Optimal Flow Control in Manufacturing Systems

Applied Optimization

Volume 18

Series Editors:

Panos M. Pardalos University 01 Florida, U.S.A.

Donald Hearn University 01 Florida, U.S.A.

The titles published in this series are listed at the end 0/ this volume.

Optimal Flow Control in Manufacturing Systems Production Planning and Scheduling

by

OdedMaimon Department ollndustrial Engineering, Tel-Aviv University, Tel-Aviv, Israel

Eugene Khmelnitsky Department ollndustrial Engineering, Tel-Aviv University, Tel-Aviv, Israel

and

Konstantin Kogan Department ollndustrial Engineering, Tel-Aviv University, Tel-Aviv, Israel

Springer-Science+Business Media, B.V.

A c.I.P. Catalogue record for this book is available from the Library of Congress.

ISBN 978-1-4419-4799-4 ISBN 978-1-4757-2834-7 (eBook) DOI 10.1007/978-1-4757-2834-7

Printed on acid-free paper

Additional material to this book can be downloaded from http://extras.springer.com.

All Rights Reserved © 1998 Springer Science+Business Media Dordrecht

Originally published by Kluwer Academic Publishers in 1998. Softcover reprint ofthe hardcover 1st edition 1998 No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner.

To our lamilies

When a ship does not arrive to a certain port

neither by due-date, nor later,

the Company's Boss exclaims: "The Devil !",

the Admiralty: "My God !"

(loosely translated from "New Jules Veme", a poem by Joseph Brodsky)

CONTENTS

Preface xv

PART I: BASIC CONCEPTS

CHAPTER 1. INTRODUCTION 1.1 Top-Down Production Planning Hierarchy 1.2 Planning and Scheduling Approaches 1.3 Why Specifically Optimal Control ?

1.3.1 Advantages 1.3.2 Limitations

1.4 Outline of the Book Bibliography

CHAPTER2. MATHEMATICAL FUNDAMENTALS OF OPTIMAL CONTROL

2.1 Calculus ofVariations in Optimization Problems 2.1.1 Nature ofDynamic Systems 2.1.2 Modeling ofManufacturing Environment 2.1.3 Optimization of a Single-Time-Step Dynamic System 2.l.4 Optimization ofa Multi-Time-Step Dynamic System with Mixed Controland State Variables Constraints

3

3

4

6

7 7

8

11

11 11 12 13

16

2.2 Maximum Principle Formulation for Optimal Control Problems 25 2.2.1 Statement of the Canonical Form of an Optimal Control Problem 26 2.2.2 Construction of aDescent Variation 28 2.2.3 Necessary Optimality Conditions 39

2.3 Maximum Principle-Based Numerical Methods 2.3.1 Shooting Method 2.3.2 Time-Decomposition Method

Bibliography

46 47

52

x

PART 11: FLOW CONTROL IN PRODUCTION PLANNING

CHAPTER3. ONE-ITEM SINGLE-FACILITY AGGREGATE PRODUCTION PLANNING PROBLEMS 59

3.1 Introduction 59

3.2 Production Smoothing with Capacity Expansion 60 3.2.1 Problem Fonnulation 61 3.2.2 Analysis ofthe Optimal Production Plan 62 3.2.3 Analytical Rules for Managing Production Cases 66 3.2.4 Numerical Algorithm for Arbitrary Demands 68

3.3 Capacity Evolution Planning: Expansion, Deterioration and Selling 70 3.3.1 Problem Fonnulation 70 3.3.2 Analysis ofOptimal Production and Capacity Planning 72 3.3.3 Optimal Control Algorithm for Production and Capacity Planning 77

Bibliography

CHAPTER4. PRODUCTION PLANNING AT DIFFERENT LEVELS OF AGGREGATION 85

4. 1 Introduction 85

4.2 Balancing Facilities in a Make-To-Order Environment 86 4.2.1 Inventory Flow through Distributed Facilities 86 4.2.2 Numerical Algorithm for Balancing Facilities 89

4.3 Dispatching in a Make-To-Stock Environment 93 4.3.1 Dispatching Control Problem 93 4.3.2 Analysis ofthe Optimal Dispatching 94 4.3.3 Analytical Solution Algorithm 97

4.4 Draining Initial Inventories in a Delivery-To-Order Environment 98 4.4.1 Draining Control Problem 98 4.4.2 Optimal Draining 99 4.4.3 Numerical Algorithm for Draining 102

4.5 Dynamic Job Assignment 4.5.1 Formulation ofthe Dynamic Assignment Problem 4.5.2 Optimal Assignment (production) Regimes 4.5.3 Dynamic Job Assignment Numerical Algorithm

Bibliography

xi

105 107 111 113

PART 111: FLOW CONTROL IN SCHEDULING

CHAPTER5. MODELING PRODUCTION SYSTEMS WITH MULTI-LEVEL BILLS OF MATERIALS 119

5.1 Introduction 119 5.1.1 Production Rate Control: Machine Loading and Setup Change 119 5.1. 2 Example of Controllable Production and Setup Rates 120 5.1.3 Known Approaches to Solving Scheduling Problems 120

5.2 Modeling Assumptions 124

5.3 Production Process with Bill ofMaterials 124

5.4 Setup Process 5.4.1 Negligible Setup Times 5.4.2 Significant Setup Times 5.4.3 Sequence-Dependent Setup Times 5.4.4 Special Case ofldentical Machines

5.5 Demand Profiles 5.5.1 Discrete Orders 5.5.2 Piece-wise Continuous Demand Flow

5.6 Types ofSubcontracting 5.7 Relevant Performance Measures

5.7.1 Cost ofProduction 5.7.2 Cost of Setup 5.7.3 Cost of Subcontracting and Purchasing

Bibliography

126

126 126 129 130

132

133 133

133

134

134 135 136

xii

CHAPTER6. NECESSARY OPTIMALITY CONDITIONS FOR SCHEDULING PROBLEMS

6.1 Introduction 6.2 Production Systems with Negligible Setup Times

6.2.1 Flexible Manufacturing Cells with Identical Machines 6.2.2 Individual Machines

6.3 Production Systems with Significant Setup Times 6.3.1 Single Machine 6.3.2 A Generalized Plant with Different Models for Setup Costs

139

139

140 140 149

155

155 167

6.4 Production Systems with Sequence-Dependent Setup Times 177 6.4.1 Partially Sequence-Dependent Setup Times 177 6.4.2 Flexible Cells with Identical Machines 181 6.4.3 Individual Machines 186

6.5 Case Study: Sikorsky Aircraft Corporation's Manufacturing Cell 200 Bibliography

CHAPTER 7. SOLUTION METHODS 207

7. 1 Introduction 207 7.2 Analytical Methods for Scheduling under Constant Demand 209

7.2.1 Limit Cycles 210 7.2.2 Switching Surfaces in State-Co-state Space 214 7.2.3 Approximation ofSwitching Surfaces 217

7.3 Numerical Methods for Scheduling with Negligible Setup Times 220 7.3.1 Straight-forward Time Decomposition 220 7.3.2 Combined Time Decomposition Method for Scheduling under Piece-wise Constant Demand 222 7.3.3 Time Decomposition Method for Discrete Event Control ofIndividual Machines 226

7.4 Numerical Methods for Scheduling with Significant and Sequence-Dependent Setup Times 232 7.4.1 Shooting-based Scheduling Method 232 7.4.2 An Enhanced Modification to the General Shooting-based Method 234

7.4.3 Heuristic Setup Localization 7.4.4 Juggling Strategies

Bibliography

xiii

239 257

PART IV SCHEDULING IMPLEMENTA TION

CHAPTER 8.

8. 1 Introduction

IMPLEMENTATION METHODOLOGY

8.2 Modeling ofManufacturing Systems 8.2.1 Model Classification 8.2.2 Aggregation and Decomposition of Scheduling Models

8.3 Performance Factors of Scheduling Models 8.3.1 Machines and Products 8.3.2 Bill ofMaterials 8.3.3 Setup 8.3.4 Time Variability ofDemand

8.4 Input Data 8.5 Solution Method 8.6 Structure ofOutput 8.7 Discretization and Rounding Bibliography

CHAPTER9. CASES OF PRACTICAL SCHEDULING

271

271

273

273 274

275

276 277 277 278

279

281

282

284

287

9. 1 Introduction 287

9.2 Spectra Final Assembly Area 287

9.3 The Rigid Panel Manufacturing Process at Albany International 296

9.4 Polariod's Battery Production Plant 305

9.5 Avionic Harness Manufacturing System at Sikorsky Aircraft 316

9.6 A Flexible Machine at Remington Arms Co. 324

9.7 Lockheed Martin Company Manufacturing System 330

xiv

9.8 Instrument Panel Manufacturing for F ord Motor 9.9 Conclusion

INDEX

336 343

345

PREFACE

This book presents a unified optimal control approach to a large class of problems arising in the field of production planning and scheduling. It introduces a leading optimal flow control paradigm which results in efficient solutions for planning and scheduling problems. This book also introduces the reader to analytical and numerical methods of the maximum principle, used here as a mathematical instrument in modeling and solving production planning and scheduling problems.

The book examines control of production flows rather than sequencing of distinct jobs. Methodologically, this paradigm allows us to progress from initial assumptions about a manufacturing environment, through mathematical models and construction of numerical methods, up to practical applications which prove the relevance of the theory developed here to the real world. Given a manufacturing system, the goal is to control the production, subject to given constraints, in such a way that the demands are tracked as closely as possible. The book considers a wide variety of problems encountered in actual production planning and scheduling. Among the problems are production flow sequencing and timing, capacity expansion and deterioration, subcontracting and overtime. The last chapter is entirely devoted to applications of the theory to scheduling production flows in real-life manufacturing systems. The enclosed disk provides software implementations of the developed methods with easy, convenient user interface.

We aimed this book at a student audience - final year undergraduates as weIl as master and Ph.D. students, primarily in Industrial Engineering and Management Science. It can also be used in control type ElectricallMechanical Engineering courses. The book is also useful for practicing engineers interested in planning, scheduling and optimization methods. Since the book addresses the theory and design of computer-based production planning methods, applied mathematicians and computer software specialists engaged in developing production control software for industrial engineering and management problems will find the methods developed here very efficient to embed in large applications.

The comments of many colleagues and students have been helpful during the writing of the book and teaching some of the material in undergraduate and graduate courses in the Industrial Engineering Department of Tel-Aviv University. We would especially like to thank our esteemed colleagues Profs. Stanley Gershwin, Michael Caramanis and Alexander Ilyutovich for fruitful and important discussions. We are also grateful to numerous students who initially tested the software. In particular, we thank Evgeni Sheinin and Wei Hua for technical assistance in working with the software.

o. Maimon, E. Khmelnitsky, K. Kogan Tel-Aviv, January 1998