish to use a simulation model of a factory to evaluate the performance of a clearing function model with an iterative linear programming/ simulation approach. The two models have completely different conceptions of how the WIP inventory is treated, rendering a direct comparison of objective function values of little use. Both planning models operate with discrete planning periods, so their decisions must be disaggregated in some fashion in order to be implemented on the shop floor. There is no obviously correct way of doing this disaggregation – should the work releases recommended by the planning models be distributed evenly over the entire period, at certain intervals within the period (such as Monday mornings, for example), or made available at the start of the period? Different dispatching and scheduling policies on the shop floor may affect the performance of the different planning solutions in different ways, but a comprehensive enumeration of all possibilities, or even a reasonable subset, is simply not possible. These observations seem to suggest that an elegant, unifying solution such as Maxwell’s equations brought to electricity and magnetism (Mahon 2003) is indeed unlikely to be achievable for the planning problems of the extended enterprise in the near future. However, having said this, it is worth taking the analogy with physics a step further. Maxwell’s work was made possible by the extensive empirical observations and mathematical modeling efforts of researchers over several centuries prior to the Laird of Glenlair’s advent upon the scene. Perhaps, this suggests a way forward for the field of planning in the extended enterprise: to complement the rigorous mathematical and quantitative approaches that make up the bulk of this volume, and have proved so successful so often, with more exploratory, experimental studies, each of which provides limited but useful learning that can eventually, with hard work and good fortune, help us to see larger patterns. As McKay has suggested, more effective collaborations with our colleagues in the social and behavioral sciences are also needed to enhance our insight as to how the techniques and tools we propose may be used within the large, complex organizations that dominate the new global economy. The high road of mathematical rigor and powerful computation has brought us very far indeed since Modigliani and Hohn proposed their model in 1955; with the additional perspective and insight from these complementary approaches, the mathematical base that is currently developing is likely to serve us well far into the future.

References Ankenman BE, Bekki JM, Fowler J, Mackulak GT, Nelson BL, Yang F (2010) Simulation in production planning: An overview with emphasis in recent developments in cycle time estimation. In: Kempf KG, Keskinocak P, Uzsoy R (eds) Planning production and inventories in the extended enterprise: a state of the art handbook, vol 1. Springer, New York, pp 565–592 Artiba A, Dhaevers V, Duvivier D, Elmaghraby SE (2011) A multi-model approach for production planning and scheduling in an industrial environment. In: Kempf KG, Keskinocak P, Uzsoy R (eds) Planning production and inventories in the extended enterprise: a state of the art handbook, vol 2. Springer, New York, pp 489–530

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Aydinliyim T, Variraktarakis GL (2010) Sequencing strategies and coordination issues in outsourcing and subcontracting operations. In: Kempf KG, Keskinocak P, Uzsoy R (eds) Planning production and inventories in the extended enterprise: a state of the art handbook, vol 1. Springer, New York, pp 269–320 Birge JR, Louveaux F (1997) Introduction to stochastic programming. Springer, New York Carlson R, Rafinejad D (2010) The transition to sustainable product development and manufacturing. In: Kempf KG, Keskinocak P, Uzsoy R (eds) Planning production and inventories in the extended enterprise: a state of the art handbook, vol 1. Springer, New York, pp 45–82 Chandrashekar TS, Narahari Y (2011) Procurement network formation: a cooperative game approach. In: Kempf KG, Keskinocak P, Uzsoy R (eds) Planning production and inventories in the extended enterprise: a state of the art handbook, vol 2. Springer, New York, pp 185–206 Cheng F, Ettl M, Lin GY, Tonner M, Yao DD (2011) Designing flexible supply chain contracts with options. In: Kempf KG, Keskinocak P, Uzsoy R (eds) Planning production and inventories in the extended enterprise: a state of the art handbook, vol 2. Springer, New York, pp 207–230 Dessouky MM, Leachman RC (1997) Dynamic models of production with multiple operations and general processing times. J Oper Res Soc 48(6):647–654 Dionne L, Kempf KG (2010) Data in production and supply chain planning. In: Kempf KG, Keskinocak P, Uzsoy R (eds) Planning production and inventories in the extended enterprise: a state of the art handbook, vol 1. Springer, New York, pp 167–184 Elmaghraby SE (2010) Production capacity: Its bases, functions and measurement. In: Kempf KG, Keskinocak P, Uzsoy R (eds) Planning production and inventories in the extended enterprise: a state of the art handbook, vol 1. Springer, New York, pp 119–166 Ettl M, Sourirajan K, Huang P, Ervolina TR, Lin GY (2011) Supply and demand synchronization in assemble-to-order supply chains. In: Kempf KG, Keskinocak P, Uzsoy R (eds) Planning production and inventories in the extended enterprise: a state of the art handbook, vol 2. Springer, New York, pp 81–104 Fischbein S, Yellig E (2011) Why is it so hard to build and validate discrete event simulation models of manufacturing facilities? In: Kempf KG, Keskinocak P, Uzsoy R (eds) Planning production and inventories in the extended enterprise: a state of the art handbook, vol 2. Springer, New York, pp 271–288 Floudas CA, Lin X (2005) Mixed integer linear programming in process scheduling: modeling, algorithms and applications. Ann Oper Res 139:131–162 Garg D, Narahari Y, Foster E, Kulkarni D, Tew JD (2011) A mechanism design approach for decentralized supply chain formation. In: Kempf KG, Keskinocak P, Uzsoy R (eds) Planning production and inventories in the extended enterprise: a state of the art handbook, vol 2. Springer, New York, pp 153–184 Graves SC (2010) Uncertainty and production planning. In: Kempf KG, Keskinocak P, Uzsoy R (eds) Planning production and inventories in the extended enterprise: a state of the art handbook, vol 1. Springer, New York, pp 83–102 Grinold RC (1983) Model building techniques for the correction of end effects in multistage convex programs. Oper Res 31(3):407–431 Heching A, King AJ (2010) Financial uncertainty in supply chain models. In: Kempf KG, Keskinocak P, Uzsoy R (eds) Planning production and inventories in the extended enterprise: a state of the art handbook, vol 1. Springer, New York, pp 185–204 Karaesmen I, Scheller-Wolf A, Deniz B (2010) Managing perishable and aging inventories: review and future research directions. In: Kempf KG, Keskinocak P, Uzsoy R (eds) Planning production and inventories in the extended enterprise: a state of the art handbook, vol 1. Springer, New York, pp 393–436 Katircioglu K, Gallego G (2011) A practical multi-echelon inventory model with semiconductor manufacturing application. In: Kempf KG, Keskinocak P, Uzsoy R (eds) Planning production and inventories in the extended enterprise: a state of the art handbook, vol 2. Springer, New York, pp 133–152 Lefeber E, Armbruster D (2010) Aggregate modeling of manufacturing systems. In: Kempf KG, Keskinocak P, Uzsoy R (eds) Planning production and inventories in the extended enterprise: a state of the art handbook, vol 1. Springer, New York, pp 509–536

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Levi R, Roundy RO, Shmoys DB, Truong VA (2008) Approximation algorithms for capacitated stochastic inventory control models. Oper Res 56(5):1184–1199 Maddah B, Bish EK, Munroe B (2010) Pricing, variety and inventory decisions for product lines of substitutable items. In: Kempf KG, Keskinocak P, Uzsoy R (eds) Planning production and inventories in the extended enterprise: a state of the art handbook, vol 1. Springer, New York, pp 367–392 Mahon B (2003) The man who changed everything: The Life of James Clark Maxwell. Wiley, Chichester McKay KN (2010) Field-based research on production control. In: Kempf KG, Keskinocak P, Uzsoy R (eds) Planning production and inventories in the extended enterprise: a state of the art handbook, vol 1. Springer, New York, pp 205–232 Missbauer H, Uzsoy R (2010) Optimization models for production planning. In: Kempf KG, Keskinocak P, Uzsoy R (eds) Planning production and inventories in the extended enterprise: a state of the art handbook, vol 1. Springer, New York, pp 437–508 Missbauer H, Hauber W, Stadtler H (2011) Developing a computerized scheduling system for the steelmaking–continuous casting process. In: Kempf KG, Keskinocak P, Uzsoy R (eds) Planning production and inventories in the extended enterprise: a state of the art handbook, vol 2. Springer, New York, pp 461–488 Orcun S, Uzsoy R (2011) The effects of production planning on the dynamic behavior of a simple supply chain: an experimental study. In: Kempf KG, Keskinocak P, Uzsoy R (eds) Planning in the extended enterprise: a state of the art handbook. Springer, Berlin, pp 43–80 Ormeci Matoglu M, Vande Vate J (2011) Build-to-order meets global sourcing: planning challenge for the auto industry. In: Kempf KG, Keskinocak P, Uzsoy R (eds) Planning production and inventories in the extended enterprise: a state of the art handbook, vol 2. Springer, New York, pp 231–248 Ouyang Y, Daganzo C (2010) Robust stability analysis of decentralized supply chains. In: Kempf KG, Keskinocak P, Uzsoy R (eds) Planning production and inventories in the extended enterprise: a state of the art handbook, vol 1. Springer, New York, pp 537–664 Ovacik IM (2010) Advanced planning and scheduling systems: the quest to leverage erp for better planning. In: Kempf KG, Keskinocak P, Uzsoy R (eds) Planning production and inventories in the extended enterprise: a state of the art handbook, vol 1. Springer, New York, pp 33–44 Ozer O (2010) Inventory management: information, coordination and rationality. In: Kempf KG, Keskinocak P, Uzsoy R (eds) Planning production and inventories in the extended enterprise: a state of the art handbook, vol 1. Springer, New York, pp 321–366 Powell WB (2007) Approximate dynamic programming: solving the curses of dimensionality. Wiley, New York Puterman ML (2005) Markov decision processes: discrete stochastic dynamic programming. Wiley, New York Simon HA, Holt CC (1954) The control of inventories and production rates – a survey. Oper Res 2(3):289–301

Author Index

A Aarts, E., 443 ACEA. See European Automobile Manufacturers Association Acker, N., 288 Ackoff, R.L., 250 Adam, E.E., 48, 306, 318, 410, 413 Adelsberger, H., 469 Aichlmayr, M., 106 Akcay, Y., 83 Akella, R., 210 Akif, J.C., 210 Alden, J.M., 264 Aldowaisan, T., 436, 465 Allahverdi, A., 436 Allan, R., 5, 7, 9, 12 Alle, A., 437 Alter, S.L., 469 An, C., 46, 47 Anderson, D.M., 232 Angelides, M.C., 48 Angerhofer, B.J., 48 Anupindi, R., 209 Apostolakis, G.E., 109 Appelqvist, P., 493 Araman, V., 210 Arnott, D., 469 Arntzen, B.C., 46, 318 Arrow, K., 181 Artiba, A., 495, 507, 521 Askin, R.G., 434 Asmundsson, J., 5, 6, 18, 44, 53 Assaf, I., 43, 47, 48, 83, 109, 134, 232 Avinash, M.W., 507 Aviv, Y., 51 Aytug, H., 533, 551

B Babaioff, M., 187, 188 Bachelet, B., 512, 519 Badell, M., 430, 438, 512 Badgwell, T.A., 19, 407, 413, 415, 416 Bagajewicz, M., 430, 438, 512 Bajari, P.L., 185 Balakrishnan, A., 83 Ball, M.O., 83 Baosteel, N.D., 18 Barba-Romero, S., 512 Barbuceanu, M., 50 Barnes-Schuster, D., 209 Barnhart, C., 83, 89 Barrett, L., 106 Bass, E., 288 Bassok, Y., 207 Batty, M., 295 Bauer, A., 431, 434 Bayiz, M., 441 Beard, C., 290 Bellabdaoui, A., 470, 472, 473 Belt, B., 468 Benson, R., 17–20, 23, 26, 28, 53, 77, 318, 325, 338 Benton, W.C., 156 Bermon, S., 358 Berning, G., 431 Berry, W.L., 84, 430, 486 Bertrand, J., 4 Bertsekas, D.P., 441 Bertsimas, D., 21, 468 Beth, A., 203 Bhaskaran, K., 439 Bhaskaran, S., 260 Bhattacharya, K., 135 Bibner, O., 43, 47, 48, 83, 232 Biehl, M., 110 Billaut, J.C., 531

K.G. Kempf et al. (eds.), Planning Production and Inventories in the Extended Enterprise, International Series in Operations Research & Management Science 152, c Springer Science+Business Media, LLC 2011 DOI 10.1007/978-1-4419-8191-2,

571

572 Billington, C., 107 Binato, S., 447 Birge, J.R., 565 Biswas, S., 50 Bitran, G.R., 5, 135 Bixby, A., 207 Blackstone, J.H., 439 Blumenfeld, D.E., 249, 257, 259 Bollapragada, R., 431 Botta, V., 495, 507, 521 Bowden, R., 431, 434 Box, G.E.P., 284 Boyaci, T., 134 Boylan, J.E., 299 Bradley, J., 210 Brandenburg, M., 431 Brans, J.P., 509, 512 Braun, M.W., 139, 411 Bremner, B., 233 Breuil, D., 46 Brown, A.O., 135 Brown, D.E., 135 Brown, G.G., 135 Browne, J., 46 Brucker, P., 438, 440, 443 Buchanan, J.T., 547 Buckley, S., 46, 47 Buitenhoek, R., 4 Bunn, D.W., 309 Burda, R., 351 Burka, E., 531 Burkard, R.E., 437 Burns, J.F., 48, 257 Burns, L.D., 48, 257, 259 Buzacott, J.A., 257

C Cachon, G.P., 109, 158, 209, 210, 212 Candea, D., 43 Carlier, J., 507 Carlyle, W.M., 135, 411 Cavalieri, S., 47 Chambers, J.C., 292 Chandrashekar, T.S., 185 Chang, S.Y., 439 Chang, Y.L., 439 Chao, X., 430, 431, 436, 439, 449 Charnes, A., 5 Chatfield, C., 291, 295, 305, 307, 309 Chaudhry, S.S., 156 Chen, B., 48, 109, 232 Chen, C.Y., 47 Chen, F., 43, 47, 48, 83, 97, 109, 232, 563, 564

Author Index Chen, H., 468 Chen, M., 48 Chen-Ritzo, C.-H., 83 Cheng, F., 97, 109, 563, 564 Child, M., 134 Cho, M., 48 Choi, T., 49 Chong, C., 135 Chopra, S., 109 Chowdhary, P., 135 Chrétienne, P., 507 Chuang, T., 532 Church, L.K., 533 Clark, A.J., 134 Clarke, E., 176 Clement, J., 434 Coffman, E.G., 436, 448 Cohen, M.A., 134 Coldrick, A., 434 Cole, G.C., 432 Coleman, B.J., 106 Collins, J., 134, 187 Collins, L.A., 134 Conway, R., 256 Cooper, F.W., 20, 48, 107, 109, 131, 203, 306, 318, 410, 413, 446 Cooper, W.W., 5 Costy, T., 264 Cowling, P.L., 465 Coyle, R., 48, 58, 66 Cross, M., 267 Current, J.R., 156

D d’Aspremont, C., 181 Daganzo, C.F., 257 Dai, J., 20 Dallery, Y., 134 Dangat, G.S., 353 Daniels, R., 210 Dash, S., 492 Daskin, M.S., 250 Dasu, S., 135 Davenport, T., 45, 318 Davies, O., 295 Davis, T., 110 Dayama, P., 185 de Kok, A., 43 De Lit, P., 512 Degbotse, A., 312 Dejonckheere, J., 43, 48 DeJong Chad, D., 271, 567 Del Vecchio, A., 48

Author Index Delchambre, A., 512 Deleersnyder, J.L., 4 Deleris, L.A., 111 Dempsey, J., 288 den Besten, M., 444 Denardo, E.V., 4 Denton, B., 325, 332, 372 Dessouky, M., 567 Detti, P., 444 Dews, B., 351 Dhaevers, V., 512 Dietrich, B., 83 Diltz, J.D., 259 Dilworth, J.B., 250 Disney, S., 48 Dobson, G., 1, 83, 209 Dong, L., 210 Dooley, K., 49 Dorigo, M., 444 Drake, E.M., 108 Drexl, A., 440, 443 Drezner, Z., 48 Duarte, B., 47 Dubois, D., 537, 539 Duchessi, P., 318 Duenas, A., 552, 553 Duggan, J., 431, 434 Duin, C., 441 Duncan, K., 109 Duvivier, D., 512, 519 Dyer, G., 233

E Elimam, A., 493 Elkins, D., 126 Elmaghraby, S.E., 493, 564 Emelyanov, V., 495, 507, 564 Enscore, E.E., 439 Eppen, G., 209 Erhun, F., 109, 125 Ertogra, K., 210 Ervolina, T., 83, 92 Espuña, A., 430, 438, 512 Ettl, M., 46, 89, 563, 564 European Automobile Manufacturers Association (ACEA), 233

F Fan, M., 157, 158, 187, 189, 190 Fang, L., 462, 464, 470–473, 482 Fargier, H., 532 Farlow, D., 207

573 Fayad, C., 539, 543 Federgruen, A., 109 Feigin, G., 46, 89, 134 Feo, T.A., 441 Fildes, R., 290, 309 Fink, A., 441 Fiordaliso, A., 470, 472, 473 Fischbein, S.A., 567 Fishbeck, P.S., 108 Fliedner, G., 295 Flores Godoy, J.J., 42 Floudas, C.A., 566 Fogarty, D., 318 Fogel, D.B., 256 Fonlupt, C., 512, 519 Fordyce, K., 77, 319, 376, 564, 565 Forges, F., 203 Forrest, J., 43, 48, 352 Forrester, J., 325, 332, 372, 492 Foster, E., 153 Fourer, R., 264 Fowler, J., 47, 564 Fox, D., 50 Fox, M., 50 Fragnelli, V., 195 França, P.M., 443 Fransoo, J.C., 43 Fraser, J., 321 Freeman, S., 4 Frick, M.C., 257, 259 Friedler, F., 430, 438 Fröhning, M., 479 Frost, F.G., 156 Fry, T.D., 467

G Gaalman, G., 4 Gaba, D.M., 108 Galbraith, J., 332 Gallego, G., 83, 133–137, 149, 564, 566 Gamarnik, D., 83, 134, 135, 137, 149, 564, 566 Gan, G.P., 47, 135 Ganeshan, R., 43, 47, 83, 109, 134, 232, 430, 468 Gaonkar, R.S., 154, 156 Garcia-Jurado, I., 195 Gardner, E.S., 295, 305 Garey, M.R., 449 Garg, D., 158, 165, 166, 563, 564 Garrick, B.J., 108 Gary, K., 313 Gay, D.M., 264 Geiger, C.D., 439

574 Gendreau, M., 443 Geneste, L., 532 Geoffrion, A.M., 256 Gérard-Varet, L.A., 176 Gershwin, S.B., 134 Geunes, J., 83 Gferer, H., 5 Giffler, B.L., 472, 473, 476 Gini, M., 187 Ginsberg, M.L., 441 Glover, F., 318, 443 Goetschalckx, M., 46 Gokhale, A.R., 353 Golany, B., 157 Goldbreg, D., 542 Golder, E.R., 295 Goldman, S., 332 Golla, C., 533 Golovachakina, N., 210 Gomide, F., 532 Gonçalves, P., 109, 125 Goodwin, P., 290, 309 Grabot, B., 532 Grabowski, J., 436 Graham, R., 436, 448 Graves, R.J., 318, 330 Graves, S.C., 4, 18, 134, 318, 330, 565 Green, J.R., 158, 172, 176 Grey, W., 210 Griffin, P.M., 46 Grinold, R.C., 567 Grossler, A., 48 Grossmann, I.E., 470, 471, 473 Groves, T., 153, 158, 176, 181, 188 Guan, X., 257, 492, 532 Guinet, A., 495, 507, 521 Gung, R.R., 294 Günther, H.-O., 462 Gupta, J.N.D., 436, 465 Gürsoy, K., 457

H Hale, W., 83 Hall, N., 436, 533, 547 Hall, R.W., 256 Ham, L., 439 Hamming, R.W., 256 Hansen, P., 444 Hansmann, K.-W., 507 Hanssmann, F., 4 Hapke, M., 532 Harjunkoski, L., 470, 471, 473

Author Index Harker, P.T., 135 Harrison, P.J., 295 Harrison, T., 295 Harry, M.J., 166 Harsanyi, J.C, 200 Harvey, W.D., 441 Hatzl, J., 437 Hauber, W., 460 Haupt, R., 439 Hausman, W.H., 5 Hax, A.C., 4 Health & safety executive, 108 Heching, A., 97 Hedge, S., 372 Hendricks, K.B., 105 Henley, E.J., 108 Herrmann, J.W., 533 Herroelen, H., 533 Hery, W.J., 441 Hess, S.W., 4 Hibon, M., 305 Hicks, D.A., 114 Hildum, D.W., 50 Hodgson, T.J., 4 Hoeck, M., 507 Hoffman, T., 318 Hogg, G.L., 439 Holczinger, T., 430, 438 Holmstorm, B., 201 Holt, C., 4 Holweg, M., 235 Homem-de-Mello, T., 5 Hong, T., 532 Hong, Y., 492 Hood, S., 358 Hoogeveen, H., 529 Hoos, H.H., 510 Hopman, J., 105 Hopp, W.J., 44, 54 Horiguchi, K., 17 Howard, R.A., 112 Hu, C.W., 289 Huang, G.Q., 46 Huang, P., 47 Huang, W., 430 Huegler, P.A., 471 Hull, J.C., 208, 226 Hung, Y.F., 17, 23, 25 Hunt, G.C., 257 Hurink, J., 443 Hutton, R.D., 264 Hyndman, R.J., 290, 305

Author Index I Iassinovski, S., 489 IBM, 46, 47, 81, 102, 207, 312, 319–325, 342, 350, 352, 359, 362–364, 367, 376, 377, 564 Ichiishi, T., 203 India Brand Equity Foundation, 247 Ingalls, R.G., 114 Inman, R.R., 248 Interfurth, K., 134 Ishibuchi, H., 532 Ishii, H., 532, 539 Iskander, W., 439 Itoh, T., 539 Iyer, A., 209

J Jackson, C.A, 264 Jackson, E.C., 243 Jackson, J.R., 449, 450 Jaikumar, R., 19 Jain, A., 4, 437 James, R.J.W., 547 Janakiram, M., 288 Jantzen, J., 364 Jawahir, K., 313, 492 Jenkins, G.M., 292 Jennings, K.M., 106 Jiong, S., 187 Johnson, D.S., 449 Johnson, E.L., 83, 89 Johnson, L.A., 43, 53 Johnston, F.R., 299 Jones, D.T., 231 Jones, G., 318 Jordan, W.C., 106, 259 Juhasz, N.M., 430 Juttner, U., 110

K Kadipasaoglu, S.N., 307 Kalagnanam, J., 492 Kalai, E., 206 Kalelkar, A.S., 108 Kanakamedala, K.B., 441 Kane, V.E., 165 Kanet, J.J., 469 Kang, J., 17–20, 23, 25, 26, 28, 53, 77, 318, 325, 332, 337, 567 Kann, A.R., 436, 448

575 Kapoor, S., 135 Karabuk, S., 157 Karmarkar, U.S., 1, 44, 53 Karney, D., 318, 443 Katircioglu, K., 83, 133, 564, 566 Katzberg, J., 529 Keeney, R.L., 116 Kekre, S., 4 Kelly, J.P., 530 Kelton, W.D., 47 Kempf, K.G., 48, 135, 313, 318, 325, 332, 351, 403, 411, 429, 430, 563 Kendall, G., 47, 531, 542 Kernighan, B., 264 Kerwin, K., 233 Kienle, A., 430 Kilger, C., 43, 46, 47, 77 Kilgore, S.S., 134 Kim, B., 20 Kim, D.S., 264 Kim, K., 5, 7, 9, 12, 81, 135, 207, 318, 410, 430, 566 Kim, S., 20 Kim, T., 20, 48, 107, 109, 134, 203, 306, 318, 410, 413, 446 Kimball, G.E., 4 Kindt, V.T., 531 King, A.J., 97, 109, 210, 563 Kingman, J.F.C., 257 Kirwan, B., 109 Kleijnen, J.P., 47 Kleindorfer, G.B., 410 Kleindorfer, P.R., 19, 48, 110, 410 Kleinknecht, J., 210 Kletter, D.B., 262 Klingman, D., 443 Koch, C., 106 Kochenberger, G., 443 Kodama, D., 134 Koenig, L., 135, 207, 318, 430 Kohls, K.A., 264 Konopka, J.M., 4, 18, 134, 259, 318, 330, 564, 565 Korst, J., 443 Kotz, S., 165 Krappinger, R., 475 Kriebel, C.H., 19, 41, 48, 110 Kubota, R., 532, 539 Kulkarni, D., 153, 158, 185 Kumamoto, H., 107 Kuroda, M., 532 Kussi, K., 431 Kutanoglu, E., 15, 171

576 L Laguna, M., 443 Lakats, L.M., 108 Lal, A.M., 507 Land, M., 4 Laporte, G., 443 Lariviere, M., 210, 212, 222 Lasdon, L.S., 4, 307 Lasserre, J.B., 5 Latinne, P., 512 Latour, A., 105, 106 Lau, J., 46, 47, 81, 422, 430 Lawler, E., 436, 448 Lawley, M.A., 533, 551 Lawrence, M., 290, 309 Lawrence, S., 441 Lazraq, A., 489 Leachman, R.C., 17–20, 23, 25, 26, 28, 53, 77, 318, 325, 332, 337, 567 Lee, C.Y., 48, 446 Lee, D., 203 Lee, H., 48, 107, 109, 134, 318 Lee, T.S., 306 Lee, Y., 20, 48 Lehtonen, J.M., 493 Lemos, R., 106 Lendermann, P., 47, 135 Lenstra, J.K., 20, 49, 436, 448, 507, 532 Leong, T., 135 Leplat, J., 109 Leung, Y.T., 289 Leus, R., 533 Levanoni, M., 289 Levans, M.A., 243, 244 Levecq, P., 489, 564 Levi, R., 566 Li, J., 257, 492, 532 Li, L., 492 Li, S., 353 Li, Y., 532 Lin, E., 533 Lin, G., 81, 135, 207, 318 Lin, S., 5, 7, 9, 12 Lin, V., 17–20, 23, 25, 26, 28, 53, 77, 318, 325, 332, 337, 567 Lin, X., 430, 566 Lindo Systems Inc., 495, 521 Linn, R., 470 Little, J., 332, 351 Liu, C., 17, 23, 25, 77, 318, 325, 332, 337 Liu, J.Y., 462, 464, 469–473, 482, 507 Liu, Z., 532 Lockwood, C., 533 Loewenstern, D., 441

Author Index Logendran, R., 4 Lorenço, H.R., 444 Louveaux, R., 565 Lovejoy, W., 208 Lovelace, C.R., 165 Lu, Y., 46, 81, 134, 207, 563, 564 Luh, P.B., 462, 470, 471 Luo, K., 430 Lyon, P., 135, 313, 319, 320, 325, 377 Lyons, G., 431, 434

M Macal, C.M., 49 Macalister, T., 106 Mackintosh, J., 233 Magazine, M., 43, 46, 83, 210, 212, 222, 318 Makridakis, S., 290, 305 Malhotra, M.K., 467 Mallik, A., 430 Maloni, M., 243 Maloubier, H., 46 Mandelbaum, A., 468 Manne, A.S., 4 Manrodt, K.B., 305 Mareschal, B., 509, 512 Martin, O., 444 Martin-Vega, L.A., 43, 53, 135, 318, 533 Mart´ınez-de Albéniz, V., 109, 210 Mas-Colell, A., 158, 172, 176 Mascis, A., 438 Matheson, J.E., 112 Matsuo, H., 5, 135 Mauch, E., 135 Maxwell, W., 256 Mayer, E., 430 McAfee, R.S., 188 McClain, J.O., 256 McCormick, S.T., 436, 437 McKay, K.N., 429, 468, 469, 564, 568 McMillan, R.S., 185 McMullen, P.R., 441 Meacham, A., 83 Meal, H.C., 468 Meeran, S., 437 Mehta, S.V., 533, 550 Mehta, V., 431 Meloni, C., 429, 438, 441 Mendez-Naya, L., 195 Mercé, C., 5 Meskens, N., 489, 512 Michalewicz, Z., 256 Miemczyk, J., 235 Milne, R.J., 135, 313

Author Index Minelli, E., 203 Miser, H.J., 250 Mishra, B.V., 430 Missbauer, H., 5, 18, 43, 50, 53, 73, 461, 564–566 Mladenovic, N., 444 Mockus, L., 430 Modi, S., 430, 566 Modigliani, F., 4, 410, 565 Mohan, S., 533, 551 Möhring, R., 438, 440, 443 Montgomery, D.C., 53, 236 Montgomery, J., 544 Moore, J., 106, 404 Morgenstern, O., 116 Morreale, M., 313, 320, 325, 364 Morrice, D., 288 Morton, T.E., 439, 441, 443 Moscardini, A.O., 267 Mote, J., 318, 443 Muckstadt, J.A., 134 Müller, F.M., 443 Muller, H., 4 Mullick, S.K., 292 Murata, T., 535 Murphy, J.V., 106, 109 Musselman, K., 43, 46 Muth, J., 410 Myerson, R.B., 181, 200, 201, 203, 205

N Naccarato, B., 81, 135, 207, 318 Nagarajan, M., 186 Nam, S., 4 Narahari, Y., 50, 153, 185, 564 Narsimhan, R., 156 National Chamber Foundation of the U.S. Chamber of Commerce, 243 National Oceanic & Atmospheric Administration, U.S. Department of Commerce, 243 Navas, D., 134 Nawaz, M., 439 Nemeth, C., 108, 109 Nemhauser, G.L., 83, 89 Netessine, S., 83, 209 Neumann, K., 116 Newman, R.G., 106 Nisan, N., 187, 188

577 Norden, P., 318 North, M.J., 49 Nowicki, E., 439, 443

O O’Grady, P.J., 4 OECDITCS, 237 Office of Aerospace and Automotive Industries, Manufacturing and Services, International Trade, 233, 235 Olhager, J., 47 Orcun, S., 52, 565 Orlicky, J., 318 Orlov, L.M., 134 Ormeci, M., 245, 246, 564 Orzell, R., 135 Osman, I.H., 530 Ouelhadj, D., 465 Ovacik, I.M., 46, 135, 438, 564–566 Owen, G., 249 Owen, J.H., 249 ¨ ¨ 134, 137, 149, 564, 566 Ozer, O.,

P Pacciarelli, D., 429, 438 Padmanabhan, P., 48, 107, 109, 134, 306, 318, 410, 413, 446 Padmanabhan, V., 48, 107, 109, 134, 306, 318, 410, 413, 446 Pahl, J., 5, 19 Pan, A.C., 156 Panelli, A., 107 Pang, H., 492, 532 Panwalkar, S.S., 439 Papadimitriou, C.H., 134 Papadopoulos, C.T., 134 Papageorgiou, L.G., 437 Park, H., 492 Parker, R.G., 531 Paté-Cornell, M.E., 108 Pati, M.C., 372 Patil, S.P., 313, 372 Peck, H., 110 Pedrycz, W., 532 Pempera, J., 436 Pender, K., 105 Pentico, D.W., 439, 441, 443 Perona, M., 4 Persson, F., 47 Pervan, G., 469 Pesch, E., 438, 440, 443 Peters, D., 111, 125

578 Petersen, C.M., 465 Peterson, R., 240 Petrakian, R., 135, 431 Petrovic, D., 532, 539, 540, 546 Petrovic, R., 532, 539, 540, 546 Petrovic, S., 539, 546, 552, 557, 558 Philipoom, P.R., 467 Phillips, D.T., 135 Phillips, L., 135 Piachaud, B., 433 Pidd, M., 251 Pillai, D., 288 Pinedo, M., 430, 431, 436, 437, 439, 449 Pinkus, G.S., 430 Pinto, J.M., 437 Piplani, R., 135 Pollock, S.M., 267 Pomerol, J.-C., 512 Portioli, A., 4 Posner, M., 547 Potts, C.N., 533 Powell, S.G., 267, 565 Powell, W.B., 565 Prade, H., 537 Pranzo, M., 438 Prater, E., 110 Preux, P., 489, 512, 519 Promoting O.R, the science of better, 387 Proust, C., 531 Ptak, C., 45 Puah, S.A., 135 Puigjaner, L., 430, 438, 512 Puterman, W.L., 565 PWC autofacts light vehicle capacity data, 233 Pyke, D.F., 83

Q Qin, S.J., 19, 407, 415, 416 Qiu, R., 83, 135

R Raar, D., 17–19, 23, 26, 28, 53, 77, 318, 332, 337, 567 Raghavan, N., 17 Raheja, A.S., 533 Raiffa, H., 116 Raisch, H., 430 Rajaram, K., 5, 7, 12 Ramachandran, B., 83, 134, 135, 137, 149, 566 Ramaswamy, S., 289, 309 Rardin, R.L., 53 Rasmussen, J., 109

Author Index Ratner, M., 4 Rayward-Smith, V.J., 50 Reason, J., 109 Reddy, C., 533 Reddy, K.R.B., 492 Reeves, C.R., 50 Reid, R.L., 353 Rekiek, B., 512 Reklaitis, G.V., 441 Rentz, O., 479 Resende, M.G.C., 441 Rezig, W., 465 Riane, F., 489, 495, 507 Riaño, G., 5 Rice, J.B., 105 Rice, R., 313 Rinnooy Kan, A.H.G., 1, 4, 5, 7, 9, 11, 12, 44, 53 Rivera, D.E., 19, 411 Robilliard, D., 489, 512, 519 Rodriguez, A., 19 Roeder, T., 274 Rogers, R.V., 437 Romero, J., 430, 438, 512 Rong, A.Y., 4, 109, 462, 464, 469–471, 473, 482 Rosa, C., 185, 564 Rose, O., 17 Rosenbaum, B.A., 134 Rosling, K., 134, 149 Rothblum, U.G., 157 Roundy, R.O., 566 Roux, O., 512 Roy, B., 437 Rudi, N., 83 Ruiz, R., 441 Russell, D., 83, 135 Ryan, J.K., 43, 47, 48, 83, 97, 134, 232, 468 Rycroft, R.S., 291

S Saad, G., 43 Sabuncuoglu, I., 441 Sachon, M., 108 Sadeh, N., 50, 187, 431, 438 Safayeni, F.R., 429, 468, 469, 564, 568 Sahinidis, N.V., 5 Sakasegawa, H., 257 Sakawa, M., 532, 539 Sanders, N.R., 291, 305 Sanmart´ı, E., 437 Sari, J., 434 Sarker, B.R., 507

Author Index Sastry, K., 542 Satterthwaite, M.A., 205 Savelsbergh, M.W.P., 83, 89 Savit, R., 50 Scarf, H., 134, 194, 239 Scherer, W.T., 431 Schieritz, N., 48 Schmidt, G., 207 Schneeweiss, C.H., 466, 467, 484 Schrage, L., 134 Schragenheim, E., 45 Schruben, L., 274 Schultmann, F., 479 Schutten, J.M.J., 437 Schwartz, L.B., 134, 411 Schwarz, M., 97, 109, 563, 564 Schwindt, C., 436 Seidmann, A., 507 Sethuraman, J., 468 Settle, J.G., 295 Shah, N., 432, 433 Shang, K., 134 Shapiro, A., 5 Shapiro, J.P., 5 Sheffer, S., 430 Sheffi, Y., 106 Shen, Z., 83 Shenker, S., 436, 437 Sherbrooke, C.C., 134 Shi, D., 210 Shmoys, D.B., 46 Shobrys, D., 318, 321 Shugan, S.M., 256 Shumsky, R.A., 83 Sides, R., 5 Silver, E.A., 240 Simchi-Levi, D., 46 Simon, H., 351, 377, 409, 410, 485, 565 Singh, H., 372 Singhal, V.R., 105 Sivazlian, B.D., 48 Slovic, P., 124 Slowinski, R., 532 Smith, B.T., 50, 290, 318, 507 Smith, D.D., 50, 292 Smith, G.D., 50 Smith, J.M., 134 Smith, K.D., 50 Smith, M.A., 50 Smith, M.L., 50 Smith, S.F., 318 Smutnicki, C., 439, 443 Sodhi, M.S., 109 Sohn, B., 288

579 Song, J.S., 134 Sørensen, K.I., 465 Sosic, G., 186 Sotskov, Y.N., 547 Spearman, M., 4, 44, 54, 256, 257 Squillante, M., 46, 81, 89, 134, 207, 563, 564 Sridharan, V., 307 Srinivasan, R., 207 Sriskandarajah, C., 436 Stadler, W., 461 Stadtler, H., 43, 46, 47, 77 Stallaert, J., 157, 158, 187, 189, 190 Stallkamp, T., 186 Standridge, C.R., 434 Steele, D.C., 467 Steiglitz, K., 530 Sterman, J.D., 43, 48, 56, 58, 62 Stewart, R., 166 Storer, E.H., 476 Stoynoff, L.K., 156 Stützle, T., 441 Subramaniam, B., 288 Subramaniam, V., 533 Sueyoshi, T., 439 Sullivan, G., 313, 320, 325, 364 Sullivan, R.S, 439 Suman, M., 135 Sussman, R., 437 Swaminathan, J.M., 50, 83, 318 Sycara, K., 438

T T’kindt, V., 531 Tada, M., 532 Tadelis, S., 185 Tag, P., 271 Tagaras, G., 108 Talbi, E.-G., 507, 522 Tang, C.S., 4, 109 Tang, L., 462, 464, 469–473, 482 Tang, X., 353 Tarasewich, P., 441 Tayur, S., 46, 83, 318 Teghem, J., 470 Terjung, R.C., 4 Tersine, R.J., 250 Terzi, S., 47 Tew, J.D., 153 Theil, H., 302 Thomas, D.J., 46, 81 Thomas, L.J., 256 Thompson, G.L., 106, 410, 472, 473, 476

580 Tölle, F.-J., 431 Tomlin, B., 109 Towill, D.R., 48, 56, 66, 410 Trafton, L., 46, 318 Trautmann, N., 436 Treece, J.B., 105 Trigg, D.W., 295 Truong, V.A, 566 Tsai, R.Y., 294 Tsakalis, K.S., 19 Tsay, A., 208 Tsitsiklis, J.N., 441 Tufte, E.R., 114 Tukey, J.W., 114 Turnquist, M.A., 264

U Upton, R., 107 Uzsoy, R., 43, 135, 318, 438, 533, 550, 563

V Vaccari, R., 476 van Beek, P., 462 Van Dyke Parunak, H., 49, 50 Van Houtum, G.J., 134 Van Zant, P., 134 Vance, P.H., 83, 89 Vande Vate, J., 231, 564 Vander Veen, D.J., 264 VanLaarhoven, P.J.M., 443 Vargas-Villamil, F.D., 411, 414 Vasko, F.J., 471 Vassilopoulos, A.I., 309 Venkatadri, U., 83 Venkatasubramanian, V., 441 Venkateswaran, S., 17 Vickrey, W., 153, 176, 188 Vidal, C., 46 Vidal, R.V.V., 465 Vieira, G.E., 533 Vignier, A., 530 Vincke, P., 509, 512 Vinson, J.M., 430 Viswanadham, N., 154, 156 Voˇ, S., 5, 19 Vohra, R., 203, 204 Volij, O., 203 Volkema, R.J., 250 Vollmann, T.E., 84, 430, 486 von Neumann, J., 116 Voss, S., 43, 441

Author Index W Walsh, T., 441 Walsh, W.E., 154, 156, 187, 188, 190 Wang, C., 313 Wang, W., 135, 411 Wang, Z., 532 Waren, A.D., 4, 307 Waters-Fuller, N., 106 Weber, C.A., 156 Webster, S., 429 Weiss, G., 20, 49, 532 Wellman, M.P., 154, 156, 187, 190 Weng, W., 135 Werner, F., 440 West, M., 295 Whang, S., 48, 107 Wheelwright, S.C., 290, 305 Whinston, A.B., 157, 158, 187, 189, 190 Whinston, M.D., 158, 172, 176 White, K.P., 437 Whybark, D.C., 84, 430, 486 Wiers, V.C.S., 429 Wiesel, H., 207 Wight, O.W., 468 Wilhelm, W.E., 465 Wilson, R., 192, 201 Winkler, A., 440 Winston, W.L., 250 Wolf, B., 436, 437 Wolf, S., 134 Wolfson, R., 373 Womack, J.P., 231 Woodruff, D.L., 43 Wooldridge, M., 49 Woolsey, G., 318 Wreathall, J., 108, 109 Wu, C., 441 Wu, D.J., 211 Wu, D.S., 151 Wu, H., 257, 492 Wu, S.D., 46, 83, 134, 210, 476 Wybark, D.C., 84, 430, 486 X Xia, Y., 83 Xiong, Y.L., 438 Xu, Q., 135 Xu, S., 83 Y Yamazaki, A., 203 Yanasse, H.H., 5 Yang, Z.H., 462, 464, 469–473, 482

Author Index Yano, C., 135 Yao, D., 207 Yar, M., 295, 305, 307, 309 ¨ 109 Yazlali, O., Yea, S., 20, 264 Yellig, E., 271 Yen, C., 353, 372 Ygge, F., 156, 188, 190 Yoo, J., 5, 7, 12 Yu, H., 492 Yurkiewicz, J., 291, 305 Z Zadeh, L.A., 552 Zapfel, G., 5, 73, 467

581 Zazanis, M.A., 4 Zemel, E., 206 Zhang, B., 83 Zhang, J., 171, 211 Zhang, W., 470 Zhao, Z., 102 Zhou, G., 46 Zhu, B., 492 Zhu, K., 210 Zhu, X., 465 Zijm, W., 4, 134 Zipkin, P.H., 4, 134, 213 Zisgen, H., 351, 358 Zsidisin, G.A., 107 Zydiak, J.L., 156

Subject Index

A Advanced planning and scheduling (APS), 43, 46, 47, 52, 66 Aggregate capacity allocation, 156 Aggregate planning, 4, 566 Allocated clearing function, 18 Allocation of safety stocks, 21 Anticipation function, 466 Arbitrage pricing, 208 Artificial intelligence, 438, 565 Auctions, 181, 185–189 Availability, 23, 37, 38, 41, 52, 81–84, 86, 93, 99, 100, 102, 115, 133, 263, 273, 278, 282, 283, 285, 314, 344, 363, 365, 368, 371, 429, 430, 432, 446, 448–449, 451, 463, 465, 467, 470–473, 475, 476, 490, 502, 503, 523, 567

B Baseline measurement, 302 Batching, 1, 4, 16, 20, 282, 545 Bill of material, 89, 318, 329, 332, 342, 390, 396, 435 Bottleneck, 17, 18, 20, 21, 26–29, 37–39, 54, 114, 251, 266, 308, 455, 462, 465, 471, 492, 499, 507, 508, 525, 527 Buffers, 20, 26–28, 30, 107, 257, 265, 266, 316, 330, 418, 436, 437, 446, 447, 462, 468 Bullwhip effect, 43, 48, 232, 244, 246

C Capacity and lead time, 1–4, 12, 70, 119, 148 constraints, 2, 12, 63, 112, 290, 330, 350, 371, 384, 385, 395, 397, 492 expansion, 486

Category management, 308 Centralized planning, 318, 464 Chance constraint models, 5 Characteristic curves, 188 Clearing function, 2, 3, 5–7, 12, 18, 19, 50, 53–55, 61–64, 68, 70, 71, 73, 468, 566, 568 Client-server technologies, 494 Cognitive psychology, 123 Collaboration, 109, 111, 116, 374, 376, 484, 491, 528, 533, 541, 548, 556, 564, 568 Common goods, 436 Competition, 16, 112, 212, 273, 276 Compliance, 229 Consumer choice models, 235 Continuum models, 221, 222 Contract manufacturing, 106, 154 Contracting, 185, 210–212, 228 Contracts, 89, 97, 106, 109, 110, 117, 124, 154, 156, 185, 188, 192, 197, 199–202, 204, 207–228, 350, 351, 430 Control, 1, 3, 4, 15, 19–21, 26, 28, 32, 43, 44, 48, 49, 51, 55, 56, 59, 61, 62, 66, 75, 77, 111, 134, 153, 165, 189, 253, 259, 260, 265, 272–275, 292, 306, 318, 319, 326, 330, 351, 360, 372, 390, 403–426, 430, 433, 434, 454, 466–468, 485, 490, 493, 494, 507, 526, 528, 531, 565 CONWIP, 467 Coordination, 15–41, 50, 157, 207–211, 215, 220–225, 228, 315, 429, 431, 432, 445, 456, 464, 465, 484, 564 Cost allocation, 83, 89, 92, 100 Cost of ownership, 199 Cost of sustainability, 429, 432 Customer behavior, 46 Cycle time, 16–28, 30, 31, 39, 41, 45, 47, 53–56, 58, 61–63, 76, 133–135, 163,

583

584

Subject Index 277, 278, 284, 314, 322–324, 326, 327, 329–331, 333, 348–350, 352–354, 360, 372, 378, 383, 505, 506

D Data acquisition, 494, 496 gathering, 276 Decentralized planning, 157, 373 Decentralized supply chains, 52, 207 Decision space, 262, 409 Decomposition, 4, 7–11, 21, 86, 108, 113, 189, 261, 263, 364, 410, 466–468, 475 Demand fulfillment, 425 Dependent demand, 353–358, 362, 370 Deteriorating goods, 147 Deterministic models, 7 Discrete parts manufacturing, 16 Dispatching, 2, 16–20, 26, 28, 29, 31–35, 37, 38, 41, 439, 441, 499, 506–510, 512, 514–516, 520–522, 527, 532, 543, 544, 547, 552, 553, 568 Distribution, 5, 26, 28, 30, 33, 36, 37, 47, 50, 108, 109, 111–116, 118–120, 122, 123, 125, 126, 140, 143, 148, 153, 158–160, 166, 167, 170, 173, 174, 176, 181, 201, 211–213, 218, 220, 222, 260, 272, 282, 292, 296, 307, 308, 318, 343, 391, 394, 395, 404, 405, 410, 411, 417, 418, 422, 532, 551, 567 Distribution systems, 134, 307, 566 Double marginalization, 215 Drum-buffer-rope, 467 Dynamic pricing, 83 E Economic order quantity, 84, 208, 401 End of life management, 326 Enterprise data, 289, 315, 318, 319, 324, 331, 352, 358, 360–362, 373, 377 Enterprise resource planning (ERP), 45, 46, 52, 66, 445, 493–495, 527, 528, 565–567 Enterprise transaction systems, 46 Environmental responsibility, 274 Equilibrium, 157, 181, 187–189, 192, 208–210, 220 ERP. See Enterprise resource planning F Factory physics, 256 Financial instruments, 115

Financial markets, 208 Finite buffers, 260, 437 Fixed allocation, 90 Fixed lead time, 44, 53, 54, 62–64, 70, 73, 76 Flexible capacity, 207, 211, 228 Flow times, 466–468, 482 Fluid approximation, 468 Flux models, 425 Forecast accuracy measures, 66, 119, 124, 165, 232, 241–243, 245, 246, 293, 294, 296, 298, 305, 307, 309, 418 Forecasting, 46, 63, 66, 71, 122, 231, 232, 234, 236, 241, 242, 244, 289–309, 391, 411 Forecasting algorithms, 307 Formulations, 4–6, 9, 12, 17, 20, 23–28, 43, 46, 50, 63, 83, 86, 88–89, 91, 92, 142, 143, 169, 196, 197, 208, 250, 255, 259, 260, 264, 352, 357, 359, 374, 376, 380–385, 395–399, 409–417, 424, 436–438, 470, 472, 482, 500, 563, 565, 567 Frozen schedules, 260 G Games, 187, 190, 191, 193, 195, 200, 202, 203, 209, 327 Genetic algorithms, 533, 540, 542–544, 558 Graphical user interface (GUI), 400, 500, 507, 521 Guidelines, 3, 30, 257, 316, 330, 331, 341, 468 H Hardware, 362 Hierarchical production planning, 3, 4, 15, 21, 22 Hierarchies, 41, 153, 277, 295, 308, 313, 363, 468, 473, 486, 490 Human planners, 509, 510, 515, 516, 519, 520, 526 I Incentive compatibility, 181, 204 Incentive schemes, 157, 158, 176, 178, 181 Independent demand, 354, 355 Industrial practice, 17, 56, 434, 437, 563–566 Information sharing, 46, 51, 435, 454 Inventory decentralized, 11 Inventory management, 2, 106, 292 Inventory multi-echelon, 133–149 Inventory perishable, 564

Subject Index J Joint replenishment, 136, 416 Just in time, 336, 350, 501

K Kanban, 1, 4, 467

L Large scale systems, 315 Lead time, 1–13, 17, 44, 45, 53, 54, 76, 136, 141, 142, 144, 145, 148, 162–167, 210, 231, 232, 235, 238, 241–243, 245, 246, 260, 314, 403, 410, 425, 468, 474, 565 Lead time (product, replenishment, quotation), 3, 73, 117, 136, 143, 164, 214, 313, 445 Lead time distribution, 5, 120–122 Lean manufacturing, 106, 133, 148 Likelihood ratio, 107–108 Linear decision rule, 4, 17 Linear programming, 4, 5, 9, 17, 22, 23, 52, 64, 156, 189, 196, 318, 438, 472, 502, 568 Little’s Law, 20, 25, 26, 54, 58 Logistics, 4, 83, 105, 154, 160–167, 176, 240, 243, 244, 253, 384, 414, 489 Long-range forecasting, 285 Lot-sizing, 1, 359, 541, 542, 545–546

M Maintenance, 115, 148, 154, 262, 271–273, 282, 292, 294, 297, 309, 400, 434, 437, 447, 456, 490, 495, 496, 500, 523, 528, 551 Make to order, 123, 162, 289, 492, 497 Make to stock, 21, 289, 292, 492 Manufacturing, 1, 15, 45, 83, 105, 133, 153, 186, 207, 231, 253, 271, 289, 314, 389, 401, 430, 462, 489, 532, 564 Manufacturing control systems, 406, 422, 466, 467, 485 Manufacturing control technology, 16, 33, 241, 249, 285, 324, 404 Manufacturing execution systems (MES), 275, 283, 493–496, 527 Manufacturing planning and control, 466, 467, 485 Markov chain, 565 Markov decision process, 565 Martingale, 208 Master data management, 353, 479 Master production schedule, 292, 410

585 Material balance equation, 331, 360 Material requirements planning (MRP), 17, 95, 318, 352, 354–358, 367, 369, 527 Mathematical programming, 43, 81, 83, 85, 86, 256, 257, 259, 352, 437, 565 Measurement of capacity, 408 Merchandising, 289 Metaheuristics, 441–444, 472, 476 Metamodeling, 489 Metrics, 253, 277, 278, 282, 418 Model calibration, 55, 434, 506 Moment expansions, 265, 319, 486 MRP. See Material requirement planning Multi-location inventory systems, 262 Multiple comparisons, 274, 512, 568 Multiple organizations, 496 N Nash equilibrium, 157, 181, 189, 220 Near-myopic heuristics, 441 Neighborhood, 439, 440, 443, 444, 449 Network structure, 116 Newsvendor model, 211–215, 217, 218, 220, 223 Nominal capacity, 413, 417 Non-cooperative subcontracting, 3 Noninteger lead times, 17 Nonlinear behavior, 50 Nonuniformity, 49 Novelties, 50, 84, 107, 567 O Object models, 266 Operational capacity, 492 Operational risk, 112 Opportunity cost, 405 Options, 97, 114, 115, 185, 190, 207–228, 231, 233–235, 241, 242, 282, 353, 358 Order point, 120, 162, 527 Order pool, 369 Order release, 18, 314, 468, 527 Ordering costs, 208 Ordinary differential equations, 49, 411, 564 Organizational changes, 113 Outsourcing, 232 P Partnership, 186 Percentile estimates, 296 Perishable inventory, 334, 336, 337, 564 products, 334, 336

586 Permutation games, 465 Perturbation analysis, 443, 444, 494, 528 Pharmaceutical industry, 429–456 Physical manufacturing control, 259 Planned capacity, 17 Planned lead times, 4, 18 Planners, 44, 107, 108, 111–113, 115–117, 119, 123–126, 153–155, 158, 174, 249, 267, 324, 343, 348, 362, 363, 365, 369, 371, 372, 376, 384, 414, 445, 454, 465, 469, 479, 482, 484–486, 509, 510, 515, 516, 519, 520, 526 Planning, 1, 15, 43, 81, 114, 133, 157, 231, 249, 272, 289, 313, 389, 406, 429, 461, 489, 531, 563 Planning algorithms, 19, 48 Planning period, 2, 6, 18, 48, 55, 56, 66, 82, 360–361, 391, 396, 566–568 Point forecast, 92 Portfolios, 81, 83, 85, 86, 93, 102 Precision-based design, 139, 166, 283 Predictive distributions, 404, 417 Price dependent demand, 212, 391 Price mechanisms, 186, 189, 208, 210, 211, 228 Pricing, 83, 92, 207, 208, 210, 211, 218–220, 228, 290, 564 Procedural manufacturing control, 400 Process industries, 410, 412, 462, 468 Product perishable, 436 substitution, 81–86, 89, 91, 95, 99 Product density, 212 Product information management, 86, 95, 109 Product mix, 16, 18, 111, 272, 274, 279 Production planning, 1, 15, 43, 81, 135, 231, 249, 272, 290, 389, 429, 464, 489, 531, 563 Production planning software, 43, 48, 106, 241, 264, 489 Proportional allocation, 20 Push-pull point, 433

Q Quadratic costs, 4, 178, 410 Queueing, 20, 44, 53

R Random lead time, 142 Random search, 443 Ranking and selection, 86, 89, 91, 95, 156, 185, 278, 298, 301, 409, 414, 439, 440,

Subject Index 443, 479, 491, 509–512, 514, 515, 517, 519, 543, 544, 555, 563 Re-entrant factory, 16, 19, 20, 233, 272, 326, 411 Realized capacity, 417 Recourse actions, 197, 210, 565 Replenishment, 46, 56, 136, 139, 142, 143, 210, 410, 416 Resource load, 1 Response surface methodologies, 281, 490 Retail, 51, 186, 212, 289, 294, 314, 410, 564 RFID, 495 Risk capacity, 115, 207 inventory, 109 quality, 113 Risk assesment, 105–128, 314 Routing, 28, 43, 126, 259, 265, 281, 328, 434, 439, 463, 467, 472, 476, 479, 482 Rule-based forecasting, 466

S Safety stocks, 1, 2, 12, 20, 139, 148, 149, 240 Sales and operations planning, 49, 289, 290, 324, 413, 414 Sales/marketing, 391 Sample path optimization, 301 Scatter search, 522 Scheduling, 1, 15, 43, 135, 156 Scrap, 46, 60, 61, 65, 67–70, 72, 73, 75, 344, 462, 482 Semiconductor manufacturing, 15, 20, 77, 133–149, 315, 318, 324, 325, 337, 406, 411 Sensitivity analysis, 119, 120, 127, 128, 271, 517, 534, 546–548 Serial system, 256, 492 Setup time, 262, 401, 432, 434, 435, 446, 448, 453, 498, 534, 535, 538, 545, 546 Shapley value, 205 Shelf space allocation, 320 Short range forecasting, 19 Signaling, 49, 232, 238, 241, 261, 262, 295, 316, 374, 408, 413, 416, 417, 419, 421 Simulated annealing, 443, 513 Simulation, 15, 17–19, 22–24, 27–31, 34, 41, 43–50, 52, 83, 109, 110, 114, 120, 122, 144, 189, 238, 243, 246, 255, 265, 271–287, 411, 417, 422–424, 456, 468, 469, 489–491, 493–496, 499, 506–512, 516, 521, 522, 526, 553, 565, 567, 568 Simulation on demand, 273, 274 Simulation optimization, 264, 494

Subject Index Simultaneous perturbation, 494 Six sigma, 148, 156, 166 Smoothing, 305–307 Societal responsibility, 237 Spreadsheet applications, 125, 253, 257, 272, 279, 282, 285, 286, 289, 299, 306, 309, 364, 394 Stability conditions, 20, 119, 191, 192, 195, 245, 320, 350, 353, 406, 455, 534, 553–555, 557–559 State equation, 108, 202, 411, 413 Stochastic approximation, 5 Stochastic dynamic programming, 565 Stochastic programming, 210, 456, 565 Subassemblies, 115–119, 122, 123, 125, 190, 194, 241 Subcontracting, 1, 3 Substitution, 81–87, 89, 91–93, 95, 97, 99–101, 314, 329, 330, 332, 337–344, 348–350, 352, 353, 358–360, 364, 365, 369, 372, 380–382, 384 Supplier behavior, 45, 51, 211 Supply chain management, 45, 46, 106, 107, 124, 157, 291, 292, 294, 298, 301, 305, 309, 404, 410–412, 416, 419 Sustainable manufacturing, 432 Sustainable product development, 429 System dynamics, 43–45, 47–49, 52, 53, 55–62, 66, 75, 77, 257, 287, 352, 494, 565

T Tabu search, 443, 513, 520, 521 Tactical decisions, 50, 106, 411, 416, 420, 421 Technological situations, 133 Theory of constraints, 431, 470 Toyota, 232, 235, 238, 241 Tracking, 283, 295, 309, 376, 410, 414, 496 Transient behavior, 272 Transportation, 49, 111–113, 115–117, 119, 121, 123–125, 127, 128, 160, 176, 190,

587 232, 239, 240, 253, 259, 320, 382, 395, 400, 401, 403, 405, 407, 413, 414, 417, 425, 483, 489, 497 Transshipment, 160, 176, 239, 240 Types of uncertainty, 560, 566

U Uncertainty, 1–13, 47, 51, 72, 77, 83, 113, 138, 147, 148, 208–210, 228, 326, 332, 350, 376, 405, 411, 417, 468, 531–560, 565, 566 Utilization, 1, 17, 18, 20, 21, 29, 44, 54, 55, 62, 71, 73, 76, 133, 240, 262, 278, 316, 360, 431, 498, 499

V Value at risk, 108, 112, 114, 118, 120, 126, 216 Value of information, 19, 112, 114, 153, 155, 170 Variance based designs, 139, 171, 257 Varying efficiency, 557

W Weak sustainability, 564 Work in process (WIP), 1, 13, 16, 106, 136, 139, 141, 256, 264, 272, 468 Workload control, 467 Workload-dependent lead time, 1–13 Workload-independent lead time, 4, 542

Y Yield, 5, 38, 44, 52, 56, 58, 75, 91, 95, 97, 100, 108, 112, 118, 128, 133–135, 138–144, 146, 148, 209, 242, 260, 283, 306, 307, 309, 314, 316, 326, 327, 329–331, 348, 352, 353, 360, 362, 369, 378, 392, 393, 395, 403, 406, 407, 413, 420, 425, 465, 471, 473, 476, 480, 481, 545, 559

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