Product warranty handbook 9780824789558, 0824789555

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Product Warranty Handbook edited by Wallace R. Blischke University of Southern California Los Angeles, California D. N. Prabhakar Murthy The University of Queensland Brisbane, Queensland, Australia

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Library of Congress Cataloging-in-Publication Data Product Warranty Handbook / edited by Wallace R. Blischke, D. N. Prabhakar Murthy. p. cm. Includes index. ISBN 0-8247-8955-5 (alk. paper) 1. Quality of products—Handbooks, manuals, etc. 2. Warranty— Handbooks, manuals, etc. I. Blischke, W. R. II. Murthy, D. N. P. HF5415.157.P764 1995 658.5'6—dc20 95-40479 CIP The publisher offers discounts on this book when ordered in bulk quantities. For more information, write to Special Sales/Professional Marketing at the address below. This book is printed on acid-free paper. Copyright © 1996 by Marcel Dekker, Inc. All Rights Reserved. Neither this book nor any part may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, microfilming, and recording, or by any information storage and retrieval system, without permission in writing from the publisher. Marcel Dekker, Inc. 270 Madison Avenue, New York, New York 10016 Current printing (last digit): 10 9 8 7 6 5 4 3 2 1 PRINTED IN THE UNITED STATES OF AMERICA Start of Citation[PU]Marcel Dekker, Inc.[/PU][DP]1996[/DP]End of Citation

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PREFACE Warranty has a very long history, going back several thousand years. Over the last few decades, the study of warranty has received a good deal of attention from researchers and has been pursued from several disciplines. Some of the studies have their origin in practical application; others have been mainly theoretical. A two-way interaction between theory and application is critical for the nurturing and growth of warranty research. Unfortunately for both researcher and practitioner, a survey of the literature on warranty reveals two glaring deficiencies: 1. The vast literature is disjoint, and investigators from a particular discipline are often unaware of the research carried out by those from other disciplines. A large gap exists between researchers from different disciplines. 2. The gap between theory and practice is even larger, with very few papers from practitioners appearing in the open literature. Start of Citation[PU]Marcel Dekker, Inc.[/PU][DP]1996[/DP]End of Citation

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The editors became aware of this during their research and preparation for their first book on warranties (Warranty Cost Analysis, Marcel Dekker, Inc.). This led to a three-part survey paper on warranty management that appeared in the European Journal of Operations Research in late 1992. These papers presented a taxonomy for classifying the different types of warranties and proposed a unified approach to the study of warranty. The genesis of this book was an idea on the part of the editors to attempt to close the two gaps mentioned. The original plan was to have over 45 chapters written by researchers from different disciplines and by practitioners from different industry and government sectors. In due course, the editors realized that they had undertaken a nearly impossible task. Bridging the first gap would require a multivolume handbook with separate volumes for each of the three different types of products—consumer durables, commercial and industrial products, and government acquisition. The warranties and the issues involved are different for each of these categories. Reducing the second gap proved to be even more difficult, for two important reasons: (1) many practitioners are unaware of the theoretical literature and lack the skills to fully understand it and (2) the great majority of practitioners (or their employers) are very reluctant to release relevant warranty information because of commercial sensitivity. In light of this we revised our original plan and decided to focus on bridging the first gap and, as far as possible, the second, in the context of consumer durable products. The book deals with consumer product warranties viewed from different perspectives, with each chapter written by an expert from the relevant discipline. Included are chapters on basic warranty concepts and techniques for analysis; history of warranty; warranty legislation and legal actions; statistical, mathematical, and engineering analysis; cost models; and the role of warranty in marketing, management, and society. In focusing on warranties on consumer products, we are dealing with situations in which there are typically a large number of buyers and few or a modest number of manufacturers. In such cases, individual consumers have no power in setting warranty terms; they are dictated by the manufacturer and market forces. In addition, the buyer has little direct information about product quality and related issues and no opportunity to obtain such information. On the other hand, the manufacturer often has data and tools for analysis of warranty costs (but this information is seldom fully utilized). Because of this inequity with regard to information and analysis, consumer warranties have become a public policy issue. Many of the chapters of the Handbook are somewhat theoretical; however, all deal with application issues as well. As such, this handbook can be viewed as an attempt by theory-oriented researchers to address practiStart of Citation[PU]Marcel Dekker, Inc.[/PU][DP]1996[/DP]End of Citation

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cal issues. In this sense, this compilation differs from many handbooks, which are simply collections of techniques, formulas, and tables meant for the practitioner, that provide little or no depth of understanding. Our intended audience includes both academics and practitioners, including specialists in the many fields in which warranty issues are addressed—engineering, production, marketing, management, accounting, reliability, statistics, consumerism, law, economics, and public policy. It is our hope that those involved in theoretical studies will benefit from the broader perspective gained from the book. We hope that practitioners will also find this book of significant relevance and will, in fact, be motivated to contribute to a future volume that would not only deal with the application of theory to real problems, but also trigger further new theoretical research into warranties for consumer durable products. It is hoped that this book will form the basis for other articles and/or volumes dealing with commercial and industrial products and government acquisition. The aim of the Handbook, then, is to give a broad overview of the study of consumer product warranty carried out by researchers from different disciplines, linking the different research areas. The focus is on concepts to bridge the first gap and guidance to practitioners toward bridging the second. Each chapter concludes with advice and suggestions for practitioners. In addition, areas for future research are indicated. In presenting the results, particularly those of a more theoretical nature, many technical details are omitted. For details, interested readers may consult the references cited by the authors of the chapters. Also, a bibliography on warranties including many additional references, as well as those given in the chapters, is provided in Chapter 33. The text of the book is structured into eight parts (Part A through Part H). Each part deals with a particular aspect of warranty and consists of two or more related chapters and an introduction tying them together. The eight parts are shown in the figure on the following page. The book is aimed at a very wide audience, so paths that may be followed by readers from a few particular perspectives are indicated in the figure. Other suggested paths are indicated in the introductions to each section. Specific parts that would be of particular interest to different professional groups are as follows: Lawyers: Parts B and G Statisticians: Parts C and D Operations Analysts: Parts C and F Engineers: Part F Accountants: Parts G and H Marketing Managers: Parts E and H Public Policy Analysts: Parts B and G Warranty Managers: Parts D, E, and H Start of Citation[PU]Marcel Dekker, Inc.[/PU][DP]1996[/DP]End of Citation

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The diagram shows sequential linkings between different parts. Starting with Part A, one can follow the appropriate path depending on the aspect of interest. Thus, for example, for the Public Policy Analyst interested in "Warranty and Society" (Part G), the recommended sequence is Part A  Part B  Part G. Similarly, a reader interested in "Warranty Management" (Part H) may wish to follow the path Part A  Part D  Part E  Part H (with Part C included as an option) in order to fully understand and appreciate warranty and to manage it effectively. The introduction to each part gives a diagram similar to this one showing the linking between the chapters in the part. These are meant primarily to assist the reader in learning aspects of warranties of particular relevance to a specific information objective. Topics not covered or mentioned only briefly in the Handbook include warranties on commercial/industrial products; warranty in government acquisition; warranties on service; warranties on software; group, fleet, or cumulative warranties; reliability improvement warranties; and multifaceted warranties on complex equipment. In many applications of some of these types of warranties, the rules of the game are quite different; Start of Citation[PU]Marcel Dekker, Inc.[/PU][DP]1996[/DP]End of Citation

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the buyer often has more power than the seller (e.g., the federal government) and may have as much or more information as well. The editors thank the contributors to the Handbook for their high quality technical contributions and for responding positively to the various comments made on their earlier versions and patiently carrying out revisions. The editors have made a significant attempt at ensuring consistency and smoothness in the flow, but in any volume with numerous authors this is a difficult task. It is hoped that despite the differing styles of presentation and levels of theory, the readers, both academics and practitioners, will find the book interesting and useful. WALLACE R. BLISCHKE D. N. PRABHAKAR MURTHY Start of Citation[PU]Marcel Dekker, Inc.[/PU][DP]1996[/DP]End of Citation

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CONTENTS Preface

iii

Contributors

xxi

Part A Introduction 1. Warranties: Concepts and Classification Wallace R. Blischke and D. N. Prabhakar Murthy

5

1.1. Introduction

5

1.2. Definition of Warranty

6

1.3. The Role of Product Warranty

8

1.4. Warranty Structure and Terminology

9

1.5. A Taxonomy for Classification of Warranties

12

References

28

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2. Historical Perspective on Warranty Avrinder P. S. Loomba

29

2.1. Introduction

29

2.2. Origin of the Word "Warranty"

30

2.3. The Early Civilizations

31

2.4. The European Period

35

2.5. The Middle Ages

38

2.6. Industrial Revolution Era and Beyond

40

2.7. Conclusion

43

References

44

3. A Framework for the Study of Warranty D. N. Prabhakar Murthy and Wallace R. Blischke

47

3.1. Introduction

47

3.2. Stakeholders in Warranty

48

3.3. Systems Approach to the Study of Warranty

49

3.4. Simplified System Characterization

49

3.5. Detailed System Characterization (Consumer/ Manufacturer Perspectives)

57

References

72

Part B Warranty and Law 4. Warranty Legislation Craig A. Kelley

79

4.1. Introduction

79

4.2. Warranty Legislation

80

4.3. Literature Review of the Effects of Warranty Legislation

90

4.4. Implications of Warranty Legislation for Business and Consumers

92

4.5. Future Research of Warranty Legislation

94

References

95

5. Warranty and the Courts Rachel S. Kowal 5.1. Introduction

97

97

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5.2. Warranty Litigation Under the Magnuson–Moss Warranty Act

97

5.3. Warranty Litigation Under the Uniform Commercial Code and Other State Laws

116

5.4. Conclusion

123

References

125

Part C Mathematical and Statistical Techniques for Analysis of Warranty 6. Modeling for the Study of Warranties D. N. Prabhakar Murthy

133

6.1. Introduction

133

6.2. Mathematical Model Building

134

6.3. Modeling Based on a Simplified System Characterization (Consumer and Manufacturer Perspectives)

136

6.4. Modeling for Study from Public Policy Perspective

152

6.5. Modeling Based on a Detailed System Characterization

155

References

155

7. Mathematical Techniques for Warranty Analysis Jeffrey J. Hunter

157

7.1. Introduction

157

7.2. Warranty Claims Modeled as Point Processes

157

7.3. Classification of Point Process Formulations

160

7.4. Analysis of Poisson Processes

164

7.5. Analysis of One-Dimensional Renewal Processes

165

7.6. Analysis of Two-Dimensional Renewal Processes

176

7.7. Other Process Formulations Relevant to Warranty Analysis

186

7.8. Conclusions

187

References

188

8. Statistical Techniques for Warranty Cost Analysis Wallace R. Blischke 8.1. Introduction

191

191

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8.2. Probability Distributions and Related Concepts

194

8.3. Parameter Estimation

203

8.4. Confidence Interval Estimation

214

8.5. Estimation of Renewal Functions

221

8.6. Applications in Warranty Analysis

223

8.7. Recommendations and Conclusions

227

References

229

9. Statistical Analysis of Warranty Claims Data J. D. Kalbfleisch and J. F. Lawless

231

9.1. Introduction

231

9.2. Simple Age-Specific Claims Analysis

232

9.3. Adjustments for Reporting Delays

240

9.4. Examples

241

9.5. Adjustments When Sales Are Estimated

247

9.6. Prediction

248

9.7. Covariates and Regression Analysis

250

9.8. Calendar Time Effects

251

9.9. Analysis of Costs

252

9.10. Estimation of Field Reliability

253

9.11. Concluding Remarks

256

Appendix

257

References

259

Part D Warranty Cost Models 10. The Basic Free-Replacement Warranty and Related Rebate

265

Warranties Wallace R. Blischke 10.1. Introduction

265

10.2. Assumed Life Distributions and Warranty Parameters

268

10.3. Modeling Seller's Per-Unit Warranty Cost

271

10.4. Modeling Buyer's Expected Cost of Warrantied Items

279

10.5. Present Value Cost Models for the FRW

282

10.6. Life-Cycle Cost Models for the FRW

283

10.7. Indifference Prices for the Nonrenewing FRW

285

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10.8. Statistical Estimation of Cost Models

287

10.9. Concluding Remarks

288

References

290

11. The Basic Pro-Rata Warranty Jayprakash G. Patankar and Amitava Mitra

293

11.1. Introduction

293

11.2. Modeling the Pro-Rata Warranty

298

11.3. Estimation of Expected Warranty Costs

305

11.4. Results

308

11.5. Conclusions

316

References

317

12. Combination Warranties Wallace R. Blischke

319

12.1. Introduction

319

12.2. Modeling Seller's Per-Unit Warranty Cost

324

12.3. Per-Unit Cost to Buyer

330

12.4. Present Value Cost Models

332

12.5. Life-Cycle Cost Models

334

12.6. Estimating Cost Models

337

12.7. Recommendations and Conclusions

338

References

339

13. Two-Dimensional Free-Replacement Warranties Herbert Moskowitz and Young Hak Chun

341

13.1. Introduction

341

13.2. Two-Dimensional Warranty Policies

343

13.3. Assumptions

344

13.4. Modeling Two-Dimensional Warranties

346

13.5. Cost Analysis of Two-Dimensional Free-Replacement Warranty Policies

347

13.6. Implications for the Manufacturer

349

13.7. Concluding Remarks

357

Appendix

358

References

362

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14. Two-Dimensional Pro-Rata and Combination Warranties Richard J. Wilson and D. N. Prabhakar Murthy

365

14.1. Introduction

365

14.2. Pro-Rata Warranty Policies

366

14.3. Combination Warranty Policies

369

14.4. Model Formulation

372

14.5. Cost Analysis: Pro-Rata Policies

374

14.6. Cost Analysis: Combination Policies

379

14.7. Conclusions

386

References

387

Part E Warranty and the Marketplace 15. Marketing and Warranty V. Padmanabhan

393

15.1. Introduction

393

15.2. The Conventional Wisdom

394

15.3. Theories of Product Warranty

395

15.4. Warranty Provision and Postpurchase Behavior

398

15.5. Quantifying the Warranty Elasticity

399

15.6. Warranty and Distribution Channel Structure

400

15.7. Extended Service Contracts

401

15.8. Warranty and Industrial Goods Markets

402

15.9. Warranty and Packaged Goods

402

15.10. Summary Discussion

403

References

406

16. Warranty and Consumer Behavior: Product Choice Craig A. Kelley

409

16.1. Introduction

409

16.2. Warranties and Product Attributes

410

16.3. Explanations of the Impact of Warranties on Product Choice

411

16.4. Impact of Warranties on Postpurchase Behavior

414

16.5. Directions for Future Research

417

References

418

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17. Warranty and Consumer Behavior: Warranty Execution Jayprakash G. Patankar and Amitava Mitra

421

17.1. Introduction

421

17.2. Model Development

423

17.3. Analysis and Results

431

17.4. Conclusions

437

References

438

18. Extended Warranties V. Padmanabhan

439

18.1. Introduction

439

18.2. The Conventional Wisdom

440

18.3. Research on Consumer Perceptions of EW

441

18.4. Theories of Product Warranty and EW

442

18.5. Recent Institutional Developments

446

18.6. Design of Alternative EW Policies

448

18.7. Conclusion

450

References

451

19. Warranty and Product Distribution Arvinder P. S. Loomba and K. Ravi Kumar

453

19.1. Introduction

453

19.2. Warranty/Servicing Practices in Distribution Environments

456

19.3. Product Distribution Overview

458

19.4. Channel Design for Distribution and Warranty/Servicing

463

19.5. Optimal Distribution and Warranty/Servicing Models

465

19.6. A Numerical Example

472

19.7. Conclusions

475

Appendix

476

References

480

20. Design of Warranty Policies to Balance Consumer and Producer Risks and Benefits Richard Marcellus and Ba Pirojboot

483

20.1. Introduction

483

20.2. The Warranties Considered

484

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20.3. Warranty Design

485

20.4. Risks and Benefits of Offering Warranty

486

20.5. Consumer's Expected Costs Without Warranties

493

20.6. Producer's Expected Profits Without Warranties

493

20.7. Nonrenewing Free-Replacement Warranty

493

20.8. Renewing Free-Replacement Warranty

498

20.9. Renewing Pro-Rata Warranty

503

20.10. Conclusion

508

References

509

Part F Warranty and Engineering 21. Warranty and Design D. N. Prabhakar Murthy

515

21.1. Introduction

515

21.2. Design Process

517

21.3. Reliability Choice and Allocation

518

21.4. Redundancy

526

21.5. Product Development

534

21.6. Other Design Issues

537

21.7. Implementation Aspects

538

21.8. Summary and Conclusions

539

References

539

22. Warranty Analysis for Complex Systems Stefanka Chukova and Boyan Dimitrov 22.1. Introduction

543

543

22.2. Warranty Policies

545

22.3. Modeling Failures (Warranty Claims)

546

22.4. Warranty Cost Analysis: Independent Failures

561

22.5. Warranty Cost Analysis: Dependent Failures

566

22.6. Conclusions and Recommendations

582

References

583

23. Warranty and Manufacturing D. N. Prabhakar Murthy

585

23.1. Introduction

585

23.2. Modeling Product Quality Variations and Control

586

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23.3. Quality Control Models for Type A Nonconformance

592

23.4. Quality Control Models for Type B Nonconformance

599

23.5. Presale Testing

614

23.6. Implementation Aspects

618

References

619

24. Warranty Servicing D. N. Prabhakar Murthy

621

24.1. Introduction

621

24.2. Warranty Reserves (Nonrenewing PRW Policies)

622

24.3. Demand for Spares (FRW Policy)

629

24.4. Demand for Repairs

632

24.5. Repair Versus Replace

639

24.6. Cost Repair Limit Strategy

644

24.7. Some Additional Topics

646

24.8. Implementation Aspects

651

References

652

Part G Warranty and Society 25. The Economic Theory of Warranties Nancy A. Lutz

659

25.1. Introduction

659

25.2. Attributes of Any Economic Model of Warranties

660

25.3. Economic Reasons for Warranties

664

25.4. Conclusion

672

References

672

26. Financial Accounting and Reporting for Warranties Richard A. Maschmeyer and Kashi Ramamurthi Balachandran

675

26.1. Introduction

675

26.2. Financial Accounting and Reporting Requirements

676

26.3. Current Reporting and Disclosure Practices

684

26.4. The Need for Disclosure Reform on Warranties

689

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26.5. Future Research on the Accounting and Disclosure of Warranties

692

References

694

27. Warranties and Public Policy Gregory C. Mosier and Joshua Lyle Wiener

697

27.1. Introduction

697

27.2. Magnuson–Moss: An Overview

699

27.3. Warranties as Signals of Reliability

705

27.4. Warranty Coverage and Quality

710

27.5. Warranty Competition

713

27.6. Summary and Implications

713

References

715

28. Warranty Protection: A Consumerist Perspective John R. Burton

719

28.1. Introduction

719

28.2. The First Consumer Movement

721

28.3. The Second Consumer Movement

723

28.4. The Third Consumer Movement

725

28.5. Consumer Entrepreneurs and the Passage of Magnuson–Moss

734

28.6. Evaluation of Magnuson–Moss from a Consumerist Perspective

737

28.7. Summary of the Evaluation of Warranty Policy

747

28.8. A Consumerist's Suggestions for Improvements in Magnuson– Moss

748

28.9. Possible Future Legislation

751

References

752

Part H Warranty and Management 29. Cost Management Planning and Control for Product Quality and Warranties Richard A. Maschmeyer and Kashi Ramamurthi Balachandran 29.1. Introduction

763

763

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29.2. External Versus Internal Accounting

764

29.3. The Role of Warranties in Product Life Cycles

766

29.4. The Relationship of Product Warranty Cost to Other Quality Enhancement Costs

769

29.5. Strategic Management of Warranty Costs

775

29.6. Estimation Models for Planning and Control

781

References

787

30. Warranty As a Contingent Claim Edward W. Frees

789

30.1. Warranties and Financial Risks

789

30.2. Management Perspective

790

30.3. Actuarial Models of Individual Contracts

793

30.4. Actuarial Models of Groups of Contracts

800

30.5. Summary and Concluding Remarks

801

References

801

31. Forecasting Warranty Claims Jingxian Chen, Nicholas J. Lynn, and Nozer D. Singpurwalla

803

31.1. Introduction

803

31.2. Dynamic Linear Models

808

31.3. Dynamic Linear Models with Leading Indicators

810

31.4. Results of Data Analysis

812

31.5. Concluding Comments

815

References

816

32. Multicriteria Models for Determining Warranty Parameters Amitava Mitra and Jayprakash G. Patankar 32.1. Introduction

819

819

32.2. Multiple Criteria in Modeling

822

32.3. Model Formulation and Application

832

32.4. Conclusions

836

References

837

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Part I Warranty Bibliography 33. Bibliography on Warranties I. Djamaludin, D. N. Prabhakar Murthy, and Wallace R. Blischke

839

33.1. Introduction

839

33.2. Categories for Classification

840

33.3. Classification of References

842

33.4. Warranty References

850

Index

917

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Contributors

Kashi Ramamurthi Balachandran. B.E.(Hons.), M.S., Ph.D.

Depart-

ment of Accounting, Auditing and Business Law, New York University, New

York. New York Wallace R. Blischke, B.S., M.S., Ph.D. Department of Informat ion and Operations Management . University of Southern California. Los Angeles. Cal ifornia

John R. Burlon , B.S., M.A., Ph .D.

Department of Consume r Studies

a nd Family Economics. U ni ve rsity of Utah, Salt Lake City, Utah

J ingxian Chen. D.E., M.S. , D.Se. Institute for Re liability and Ri sk Anal ysis. The George Washington University, Washington. D.C.

Steranka Chuko"a, Ph.D. Department of Science and Mathematics. GM I Engineering and Management Institute, Flint . Michigan Young Hak Chun, Ph.D. Department of Information Syste ms and Decision Science. College of Bu sinc ss Adminislralion . Lou isiana State Uni versity, Balon Rouge . Loui siana xxi

XXII

Conrribu rOfS

Boyan Dimitrov, Ph.D. Department of Scie nce a nd Mathe matics. GMI Engineering and Management Institute , Flint , Mic higan isliana Djamaludin , M.Eng.Sci., Ph.D. Tec hnology Ma nagement Ce ntre. The Un ive rsilY of Quee nsla nd . Bri sbane. Queensla nd . Au srr.:dia Edward W. Frees, Ph.D. School of Busine ss and Depart ment of Statistics, Universit y of Wi sconsi n. Madison. Wisconsin Jeffrey J . Hunter, M.Sc.(Hons.), Ph.D. Department of Statistics. Massey Uni versity. Palmerston North. New Zealand

J. D. Kalbneisch, B.Se., M.Math. , Ph .D. Depa rtment of Statist ics and Actuarial Science, and Dean, Faculty of Mathematics, University of Waterloo. Waterloo. Ontario. Canada Craig A. Kelley, Ph.D. Department of Manage ment. California Slate Universit y. Sacramento. Sacramento, Californi a Rachel S. Kowal, J.D. Department of Accounting, Taxation and Business Law. Leonard N. Stern School of Business. New York University . New York. New York K. Ravi Kumar, Ph.D. Department of Information and Operations Managemenl, University of Southern California , Lus Angeles. California

J. F. Lawless , Ph.D.

Department of Statistics a nd Actuarial Science. University of Waterloo. Waterloo . Ontario. Canada

Arvinder P. S. Loomba, B.Engg., M.B.A. , Ph.D. Departme nt of Ma nage· ment. University of Northe rn Iowa. Cedar Falls. Iowa Nancy A. Lutz, Ph.D. Department of Economics. Virginia Polytec hnic Institute a nd State Unive rsity. Bl acksburg. Virginia Nicholas J. Lynn, B.S., M.S. Institute for Reliabilit y and Ri sk Analysis. The George Washington Universit y. Washington . D.C. Richard Marcellus, Ph.D. Industrial Engineeri ng Departme nt. Northern Illinois Universit y. Dekalb. Illinois Richard A. Maschmeyer, B.S. , M.Ac. , D.B.A. Departme nt of Accounting. School of Bus iness. University of Alaska Anchorage. Anchorage. Alaska Amitava Mitra, Ph.D. College of Business and Departme nt of Management. Auburn University. Auburn. Alabama

CanlTibulon

lIlIiii

Gregory C. Mosier, J .D., Ed.D. Department of Economics and Legal Studies in Business. College of Bu siness Administration . Oklahoma State University, Stillwater, Oklahoma Herbert Moskowitz, Ph .D., M.B .A., B.Sc.CMech.Eng.) Krannert School of Ma nageme nt, Purdue University, West Lafayette. Indiana D. N. Prabhakar Murthy, D.E., M.E., M.S., Ph.D. Technology Manage· ment Centre, The University of Queensland , Brisbane. Queensland, Australia V. Padmanabhan, B.Tech., M.S., Ph.D. Graduate School of Business. Stanford University, Stanford. California Jayprakash G. Patankar, Ph.D. versity of Akron , Akron , Oh io

Department of Management , The Uni-

Ba Pirojboot, M.S. Industrial Engineering Department, Northern Illinois University, DeKalb. lUinois Nozer D. Singpurwalla, Ph.D. Depart ment of Operat ions Research. The George Washington University. Washington, D.C . Joshua Lyle Wiener, Ph .D. Department of Marketing, Oklahoma State University, Stillwater, Oklahoma Richard J. Wilson, B.Sc. Ph.D.(UNSW) Department of Mathe matics. The Unive rsit y of Queensland, Brisba ne, Queensland , Australia

Part A Introduction

, PART A INTRODUCTION

I

CHAPTER 1

WARRANTIES: CONCEPTS AND CLASSIFICATION

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CHAPTER 2 HISTORICAL PERSPECTIVE

CHAPTER 3 FRAMEWORK FOR STUDY OF WARRANTIES

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Introduction

Part A serves as an overall introduction to this Product Warranty Handbook. It provides background information that is useful in understanding the concept of warranty . its history. and some of the methods of analysis used in the study of the warranty process from the many perspectives represented in the Handbook. Chapter 1, writlen by the editors. is concerned with the basic warranty concept. Warranty is defined and its roles in various types of transactions are discussed. There are a large number of possible warranly policies: for illu stration, 35 different policies are defined in this chapter. To provide structure for this wealth of possibilities. a method of classification is proposed. This results in a taxonomy of warranties useful as both an overv iew of the warranty process and a starting point in the analysis of warranties from any of the many perspectives. In Chapter 2, A. P. S. Loomba presents a most interest ing study of the hi story of warranty. beginning in ancient times. Warranty of a sort has been involved as an aspect of trade for a very long lime in certain J

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(',lit A

civilizations. Loomba cites ev idence dati ng back over 4000 years. His survey includes carly civili zatio ns, the Ro man era a nd subsequent European periods. the Middle Ages . and the prc- a nd post-industrial era. According to local expert s o n the history of commerce, this may be the fir st such study uodenaken.

The third chapte r of the Handbook . a lso writlcn by t he cd ilOrs, is concerned primaril y with thc systems approach as a fra mework for the study of warranty . from the point s of view of the stakeholders in the process a nd of the many disciplines in volved in warrant y analysis. Stakeholders include manufac tu rers, consumers. and soc iet y: and the model s used (or analysis from these differe nt perspec ti ves ma y vary both in sl ruc· ture and conten!. A number or s uch models are given. In addition , the models are pre se nted in the context or the man y di sc iplines in vol ved- engineering. production . accounting, marketing, consume rism , and ot hers.

1 Warranties: Concepts and Classification Wallace R. Blischke University of Sou/hem California Los Angeles, Californ ia

D. N. Prabhakar Murthy The University of Queensland

Brisb.me, Queensland, Australia

1.1

INTROOUOION

In loday's markel , product warranty plays an increasingly impo rtant role in both consumer and commercial transactions. The use of warranties is widespread a nd they serve man y purposes, induding protec tion for manufacturer, seller, and buyer ; signal s of qualit y; elements of marketing strategy: factors in dispute re solul ion ; and consideration s in public policy and legislation .

Because of this di vers it y of purpose, warranties have been st udied by researchers in many different di sciplines. including the following: engineering, management , operation s analysis, marketing , economics. law . public administration , consumer affairs. statistics, accounting , and qualit y control. As a result , a vast and disjointed literature on warrant y has developed. The major objectives of this book are to structure and unify the many important results which are scattered throughout this literature and to thereby provide an integrated framewo rk for both the study of warranty 5

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policies from these different perspectives and the application of the results in real-life situations. St ructure for the many important results on warranty is provided in the Handbook by organization of the chaplers according 10 both discipline and area of application . The unifying theme of the book is the systems approach to analysis of warranties. In the analyses. three points of view are recognized : thai of the manufacturer (i ncluding distributor, retailer, and so (orth). that of the consumer (individual. corporat ion . or government agency), and that of society (including legislators, consumer affairs groups, the courts, and public policy decision-makers). In this chapter. the subjec t is introduced by means of a di sc ussion o f the basic warranty concept and presentation of a taxonomy for classification o f warrant ies. Some definitions of warranty and its relation to other concepts are discussed in Section 1.2. Section 1.3 deals with the role of warrant y in consumer. commercial. and government transactions. The discussion he re is introductory in nature and is intended to set the sce ne for the more detailed, in-depth presentations given in later chapters of the H andbook. Sect ion 1.4 deals with the structure of warranty. that is, the various warrant y terms that might be otTered. and some o f the termi nology used in de sc ribing warran ties. Finall y. Section 1.5 is devoted to a warranty taxonomy. There. the most commonly used warranties. as well as many warranties having s pecial or unique feature s, will be ex plicitl y defined and the many types of warranties wi ll be categorized. This will provide an important foundation for further discussion . particularl y for the chapters on cost models and warrant y marketing and management. 1.2

DEFINITION OF WARRANTY

A warrdnty is a seller's assurance to a buyer that a product or service is o r shall be as represe nted, It may be considered to be a con tractual agreement between buyer and selle r (or manufacturer) which is entered into upon sale of the product or se rvice . A warrant y may be implicit or it may be explicit ly stated. In this context, by "seller" is mean! the party responsible for assuring that the warranty term s are met. This is ordinarily the manufacturer of the product o r a dealer or other retail or di scount outlet or the provider of the service . The "buyer" is ordinarily the ultimate paying consumer. In some cases, other parties- for example , an insurer or a n indepe nde nt repair fa cilit y- may be involved as well. In broad terms, the purpose of a warrant y is to establish liabilit y among the:se parties in the event that an item fail s o r i:s unable to pe rform

Warranties: Concepts and Classification

7

its intended function when properly used. or when a service is improperly performed. The contract specifies both the performance that is to be expected and the redress available to the buyer if a failure occurs or the performance is unsatisfactory. In this handbook . we shall be concerned almost exclusively with prodllcl warranties , although warranties on services (for example . auto repairs. home improvements or repairs. delivery services, etc.) will occasionally be mentioned. Furthermore. the emphasis will be on manufactured products (e.g., consumer durables. large purchases suc h as automobiles. commercial purchases such as aircraft. and so forth). for which failures are reasonably well defined and the time at which a failure occurred can be established with a reasonable degree of accuracy . In these cases. the warranty is intended to assure the buyer that the product will perform its intended function under normal conditions of use for a specified period of time. Many of the result s to be presented in the Handbook arc also applicable in situations where failures are not as well defined or easil y detected. for example. warranties on computer software . In such cases. the main difficulty would be in the application of analytical results . such as the cost models . In fact. special care has to be taken even in defining the concept of a failure and "time" between failures. To date. little analysis ofwarranties on software has been done. and these are not covered explicitly in the Handbook . The terms warranty and guarantee are often used synon ymousl y. The distinction is that a guarantee is defined to be a pledge or assurance of something ; a warranty is a particular type of guarantee. namely. a guarantee concerning goods or services provided by a seller to a buyer. Another related concept is that of a service contract or "exte nded service" contract. The difference between a warranty and a service contract is that the latter is entered into voluntarily and is purchased separately-the buyer may even have a choice of terms-whereas a warranty is a part of lhe product purchase and is an integral part of the sale . Service contract s are covered in the Handbook but are not li sted separately in the warranty classification scheme given later in this chapter. At present, nearly everything purchased or leased . whether by an individual. a corporation , or a government agency. is covered by warranty, either express or implied . An express warranty is one whose terms are explicitly stated in writing. whcreas an implit!d warranty is a contract that is automatically in force upon purchase of an item from a purveyor of such goods. Sales of this type are taken to imply ·'merchantability" and "fitness for a particular purpose." The dist inct ion between express and implied warranties is specified in the United States in Article 2 of the

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Un iform Commerc ial Code (Vee). For additio nal details a nd discussion . see Chaplers 4 and 5 of the Handbook a nd Chapter [ of Ref. I. In most of Ihc preceding . Ihc emphasis has bee n o n co n .HW1(" warranties, Ihat is. warranties offered (or implied ) in purchases by indi vidual s o f one or a few of a particular item. for example . a toas ter. te levi sion . sel of dishes. o r pair of replacement automobi le tires. Other transactio ns involving warranties are commercial and government purchases. A commercial transact ion is one in volving two (or more ) private organization s (usuall y corporations) , for example. an auto manufacture r purchasing machine tools. an a ircraft manufacturer purchas ing jet engines. or an airline purchasing jet aircraft. Government purchases are purchases made by a gove rnment age ncy from a corporation . Many of the princ iple s and method s of Ihe Handbook appl y to all three t ypes of transac tions. Ot hers are unique to the partic ular type of application. however. and this will be noted in the remainder of thi s ehapler and in the chapters 10 fo llow. 1.3

THE ROLE OF PRODUCT WARRANTY

Warranties arc an integral part of nearl y all co nsume r and commercia l and many government transactions that invol ve product purchases. In such tra nsactions. warrant ies serve a somewhat different purpose fo r buyer and se ller. 1.3.1

Buyer' s Point of View

From the bu yer's point of view. the main role o f a warrant y in these tra nsactions is pro tectional; it provides a mea ns of redress jf the it em. when properly used, fail s to perform as intended or as specified by the selle r. Specifi call y. the warrant y assures the buyer t hat a fault y it em will e ither be repa ired or replaced al no cosl o r al reduced Cos i . A second role is in fo rmat ional. Many buyers infer that a produc t wit h a relati vel y long warrant y period is a illorc reliable a nd lo ng· last ing product than one with a shorter warranty period . 1.3.2

Seller's Poinl of View

One of the mai n roles of warranty from the se ller's point of view is also protectional. Warrant y terms may and ofte n do spec ify the usc and conditions of use for which the produc t is inte nded and provide for lim ited coverage or no coverage at all in the event of misuse of the product. The se ller may be provided further protection by specifi cation o f requi re ment s for care a nd mainte nance of the product.

Warranries: Concepls and Classification

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A second important purpose of warranty for the seller is promo-. tional. As buyers often infer a more reliable product when a long warranty is offered , this has been used as an effective advertising tool. (See Chap· ters 22 and 23.) This is ofte n particularly important when marketing new and innovative products , which may be viewed with a degree of uncer· tainly by many potential consumers. In addition, warranty has become an instrument. similar to product performance and price, used in competi· tion with other manufacturers in the marketplace . 1.3.3

The Role of Warranty

in Government Contracting

In simple transactions involving consumer or commercial goods. a government agency may be dealt with in basically the same way as any other customer. obtaini ng the standard product warranty for the purchased item. Often. however . the government . as a large entity wielding substantial power as well as a very large consumer, will be dealt with considerably differently . with warranty terms negotiated at the time of purchase rather than specified unilaterally by the seller. The role of warran ly in these transactions is usually primarily protectional on the part of both parties. In some instances . particu larly in the procurement of complex mili· lary equipment. warranties of a certain type playa very different and important role , that of incentivizing the seller 10 increase (he reliabilit y of the ilems after they are put inlo service. This is accomplished by requiring that the contractor service the items in the field and make design changes as failure s are observed and analyzed. The incentive is an increased fee paid the contractor if it can be demonstrated that the reliability of the item has. in fact. been increased . Warranties of this type are called reliability improvement warranties (RIW) and these are not discussed in this handbook. 1.4 WARRANTY STRUCTURE AND TUMINOLOGY

In order to classify warranties, which will provide st ructure to their analysis and to the discussions throughout the Handbook, it is necessary to identify specific warranty feature s. This will be further useful to the practi(ioner in providing a list of the various options that might be available in defin ing product warranty terms . 1.4.1

Basic Consumer Warranty Terms

The first important characteristic of a warranty is the form of payment to the customer on failure of an item. The most common forms arc (I) a

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lump-sum rebate (e.g .. "money-bac k guarantee"), (2) a free replacement

of an item identical duced

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the failed item. (3) a replacement provided at re-

the bu yer. and (4) some combination of the preceding terms. Warrantie s of type 2 are called free-replacement warranties ( FRW) . COS I (0

See Chapter 10 for cost models and additional discussion . For warranties of type 3, the amount of reduction is usuall y a funclion of t he amount of service received by the buyer up to the lime of fai lure . with dec reasi ng discount as time of service increases. This di scounl may be a straight percentage of the purchase price that changes one or more times during the warrant y period (e.g .. .50% discount during the tirst half and 20% discount during the remainder of the warranty period) or il ma y be a continuous function of the time rema ining in the warranty pcriod. The lattcr is called a pro-rata warranty ( PRW) and is disc ussed in more detai l in Chapter I J. The most common combination warranty is o ne that pro vides for a free replacement up to a s pecified time and a replacement at prorated cost during the balance of the warranty period . This is called a combinat ion FRW!PRW and is analyzed in Chapler 12. Note that there arc many va riations of these basic wa rranties and many possible combinations. First, there arc rebate form s of all of these. under which the buye r is given a partial o r full refund. Thi s mayor may not be used toward the purc hase ora new item, at the buyer' s discret ion. Under the nonrebate fo rm s, the buyer doc s not have a choice: the value uf the refund may onl y be applied as a discount on the purchase of a replacement item . Second, in the case of repairable items. the item ma y be repaired rather than re placed or onl y some parts replaced or repaired . Again, this may be Jone at no cost to the buyer o r al it reduced cosl. Warranty coverage may also be li mited in many ways . For example. certain Iypes offai lurcs o r cerlai n part s may be spec ificall y exc luded from cove rage . Coverage ma y include p,trt s and labor or paris on ly. or part s and labor for a portion of the warranty period and paris o nl y thereafter. The variations arc ncarly e ndless . It is very imporcant Chat che exacllerms or a warranty be cardully delineated. particularly when attempting to pre· dict future warranty costs .

1.4.2

Renewing Warranlies

Unde r a rl'llt-h ';lIx warranty. all rcplilccd or repaired items arc l:ov ercd under a warranty that is idcnlicalto that of the original purchased ite m. Under a flolln' IIl' lI"iflK warrant y_ the coverage ex tends on ly over the time n:nmilling in the original warranty period . For example, ir an item is sold wit h an FRW of Ic ngth I ycar and it fail s after 9 month >;, the replace ment

Warranties: Concepts and

C/assificalion

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item would carry a warranty of I year if the FRW were renewing, bUI it would be warranticd for only 3 months under nonrenewing FRW. Under a renewing warrant y, items are replaced or repaired unti l the time between two successive failures is atieast the length of the warranty period. say W . that is, ilems are replaced uOlil one is fou nd lhal has a lifetime of at least W. Under a non renewing warranty. items are replaced or repaired unlil the total service time of the original and all replace ment items is al least W. Note that the rebate form ofa warranty is automatically nonrenewing because no replacement item is supplied . It would, in effect. be renewing. however. if the buyer immediately used the rebate toward the purchase of an ident ical new item . 1.4.3 Warranty Dimension Another important characte ri stic. particularly for purposes of anal ysis. is the dimensionality of a warranty. By this is meant the number of variables specified in the warranty terms . The most common warranty is a onedimensional warranty. and the most commonl y used variable is time from purc hase. For example, a warranty may guarantee that a product be "free from defects in material and workmanship for a period of one year from date of purchase." Other examples of one-dimensional warmntics are those based on usage (e.g .. mile s driven or milime ters of tire wear) or on "cycles" (e.g .. number of take-offs and landings of an airc raft). Higher-dimensional warranties in volve more tha n one characteristic measuring product service. with guaranteed service amounts specified for each. The most common two-dimensional warranty is one based on both calendar time and usage. Thi s is particularly common in the automobile industry, where cars are typically sold with warranties limited by time from purchase as well as mileage (e.g .. 2 years or 24,000 mi les. whichever occurs first). Many other products are sold with warranties of th is type. Three- and higher-d imensional warranties wou ld involve additional service measures. An example would be calendar time. flight hou rs. and la ndings of an aircraft. Because of their increasing complexity. these arc used only in a few speciali zed applications. 1,4.4 Multicharacteristic Warranties Rat her than using multivariate measures of service, man y of the more complex warranties guarantee two or more characteristics. Each characteristic may be quantified by mean s of one or more of the variables di scussed previously. and these variables mayor may not be related . For

W. R. B/iKhke and D. N. 1'.

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MUf{h ~'

example . an engine may be guaranteed to operate wit ho ut failure for a s pecified pe riod of lime a nd to provide a specified level of fuel effic iency. On ce rtain types o f products. separate warranties may be provided o n individual parts or subsystems . Thi s is true. for example, of many appliances, fur example (he;: compressor of a refrigerator or a ir condit ioner carrying different warranty terms from those of rest of the unit . Tele vision picture tubes usually carry a separate warrant y. On complex equipment. this is ncarty always the case. For some of these. where part s or subsystems arc obtained from sources other tha n the equipment manufacturer, the suppliers. themselves, onen warranty the items they suppl y. An example of a consumer warm nty of this type is a n automobi le warrant y. T ypica lly . items such as tires and batteries arc warrantied by the manufacturers o f these items rdther than by the automobile manufacturer.

1.4.5

Other Types of Warranty

A numbe r of different warranties have been suggested in the context of commercial and government purchasing. One of these is the RIW . mentioned prev iously. Reli abilit y improvement warranties are typically multivariate and multiattribute warranties. guaranteeing a numbe r of features in addition to se rvice . For example. RI Ws usually guarantee not just replacements o n failure of a warm nticd ite m but t he mean time to failure of a batc h of items. Another type of warrant y that has been proposed for use in sit uations of this type is the Clwlllfal;ve warra nty. Under thi s warranty. an entire batch of items is guaranteed to provide a specified total amou nt of service. without specifying a guarantee on a ny individual item. For example. rather than guaranteeing that each item in a batch of 100 will operate wi thout failure for 2000 hours. the batch as a whole is guaranteed to provide at least 200.000 hours of service. If afterthe last ite m in the batc h has failed . the lotal service lime is less than 200,(XX> ho urs. items arc provided as spec ified in the warranty (e .g .. free of charge or at pro-rata cost) until such time as the total o f 200.000 hours is achieved. Warranties o f this type are not discussed in the Handbook. 1.5

1.5.1

A TAXONOMY FOR CLASSIFICATION OF WARRANTIES

Basis of Classification

The first criterio n for classification of a warranty to be used in the taxono my to be presented in this section is whethe r o r not the warrant y require s development after the sale of the product. Policies which do not involve

Warranties: Conceplj and Classification

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product developme nt can be furth er divided into two groups-Group A. consisting of policies applicable for single ite m sales. and Group B, policies used in the sale of groups of items (call ed lot or balch sales). This division and the remainder of the taxonomy are shown in Figure 1.1. Policies in Group A can be subdivided into two subgroups. based on whether the policy is re newing or nonrenewing. A furth er subdi vision comes about in that warranties may be classified as "sim ple" or "combination." The free-replacement (FRW) and pro-rata (PRW) policies discussed previously are simple policies. A combinat ion policy is a simple policy combined with some additional feature s. or a policy which combines the terms of two or more simple policies. The resulting four types of policies under category A are labeled A I - A4 in Figu re 1.1.

I

I

I

,

I

SIl1C Ie Item

I

Warranty Policies

Not IR\OOMng Prod"CI Development

I

I I'

I

L

c

c..... p 01 hems

I sm,. I

!

I

CombinalOon

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'-

Pfoduct Development

I

"

I

I Renewine I

1_'·..... 1

I I

LS,,:'"J

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Figure 1. 1

Ta:ltOnomy for warranty policies.

Combinalion

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Lsm--",~ I "

Combination -'

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W. R. Bli5Ch h ' ,md O. N P. Murlhy

Each of these four groupings can be further subdi vided into two subgroups based on whethe r the policy is one dimensional or two (or more) dimensional (nol shown in Figure 1. 1). Examples of policies of both types will be given in Section 1.5 .2. Policies in Group B can be subdividell into (wo categories based on whethe r the policy is "simple " or "combination." These arc labeled B' and 82 in Figure 1.1. As in Group A. B I and B2 can be further subdivided based on whether the policy is one dimensional or two dimensional. Finally. policies involving product development are labeled Group C. Warranties of thi s type are typically part of a maintenance con lraet and are used principall y in commercial applications and governme nl acquisition of large. complex items-for example. aircraft or military equipment. Nearly all such warranties invol ve time and/or some fun ction of time as well as a number of characteristics that may nOI involve lime. for example . fuel efficiency .

1.5.2

Examples of Warranties

A variety of warranty policie s will now be described . Typical products sold with some of (hese warranties will be indicated. The following nota· tion is used: W = length of warrant y pc.fiod C = se lling price of the item (cost to bu yer) X == time to failun: (Iift:lime) of an ilclrI

One-Dimensional Group A Policies Subgroup A I (Nonrenew;llg Simp le PoliciesI

Policie s in this group are vari,ttions of {he basic FRW and PRW . Three versions of the FRW are as follow s: Policy I FREE-REPLACEMENT POLICY (FRW) : The seller agree s to

repair or provide replacement s for failed items free of charge up to a time W from the time of the initial purchase. The warrant y expires al time W afl er purchase . Policy 2 REBA TE FR W: The seller agrees to refu nd an amount nC. where o < 0 < I. if the item fails prior to time W from the time uf

purchase.

W,manties: Concepts and Classification

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Policy 3 REBATE FRW (" MONEY-BACK GUARA N TEE"): The seller

agrees to refund the full purchase price C if the item fail s prior to time W from the time of purchase. The warra nty of Policy I is nonrenewing. Thu s. in the case of nonrepairable items. should a failure occur at age X (with X < W). then the replaced ite m has a warranty for a period W - X-the remaining duration of the original warranty . Should additional fail ures occur, thi s process is repeated until the total service time of the original it em and it s re placement s is a t least W. In the case of repairable items. repairs a re made free of charge until the total service time of the item is at least W . Policy 2 is tec hnically not an FRW because the refu nd is not suffi cient for purc hase of a replace ment. It is included he re because the structu re and analysis of the policy a re bas icall y the sa me as those of the nonrencwing FRW. Policy 3 can be considered an FRW because the bu ye r has the opt ion of using the rebate for purchase of a replace me nt ite m. Typical applications of these warranties are consume r products. ranging from inexpensive items such as photographic fi lm to relati vel y expensive repaimble ite ms such as automobiles. refrige rators. largescree n color' TVs. and so forth . and on ex pensive nonrepairable it ems such as microchips and ot he r electronic compone nt s as well. The basic nonrcnewing PRW is as follows: Policy 4 PRO-RA TA R EBA TE POLIC Y: The se lle r agrees to refund a frac-

tion of the purchase price should the item fail before time W fro m the time of the initial purchase. The buyer is not constrained to buy a replacement item. The refund depe nd s on the age of the item at fai lure (X) and it can be either a linear or nonlinear fun ction of W - X. the remaining time in the warrant y period . This defines a fa mily of pro-rata policies whic h is charac teri zed by the form of the refund fun ction. Three members of this famil y are as follows: Policy 5 LINEA R PRW: The sell er agrees to refund an amount [( W - X)I Wj C should the ite m fail prior to W afte r the time of purchase . Policy 6 PRO PORTIONAL LINEAR PRW: The seller agrees to refund a n amount 10::( W - X )/ WjC. where 0 < Q < I. should the it em fait prior to W after the time of purchase .

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W. R. 81ischke and O. N. P. Murthy

Policy 7 NONLlNEA R PRW: The seller agrees to refund an amou nt l(W X)/W]2C shou ld the item fail prior to W after the time o f purchase.

Policies such as 5 a nd 6 are sometimes offered o n relat ive ly inexpensive nonrepai rable products such as hatteries. lire s. ceramics. and so o n. Poliey 7 feature s a Quadratic rebate fun ction a nd is an alternat ive for which the rebate decreases more rapid ly. Subgroup A2 (Nonrenew;ng Combined Po/iciesJ

Warranties of this type typically fealure term s whic h change one or more limes during the warranty pe riod (e.g., full to lim ited warranty. o r rRW to PRW). The most common policy of this type is

Policy 8 COMBINA TIO N FRWIPRW: The seller agrees to provide a replacement or repair free of charge up to time W, from the initial purchase: any fai lure in the interval W, to W (where W, < W)

result s in a prorated refund . The warrant y does nol renew. The proml io n can be e ither linear o r nonlinear. Again , depe nding on the fo rm of the pror.ilion cost fun ction, we have a fa mil y of combined freereplacement and pro-ra ta policies similar to thaI fo r the PRW . Warranties of thi s type are sometimes used to cover replacement parts or components where the original warranty covers an entire syste m. They are a lso widely used in sales of consumer products. Multistage warranties suc h as the fo llowing are also included in this famil y: Poliry 9 THREE-STAGE WARRA NTY: The se lle r agrees to provide replaceme nts or repairs free of charge li p to liFile W , a ft er init ial

purchase. at cost C if the fa ilure time X is in the interva l (W" W: J, and at cost C 2 if X is in the inte rval ( W!. W I. where 0 < W, < W 2 < Wand C , < C 2 • The warntnty expires at time Wafter purchase. Examples are telev ision sets or a ppliances for whic h full coverage is provided for an initial period a nd only partial coverage (e.g ., some part s and! or labor) in later period s. The geneml form of this type of policy is as follow s:

Warranties: Concept5 and Classificafion

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Policy 10 MULTISTAGE REBATE WARRANTY: The seller agrees to provide replacements or repairs free of c harge up to time WI after initial purchase. at cost aIC(X) if the failure time X is in the interval (WI, Wl ). at cost a lC(X) if X is in the interval (W2 • W3J. and so forth, up to cost akC(X) if X is in the interval (Wk , WJ. where (Xl < (Xl < ... < Ok are the proportions of the current price C(X) at time of failure for failure s in each of k time intervals. An example-a s tand -alo n~ fireplace warrantied for 25 years-is di scussed in Ref. I, Chapters I and 2. Another version of this is a combi nation lump-sum rebate warrant y. under which a refund which is a declining proportion of the original purchase price is given rather than a replacement item or a repair. Thi s policy is given by the following: Policy 11 COMB INATION LUMP-SUM REBATE WARRANTY: A rebate in the amount of (1\ C is given for any item that fail s prior to time WI from the time of purchase ; the rebate is a 2C for items that fail in the interval (WI . W21 . a l C for items that fail in the interval (W2 • WJ 1. and so forth. up to a final interval ( W k _ I . WJ. in which the rebate is akC. with I ~ a l > al > ... > a k > O. Still another policy of this type is a combination lump-sum and prorata refund . given in the following:

Policy 12 REBATE COMBINA TION FRW!PRW: The seller agrees to provide a full refund of the original purchase price up to time WI from the time of initial purchase; any failure in the interval from WI to W ( > Wd results in a pro-rata refund . The warranty does not renew. The following combination policy is a modification of the FRW thai is particularl y appropriate for items which are sold as spares and hence not used immediately after purchase. Policy 13 WARRANTY WITH STORAGE LIMITATION: The policy is characterized by two parameters wand W . with w < W . Let S denote the time. subsequent to its purchase, that the item is kept unused in storage before being put in operat ion and X the time to failure of the item after being put into service. The item, upon failure . is covered under warra nt y only if X < It' and X + S
Wd is replaced at a prorated cost. The proration can be either linear or nonlinear. In either case. the replacement item is offered with a new warranty iden tical to the original one. Policy 17 PARTIALLY RENEWING COMBINATION FRW/PRW, The

seller agrees to provide a replacement free of c harge up to time WI from the time of the initial purchase. Replacement item s in this time period assume the re maining warrant y coverage of the original item . Failures in the interval WI to W( > Wd are replaced at pro-rata cost. Replacement items in this interval are provided warranty coverage identical to that of the origi nal item . Policy 18 PARTIALLY RENEWING COMBINATION FRW/PRW; The

seller agrees to provide a replacement free of charge up to time WI from the time of the initial purchase. Replacement items in this time period are provided warranty coverage identical to that of the original item . Failures in the interval WI to W ( > WI ) are replaced at prorata cost. Replacement items in this interval are covered by pro-rata warranty up to time W from the time of the lasl replacemenl . Policy 16 is the most commonl y used version of the combination warrant y. In multiperiod policies, many add itional versions are possible . Under Policy 17, the warranty renews only in the PRW period. This means that upon failure of an item at, say , x :S WI. a free replacement is provided and this replacement item is covered under FRW until time W, - x. and then under PRW until time W from the initial time of purchase. On the other hand, if the failure occurs at time x', with WI < x' S W . the replacement is provided at prorated cost to the buyer and the warranty

LO

W. K 8fischke and D. N. P_Mvrlh y

begins anew. Policy 18 feature s renewal only in the FRW period and is rare ly used.

Two-Dimensional Croup A Policies In the onc-d imensional case, discussed in the previous subsection . a policy is c haracterized by an interval. called the warranty period. which is defined in terms of a single variable-for example. time. age, usage . In the case of two-dime nsiona l warranties. a warran ty is c haracterized by a region in a two-dimensional plane with one axis represe nting time or age and the o ther representing item usage . As a result. man y different type s of warranties . based on the shape of the warrant y coverage regio n. ma y be defined . Subgroup AI (Nonrenewing Simple Two-Dimensional Polic ies)

The fo llowing are five two-dime nsional polic ies of type A I. The warranty coverage regions in the two-dimensional plane for the se five policies are indicated as shaded regio ns in Figure 1.2. The policies are as foll ows: Policy 19 TWO-DIMENSIONAL FRW: The seller agrees to repair or provide a replacement for failed items free of charge up to a time W o r up to a usage U. whichever occurs fir st. from the time o f the initial purchase . W is called the warranty period and U the usage limit. The warrant y region is the rectangle shown in Figure 1.2(a l.

Note that under this policy, the buyer is provided wammty coverage for a maximum time period Wand a maximum usage U . If the usage is hcavy. the warranty can ex.pire we ll before W, and if the usage is very light. then the warranty can expire well before the limit U is reac hed. Should a failure occ ur at age X with usage Y. it is covered by warranty only if X is less than Wand Y is less than U. If the item is replaced by a new one. the rcpla(;cmenl item is warrantied for a lime period W - X and for usage U - Y. This type of policy is offe red by nearly all auto manufacturers. with usage corresponding to di stance driven . The second two-dimensional Group A I polic y is the following: Policy 20 TWO-DIMENSIONAL FRW: The seller agrees to repair or provide a replacement for failed items free of charge up 10 a minimum time W from the time of the initial purchase and up to a minim um total usage U. The warranty region is give n by two strips. as shown in Figure 1.2(b).

Z1

Warranties: Concepl5 and Class ification

U

::\."""""""""~ ~ ~

~ ~

~"""""""""'~w (b) Po licy 9

w (dJ Policy 11

(d Policy 10

(e' Policy 13

w, Figure 1.2

w,

Warranty regions for two-dimensional policies (horizontal uis: time ; venical axis: usage) ,

11

W R. BliKhke and D. N. P. Murt hy

Note that. under this policy. the buyer is provided warranty coverage for a minimum time period Wand for a minimum usage U. If the usage is heavy. the warranty will expire allimc W. and if the usage is very light. the warranty wi ll expire o nl y when the total usage reaches the limit U. Policy 19 tends to favor the manufaclUrer because it limits the maxi -

mum lime and usage coverage for the buyer. For a bu yer who is a heavy user. the warranty expires before lime W due 10 usage reac hing U. Similarly, for a buyer who is a light user, the warranty expires at time W wi th the total usage below U . In cont rast. Policy 20 fa vors the buyer . Here . a heav y user is covered for a time period W by which lime the usage would have well exceeded the limit V and a light user is covered well beyond W. for the policy ceases only when the total usage reaches V . Thi s implies that in the latter case. the manufacturer has to carry spares or replacement units for a time period we ll beyond W. The following two policies achieve a compromise between these two extremes: Policy 21 TWO·DIMENSIONAL FRW: The seller agrees to repair or pro-

vide a replacement for failed item s free of charge up to it time WI from the lime of the inilial purchase. provided the tolal usage at failure is be low V2. and up to a lime W 2. provided the total usage at fai lure does not exceed VI_ The warra nt y region is gi ven by the region shown in Figure 1.2(c) . Note that under this policy, the buyer is provided warranty coverage for a minimum time pe riod WI and for a minimum usage VI. At the same time. the manufacturer is obliged to cover the item for a maximum time period W2 and for a maximum usage V 2 • The second policy whic h is a compromise between Policies 19 and 20 is the following: Policy 22 TWO-DIMENSIONAL FRW: The seller agrees to repair or pro·

vide replacements for failed ite ms free of charge up to a maximum time W from the time of the initial purc hase and for a maximum tolal usage of V. Let X be the time since purchase and Y the tota l usage at failure. The ilem is covered under warranty if [Y + (VI W)Xl < U. If [Y + (UI W)XI 2: U. the n the item is not covered by the warranty_ The waminty region is given by the triangle shown in Figure 1. 2(d). Note that under this policy. warrant y coverage extends fo r a max imum

W,manl ;e5: Concepfs and Class ification

13

time period Wand a maximum usage U. The time instant at which warranty ceases depends on the usage rate . The following are two-dimensional Group A I policies based on the

PRW , Policy 23 TWO- DIMENSIONAL PRW: The seller agrees to refund to t he buyer a fraction of the original purchase price should the item fail before lime W from the time of the initial purchase and the total usage at failure is below U. The fraction refunded depends o n W - X and U - Y. As in the case of Policy 4, this leads to a family of policies, bascd on thc form of the funct ion which determines the refund. This function can be either linear or nonlinear. Three examples are as follows: Policy 24 TWO-DIMENSIONAL PRW: The seller agrees to provide a refund in the amount of (I - XIW)(1 - YI U)C if the item fails prior 10 time W from the lime of the initial purchase a nd the total usage is less than U. where X is the lifetime of the item a nd Y is the total usage. Policy 2S TWO -DIMENSIONAL PRW: The seller agrees tu provide a refund in the amount of min{1 - XIW, I - YI U}C if the item fa ils prior to time W from the time of the initial purchase and the total usage is less than U, where X is the lifetime of the item and Y is the total usage. Policy 26 TWO -DIMENSIONAL PRW: The seller agrees to provide a refund in the amount of (I - XYI UV]C if the item fail s prior to time W from the time of the initial purchase and the total usage is less than U. where X is the lifetime of the item and Y is the tolal usage . Under Policy 24, the proportion of the purchase price refunded is the product of the proportion of time since the sale to the guaranteed time and the proportion of usage Y to guaranteed usage U. Under Policy 25 . it is the minimum of these two proportions . Under Policy 26, it is 1 minus the product uf these two proportions . The rebate increases substantiall y over these three policies. The warranty coverage region for these three policies is given by the rectangular area shown in Figure 1.2(a). Pro-rata policies for warranty

W. R. Blischke and D. N. P. Murrh y

regions shown in Figures 1.2(b)-1.2{d) are defined similarly. In addition . all of these policies may be fully or partiall y renewing. Because there are many choices for each of these factors. a very large number of policie s of this type are possible . Although few of these have been used in prac tice . this family provides a ric h source of possibilities for the practitioner. Cau· tion should be used. howe ve r, as a careful cost analysis is necessary before such policies are offered. Subgroup A2 (Nonrenewing Combinalion Two-Dimensional Policies)

Many two-dimensional combined versions of the simple warranty policies are possible. The following is one example of thi s type : Policy 27 TWO- DIMENS IONAL COMBINATION FRW/PRW: The se ller agrees to provide replace me nts for failed item s fre e of charge up to a time WI from the time of the initial purchase provided the total usage at failure is below VI. Any failure . with time at failure greater than WI but less than W2 and/or usage at failure greater than V I but less than V 2 • is replaced at a prorated cost. The proration can be either a linear or nonlinear functi on of the warrant y parameters and the age and usage at failu re. The coverage a rea is ide ntified in Figure 1.2(e) . Many other such polic ies are possible . Two-dimensional version s of Group A3 and A4 policies may be defined similarl y. One-Dimensional Group 8 Policies

The policies in thi s group are called cumulative warranties and are applicable onl y when items are sold as a single lot of n items and the warrant y refers to the lot as a whole . The policies a re conceptuall y straightforward e",te nsions of the nonrenewing free-replacement and pro-rata warranties discussed previously. Under a cumulative warranty. the lot of f! item s is warranted for a total time of 11 W. with no specifi c service time guarantee for any individual item . Cumulative warranties would quite clearly be appropriate on ly for com mercial and governmental transactions. as individual consumers rarely purchase items by lot. In fact. warranties of Ihis type have been proposed in the United States for use in acquisition of military equipment. The rationale for such a policy is as follo ws. The advantage 10 1111;: buyer is that multiple-item purchases can be dealt with as a unit rather than having to deal with each item individually under a separate warranty contrac!. The ad vantage to the seller is that fe we r warranly claims ma y be expected because longer-li ved items can offset earl y failures.

25

Warra nties: Concepts and Cla ss ;ficat;on

There are some conceptual difficuhies in implementing cumulative warranties and they have not been widel y used . (See Ref. I.) The policies given below are based on the work of Guin [21 . The following nOlalion will be useful for expressing Ihe terms of these policies:

Xi

=

service life of item i , i

=

I, 2, . .

Subgroup BllCumulalive Simple Policies /

The following two policies are cumulative versions of the FRW. Policy 28

CUMULATIVE FRW: A lot of 1/ items is warranted for a total (aggregate) period of nW. The 1/ items in the lot are used one al a time . If S n < nW. free-replacement items are supplied , also one at a time. until the first instant when the total lifetimes of all failed items plus the service time of the item then in use is at least 1/ W.

This type of policy is applicable to component s of indu strial and commercial equipment bought in lots as spares and used one at a time as items fail. Examples of possible applications are mechanical component s such as bearings and drill bit s. The policy would also be appropriate for military or commercial airline equipment such as mechan ical or electronic modules in airborne unit s. In the following, it is assumed that more than one item is in use at a given time: Policy 29 CUMULATIVE FRW: A lotofn item s is warranted undercumulative warranty for a total period of nW. Of these, k « n) are put into use simultaneously . with the remaining n - k items being

retained as spares . Spares a re used one at a time as failures occur. Upon failure of the nth item , free replacements are supplied as necessary until a total se rvice time of nW is achieved. The follo wing are two cumulative pro-rata warranties : Policy 30

CUMULATIVE PRW: A lot of n items is purchased at cost nC and warranted for a total period nW. The n items may be used either individually or in batches. 5". the total service time. is calculated after failure of the last item in the lot. If S" < 1/W. the

16

W. R. Biischk(' ,md O. N. P. Murth y buyer is give n a refund in the amount of C(II - S"IW ). w here C

is the unit purchase price of the ite m. Rather than wa iting for the last fai lure in the 10 1 before a settlement is made (whic h could occasion:lll y take a ve ry long lime). in the fo llowing policy a settlement is made after eve ry Klh item fai lure . ( K is an integer agreed upon by the buyer and Ihe ma nufa(,; ture r a nd prcrcrabl y is an integral di visor of fl .) Policy 31 CUMU LA TlVE PRW: A 101 o f n items is warranted for a to tal time II W. The lot is di vided inlo s ubsets of size K . Under a c umulati ve pro-rata warrant y. the refu nd to the buyer at the instant of the Klh fai lure in each subset is gi ven by max{O. C(K - S,.../W) }. where SI{ is the sum of the se rvice limes of the K failed items in the subset and C is the unit purchase price of the item. Subgro up 82 (Cumulalive Combination Policies)

Again . man y combinatio ns could be de vised . The fo llowing illu strates the many possibilities : Policy 32 CUMULATIVE COMBINA TIO N FRWIPRW: A lot uf" items is warmnted for a total time of "W. Upon failure of the final item in the group. the total service ti me S" is calculated. If 5" < IIW , . where WI < W is a prespecifi ed age. free replacement s arc provided until a total se rvice ti me of "W I is achie ved . say with it em " + J. Upon failure of thi s ite m, the buyer rece ives a rebate in the amount of C(maxN. (II - S". JIW H]. If "WI < S " < nW_ the buyer rece ives a rebate of C( fl - S,./W).

Cro up C Policies Reliabilit y Improvement Warranties

The basic idea of a reliabilit y improvement warrant y (RIW) is to extend the notion of a bas ic cons umer warranty (usually the FRW) to include guarantees o n the reliabi lity of the item and not jus t on it s immediate o r short-term performance . This is particularly appropriate in the purchase of complex, repai rable equipme nt Ihat is inte nded fo r relative ly lo ng usc . The intent of reliabilit y improve ment warranties is to negotiat e warrant y terms that will mot ivate a manufaclUrer to continue improvements in reliabilit y aCier a product is delivered .

Warranties: Concepf5 and Cla n ificalion

Under RIW, the contractor's fee is based on his abil ity to meet the warranty reliability requirements. These often include a guaranteed MTBF (mean time between failures) as a part of the warrant y cOnlrac!. (See Chapter 6 of Ref. I.) The following will illustrate the concept: Policy 33 RELIABILITY IMPROVEMENT WARRANTY: Under this pol-

icy, the manufacturer agrees to repair or prov ide replacements free of charge for any fa iled parts or units until time W after purchase . In addition, the manufacturer guarantees the MTBF of the purchased equipment to be at least M . If the computed MTBF is less than M , the manufacturer will provide. at no cost to the bu yer. (I) engi neering analysis to determine the cause of failure to meet the guaranteed MTBF requirement. (2) engineering change proposals. (3) modification of all existing units in accordance with approved engineeri ng changes. and (4) consignment spares for buyer use until suc h time as it is shown that the MTBF is alleasl

M 131. The following RIW 14] provides for an initial period during whic h no MTBF guarantee is in etTect. followed by successive periods in which specific improvements in MTBF are required . Policy 34 RELIABILITY IMPROVEMEN T WARRANTY: Undcr this pol-

icy . the manufacturcr agrees to repair or provide replaceme nts for any failed parts or units until time W after purchase. In addi· tion . the manufacturer guarantees the MTB F of the purchased equipment to be as follows: No MTBF is guaranteed unt il time WI after the date of first production delivery ; during the period from WI to W:z after first delivery. the MTBF is guaranteed 10 be at least M I ; from W:z to W). the MTBF is guaranteed to be at least M 2 ; a nd from W) to W . the MTBF is guaranteed to be at least M ) (with 0 < WI < W:z < WJ < Wand 0 < MI < M:z < M 3)' If during any period the MTBF guarantee is nol met , the manufacturer will provide , al no cost to the buyer, engineering changes and product modifications as necessary to achieve the MTBF requirements . A variation of this (5] allows the manufacturer some "frec" fai lures at the oulset:

18

W. R. B/;schke ,md D . N . P. MUrlhy

Policy 3S RELIABILITY IMPROVEMENT WARRANTY: A lot of n items is purchased with indi vidual warrant y periods W. Items which fail prior to W arc repaired or replaced al buyer's expense until k such failures occur. after which the manufacturer will repair or replace failed items unti l each oflhe n items in the 101 and it s replacement s ach ieves a tOlal service time of w.

Cost models for many of the Iype A (consumer durable) warranties li sted in this chapter are given in Part D of the Handbook. Not included are RIW and cumulati ve warranties. For cosl models for these. see Ref. I.

REFERENCES Blischke. W. R .. and Murthy, O. N. P. (1993). Warranty Cost Analysis. Marcel Dekker, Inc., New York. 2. Guin , L. (1984). Cumulative Warranties: Conceptllali;;mion Ilnd Anlllysis. Doctoral Dissertation. University of Southern California. Los Angeles. CA . 3. Gandard , A.. and Rich, M. D. (1977). R elillbililY ImprrJl'emenl WllrfantieJ/m Mililary Pwcurement . Report No. R-2264-A F, RAND Corp .. Santa Monica. CA. 4. Balaban. H. S. (1975) . "'Guaranteed MTU F for Military Procurement. " Pw('. 10th Int . Logislit'S Symp .. SOLE. 5. Kru vand, D. H . t 1987). Army aviat ion warranty concepts. 1987 Pw c. Annllal Reliab. lind Main/ . Symp .. 392-394. I.

2 Historical Perspective on Warranty Arvinder P. S. Loomba University of Northern Iowa Cedar Falls. Iowa

2.1

INTRODUCTION

In order to look at the hi story of warranty, one has to start by looking at the history of trade itself. Trade. one of the oldest and most predominant of human activities. can be defined as "3 business. especially mechanical

or mercantile employment, as opposed to profession, carried on as a means of livelihood or profit. " The concept of trade certainly originated as a vehicle to satisfy human needs for numerou s goods. Primitive man did Ify the easy way of need gratification: with the use of brute force. He took what he wanted from those who had it. Over time . the human population increased considerably and brule force techniques gave way to more con servative trade practices that included various modes of product and service exchange, such as the barter system, product sale and purchase using acceptable monetary unitS. and so on. ThiS evolution in trade practices created a need for some standards . which were developed in some part inadvertently and 29

A p_ 5_Loomba

10

the rest in a systemat ic and preconceived manner. Among these. o ne we ll establis hed trade practice that has been evolved over li me is known as produc t warranty. "Prod uct warranty" can be conside red a s a statement (written and} or oral) about the nalUre or qualit y of the product and it s condition or

what the warrantor (gener.tll y t he manufaclUrer) will do if anyt hing is or goes wrong with the product. A more precise legal definition of product wa rrant y under Magnuson- Moss Warran ty-Fede ral Trade Com mi ssion Improvement Act of 1975 is offered by Kelley in C hapter 4. In rece nt li mes . most ma nufacture rs offer some form of warra nty on their products. usuall y at no cost 10 the co ns umer. However. there are s ubstantial varialions in provi sions for protection. durat io n. and serv ice support manageme nt of such warranty offerings as arc discussed in greater detail by Blischke and Murthy in C hapter I.

2.2

ORIGIN OF THE WORD " WARRANTV"

The concept of warrant y is ancient in origin . At t he ve ry out set o f mankind' s soc ial development. one t ribe traded with another. and the qualit y d isputes that arose were similar to the ones still bei ng debated in prese nt day l.:uurlroOllls. which is di scussed in delail by Kowal in C hapter 5. Before looking at the hi storical perspective of warranty. o ne needs to understand the meaning and origins of the term . The words warranty and ~u/(/rtlntf' f'. known to lingui sts as " doublets ," are derived fro m same original source bUI trave ling to today's E nglis h la nguage by different routes. The origins of the word warranty can be traced bac k to the Old North French word waUl'" and w{lrantie . to the Old High German word II'l'fcnto meaning " protector" (I. 2), During the Midd le Ages. the original expressions used included huc t' x ('o ndici o tll', It'arrllll l iZeH'it, promi",i t , and .\"IIh IlIIi piel'intl 13, p . 1161), Warrant y has many definitions. dcpc mJing o n the;: I,;un!t:xl in whidl it is used , Accordi ng to o ne source: , (i) La w in a cont rac t. a promise or binding statement which is non-essential to the main purpose of the contract. so that a fail ure to honor it does not cause the contract to be ended but may give the ol her part y good rcason to claim damages for breach of warrant y. (ii) insurance , a statement made by the ins ured declaring that facts given by him are t rue and that insurance contrdct may be void if an y of these fac ts prove to be untrue, (i ii ) Commercilil. a promise o r s tatement by the sell er o r the buyer

HiSlorical Perspective on Wa rranl y

31

concerning the quality of goods or their fitn ess for a particular purpose. Without warranty, the goods are being sold on the condition that the seller has no responsibility for any fault s or im perfections in the goods. and the buyer has no right to return them or claim damages or any other remedy .. .. [4] From a legal standpoint , as discussed by Kelley in Chapter 4 a nd by Kowal in Chapte r 5. the issues of negligence, fault. and/or due care are significant in tort liability cases. In contrast. in the case of breach of warranty. these matters are irrelevant. If the goods have some late nt defect. the seller is held accountable even though he did not know of and had no means of ascertaining the existence of the defect. Product warrant y has demonstrated its strategic significance to product manufacturers competing in today's industrial marketplace . To acquire a better appreciation for framework to stud y product warrant ies. as presented by Murthy and Blischke in Chapter 3. it is imperative that we understand the chronological evolution of thi s trdde prdctice . In the rollowing sections , we review the development of warrant y policies and procedures over ti me from a historical perspective. 2.3

THE EARLY CIVILIZATIONS

2.3 .1

The Babylonian and Assyrian Era

Over the course of human civilization, the issue of warrant y has been raised on a consistent basis over a wide variety of products and services ranging from cattle and slaves in the ancie nt times to automobiles , office equipment, and complex weapon systems in modem times. Some of the earliest documented accounts about trade and customer complaint have bee n found on the clay tablets from the Ur III dynasty in Babylonia (2128- 2004 B.C. ). The tablet reads as follows: . .. Tell Ea-nasim : Nanni sends the fo llowi ng message: When you came, you said to me as follow s: " ) will give GimilSin (when he comes) fin e quality copper ingot s." You left then but you did not do what you promised . . . What do you take me for. that you treat somebod y like me with such contempt ? . How have you treated me for that copper? You have withheld my money bag from me in enemy territory; it is now up to you to restore (m y money) to me in full. . . . Take cognizance that (from now on) I will not accept here any copper from you that is not of fine qualit y. I shall (from now on) select and take the ingot s indi viduall y in my own yard, and I shall exercise against you my

32

A P. S. Loomba

right of rejection because you have treated me w ith contempt. ... [5)

One of the first Cases of warranty of fitne ss has been found recorded in the Hammurabic Code (c irca twentie th century H. C. ), dealing wi th prod ucts and services. It provides fo r an eye-!or-llll-eye type of compe nsation . For instance. a house builder. "who has nOI made st rong hi s work" causing the house to collapse thereby killing the owner, is put to death for his negligence. If the defecti ve workmans hip results in the death of t he son of the owner, then the son of the builder was put to death . However. if any slaves gol killed due to same reasons, the builder has to replace them and rebuild the house at his ex.pense 16. pp. 83-951 . The code a lso provided compensat ion in the form of "money-back guarantees" for defects di scovered in a slave (in this case. t he product) after its sale: , , If a man has brought a male or femal e slave and the slave has not fulfilled his month , but the bemlli di sease has fa llen upon hi m. he (the buyer) s hall return the slave to t he se ller and the buyer shall take back the money he paid .. , , 17 , Section 278. p. 67[

And: ... If a man has bought a male or female sla ve and clai ms bee n .17, Section 279 , p. raised. the seller s hall a nswe r t he claim. 67[

Subsequent to the Hammurabi rule, a nother code appeared around the second year of the reign of Ashurbanipal. king of Babylon. O ne frag ment of the tablet containing the code prov ided the following law: . The man. who has sold a fe male slave and has had an objection made concerning her, shall take her back. The seller shall give to the buyer the price named in the deed of sale to it s exact amount. and shall pay half a shekel of si lver for each of the children born to her. , .. 17, Law B. Col. II. 15-23, p. 70J For these codes, the time period within whic h a warrant y claim (on a ckfec(i\'(' slave) could be made varied conside rabl y, During the later Babylo nian and Assyrian periods. clauses were inserted in t he sa le contracts of slaves whic h prov ided up to a month to 100 days for fever or se izures to de velop. But the slaves could be returned at any time on grounds of a .wrlll (v ice) plea [7. p. 234).

Historical Perspective on W.manty

JJ

2.3.2 The Egyptian Era In spite of the highly developed conceptions of ownership and possession. there does not see m to have ex isted a correspondingly deve loped theory of sale. such as the one that ex isted in Rome. One pl alI~ihlc explanation for this comparative indifference for sale contracts under Egyptian law is the ve ry small amollnt of commerce which was transacted. Contracts of sale. as prevalent in Egyptian e ra (5702-3 180 B.C.). were esse ntially unilateral . In contrast to Roman law, where the purchaser took possession, in Egypt the vendor transferred possession (however . in both cases. the emphasis was placed o n the act of o ne part y). Under Egyptian Jaws, the contracting party, whether in a sale or any ot her form of contract, detaches from himself a right and conveyed it to the other party. The vendor is obliged to warrant and defend the tit le whic h he conveyed. An expression in the warranty requiring the seller to defend the transaction against thi rd-party challenges by " clearing/cleaning " is claimed to origins in Asia and was firs t introduced into Egypt ian formula ry in the middle of first millennium B.C. [8, p. 90]. The earliest documents attesti ng the Arabic sa les formu laries can be traced back to about A.D. 7 17-823. The Copt ic arbit ration documents affirm that in the A.D. 730-740. Arab offi cials were arbitrating claims betwee n native Egyptians, which included dispute over title to shares in inherited reside ntial property [9]. 2.3.3

The Ancient Hindu Period

The law of India is founded in the first in stance o n the Vedas, which are essentiall y religious books of the a ncient Hindu period. Closely connected to the Vedas are the Dharma-surra, which form the fi rst order of law books, Which, in turn , are transformed in Dharma-sastras or legal handbooks used in practice. Among these, o ne, Manava-dharma·sastra o r The Ordinance of Manu , seems to have attained spec ial prominence . Manava-dh arma-sastra we re origi nall y transcribed in ancient Indi a around the year A.D. 500. In regard to sales, it provided that goods a nd wares that are damaged, deficient, mixed with another, far away , or concealed should never be sold [10, Section 203, p. 2111. In contrast, the warranty provisio ns which were provided to a di ssatisfi ed buyer were rather limited in nature: Whoe ver feel s regret in this world after buying o r se lling anything within ten days give (back) or take (back) the goods. But after the period of ten days is passed, he may neithe r give them

A. P. S. Loomb

xl

If F(x) is differentiable almost everywhere . then i(x) = dF(x)/dx exists almost everywhere and is called the density function associated with F(x) . Another important associated concept is the failure rate r(x) (associated with a distribution function F(x)J and defined as r(x) = f(x)fF(x) . r(x)3x is interpreted as the probability that the item will fail in (x. x + ax) given that it has not failed prior to x. In other words. it characteri zes the effect of age on item failure more explici tl y than the failure distribution ordensity function . The failure rate r(x), density function f( x ). and distribution F(x) are related to each other as follows : F(x)

=

J -

exp{ -

{~ r(1) dl}

and i (x)

"x)

exp

!-1: "I) dt)

O. N . P. Murth y

138

One can classify the distribution function F(x) into many categories based on the failure rate r{x). The three most important such categories are the following: Definition : F(x) is said to have an increasing failure rate (lFR) if r(x) is increasing in x ~ o. Definition; F(x) is said to have a de('Tt'asing failure raIl' (DFR) if r(x) is decreasing in x 2: O. DejinjJion: F(x) is said to have a ('on slanl failure rate if r(x) is constant for all x 2: O. Some additional categories are the following: Definition: F(x) is said to be new worse Ihan used (NWU) if for all x. )'2: 0

F(x

+

y) 2: F(x)F(y )

F(x) is said to be nell' beller than used (NBU) if fo r all x. )'

DeJinition:

" 0 Definition: F( x) is said 10 be new worse than It sed in t':r:peC"llllion (N W UE) if

has finite mean .... 2: ....F(x) fo r all x

I.

F(x)

2.

I,'" F(n dl

2:

0

Definition : F(x) is said to be nell' better than used in e:cpe('lation (N BUE ) if

I. 2.

F(:c) has finit e mean .... dt .:S ....F(x) for all x

I.."" F(n

2:

0

Definition: F(x) is said to have increa sing f ailure rate u l'('rage OFRA ) if - (l /x) log F(:c) is increasing in x 2: O. Definition : F( x) is said to have decreasing f ailure rare allerage (D FRA ) if - (I /x ) log F (x) is decreasing in x 2: O. For furthe r details and a chain of implications that link these different concepts. see Ref. 2.

Example 6.1 (Ex ponential Distribution ) The density function a nd the failu re rale are given by [ (x) "" A exp( - Ax).

and ~x)

= A

O :S;:c
I impl ies an increasing failure rate (lFR). Example 6.3 (Wei bull Distribution) The di stributio n fun ction a nd the fa ilure rate are as follows: F(x ) = F(x ; A.

1.1)

=

I - e xp( - (.v)IJJ,

0

~ x

< x. 1.1 and A > 0

and

2 .0

1 .5

-f

"-

•

~

~

!' 1.0

~

V-

L 0 .05

o

/V (j

,/

./ o

2

-,

.-, /'

.-.

,

T1mo

figure 6.1

6

•

Failure rate for the Gamma distribution as a function of 13 .

D. N. P. Murth y

/40

Figure 6.2 shows r(x) for different val ues of 13. 13 = I corresponds to constant failure rate ; 13 < I implies OFR and 13 > I implies IFR . The Gamma a nd WeihuJl distributions both reduce to the exponential distribution when ~ = 1. They are the most commonly used di stributions for characterizing time to failure in the black-box approach. In addition. many physical failure mechanisms lead to these distributions as well. Several additional failure distributions are given in Chapter 8. For a comprehensive list of di stributions, see Ref. 3. Many electrical and mechanical products exhibit a failure rate which has a "bathtub" shape (see Refs . 4 and 5) . It is characterized by a decreasing failure rate from zero to some point XI. a nearly constant failure rate over a range XI to X2 and an increasing failure rate beyond X2. as shown in Figure 6.3. Failures during the initial period are mainly due to defective material and/or poor manufacturing processes. In the case of repairable items. such failures are called " teething problems " and may often be fix.ed (i.e., discovered a nd repaired) through some form of test ing program . Failures over the middle period a re due purely to chance and, hence . are nol influenced by age . Finally. failure s over the last period reflect a true

,

.. ,

3

./

V

V /

V . 1,

V

il

..,

,

., o

o

Figure 6.2

,

6

•

Failure rate for the Weibull distribution as a function of 13 .

Modeling for the Study of Warranties

figure 6.3

141

Bathtub failure rate .

aging process which results in the failure rate increasing with age. Note that in some instances XI can be zero and/or equal to x~. Physically Based Models In the physically based modeling approach . one mod els explicitly the mec hanism which causes item failure. See Ref. 6 for more on such models. Two illustrative examples are given below. Example 6.4 (Shock Damage Model) Suppose that an item is subjected to shocks which occur randomly over time . Suppose further that the magnitude of the shock is also a random variable and results in a short-duration stress on the item with the magnitude of the stress related to the intensity of the shoc k. Failure is assumed to occur at the first instant the stress exceeds a critical value. In other words, the item fails at the first time instant the shock magnitude exceeds some critical level. As a result. the time to failure is a random variable . The distribution function for this random variable can be obtained using a model involving a marked point process. where the point process characterizes the occurrence of shocks and the assoc iated mark characterizes the magnitude of the shock. A typical example of a failure phenomenon of this type is an electronic component failing due 10 current surges in a network.

'41

O. N . P. MUrfhy

Example 6.5 (Cumulative Damage Model) Here the item is subjected to shocks as in the previous model. Each shock does a certain amount of damage to the item and the damage is cumulative. The item fail s at the firsllime instant the cumulative damage exceeds some critical value. Here again. the time to failure is a random variable. The distribution function for the time to failure can be obtained using a cumulative point process. Typical examples of item failure due to cumulative damage are crack growth in metals and tears in a conveyor belt. Multicomponent Items

Most items and s ystems are made of morc than one componen t. One

approach to modeling multicomponent item failure is to model each component failure separately e ither as a " black box" o r based o n the ph ysical mechanism causing the failure and then to relate component failure s to item failure . The di stribution for item failure would depend on how the compone nts are interconnected and the eITect of component failures on item failure . When component failures are statistically dependent. one cannot model each separately. In the black-box approach. one needs to model component failures by a multidimensional di stribution fun ction and from this obtain the distributio n of time to item failure. The analysis of such formulation s is. in general, fairl y involved and difficult. Often when a component of a complex item or system fails. it can either induce failure of o ne or more components or cause damage so as to weaken them and hence accelerate their failure . Such types of failure s are termed "failure interactions" Pl. Complex items and systems are built with modular st ructure. with a module being a collection of components. For such ilems. failure of a component results in a module failing l8J. Thus. o ne needs to relate component failure to module failure first and then module failure to item failure in order to obtain the distribution function for it em failure. These models are even more complex and usuall y require computer ana lysis o r simulation to obtain even approximate results. Modeling Failure o f Hems Used Intermittently

Some products are used continuously-clocks. pumps in industrial operations. air conditioning in large. closed buildings. a nd so forth . Ma ny other products are used intermittently or only occasionally (e .g .. a di shwasher in a home . the compressur in it re frigen:llor. an e levator in a building.

Modeling for Ihe Study of W,manties

143

special equipment such as an emergency generator in a hospital. Here . usage and idle periods for the item alternate and the failure rate during usage will ordinarily be different from that when idle . In this case. in order to obtain the distribution function for the time to failure , it is necessary to model the usage pattern . In this case, the usage and idle periods are of random duration and can be modeled by a two-state, continuous·t ime Markov chain (a spec ial type of stoc hastic process). for further details, see Ref. 9. When the duration of each usage is very small in relation to the time interval between usages, one can model usages as rando m points along a time continuum 1101. Ju stification for the Black·Box Approach Although modeling item s in terms of components and component failure in term s of the mechanisms of failure results in models which may be more realistic , this approach is not appropriate for study of warranty for two reasons: I. 2.

The resulting models become extremely complex and very difficult to analyze. The validation of such detailed models requires a large a mount of data a nd , in general , it is very difficull, if nOI impossible, to obtain sufficient data of the types needed for model validation.

Consequently , most of warranty models for analysis are based on the black-box approach. The starting point is the c haracterization of the first time to failure through a d istribution function se lected either on an intuitive basis or the basis of on an analysis of available failure data. Thi s is an approximation to the real world and is in the spirit of model ing, for model selection must be based on a sensible trade-off between the sometimes conflicting factors of realism, complex ity . solvabilit y, and ve rifiab ility, 6.3.2

Modeling First Failure (2-D Formulations)

As in the previous section , one can either take a "black box" or a " physically based" approach to modeling the first failure. The discussion will be confined to the form er case for reasons disc ussed in the previou s subsection . Black-Box Approach Lei (XI. Yd denote the age of an item and its usage at first failure. Because failure occurs in an uncenain manner and the item usage is also uncenain,

O. N. P. Murlh y

'44

XI and y, are non-negative random variables. We can model (XI , Y.) by a two-dimensional distribution F(x, y). defined by F(x. y)

= P{X,

::!!;; X,

Y, ::s: y)

Analogous to the one-dimensional case, under appropriate assumptions on F(x. yl, onc can characterize the age and usage at first failure by a density function [(:c. y) which is related to F( x. y ) by the relation f( x,y

) ~ iJ' F(x, y) iJxiJy

or through a in stantaneous failure rate fix. y) given by

>ix , y)

f(x, y) F(x, y )

where

F(x. y) = P{X, >

:C.

y.

> y}

r{x, y) ax 8y is essentially the probability that the first failure will occur with (X I , Yd E (x , x + ax) x [ yo y + 8y) given that XI > x a nd

Y. >

y.

Allernately, one can model the time to first failure , XI . by a onedimensional distribution function FI(.t) as discussed in Section 6 .3. 1. The ite m usage at first failure is modeled by a conditional distribution function F 2 (y !x) wit h

F,( ylx) ~ P{ Y, " ylX, ~ xl The product of F \( x) and F 2 (y !x) is the two·dimensiona l di stribution fun ction F( x. y ). As Y 1 represent s the item usage at failure. it is reasonable 10 assume that E{ r dx.) is an increasing function of X I ' As a result . one must choose distribution functions F(x , y ) whic h have this pro perty . The following are three such distribution function s. Example 6.6 (Beta Stacey Distribution) The density function I(x . y) for (X\. Y.l is given by bx"b -

f(x.

Y) =

I), - 0,

(yI4l)O, -

1

(x _ ylq,)O, - , exp( _ tla )b

r(a)B«(h. 62 )a4>b O. 0 < y < ( ifx S I

because if the fir st event occurs at x :s r, the n over the interval (x, I ) events are generated by an ordinary renewal process associated wit h the distribution G(x}. Thus .

f o' il + F(t)

+

MIl(1 - x) Jf( x)

I:

dx (7 . I J)

M.(I - x)!(x) dx

which, in the case of an ordinary renewal process, reduces 10 Equation (7. 12), Equation (7. 13) differs from Equation (7. 11), but the equivalence can be establi shed using Laplace transform s and the appropriate form fo r M ",(s ) from Equation (7 . 10) with F(x) replaced by G(x) . Note that , for an ordinary renewal process , the renewa l function M(t) implies complete knowledge of all aspects of the renewal proce ss. This reaso n is contained in Equation (7 . 10) whic h shows that F(t ) is immediately deducible from M (t). As a further illustration, M ( t) contains sufficient information to determi ne all the higher moments of NU). Consider the deri vatio n of M 2(t) '""' £IN(I)J2. From Eq uation (7.3), it can be shown that M 2( t ) =

L

"-,

(2n -

1)f1" I( t)

from which we can deduce, via Laplace transforms , that M 2(t) == M(t) +

210' M(r -

x) dM(x)

169

Mathematical Techniques for Warranty Analysis

The variance of N(t) is given by ~ M,(t) -

var[N(t)1

[M(t»'

Another function of interest is the renewal densit), junction, met), given by dM(t)

m(t)

dt

For ordinary renewal processes , met ) =

f(t) +

f

met - xl/ex) dx

with the Laplace transform given by riI(s)

~

j(~) I - [Is)

The renewaJ density function has an interpretation akin to the intensity function of a nonstationary Poisson process. To order 0(01 ). m(t)ol is the probability that a renewal event occurs in the interval (t. I + otl. Two other Quantities are often of interest. Consider the item currently in operation at time f . The last replacement would have occulTed at time S NIt ) and the next failure will occur at time SNl I) .. I . Thus. AU)

=I

-

S NI t)

and B(I) ;;;; SNW '"

- I

are respectively the age and the excess (or residual) /ife of the item in use at time 1. (AU) is sometimes referred to as the baCKward recurrence time at t and 8(1) as the forward recurrence time at r.) It can be shown (see Ref. 2) that for ordinary renewal proce ss P{ A(t) "

xl

~

{F(t) -

L-'

[I - F(t -

y)1dM( y),

I,

Q !Sx!S t

x>t (7. 14)

P{B(t) "

xl

F(!

+

x) -

L

[I - F(r

+x

- y)J dM(y) .

x

2:

0

(7.15)

These distributions are, in general . difficult to derive explicitly.

J. J.

170

7.5.2

HUnlef

The Renewal Fundion

The renewal funclion plays an important role in warranty analysis but . unfortunately, there are only a few special situations where analytical expressions can be obtained for M(t). As a consequence, invariably some computatio nal procedure is required . We survey, successive ly. asymptotic results. bounds. approximat ions. numerical integration, and simula· tion procedures that have been used to give useful estimates of M (r) . Unless o therwise specified . we lisl results for the renewal function associated with an ordinary re newal process.

Analytical Results We li st exam ples. relevant to warranty anal ysis. where specific mathematical ex pressio ns have been obtained for M (t). Invariably , the procedures used to derive such result s have bee n based on the Laplace tran sform M (s), given by Equation (7. 10). Example 7.1 (Exponential Distributio n) If f(t) ::. Ae - ).,', t > 0, then j(s) == AI( A + s). M( s) = AI s2 impl yi ng thai

t>0

M (t) == At,

as to be expected, as the re newal process is a statio nary Poisson process. with intensit y". (Erlang Distribution with Pa rameters k and ;q }...'-t* - Ie - )"'/(k. - I)L t > 0, then j(s) = 1M}... + s)Jk, M (s) " ''Is{(s + ,,)Ie - "*J. yielding

Example 7.2

If f(t)

""

I*2:-Il- -.

M (t) ~ -At

+ -k

k.

where

a=

j

_

II' } {l - CXp! - A,(1 - &')1} r 1 - 9'

eltp(21Tilk) with i :::: .J=1 (7).

Example 7.3 (U niform Distribution) If f(t) = 1, a :S t :S e + I, then M (t)

~

It18J minU.I, - jtlll ( _ I)i{ t -

2:

2: i_O

} _ I

(i

+

jaW

=

'-"-'C ., "( .c-"--",.)",. I. )

with the conve ntion that the first sum is zero whenever the lower summation lim it exceeds the upper (i.e., I < 6) 18-101 In the special case whe n a == ({51. p. 385),

°

M(t) = e'

Lm(i_ -e-')' "7jI(. -

;_ 0

I , m :S I :S 1m

+

1 (n/ = 0,1,2 •... )

Mathematical Techn iques fOf Warranty Analysis

17'

Explicit expressions for M(I) can also be derived for the cases (I)X 1 = Z" yl''''. where Z" has a positive stable distribution. Y is exponentially distributed, and Zo and Yare independent [Ill. (2) the shifted exponential «21. p. 288), (3) the truncated exponential [12] . (4) mixed exponentials [131. and (.5) renewal processes of phase type (l4J. Asymptotic Results The Elementary R enewal Theorem 0.5] states that lim M(I) = ~ /-+ >
I. If P is a positive integer, the distribution is also called the Erlaflg or Erlangiafl distribution. If = v/2, where v is a positive integer. and A = 2, the distribution is a chi-square (X 2 ) distribution with v degrees Qffreedom. The mean and variance are

a

(8.8) (8.9)

The Gamma distribution is plotted in Figure 8.3 with P = 0.5, 1.5, and 2 and.\ values selected so that ~ = 0.5, 1, and 2. This di stribution is also used as an alternative to the exponential in life-testing applications;

Statistical Techniques for Cosl Analysis

197

"

p- .5

•..

.r.~~::::S:~::::==~"......~....~,.~....~....~,,~....~

,..

" ~

...

".

",,2

"

~=. ,

=4

"

.

.

,

~

Figure 8.2

1.00;

(c) ~ ""

Weibull dist ribution; p

2.00.

2 ,~

,.

0.50, 2.00, 4.00. (a)

~

0.50; (b)

~

w. R. 81ischke

' 98

....,

..

~

,~

,~

. ~=l

••• i

,..'

. 0 ,~

.~

,~

,~

.~

"

50

,...=2

" l!

,~

" "o.

p.,

"

0 0

Figure 8.3

Gamma distribution; 13

1.00: (c ) ,... ". 2.00 .

0.50, 1.50, 3.00. (a) ,...

0.50: (b) JL

199

Slc1tisticai Techn iques for Cost Ana/ysis

its shape is somewhat different from that of the Weibull distribution. For more on applications, see Ref. 4. The Mixed Exponential Distribution The mixed exponential di stribution is a we ighted sum of exponent ials with distinct parameters Al and A! a nd weights p (0 < P < I) and q = 1 - p . The formula is if x < 0 ifX 2: 0

(8. 10)

The mean and variance are. respect ive ly. ~ =

a2 = 4 ( ~1

PAl - I p)q - 2

+ qAl - 1 +

qA2 - 2

(8. 11 )

+

pq(AI - 1 -

A2 - 1)2

(8. 12)

Plots of the mixed ex ponential di stribution with p = 0.05 and AI = = 0.25) and A2 selected so that ~ = 0.5, I. and 2. are given in Figure

8.4. The mixed exponential distribution is used to model situations where a process produces two kinds of item s. each of which has a constant fa ilure rate. Examples are two machines or assembly lines or a single operation that may be in or Oul of control. The lognormal Distribution The lognormal distribution with parameters " ( -Xl < " < x) and 6 ( > 0) is given by

2.'

figure 8.4

Mixed exponential distribution ; P = 0.05 . AI = 4.

W. R.

200

fIx;

~, 9) ; {O 1

xO J 2-rr exp

((lOg x _ ~)') 20 2

Blischke

if x < 0

if x

~

0

(8.13)

The mean and variance are, respectively, j.L :::

eTl •.• •/2

0"2 ::: e2Tl+8\etll -

(8.14) 1)

(8.15)

If X is lognormally distributed . then eX is normally distributed with mean 1'1 and variance 02 • (See Section 8.2.2.) Plots of the lognormal distribution

with e = 0.25. 0.50, and I and T! chosen so that j.L = 0.5. I. and 2 are given in Figure 8.5. Applications in reliability and quality control for manufactured products are discussed in Ref. 4 and in many of the references cited. The distribution is appropriate when failures are caused by material Fracture or breakage. The Inverse Gaussian Distribution

The inverse Gaussian distribution with parameters 11 (> 0) and

e(>

0) is

given by if x < 0 f(x:~,

6)

if x

~

0

(8.16)

Here. 'Tl is a location parameter and lIe is a scale parameter. The mean and variance are, respectively, (8.17) (8.18)

The inverse Gaussian distribution is plotted in Figure 8.6 with Tt = 0.5, I, and 2 and e = I. The distribution is also called the Wald distribution [4] because of applications in sequential analysis. It also has important applications in diffusion processes and has been used as another aIterna· tive life distribution in reliability applications where the previous distribu· tions do not provide an adequate fit 10 data. 8.2.2

The Normal Distribution

The normal distribution with mean ... (- 00 < ...
0), which are also the parameters of the distribution . is given by

(12

Statistical Techn iques for Cosf Analysis

"

20'

0,, 1

" "

"

11-=1

..•

" o. 0

••

0

.. 0.'

",,2

0.' ~ ~

0.'

"

••

u

Figure 8.5

1.00; (el

Lognormal distribution ; 9 ~ 0.25, 0.50. 1.00. (al II-

II- = 2.00 .

=

0.50; (b) II-

202

W.

R. B/ischke

35

" ~

~:.s

"

,.,.

"'

0 o.~

,~

,~

Figu re 8 .& Inverse Gaussian distribution : 6

. [(x.

j.L.

(T

,

)

_

-

1

"

,~

"=

(X 20'2 - 1'>')•

$U cxp

,

.00.

-"'C

< X
. represents the failure rate . Its mean is given by I/x and its variance by l /x2 . Truncated Exponential Distribution

The truncated exponential distribution is similar to the exponential. wit h the exception that time to failure is truncated at some chosen value. T. This may apply to products that undergo preventive maintenance at a certain selected time period . Its densit y functi on is given by f ( t)

'=

1 - ~f '1--"e"-Ci'TT Xe . 0

._

. O.

Wj _ I ~ X ::5 Wi. i = 1•. otherwise

. •k

( 12 .1)

where Wo = 0, W.l W , and the Oi are as indicated in the policy statement. Expected cost s to both buyer and seller can easil y be dete rmined as function s of the expected rebate . This is given by 0::

J

w

£lq(XjJ =

o

•

q( x)dF(x) =

c.L a;[F(W;) ,_I

- F ( W; _ ,)[

( 12.2)

where F( x) is the COF (c umulative di stribution function ) of X . We usc the vector notation 0: = (a' . . .. • a d' andW = (Wi • . .. Wd' . The seller's cost under Policy I is given by C (o: . W) = c .• + q(X ). The expected per-unit cost to the seller is

£le.(a. W)] =

c,

+

c.

•

L a ;[F(W;) i_ I

- F(W;_ ,)]

(12.3)

Example 12. t We consider three different three-stage rebate policies: (I) a 6-month warrant y with full refund on failure during the fir st month . 75% refund of the initial purchase price during the second month , and 50% refund during the remainder of the warranty period; (2) a I-year warrdnty with full refund if a failure occurs in the first month. a rebate of75% of the initial purchase price if a failure occurs in the second month . and a 50% rebate if a failure occurs in the remaining 10 months of the warranty period ; and (3) a 1year warra nty with full refund in the first 2 months, 75% in the third and fo unh months. and 50% during the remaining 8 months. The warranty parameters are k = 3, 0.1 = 1.0.02 = 0. 75. and 0.3 = 0. 50. and the warrant y period s for the three cases are ( I) WI = 0.0833 . W 2 -= 0. 1667, and W -= W } -= 0.5 years, (2) W, = 0.0833, W 2 -= 0. 1667. and W = 1.0 year. and (3) WI -= 0. 1667. W 1 = 0.3333, and W = I year.

JZ6

W , R. 81ischke

We consider IJ. = 1.0, 1.5,2.0, and 2.5 years and assume exponential. WcibuJI . Gamma, and lognormal distributions for time to failure. (See Chapter 8 for formulas a nd discussion of these distributions .) Parameter values for the distributions are as follows: The exponential parameter is determined by the assumed mean , because the parameter is A. ::: 1IjJ.. For the Weibull dist ribution. we use values of the shape parameter of ~ = 0.5, a decreasing fai lure rate (DFR) distribution . and 13 = 2.0, an increas· iog failure rate (lFR) distribution. Parameter values for the Gamma and lognormal distributio ns are chosen to give the same means and variances as those of the correspondi ng DFR and IFR Weibull distributions. (See Chapter 10.) This leads to Gamma shape parameters of 0.2 and 3.66, and lognormal shape parameters ofe = 1.339 and 0.4915. In Table 12 . 1, these distributions and designated Weib I , Weib2. and so on. Subst itution into Equation (12.3) will provide the seller's expected cost for this warranty . The results are given in Table 12. I. The factor tabulated is the multiplier of Cb in the cost model of Equation (12 .3). Thus. for Case I. assumi ng the exponential distribution with f.L = 1.5. the expected cost to the seller is Cs + 0.182cb. The expected cost of an item that is made for S60 and sold for $ 100 under Warranty I is S60 + 0. 182(100) = $78.20. For the DFR Wei bull distribution of the example. the cost is $60 + 100(0.444) = $104.40; for the IFR Weibull , it is $64 .50. For the

Table 12.1

Factors for Calculating Seller's Average COS I of hems Sold Under Warranty 1 Distribut ion

Case I

Case 2

Case 3

" 1.0 L5 2.0 2.' 1.0 L5 2.0 2.5 1.0 L5 2.0 2.'

Exp.

Weib l

Weib2

Gammal

Gamma2

Logo I

Logn2

0.255 0.182 0.141 0. 115

0.510 0.444 0.400 0.368

0.096 0.045 0.026 0.017

0.595 0.55 1 0.52 1 0.499

0.08 1 0.028 0.0 12 0.006

0.372 0.278 0.219 0.178

0.374 0.283 0.227 0. 189

0.572 0.507 0.463 0.429 0.628 0.558 0.5 10 0.473

0.279 0. 150 0.091 0.061 0.298 0. 159 0.096 0.064

0.639 0.593 0.562 0.539 0.676 0.628 0.595 0.570

0.287 0. 146 0.079 0.045 0.300 0. 151 0.08\ 0.046

0.466 0.380 0.32 1 0.277 0.547 0.444 0.372 0.319

0.061 0.012 O.cX)25 0.0006 0.299 0.141 0.06\ 0.026 0.304 0. 141 0.06\ 0.026

0.425 0.3 19 0.255 0.2 12

Combination Warrantie5

327

OFR Gamma. the cost is $115 . 10; for the IFR Gamma. it is $62.80. For the two lognormal examples. the costs are $87 .80 and $61 .20. Note the importance of the distributional assumptions in assessing average costs . (The OFR dist ributions have the same mean and variance . as do the IFR distributions.) 12.2.2

Seller's Cost per Unit Sold Under Warranty 2

Combination warranties which feature free replacement or full refund up to time WI from purchase, followed by linear pro-rata coverage from WI until some later time. W2 = W. are the most common combination warranties. In this section. we consider Warranty 2, which is the rebate form of this warranty . For this warranty. with linear proration. the rebate function is

0 :5 t < WI

c

_ c:(W -

q(t) -

{

W

t)

WI '

O.

WI .:s; I


-,xS)g(S) dS

'sY

+ X,y)g(S) dS

u(,N)f(NI" = >-, x

- ~.

uVN)f(NI" =

>-

N _O

( 13.4)

The classical single-dimensional warranty model can be viewed as a special case or the two-dimensional warranty model in which the restriction on one of the two warranty variables is released. The mathematical form of the single-dimensional warranty model obtained by unrestricting y (i.e., setting y to 00) is u(W)

~

r- ~

Jo

u(rN)f(NjJ.L

= A.. x

+ AyxS )g(S) dS

(13 .5 )

N _O

Based on these mathematical models with specific functional form s and parameter values, the two-dimensiona1 wammty policy will be compared in the next section with the single-dimensional warranty policy for the cases in which a producer is risk neutral or risk averse. The isowarranty cost curve will also be developed and analyzed for the Iwo-dimensional warranty policy.

13,6

IMPLICATIONS FOR THE MANUFACTURER

13.6.1

Numerical Example

Risk Preference Pratt [18] proposed a measure, called absolute risk aversion r(Z}, which captures a decision maker's risk preference with respect to cost Z. If u(Z) is a twice differentiable decreasing utility function, r{Z) is expressed as u"(Z)

~Z) = u '( Z)

(13.6)

where u ' and u" are respectively the first· and the second-order derivatives

J50

H. Moskowitz and Y. H. Chun

orthe utility function u. The relationships between r(Z) and a producer" s risk attitude toward cost Z is summarized as Decreasingly ri sk averse r ' (Z)

0

Constantly risk averse , ' (Z) "" 0

(13.7)

=0

, '(2) > 0

Risk prone r(Z)




117.3)

w

The value of the parameter & in (17.3) is influenced by factor s such as the rate of warranty attrition due to some of the reasons discussed previously . The larger the value of fl. the slower the rate of attrition . In the examples considered. the values orb were selected to be 0.5. 1.0, and 1.5. This allows us 10 gauge the effec t of the parameter & on the expected warranty reserve costs. The effect of the market characteristic s, such as compcl ilive offerings o ver a period of time, can be modeled by such all execulion functio n. A manager could obtain data on the markel share of the company versus that of a competito r over a period of time and e sli male an attrition rate. This could subseq uentl y be used to construct the war· rdn ty execut ion fu nction. The fourth fo rm of the warranty executio n function is shown under WEF 4 in Figure 17.1. Here , the warranty weight fun cl ion has a value of 1.0 up to a I.:t:rtain lillie WI' For Ihis segment of Ihe failure time (r :s: " ' 1 ), il is similar 10 WEF I and WEF 2. For the period after Il'\ . the warrant y execution function decreases exponenliall y, simi lar to WEF 3. The warranty execution func tion is given by I. ( (I _ exp & [

D.

W) ) •

0 " 1" 11 ' \ S

I

W, :s:

II '

(17.4 )

r > ,,·

For some of the reasons described under WEF I and WEF 2. the just ification for full execution up to time 11 ' 1 can be made. After time 11' 1. cuslomers are more prone to switc h brands or lose interest in the prod uct . and the attrition rate increases . The value of &will be influe nced b y sim ilar factors as described under WEF 3. One particular situation that could be modeled by this execution function is the issue of product class and warranty attritio n due to product dissatisfact ion or preference to switch brands. For expensive prod ucts , which offer a linear pro-rata rebate plan. it is possibl e that for failure wi thin the period (0. w\), full exec ution would take place . The utility assoc iated with either getting a cash refund or having the product replaced or repaired far ex.ceeds the discomforts and costs of ex.ecuti ng the warrant y. Now. afte r time 11'1. because of the de-

4]7

warranty and Consumer: Wa rrant y Execution

creased cas h rebate (if the plan offers that) . the customer may be less inclined to exercise the warrant y. A large value of & would impl y a slower rate of attritio n. In the a nalysis. the value of IVI is selected as o ne-half of w. and the values of 3 are 0. 5, 1.0. and 1.5. respect ively. II is possible for the form of reimbursement of the warrant y to influence the warra nty execution function. Depending o n whether the warra nt y gives cash. repai rs the prod uct. o r provides a replace ment fo r the prod uct . the execution fun ction could vary. Fo r insta nce. if the warrant y prov ides ca sh. candidate s WEF I. WE F 2. and WEF 4 might be fea sible executio n functions. Here . more customers would tend to exe rcise their warrant y in the early stages. Warrant y fu nct ions WE F 2. WEF 3. and WEF 4 might be appropriate if the warranty calls for repair of the prod uct . In this situation. some proponio n of the c ustome rs would choose 10 no t exe rcise Iheir warranty because of product dissati sfaction and brand switc hi ng . 17.2.2

Product Failure Distributions

The exponential distributio n is used as one of the fai lure distri butions . It is used to describe the period of useful life of products whe n the failure rate is constan t. For elec tronic components. this might be a feasible dist ri bution . The failure density function for the e xponential distri bution is given by

I(tl

he - ""

( 17.5)

where;\ denotes the fai lure rale and m is the mean life of the prod uct. In the a nalysis. the values of h are selected to be 0.5. 1. 11 . a nd 2. The value of h =:: 1.1 1 yields a mean life of 0.9. This value was selected as a guideli ne fo r comparison purposes as used in other studies \6J . For mecha nical compone nts, however. the un its may become mo re prone to failure with increasing time. This may happen because of deterioration and aging of the product. The failure rate for these products would increase with time . The Weibull distribution is used to model these situations. It s densit y function is given by

f(t) "" O;Ptll - 1 e xp( -

0; ( 11 ) .

0;

> O. I s P
yields a unique failure di stribution. These combinat ions were chose n to provide a comparison of our results with the case whe n there is full execution. 17.2.3

Warranty Rebate Plan

The rebate func tio n considered in Ihis c hapler is the linear pro-rata plan , which is given by s(t) =

17.2.4

C(I - '-). 0" I " {O. w I> W

W

117.7)

Expeded Present Value of Warranty Reserve Costs

For a give n product failure distribution and a warranty execution function . the expected present value of the warrant y reserve costs are found . In determining present values, a continuous form of discounting is used. The expected present value of warranty reserve costs. in general. is calcu lated from the expression ElPV(R)J = N

f'·

s(t)f(t)g(t)e - Ht ... 4», dl

117.8)

The warranty reserve cost per unit (r) can be est imated by determining ElPV(R))IN, from which the warrant y reserve cost pe r un it product price. ric, can be found . We now provide expressions for ElPV(R)] for some selected combinations of the failure distribution a nd the warranty execution function . For some of these. it is diffic ult to arrive at closed-form expressions . Thus. for the analysis section. numerical integration procedures are used . As a baseline case for comparison purposes, ElPV(R» is found for fuJI execution. using gel) "" I for 0 :"i I s: w. The value of the firm 's discount rate plus the inflatio n rate (0 + 0; u < - (I + ~ + '); (I + A) < a, 'Y < - I

o
0; 0 < 6 < + I;

V~ .

(19.2)

Mod.1 CC

Model CC depicts a channel structure with Centralized Distribution and Centralized Service Support. As illustrated in Figure 19.2. this channel structure is comprised of only the manufacturer and the consumer. In this channel environment . the manufacturer centralizes both product distribution and service support functions and deals directly with the end can· sumer.

Here. for a prespecified level of product quality and for a given consumer-demand function, the manufacturer derives optimal values of product price , warranty duration , repair service charge, service-contract

469

Warra nly and Producl D iSlribulion

MANUFACTURER

O(P, T,S, F, l )

CONSUMER

figure 19.2

CC channel structure .

fee, and effective serviceable life span associated with the product. For example , companies such as Xerox Corporation follow the CC channel strategy. Xerox centrali zes its product distribution by selling copiers directly to the consumers through its outlets and exclusive dealerships. It abo centralizes Ihe service support funClion by offering warranty . repairs , and service-contract provisions to the consumer through its service center network . Here, tbe manufacturer wishes to maximize his profit function, OM . which depends on the per-unit profit margin , MM. and the quantity demanded , Q. The per-unit profit margin can be expressed as follow s: Per-unit profit margin = (Per-unit selling price) - (Per-unit manufacturing cost) + (Per-unit service support revenues) - (Per-unit cos! of providing service support). Mathematically speaking. MM

~

(PI - ('+

A

+

I)

(19.7)

(19 .8)

I)

Proof. See the Appendix . 19.5.5

Model DO

Model DO depicts a channel structure with " Decentralized Di stribution and Decentralized Service Support ." As illustrated in Figure 19.3, this c hannel structure is comprised of the manufacturer, the retailer, the servicer, and the consumer. In thi s channel e nvironment , the manufacture r decenlraJizes both product distribution and service support fu nclions. Whereas the retailer manages all product distribution activities. service

MANUFA CTURER

P"(W)

W' (P' , T' , S' , F' , l ' )

SERVlC£A

RETAILER

T"

OIP,l,S, F, l l

P"

S' , F', l'

CONSUMER

Figure 19.3

S ' (W) , f" {W) . l· (W)

DO channel structure.

alP, T.S. F. l)

Warranty and Product Distrjbution

471

support operations is handled by the servicer who deals with the consumer in this regard. The manufacturer, however. is still responsible fo r offering basic warranty on the product to the consumer. Here. for a prespecified level of product quality and for a given consumer-demand function, the retailer derives opt imal value of product 's retail price as a function of product's wholesale price , which is controlled by the manufacturer. In addition , the servicer derives optimal values of repair service charge, service-contract fee, and effective product life span, all as a function of product's wholesale price. The manufacturer, in !Urn , derives optimal value of the product's wholesale price and conveys it back to both retailer and servicer, who then optimize their respective profits . For example, companies such as Dell Computers follow a DO strategy. Although the company distributes it s computer products through local dealers and retailers, it has contracted out its installation and service support to Xerox Corporation (34] . Now, the manufacturer' s profit maximization objective is given by max W .T

n

M

~

~ (9)

(W -

( 19.9)

- R9T]Q

The retailer's profit maximization objective is given by max p

nJ{

(P - W)Q

=

( 19.10)

Finally , the seJVicer's profit maximization objective is given by max n s

S Y .L

~ (~S.(L

- T) + ( I -

~)F

- R9( 1. - T)JQ

( 19. 11 )

Here, the value of Q is specified in Eq. (19.1). The following proposition outlines the optimal pricing and service support policies in a DO channel structure . Proposition 19.2. In a DD channel structure, the optimal pricing and service support policies are given by W. "" (a

P'

+

a + , + 1)~(9) T' ~ nR;;:9('"~-+""'I,,)('-a-'+:-';;p":'+7-'->-'+"-::;,c::c..+"" I)

S'

6R ~(6+>+1)

(19.12) (19.13) (19.14) (19 .1 5)

A. P. S. toomba and K. R. Kumar

472

F'

~yh + I)(P + y + " + 1)'11(9) = ~(I~~~~~~~~~~~~~~--~~~ ~)(P + 1)(5 + y + I)(y + " + IXa + P + y + " + I) (19.16)

- P),(Jl + 'Y + ).. + 1)'€(9) L' = R9(~ + I)h + " + I)(a + ~ + y + " +

( 19. 17)

I)

Proof See the Appendix . 19.6

A N UMERICAL EXAMPLE

The model results can be best illustrated by employing a numerical example. In our example. we con sider the fo llowing values of model parameters :

a = - 2.50. A

=

+ 0 .30.

P = + 0.35, K

= 1.00

X

5

10 16 ,

- 1.25 ,

y =

R = $200.00,

- 1.25,

*

= 0. 75

Also. we assume 1, + 1)1W - ~ (9)J ,,)(P + 1)(6 + , + 1)(, + • + I)

(I

P'I W - '€( 9)] R6(13 + I)("'t + },. + I)

Here , it is interesting to note that optimal value of repair service cha rge. S·. is a function of average repair cost. R . only . On the other hand. the optimal values of p •. T*. F* and L * are all functions of the selling price charged by the manufacture r, W. and product quality, 6. Therefore , after substituting the optimal values of decision variables back in the profit equation, we can rewrite the manufacturer's profit equation as a function of Wand 6: nM( W, 9) ; (

W - '€(9»)

P+

I

Q

The optimal selling price charged by the manufacturer. W. can be determined by solving the following first -order condition:

.nM

;

aw

.!.- (W- '€(9») Q ; aw

a+

0

I

After simplification , we obtain W.

::=

0.,«(6) o: + j3+-y + ).+

Substituting, the value of W from the above equation , the optimal values of decision variables can be rewritten as function of 6: o:2 O. Then the surcharge fora warranty is proporlionalto the length of the warranty. Another possibilit y is to let C A (II') "'" B( w)C,,(O), where B(w ) > I. Then the product cost with warran ty magnifie s the cost without a warranty. The magnification factor. B(w ). should increase as the lengt h of the warranty increases. perhaps tapering otT to a constant .

20.4.3

Producer Risks and Benefits

The producer' s principal ri sk for olTering a warranty is the potential cost for any necessary replace ment s that occur before warranties expire. There may al so be significa nt administrative costs invol ved in maintaining a warranty program. The benefits include protection from litigation through definition and limitation of liabilit y, acquisition of information about the consumers through collection of warranty cards. feedback about qualit y and production problem s through warranty claims . and increased sales volume and market share through increased perceptions of quality and reliability. Issues of this sort are discussed by Ude ll and Anderson (17). Thomas 14J has sugge sted a way to quantify the producer's benefits in life-cycle costing. 20.4.4

The Producer's Cost of Providing a Single Item

Life-cycle costing models require the producer' s amorti zed cost per single item of manufacturing a nd marketing the product . excluding the costs due to replacement ofilcms thai fail while under warranty. This cost is denoted e M ( \1'), with C",(O) denot ing the similar cost for product s offered without warranty . If it is trul y profitable to otTer the warranty, then the producer's peritem cost of production. marketing. and servicing all warrant ies may very well be smaller when the product is offered with warranty than when the product is offered without warranty. This is because out of all the cost s

488

R. Marcellus and 8 . Pifo/bool

involved in offering the product-advertising, marketing, manufacturing, and possible future liabilities-some are fixed. If the sales volume is in· creased. the fixed cost per item will decrease. If there are economies of scale in the manufacture of the item, the variable cost per item will also decrease with increased sales volume. Offeri ng the warranty may increase sales volume and therefore decrease the cost per item of selling the prod· uct. Glickman and Berger {51, Murthy PI, and Blischke and Murthy (3 ] have proposed models or total sales as a function of warranty length . Mitra

and Patankar [8J have proposed a model for market share as a funclion of warranty length . In these models , more attractive warranties increase sales and market share. If the producer' s cost amortized per item is less when the product is provided with a warranty, then CM(w) is less than CM (O). In the example s, the value of CM(w) is set to -y(w)CM(O), with 0 < -y(w) s 1. The n the product cost per item is decreased by a reduction factor -yew) that depends on the warranty length. There is no well-recognized way to construct the function -yew). The actual values of-y(w) should be set by managerial judgment. One possibilit y is to define -yew) in a piecewise fashion , assuming the following: Short warranties, say from time 0 up to I" would not reduce the producer's per item cost at all. 2. Warranties of length I, would reduce the producer's per item cosl by a certain factor , say -Yr . 3. The reduction factor would decrease linearly from "VI to -Y2 for warranties from length t, up to time h. 4. For all warranties greater than h lime units. the reduction factor would be -Y2. I.

Then -yew) can be defined as fo r 0 5

1.0

-yew) =

"v,

1"

(w -

t,)b , Iz

I,

-yz)

for

I, S

for h

:s;

W

< t,

w< h w

An example is shown in Figure 20.1. (This is the function used in Example 20.1.) Another possibility is to use a backward-S-shaped curve such as

489

Warranty Policies: Risks and Benefits

:

10.95

~

~

•g,

09

~

j {0.8S

,

o

,

w (Wamlnt~ r'edooJ

Figure 20.1

.

,

,

Producer's cost reduction factor defined piecewise ,

This function satisfie s -V(O) = 1.0 and decreases toward 'Ym.n as If' increases. Thus. the minimaJ reduction factor is 'Ymin. An example is shown in Figure 20.2. (This is the fun ction used in Examples 20.4 and 20.6.) Of course. if 'Y(w) ;;;;; t . then the producer foresees no particular benefit from offering warra nties. 20.4.5

Warranties Beneficial to Both Consumer and Producer

Sections 20.7 through 20.9 contain conditions that can be used to find warranties beneficial to both consumer a nd producer. The formul as in these conditions are based on the life-cycle model described in Section 20.3. It is assumed that the product is required over a n essentially infinite life cycle. that the consumer will always purchase the same type of warranty from the same producer, that the producer will always be available to replace the product , that there is negligible cost involved in claiming

R. Marce/lus and 8 . PifOibool

490

~

.•

sO.95

1 ~

,

£ 09

~

i ~

EO.8S ~

0.'

,

o w

Figure 20.2

,

,

(W;manly redod)

Producer's cost reduction factor as backward-S shape.

and collecting the warranly, and that [here is negligible lime involved in replacing the product. If Ihe product fail s while the warranty is in force , the producer bears

part or all of the cost of the replace ment . thus reducing the profit for the individuallransaction . For fail ures thai occ ur after the warranty ex pires. the consumer bears the full cost of the replacement. In this case, the producer's profit is the consume r's cost minus the per item cost of offering the product with warrant y. Specifically. the consumer incurs a cost of CA(w) fo r the inilial purchase and for each subsequent purchase resulting from a failure while the warranty is not in effect. Otherwise . for a failure that occurs while the warrant y is in effect. the consumer incurs no cost for the FRW and cost (XIII')C,,(w) for the PRW . The producer's profit is C,,(w) - CM(w ) for a purchase that occurs while the warranty is nol in effect. AI each transaclion that occurs while a warranty is in force . the producer's profit is reduced by CA(I1') for the FRW and by (1 - X I I1' )C,,( w) for the PRW .

Warranty Policies: Risks and Benefits

49 '

A warranty is considered beneficial to the consumer if the life-cycle cost without warranty is greater than the life-cycle cost with the warranty. A warranty is considered beneficial to the producer if the life-c ycle profit without warranty is less than the life-cycle profit with the warranty. Although, in many cases. the decision to purchase with or without warranty is not available to the consumer, it is still advantageous for the producer to market a warranty attractive to the consumer . If consumers recognize a disadvantageous warranty. the product' s reputation will be degraded. with possible reduction in sales volume. In a competitive market. the producer most successful at finding warranties attractive to consumers will gain a competitive advantage. Two ways are presented for evaluating the cost or profit per consumcr-producer rclation ship. They are the long-run-timc-avcragc cost or profit and the expected discounted value with minimal attractive rate of return p. Formally, the long-run-time-average cost or profit is defined as

where Ko is the cost or profit at the initial purchase. N{ T} is the number of replacements that occur from time zero to T. and K 1 • K 2 • K 3 • • . are the costs or profit s that occur at the replacements. The expected discounted value with minimal attractive rale of return p is defined form all y as the expected value of

2: j ..

Kj

exp( - pTj )

o

where Ko is the cost or profit at the initial purchase. To is time zero, K I. . are the costs or profits tha t occur at the replacements. and T 1 • T2 • T) . ... are the times at which the replacements occur. Sections 20.5 and 20.6 give formulas for these two quantities. under the assumption that warranties are not purchased. Each of Sections 20.7 through 20.9 gives formula s for a specific warranty policy, as well as condit ions for warranties beneficial to both consumer and producer . The formulas have two sources. The formulas for long-run-time-average cost or profit are based on the formulas in Ref. 12. The formulas for expecled discounted cost or profit a re based on the formulas in Ref. 13. K 2 • K)..

20.4.6

Using the Inequalities to Design Warranties

Formulas (20. 14), (20.15). (20.24), (20.25). (20.34). and (20.35) contain conditions for warranties benefi cial to both consumer and producer. Each

4.2

R. Marcellus and B. Piroiboo1

of these formulas contains two inequalities involving three expressions. The expression on the extreme right is associated with the consumer and is called the "consumer's curve." The expression on the extreme left is associated with the producer and is called the "producer's curve." The expression in the middle is called the " determining curve:' Warranties acceptable 10 both consumer and producer are those for which the determining curve lies between the consumer's curve and the producer's curve. To find these warranties, the curves can be plolled using readily available software. The desired warranties can then be esti· mated from the graphs, as in the examples. The graphs that accompany the examples were produced via MATLAB (see Ref. 18). 20.4.7

Summary of Notation

To use the formulas presented in the sections below. it is nece ssary to know or estimate the probability distribution of the product's lifetime . Methods for finding or choosing this distribution are presented in Chapter 6 of this volume and also in Chapter 2 of Ref. 3. Some of the formulas require relatively sophisticated functions associated with the probability distribution of the product's lifetime. These are its renewaJ function and its renewal density (the derivative of the renewal function) . These are discussed in Chapter 7 of this volume and also in Chapter 3 of Ref. 3. The following quantities are used in Sections 20.5 through 20.9: X

:;0;

fIx)

z;;

F(x) ...

ElXj ... [(x) == M(w) a

m(w) ,. p ;::;;: C,\(O)

:;0;

C,,(w) ~

C",(O)

EO

Cu(w)

=

the product lifetime (a random variable) the probability density function of the product lifetime the cumulative distribution function of the product lifetime the expected value of the product lifetime the ordi nary Laplace transform of the probability densit y functi on the renewal function corresponding to the distribution of the product lifetime the renewal density corresponding to the distribution of the product lifetime the minimal attractive rate of return (MARR), assumed the same for both consumer and producer the consumer 's acquisition cost for a single item without warranty the consumer' s acquisition cost for a single item with a warranty of length w the producer's cost for providing a single item wit haUl warranty the producer 's cost for providing a single item with a warranty of length II ' , excluding the cost due to replacement of items thai fail while under warranty .

493

W,:uranty Policies: Risks and Benefits

CONSUMER'S EXPECTED COSTS WITHOUT WARRANTIES

20.5

The formulas in this section are to be compared with the corresponding formulas in Sections 20.7 through 20.9. The conSllmer's long -rlln -time-a vera ge cost without warr(lnties is (20. 1)

The consumer's expected discounted cost without warranties is C.(O)

(20.2)

I - l(p)

20.6

PRODUCER'S EXPECTED PROFITS WITHOUT WARRANTIES

The formula s in this section are to be compared with the corresponding formulas in Sections 20.7 through 20.9 . The producer's long-run-t;me-average profit withollt warranties is

120.3) The producer's expected discounted profit withoul warranties is C.(O) - CM(O) I - j(p)

20.7

(20.4)

NONRENEWING FREE-REPLACEMENT WARRANTY

20.7.1

Consumer's Expected Costs

The consumer's long-run-lime -average coS( for nonrenell'ing !rei'-replaeemen, warranties of length w is

II + M (w)]ElX ]

(20.5)

For warranties beneficial to the consumer, this should be less than formula (20.1). A few algebraic steps lead to the following condition for the longrun-time-average cost to be less than the corresponding cost for no warranties: C.(O)

..,.-'''''...., < - I + M(w) C.(w)

(20.6)

The consumer's expected discounted cost fo r nOllrenewillg free-replacement warranties is :

494

R. Marcel/us and B. Piroiboo,

(20.7)

G(w.o)

where G(II'. p) =

J. ~ e - P-~f(x) dx + 10'" f. ~_ ,.

e - plH YI/ ( x) dx m( y ) dy

(20.8)

For warranties beneficial to the consumer, this shou ld be les s than formula

(20.2). A few algebraic steps lead to the following condition for the expected d iscounted cost to be less than the corres ponding cost for no warranties:

- i(o ) C. (O) , '---",,=,::-; < -G(w. O) C.(w) 20.7.2

(20.9)

Producer's Expecled Profit

The producers long-rtln-time-Qverage profit for "onrenewing j ret' -rt'placement warranties of length I\' is C,, ( w) -

[I

C~.,(IV){I + M(wH + M(w)IElX]

(20. 10)

For warranties beneficial to the producer. thi s should be greater than formula (20.3). A few algebraic steps lead to the following condition for the long-run-lime-avemge profit 10 be greater than the corres ponding profit for no warrdnties: (20 . 11 )

The

producer' ~'

expected discollnted profit fo r II ullrelle w illl.( fre t' -

replacement 1I'(lrr(lfIlies is

c..dll') -

C",(II')

I

[I + L'" G(II', p)

e - I" m (x)

d.r] (20 I2J

For warranties beneficial to the producer, this s hould be greater than formula (20.4). A few algebraic s teps lead to the fo llowing condition for the e}l;pccted di scounted profit to bt: grea ter than the I:orresponding profit for no warranties: (20.13)

495

Warranty Policies: Risks and Benefils

20.7.3

Warranties Beneficial to Both Consumer and Producer

For the long-run-t ime-average cosl and profit . a nonrenewing FRW is acceptable to both consumer a nd producer if

C,,(O) -

CM(O)

+

CM(W)


W I .X2 > W l } = F(W , • W 2 ). and the probability of having a claim is I - F( WI. W 2 ). The discussion of these cases continues in Section 22.5.1.

550

S. Chukova and B. Dim ilro v

Example 22.1

Suppose that the freezing and cooling sections of a refrigerator have a bivariate normal distribution of lifetimes Xl and X 2• given by the p.d.L

with ,.... "" 5. ,...2 ;;;:; 6, (11 =- 17 2 ;;:; I. and p ::;;:: 0.6 (p is the correlation coeffi cient between the two lifetimes), The two section s al so have a common 4-year FRW (Policy I), The marginal lifetimes have univariate normal distributions with para meters 1. Then the computational algorithm and formula (22 .25) hold with the recursive expression (22.26) replaced by the new relation [I3J Al

=

f(o. + I);

At

=

r(ak + I) k!

k~1 r(lIo. £.J

~ .. I

+

I)A

I \I.

u o( k - ,,) k - "to'

(22.)0)

Example 22.8 For the data from Example 22.5. assume that any failure of Com pressor 1 accelerates its current time to next failure by a coefficient 131 "" 1.05. whereas failure s of Compressor 2, after being removed, accelerate the time to next failure by a coefficient Ih = 1.1. Under Policy I, with W, = 3 and W 2 = 2.5. we find M1.fJ = 0.0223411;

= 0 .0881935 + 1650M2.fJ(2.5) = M2 •fJ

hence, EC = 2700M r •6 (3) $205.84. Comparing this result to EC = $205.24 in Example 22.5 , we see that the increase is not significant. The reason is that the expected times to first failure of the components are much longer than the given warranty times. For instance . if WI = 12 and W2 = 10, then the result would be EC = 868.12 + 408.63 = $1276.75 for Example 22,S , and EC = 873 .32 + 421.18 = $1294.50 for this example . Thus . the difference between the warranty costs when repairs are "good as new" and when repairs decrease the expected life increase with an increase in walTanty times and with an increase in the coefficient p. which represents the accelerated wearing time. Policy 2 Warranties Under Policy 2 the expected warranty EC =

10 w, q,(t) dFI(t)

COStS

+ EC 2(W 2 )

are given by (22 .31)

where q.(r) is the rebate function for Component I when its actual service time at the time of failure equals I , and EC 2 ( W 2 ) is the expected warrant y cost coming from the second component. Any particular choice of q.(J) and FI(t) determines the contribution of the first component to the overall EC . The contribution of the other component to EC is specified analogously .

S. Chukovd and B. Dimitrov

564

Note . Special attention must be paid if Component I is covered under a PRW . The buyer would like to buy (with the discount) a replacement for a failed Component I in order to keep the system operational. The replacement would come with a new warranty which in no way is related to the original warranty . Example 22.9 Suppose that a car battery is sold under a PRW with WI = 4 years and that the lifetime of the battery has a Gamma distribution wit h parameters ~ :: 0.5 and n = 2.5. Suppose the battery costs ChL "" $90 to the buyer and that the rebate function is proporlio nallinear prorated. given by

The amou nt q,(I) is the discount to the buyer on the purchase of a new battery to replace the failed one. The rest of the car is sold under a FRW. that is, all failures will be removed at the seller's expense, and the car is maintained under minimal repair (with a n estimated cost of e m • l = $280 per claim) for W z = 3 years. The overall claim intensity associated with the rest of this car model is a U-shaped function ,,(I) = (OA)

(10')-,."( 1 - 10')-,.,

which renects more intensive claims for new cars and morc intensive breakdowns toward the end of the car's lifetime (he re assumed to be 10 years). Then the calculations of EC, excluding the seller's cost, give EC :

( ')

f'

• (0.8) 1 - ;;- (90)

+ 280

(0.5)'·' 1'" f(2 .5) dl

f.' (tor""' (I (OA)

36.02 + 411.48 : $447.50 Policy 3 Wa rranties For this case, we again have EC = EC.(W.) + ECz(Wz), with EC;(W;) given by the first term of (22 .3 1). where q;(I) is the rebate function for Component i, i = 1, 2. The sum will have more terms if more independent components are involved .

565

Warranty Analysis for Complex Systems

Example 22.10 A battery and four new tires are mounted on a car. The battery is sold under a FRW with the parameters given in Example 22 .9. The tires are sold with a PRW of Wz = 60,000 miles each. Wearing " time" is expressed in terms of the mileage that a tire can serve in various environmental condition s. Suppose that this is a normally distributed random variable X with mean ... = 60,000 a nd standard deviation a = 5000, that the rebate funct ion for tires is

and that the seller's cost per tire is Cs • l = $32. Then the expected warrant y cost associated with this purchase is (using the first cost component of EC for Example 22.9)

J

60 000

EC = 36.02 + 4 {32 +

x exp( -

(I -

o

•

50(1 -

6O,~)5000IJ27r

60,000)') dt )

2(5000f

= 36.02 + 4132 + (1.6623)1 = 36.02 + 134.65 = $170.67 22.4.2

Renewing Warranties

The cases considered above are extended in a natural way to renewing warranties if these are offered for some components. The only difference is in the expressions for calculating the contribution EC;{W i) of the ith component to the overall warranty cost. We obtain the following: For the FRW EC,( W,)

=

(

C. .. 1

+ 1

F,(W ,))) F

(22.32)

,(W,

where C.; is the seller's repair/replacement cost per component of type i . For the PKW EC;( W,)

=

(c." + f "" 0

q,(t) dF,(I)

)(

1

+1

F,(W,) ) F ,( W,)

(22.33)

566

S. Ch ukova and B. Oimifrov

Example 22.11 Suppose that a new battery and a new engine have been mounted o n a used car under renewing PRW and renewing FRW . respectivel y (Policy 2' ), For the engine, the warranty period is W2 : 2 years. An engine costs the service s tation Ss.2 '= $950. and it is kno wn that for this kind of e ngine , the lifetime is no rmally distributed with a mean of 4 years and a standard deviation of 2 years. The batte ry lifetime and warranty parameters are as in Example 22.9. Assume that a battery costs the seller Cs . 1 :: $40, Then . using the result s of Example 22.9 and (22 .32) and (22.33). we have

EC ~ (40 + 36.02) ( I + I

+ 950 ( I + I ~

76.02(2 .494)

G(4; 2.5 . 0.5) ) G(4 ; 2.5. 0.5)

(2 - 4)/2 ) (2 4)/2

+ 950(1.1886)

~

190.02 + 1149.18

~

SIll9. 19.

Here G( x; a. A) is the incomplete Gamma fun ct io n. and.p(·) is the standard normal c.d.r.

22.S WARRANTY COST ANALYSIS, DEPENDENT FAILURES 22.5.1

Joint Lifetime Distribution known

Policy 1 Warranties Suppose that an item consist of two nonrepairablc components and that each component is replaced by a new one without changing the joint life distribution. Then the bivariate renewal process {Nz(O . N 2 (t)} must be known in order to anal yze the system , Under Policy I. the overall EC will again be represe nted by Eq . (22.24) , The only diffe rence is that now M;{ t} =- E{N;(/)} must be calculated for dependent renewal processe s.

Example 22.12 Suppose the two components have a MO' SVE distribution with parameters AI. and An· Then {N 1 (t), Nz(t)} is the bivariate Poisson proce ss, given in (22.9), with the specific values of the probabilities q i} as given there . Mo re specifically . we now have

"'2.

567

Wilrfanty Analysis fOf Complex Systems

Thus.

EC .::

Ci s

WI()'I

+

A. 12 )

+ C 2s W2(A.2 +

(22 .34)

A.12)

where C. is the seller's cost per compone nt i. i .:: 1. 2. For thi s example , consider the followin g situation . A computer (Component I) and a monitor (Componenl 2) are sold under a FRW for WI = 2.5 years and W2 = 3 years respect ivel y. Their joint distribution is MO'BVE with parameters A.J = 0.2. A.2 "" 0. 15. and A.12 "" 0. 1. The seller's cost per computer is C is = $750 and per monitor is C2s = $350. The total se ller's cost is C s == $ 1100 per set. The expected warranty cost per set sold. from (22 .34) is EC

750(2.5)(0.2 + 0. 1) + 350(3)(0. 15 + 0. 1)

= 562.5 + 262 .5 = $825 .00 Policy 3 Warranties Consider a two-compone nt , no nrepairable syste m where the jth compo· nent has a PRW for a period Wi. The rebate function q(1 l . ' 2) gives the buyer's discount if Component I fails at time I I and Component 2 fails at time 12. Constructio n of q(1 1' 12 ) reflects a possible interpretatio n of the bivariate warrant y: ( WI . W2) is understood as the amount of service guaranteed to be provided by each component. The n {XI ::: I I. X 2 ::: ' 2} is equivalent to the amount of service provided by lime (II . ' 2). Thi s inlerpre· tation justifies the following form s of (he rebate functi on: _ min(r ,. W,

o5

t i, j =

Wd)

min(t2.

W,

W2»)

. (22.35)

1, 2

0'

_ min(t ], WI) min{12, WI Wz

o .s: I i , i

= I, 2

W2») ' (22.36)

or some derivatives of these form s . Other forms of Ee( WI . W 2 ) depend on the claim pattern. There are two cases: component failures are "visible" (i .e .. detectable) or " invi si· ble "; see Section 22.3.2. Moreover. there are two claim possibilities:

568

S. Chukova and B. Oimilrov

First. Ihe claim may be for refund of the component that has just fail ed. (In Ihis case, qi(li) could be the jlh component of the sum on the righlhand side of (22 .35), where ebi is the buyer' s cost for that component). Note that here the use r has to buy a replacement to keep the system running. but thi s is of no relevance for compu ting the expected warranty cost. Second, the claim is fo r refund orthe entire item . This is delennined by (22 .36), where the amount of work (II. 12) must be established at the time of clai m. Cbs is the buyer's cost for the system . In this way . the Ee can be classified as one of the follo wing forms: (i )

The claim is only for the component that fail s fi rst. The ot her componenl will be claimed only if it fails later within its warranty . Then

+

LW'(J~n(W,. ,}q2(12) + Qt(l IHf(I I"l) d,l ) d12 (22.37)

(ii )

At least one failure must be in warrant y limit s. At the time of first failure , if it occurs within warranty limits , the e ntire item will be claimed . The warrant y of the nonfailed component will be no longer be valid . Then ECl

:::

EC(W •• W 1) =

+ (W' ( ( '"

Jo

foW'(f~n(" . w:Iq('lo t,)/(II" 1)dI2) dll

Jm;n( w,. ,,)

q(l20 i Z)/(II "l )dl. ) dr,

where q(t" ' 1 ) can be either (22 .35) or (22.36). (iii) Invisible failures. A claim arises only when bot h component s fai l within the assigned warranty. We assume that the service time o f eac h compone nt can be precisely evaluated . Then

Ee) = Eq W, • Wz)

= foWlfow, q(l, . 12)/(110 12) dl,

dl z

(iv) In visible failures . Both components must fail as in (iii) . A refund is claimed for the e ntire item. Then

569

warranty Analysis for Complex Systems

+

L (L w,

min(W,. t"

q(t~, (2)!(I \ , 12) dl\ ) dl 2 H

Other form s may arise in practice. Policies 3' and 3 require considerabl y more complicated analyses. Example 22.13 Consider the two-component refrigerating system from Example 22.1 . In addition. let us assume that Component 1 (t he free zing) and Component 2 (the cooling section) cost Cb r = $640 and Ch2 = $380, respectively , and [hat the whole system costs Cbs == $900. Thus, for a PRW wi th WI = 4 and W2 == 4, the following rebate functions could be proposed:

q n(/"

(2) == q,(t\ )

q(I" I,)

~

+

900 [1 _

q 2( t 2)

(m;n~" 4»)(m;n~"'»)l

Warranty claims under the form s discussed above will then be as follow s: (i)

If each component is claimed separately, then by (22 .37), we obtain

EC,

~ I: (f [q,( x) + q, (Y)]!(X,Y) dY) dx + ~

(ii)

fo4(f'"

[q2( Y)

+

q,(x )Jf(x, y) dx)dY

13.49 + 0.65 = $14.14

If a claim is made for the entire syste m at the time of first failure of either of its components , then

EC,

~ I: ( f q(x,X)!(X,Y)dY)dX + I: U:q(y ,Y)f(x. Y) dX)dY ~

19.06

+

1.26

~

$20.32

S. ChukoVd and B. Dim;frov

570

(iii)-(i v)

If a claim is made for the system only when both compone nts fail within the assigned warrant y. then (for this particular exa mple)

EC, = EC. = f U: q(X, .')f(X, Y )dY) dX

+

f

U:q( Y,)')f(X, Y)dX) dy

= 0.81 + 3. 29 = $4 . 10 Example 22.14

A computer .''''>'''' W,)

(1

+ 11. ' 2

(A,

+

(22.38)

A( 2) W,

Similarl y. EC\2' . the warranty cost for Component 2. is given by (22.38) with A,. WI . and C b l replaced by 11. 2• W2 • a nd C"2. respeclivel y. Finally, a third cost componcOl EC\(2) ari ses from a simultaneous claim for both component s. This occurs if a shock from source 3 wilh inlensity 11.12 comes fi rsl and destroys bot h. Therefore , if WI < W2 and A = 11.1 + 11.2 + 11. 12• we have

(22.39)

With the numerical data for the case considered . from (22.38) and (22 .39), we o btain Eel' /)

EC I

'"

=

""

1200 (0. 2) 0.2 + 0.1

(I _ (0.2 + 0. 1)(2.5)

1 _ e - \O.2 +O.U12.S))

$23 7. 1~

1 _ e - IO.U +O.110)) 450 (0. 15) ( 1 0. 15 + 0.1 (0. 15 + 0. 1)(3)

EC, U2) = (1200 + 450) 0.2 + -

O~'I~

e - (Q-4~ )(2.s,)

(450)(0. 1)

+

(0.45)(2.5)

_ 1_

+ 0. 1 0.45

e -(O_4~)(3 - 2 .j ))

(0.45)(3

2.5)

$188 .88

$80.05

572

S. Chukova and 8. Oim;!rov

Therefore, Eel = EC\IJ + Ed2) + EC\12I = 237.19 + 80.05 + 188.88 ~ $516.12.

Oi)

Now lei us assume that each failure within the assigned war· mnly causes a claim for the entire item. Then, after similar considerations of the shock model as in 0), with the rebale function

we obtain 2 5

EC/ U

=

10

.

(O .45)q, (x, x)e - O . t} = ex p( - Jo' I p,r,(x)

+

P2h(X)] dX)

(22.50)

The expected Cosi is st ill given by (22 .40) . but with expected operating costs until Ihe first syslem fai lure . now given by Ihe expression Col e)

=

ICm,(i - P .)T,(t)

x

ex~ -

L

+

Cm2(i - P2)r2( e)]

(PIT, (X )

+

P2r2(X)}

dx)

(22.51)

578

S. Chukova and 8 . Oimirrov

The long-run expected maintenance costs per unit time can be calculated by use of the expression

Er ""

{c, + ( '" IC", .(1 X ex p(

-

L

[P l r.(x)

pdT. ( I)

+

+ C m2 (l

P zTz(x)]

- P2}r2 (t )]

dX)dt} ~y

(22 .52)

In some cases, the two component s will have separale warranlies . Under Policy I. with WI < Wz • EC(W 1 • W2) will bcobtained by combining (22.40) and (2 2.5 1). Thi s yields EC(W •. W2) = (CI"! +

x IM y ( W1 ) +

em1 L"" (I

- p z) r z( t)G(r)

dl)

My(Wdl + C M y( Wd

-

fow'" + My( W.

(22. 53)

- /)JCo(l)dl

Here (22.50) . (22.51), and (22.53) ha ve to be explicitl y specified fo r any particular probability distribution. Example 22.18 Consider the system of Example 22 . 16 under the assumptio ns thaI failure s of the picture lube do not lead to fai lure of the entire system but that they may cause simu ltaneous failure of the se rvice system with probability P2 ::= 0.4. Failures in the se rvice system can cause failure of the pic ture tube with probabilit y PI = 0. 1. The cost coefficients. warranty periods. probability distribution s. a nd numerical values of their parameters a re as in Example 22. 16. Suppose that repair of the pict ure tube costs an average of em:> =:: $120 and replacement by a new tube costs Cr :> " $ 150. From the expone ntial and Weibull assumptions. the fai lure rates of the:: twu components are found to be . for I ~ O.

rdO

=::

11.1 = 0.3;

r 2(1) = Qh"I', - 1 =

(2.5)(O.2)u, u.

r

> O. (22. 54)

The lime 10 fai lure Y is determined by the survival fun ction Ott) = e - o.IJJt (, - U.410 .2tl" From thi s. we obtain W 2 } to be

Ill' =::

2.672 years. From (22.53). we find ECl WI •

579

Warranty Ana/y$i$ for Comp/ell. Systems

EC(2 .5,3) X

= ( 150 + 120 { fo"" F2(1) dl =

~2

(23 .2)

that is, the mean time to failure fo r a nonconform ing item is much sma ller than that for a (;onfo rming item. Let Pji (1 - Pjl ) denote the probability that item i in batc h j is conforming (nonconforming) . The modeling of the three cases (disc ussed in the previous subsec tion) is as follows: Case I: Pj; = P for all i and). Although items arc produced in batches , an item for sale. or use in replacement under warranty, is picked randomly from the batch. As a result , the failure dist ribution , F(t) , of an item picked randomly for use is given by (23.3)

where P is as previously defined . Case 2: Pj; = Pj' for I s; :oS L , and Pj being a random variable varying from batch to batch . For Case 2. an alternate approach is to let 0, the parameter oCthe failure distribution function , vary from batch to batch. For Ihe j lh batch. let the parameter value (a random variable) be denoted OJ. The expected warranty cost for batch j depends on OJ. Ma nn and Saunders 12.3J use such a model fo rm ulat ion to decide on the best warranty period fo r each batch based on limited life-testing of items from each batch. Marncr 14] also uses a similar model formulation in the context of selecting opt imal warranty terms . Case 3: Pj; varies with i andj as in Type A nonconformance. 23.2.4

Quality Control Schemes

As mentioned earlier. the two different approaches to reduce the numhCT of nonconforming items reaching consumers arc as follows: I. Testing to weed out nonconforming items (a reactive approac h) 2. Preventing nonconforming ite ms from being produced (a proacti ve approach)

590

D. N. P. Murth y

Oft en . the word " inspection" is used instead of testing. One can view inspectio n as a form of testing, a nd henceforth we will use " testing" to include " inspection" in the remainder of the chapter. Testing

The case of Type-A nonconforming items is considered first. As no lifctesting is involved . testing does nol affect the failure dist ributio n of items. One can employ e ither complete ( 100%) testing o r li mited (less than 100%) testing based on so me sa mpling plan. The testing can be either pe rfect or imperfect. If the testing is perfect. the o utcome of the test reveals the true status (confo rming o r nol) of the ite m be ing tested. If the testing is imperfect . a confo rming ite m can sometimes be wro ngly classifi ed as be ing nonconforming and vice versa, A variet y of sampling plans have bee n studied a nd these ca n be fo und in any standard te xt o n qualit y control, fo r example. Refs, 5 and 6, Two such plans are as fo llows . Plan 1 (Accepfance Sampling Pla n )

The sampling plan is characterized by two integers rI (0 < n :5 L ) a nd c (0 :5 (' :5 n) . A sample of n items is selec ted rando ml y fro m eac h balc h and each item in {he sample tested to see if it is conrorming or not . Let Ii denote the number of nonconfo rming ite ms in the sample . If ;; :5 (', the n the items in the batch are released with no furth er testing. Howe ver. if Ii > c. then all the re maining items in the batc h are subjected to testing. In other wo rd s. 100% testing is used o nly whe n the sample contains more than a specified number of nonconfo rming items. The rationale fo r this is that a batc h which co ntain s few no nconfo rming items will. o n the a verage. have a small ( :5!') num ber of no nconforming items in the sample. and in this case . there is no need fo r further testing of the batch. In contrast . if a balch co ntains a large number of nonconfo rming ite ms . then . o n the average . the nu mber or nonconforming items in the sa mple is more likel y to be greater than c. and in this case. it is wort hwhile testing the whole batch to weed out all of the no nco nforming ite ms . Pla n 2 (Curtailed Sampling Pla n)

The sampling plan is c haracte ri zed by an integer n . Fo r eac h batch. ite ms are tested one at a time until a nonconfo rming item is fo und or until II items have bee n tested. If a nonconfo rming ite m is fo und, the n a ll items in the batch are te sted. If no nonco nfo rming item is fo und . then the batch is released with no furth er testing. In all sampling sc hemes. the fa te of no nco nfo rm ing items depends

Warranty and Manufac turing

591

o n whether they are repairable or not. In the nonrepairable case. they are scrapped. In the repairable case. the choice between repair or scrap depends o n the relative costs of manufacturing and repair. The optimal design of a sampling plan involves a trade-off between the expected cost of testing and the expected savings in the warranty cost. In the case of Type B nonconfonnance , weeding nonconforming items requires testing items (on a test bed) for a duration T (also called as "burn-in " period). Because a nonconforming item has a higher failure rate . it is more likely to fail during the testing period than a conforming item. Hence. the num ber of items failing in the sample tcsted can be viewed as an indicator of the qua lity of the batch-a large number impl ying a poor quality. that is, a high fraction of nonconforming items in the batch. As with Typc-A nonconfo rmance. the test ing can be e ithe r complete (IQO%,) or limited (less than IQO%,) when a sampling plan is used. A variety of sampling plans have been studied and one of them is as follows : Plan 3 (Acceptam:e Sampling Plan (n, c) Involving Life-Testing)

The sampling plan is c haracteri zed by two integers n (0 < n ~ L) and c (0 ~ c ~ n) and a real variable T (>0). A sample of n (0 < n ~ L ) items is se lccted randomly from each batch and life-tested for a period T. Let ij denotc the num be r ofi tcms from thc sample whic h fail during the testing. If ;; ~ c. then the remaining items in the batch are not tested. However. if Ii> c. then the remaining items in the batch arc subjected to life-testing for a period T. All item s which fail during testing are either scrapped (in the case of nonrepairable product) or rewo rked (in the case of repairable product) and released alo ng with the no nfailed and nontested item s. Note that the larger the value of T. the greater is the likelihood ofa nonconform ing item failing during testing and . hence. getting weeded out. However. this is ac hieved at the expense of reducing the life of items released for sale . Preventio n As mentioned earlier. the state of the process affects the probability that an item is nonconforming. This probability is low (high) when the stat e is in-control (out-of- conl rol) . As a result . the number of nonconforming items produced in a batch can be reduced (in a statistical sense) by actions which either reduce the likelihood of the state changing from in-control to o ut-of-control or minimize the duration for whic h the state is out-ofcontrol once a change occ urs . In batch manufac turing. lot sizing is used to ac hieve both of these. A model whic h characterizes the impact of lot sizing on the output

D. N. P. Murlh y

592

quality is due to Porteus {7]. In this model , the process is always in-control

at the start of each batch production. If the slate is in-control at the start of an item production , it can switch to out-oC·control with probability (I - q) or stay in--control with probability q . Once the switch occurs, the slate remains unaltered until the end of the batch production . This is appropriate when it is not desirable to interrupt the process during the

production ofa batch. All items produced are conforming (nonconforming) when the process is in-control (out-of-control). Note that both the probability that the process slate changes before a lot is processed o r while

it is being processed and the expected number of nonconforming items produced (should the process state change) decreases (increases) as L decreases (increases). Thi s model has been extended in Ref. 8 to solutions in which no t all items produced are conforming (nonconforming) when the process is incontrol (out-of-control). Rosenblatt and Lee 191 model the c hange in process state and it s effect on output quality slightly differently. In their model, the number of items produced before the state changes is modeled by a discrete di stribution function. However. the fin al effect of lot size on out put qualit y is similar 10 that in the Porteus model.

23.3

QUALITY CONTROL MODELS FOR TYPE A NONCONFORMANCE

This section deals with the case in which the no nconforming items are of Type A and presents several models for quality control. The models assume Ihal nonconforming items can be made conforming by appropriate rectification actions. Rectification of nonconforming items before they are released will be termed as "rework" and these a re carried o ut at the manufacturing plant. Rectific ation of nonconforming items returned under warranty will be termed as "servicing" a nd are carried out either at a service facility or at the plant. The per-unit cost of rectific ation under servicing is larger than that under rework . This is because of the additional costs associated with the setting up of a service ce nter andlor the handing of warranty claims. (See Chapter 24.) In the models, the focus will be on the expected warranty cost resulting from nonconfo rming items being returned fo r rectification. The expected warranty costs associated with failure s over the warranty period is the same for all items (conforming or nonconforming) once the nonconforming items have been rectified . As such , this cost is of no relevance in the context of quality control schemes of this sectio n.

Warranfy and Manufacturing

593

Let C, (c'. ) denote the rework (service) cost to rectify a nonconforming unit before (after) sale , Note that Cs > C, for reasons mentioned above, 23.3.1

Models Involving Testing

Let C, denote the cost of testing each item. Model 1 The process is assumed to be in a steady state. Hence, the items produced are statistically similar, that is, each item is conforming with probabilit y p and not conforming with probability p = t - p . If no testing is used . then the expected number of nonconfonning items released per batch is

PL.

For each batch. n (0 s: n s: L) items are tested and the nonconforming ones are rectified through rework . i he remaining items. if any . are released with no testing . As a result, the expected number of nonconforming items per batch reaching consumers is given by peL - n) . The average outgoing quality (AOQ) . the expected fraction of conforming items in a batc h after te sting. is given by AOQ = I _ ji(L - n) L

(23 .4)

This increases from p (when n = 0) to I (whe n n = L). The savings in the expected warranty cost per batch is np(C. - c,) and the cost ortesting is nC,. As a result, testing is preferred to no testing if and only if p(C. c,) > c, . When n (O:s n :s L) is a decision variable . n"', the optimal n which minimizes the expected total (warranty + testing) cost is characterized as follows . If there are no capacity constraints (see Model 2), then n'" = 0 when p(C - C~) < C, and n'" = L when p(es - C.) > C,. With capacity constraints (see Model 2) . n'" can be an interior point , that is, 0 < n" < L. Model 2

This is an extension of Model I and was proposed by Tapiero a nd Lee [10, Section 3]. The quality control program involves testing n items in each batch. All nonconfonning items detected are rectified through rework . The remaining L - n items arc released with no testing. Nonconfanning items sold are returned to the service center for rectification. As a result . the ex.pected total (rework + servicing + testing) cost per lot depends the servici ng center capacity (s) and the number tested per lot (n).

D. N. P. Murthy

594

The cost of a service center with capacity s is given by Co(s)

= Co

+ se) for s > O. where Co is the fi xed cosl and c) is the variable cost. The number of items per lot returned under warranty for servicing is a rando m variable distributed bino miall y. Let u denote the number of items per lot relUmcd under warrant y, then the cost of servicing (CS) a lot i s given by

cs

= { Co(S) Cots)

+ +

C IU

C IS

+

("2(U -

$)

ifu s s if II > S

(23 .5)

where C I (C2 ) is the variable cost per unit serviced below (above) capacity for items returned . C 2 > c. refl ects the cost of overtime oroblaining service el sewhere as the capacity is exceeded . Using a normal approximatio n. Tapiero and Lee derive an expression fo r the expected total (test + rework + service) cost per lot. It is given by i

= i (n. N. s) = n(ei + C li) + Co(s) + c2(N x

{{Iug(u. N _ n)du + s( 1 - G(s. N -

n )jj - (c2 - e l )

nH}

(23.6)

where G( u. N - n) =

v ~

-::;r;; If."exp( -x ) dx 0

2

u - (N - n)p J (N n)pp

and g( u. N - n ) =

dG( u. N - n) du

Let s* and n* denote the optimal values of sand n which minimize i . Tapiero and Lee (IOJ obtain the fo llowing result. Proposition 23.1 Suppose c) < e2 I.

2. 3.

C I.

Ifp ~ CAc. - c,), then n* = L (1 00% testing) a nd s* = O. If Ii S C'!(cz - C,), then n* = 0 (no testing) a nd s* is given b y the solutio n to c) = (c~ - e . )( 1 - G(s . N )) . If C,!(c2 - C) < P < c;/(c . - C, ), then 0 < n* < Land s· > O.

(Note that n* and s* in case 3 above need to be obtained by numerical methods.)

Warranty and Manufacturing

595

If C2 == CI (implying no service capacity constraint as servicing can be outsourced at no extra cost) , then situation case 3 above does not arise and in this case n· == 0 o r Land s· is zero in the former case and is L in the laller case . Model 3 (10)

This is an extension of Model 2 and was proposed by Tapiero and Lee [10, Section 4) . Here the quality varies from lot to lot and as a result. pj (== I - pj ) is modeled by a density function f (p). The quality control process involves testing a sample of n items. All nonconforming items detected during testing are rectified through rework. Nonconforming items (from the items not tested a nd released for sale) are rectified under warranty at a service center as in Model 2. Tapiero and Lee derive a n expression for the expected total (rework + servicing + testing) cost per lot. It is given by J == n(C;

+

+ c.p) + yeN -

y(c i -

cl l

(fo~ (u

n)(cl P - C; - s)g(u .

C.p} +

N- n ) dU]

Co(s)

(23 .7)

where (23.8)

and g(u. N - n) is the same as in Model 2. Tapiero and Lee obtain result s similar to Proposition 23.1 , but they are more complex . Model 4 111 1

In this model, the item is a noncritical component of a complex system. The system is functional even when the item is nonconforming. The quality varies from batch to batch. For batch j, the probability that an item is nonconforming is given by pj ( = I - Pj ), a random variable with density function f(p) . Should the system fail and it is established that the item (noncritical component) was defective. then the manufacturer is held liable and incurs a cost. In other words, the c laim is under implied warranty. The quality control is based on an acceptance sampling plan of (n, d . Thus, when testing is less than IUO'1O. some systems sold can have nonconforming items which can result in such clai ms. The sampling plan is chosen to ensure that the average outgoing quality (AOQ) is above a guaranteed value . This can be viewed a s a se lf-

O. N . P. Murt hy

596

insurance (by the manufacturer) to reduce the liability risk . Maruchek {Ill then discusses the choice between quality control (involving acceptance sampling scheme) and no quality control based on the expected profit (reve nue - total cost) for two scenarios: (I) all rejected lots (lots where the number of failures in the sample tested is greater than r) are 100% tested and nonconforming items are either reclified or scrapped and (2) all rejected lots are scrapped. Note the total cost is the sum of the manufacturing cost, testing cost , and liability cost. The model involves the following variables: r: b:

the revenue generated per unit sale manufacturing cost per unit d: probability that a consumer receiving a nonconforming item will demand settlement Cl.: average cost per claim The other variables (for example . C, and Cd are as defined earlier. Then the choice between acceptance sampling and no sampli ng is given by the following proposition. Proposition 23.2 [11] Acceptance sampling is preferred to no sampling if C. ~ NO - y)p(C.Ld - C) •N - yiN - n)

for Scenario I . and

c. s •

(CLdj) - r)(1 - y)N

n

for Scenario 2. where y is the probability thai a batch is released for sale wit hout 100% testing and is given by (23.8). Maruchek carries out a detailed study of the costt rade-ofTs for quality improvement through (I) reduction in proce ss variability (measured by the standard deviation uf Pj ) and (2) improving the prOCt::SS mcan (measured by the mean of p). She provides a numerical example for the case where f(p) is a beta distribution . Model 5 112) In this model. Ihe 101 size corresponds to the number of items produced in each time period and is modeled as varying randomly . The control scheme involves a sampling plan and is characterized by a parameter ~ . The expected service cost to rectify nonconforming items released for

Warranly and Manufacturing

597

sale is obtained usi ng a formulation similar to Model 4 110) with service capacity s. Finally, the output quality is modeled by a parameter 9. which represents the probability that an item is conforming. Better manufacturing results in a highe r value for e, but thi s is achieved at a higher manufacturing cost. Hence, the decision variables in the model are ~, 9, and s. The model examines the optimal choice of these to maximize the expected profit per unit time for Iwo different sampling schemes: (I) acceptance sampling plan of (n, r) and (2) curtailed sampling plan. 23.3.2

Models Based on Prevention

The output quality goes down as the process state deteriorates . As mentioned earlier, Ihe deterioration can be either grad ual or sudden . In Ihis subsection. a model due 10 Porteus PI is outlined. Under this model, the process change is modeled as occurring suddenly, and lot sizing is used to reduce the likelihood of the change occurring as well as the duration for which the system is in out-of-control state o nce a change in state occurs. Model 617) Items are produced in lots of size L to meet a constant demand of m items per unit time. The time to produce a lot is re latively small so that it is treated as being zero. This implies that the time between the production of two successive lot s is Urn. At the start of each lot production , the state is checked to make sure that it is in-control. The c hange in process slate occurs in an uncertain manner and is c haracterized by the parameter q as discussed in Section 23.2.4. Should the process change from in-control to out-of-control. then a certain number of items are nonconfonning. As a result. the number of conforming items in a 101 (N ) is a random variable with an expected value given by q(l - '1') I - q

(23.9)

The manufacturing cost per lot is C, + CmL. where C,is the fixed cost and C m is the variable cost (material + labor) to produce a unit. Let h denote the inventory holding cost for an item per unit time . The total inventory holding cost per lot is hL 212m. Finally. all nonco nforming items are reworked at a cost of C. for each nonconforming item . Porteus derives a n expression for the expected total (manufacture + inventory + servicing) cost per unit time. It is given by

O. N. P. Murlhy

598

J(L)

(C,

+

CmL)

+

( hL'/2m)

+ ClL - q(l

q')I /(I - q)

Urn (23 . 10)

Note that as L increases, the manufacturing cost per unit time decreases and the inventory holding cost and the ex.pected rework cost per unit time increase. This implies that L"', the optimal L which minimizes J(L). achieves a sensible trade-off between manufacturing and the rework costs. Porteus use s the following approximation (appropriate when q is close to I )

0 represents the fixed cost and 41, the variable cost. The expected number of items per lot which fail and get discarded is given by [1 - amlL. where (23 . 16)

is the probability that an item picked randomly will survive the test. Let

Cd denote the cost of disposing each failed item . As a result, the asy mptotic total (manufacturing + testing + discarding) cost per item released is given by

>(n

= Cm + C, + amC, 1 a

(23.1 7)

Note that the failure distribution of conforming and nonconforming items which survive the test afC given by ft,(I) = F,(I +

n F,(n

F,m

(23. 18)

with; = 1 (2) for conforming (nonconforming) items. Finally. the asymptotic probability that an item released is conforming (nonconforming) is given by p [I - pl. where

,

p ~

pI'm a(n

(23. 19)

with a(n given by (23.16). Note that p > p and that p increases with T [since Fz( nlF1Cn decreases as Tincreasesl. The asymptotic (manufacturing + tesling + discarding + warranty) cost per item is given by

601

Warranty and Manufacluring

(23.20)

where C .•lD is the asymptotic warrant y cost per item. c..·(n depends on the type of warranty. the warranty duration (W) , failure distributions 1'\ and F2 • and fit the asymptotic probability that an item is conform ing. Testing is the optimal strategy if J(n > J(O) for some T > 0 and no testing is the optimal strategy if J(n < J(O) for all T > O. In the fonner case, the optimal testing period, T·, is the value of T which minimizes

1m. FRW Policy (with minimal repair )

The free-replacement warranty (FRW) is discussed in Chapters I and 10. Let C~ denote the cost of each repair. Then (23.21)

and W

C.•.(D = C[p

Jo

r2(t)dl + (I - p)

Jo

W

f 2 (/)dl]

(23.22)

Consider the special case where F.(t) and F:!{t) are exponential distributions with parameters AI and A2 , respectively, with A2 > AI > O. This characterization is often used for modeling failure of electronic systems (with either the whole system or a subsystem being the item under consideration) and here the failure of an item is not dependent on its age . In thi s case, the choice between testing and no testing is given by the following propositions. Proposition 23,2 (l4J Testing is the optimal strategy if (I - a}(C

+ Cd) + C;

m PP(" - .,)[F,m - F,(T)J > '-'----'='ik~=--'---'" we,

(23.23)

for some T> O.

Comments: I.

2.

The inequality is more likely to be satisfied if (I) Crn. Cd. and C; are small relative to We. (2) lI.2 - lI.l is large , and (3) p is close to 0.5 (i.e., 50% defecti ves!). If the inequality is satisfied for P :;:: PC" then it is also sat isfied for PC' :s: P :s: I - PC' .

O. N. P. Murth y

602

Proposition 23.3 (14] No testing is the optimal s trategy if

em

+

Cd

+ C; >

WC,PP{A Z -

)'1)

(23.24)

Comments:

The inequality is more likely to be satisfied if (I) em. Cd. and Ci are large relative 10 we.. (2) "-2 - AI is small. and (3) p is close (0 O. This is just the reverse of Proposition 21.1 . 2. The inequalit y is always satisfied when p is very close to zero . In this case, because the majority of the items are defective . there is no advantage in doing 100% testing to weed oul the I.

3.

defectives . The more sens ible option is to scra p the whole batch . If the inequality is satisfied for p = Pc-. then it is also sati sfied for all p :$ p ...

Example 23.2 Lei AI == 0.2 per year and A2 = 4.0 per year. This implies that the

mean time to failure is 5 years for conforming items and 0.25 year for nonconforming items . Let L =:: 100 and the remaining values of the model parameters be as follow s: p == 0.75, Cm = $300 , q,1 = 0.0, 2 = $45 per year , Cd = $5 , and C, = $100. This implies that the cost of testing a unit for a month is approximately $3.80, p = 0.75 implies that 25% oflhe items are defective. This is not an unusual figure for certain electronic systems using integrated chips where the fractiori of defectives can be unu sually high. The above situation corresponds to the case where the item is an expensive electronic system composed of three modules, each costing roughl y the same . Item failures occur due 10 one of the modules failing and the item being repairetl by n:plal,;ing the failetl mooule by a new ont: o Hence . the cost of each replacement is roughly one-third the cost of a new item . Table 23 .1 shows P and J(P) for four different warranty periods ranging from I to 4 years. Note that P = 0 implies that no testing is the optimal strategy. and T* > 0 implies that testing is the optimal strdtegy. For W = I. no te sting is the optimal strategy . For W 2: 2, testing is the optimal strategy. Figure 23 . 1 shows T* versus W . Note that T* is zero for W ~ 1.25 and increases with W for W > 1.25 . The reason for this is as follow s. Because all items released for sale are rectified through minimal repair. the warrant y servicing cost increases rapidl y with W for any nonconforming item released. As a result , as the warranty period in-

603

Warranly and Manufacturing

Table 23.1 W

Optimal Strategy. P and J( P) Under FRW Optimal strategy No testing Testing Testing Testing

2

3 4

,.• "

,.

I

T' 0.0 0.2259 0.423 1 0.5343

J(P) 41 5.00 :5 14 .30 567. 13 606.47

/

I

"

,., , Figure 23. 1

" - - -"-

w

"

P versus W ( FRW policy with minimal repair).

604

D. N. P. Murthy

creases. longer testing is needed to reduce the number of nonconfo rming items being released. T his is seen more clearly in Table 23.2 where as W increases, T* increases . and the expected number of nonconforming items

released decreases. The optima l testing period varies from 0.2259 year fo r W :< 2 years (0 0.5343 year for W "" 4 years. Obviously . it is not possible to test items for such long periods. However, life-tcsting is usually carried oul in accelerated manner . Under accelerated life-testing , the item is subjected to a harsher environ ment which hastens the aging process. As a re sult . testing for one unit of lime in the accelerated mode corresponds to 13 ( > 1) units of testing under normal conditions. This implies that testing for TIP units of time in the accelerated mode corresponds to testing for T units under normal conditions. If the testing in accelerated mode is done with P 50. testing for 0.5343 year in norma l mode would require accelerated testing for only approximate ly 4 days . This practical solution corresponding to the optimal period may be obtainable . Figure 23 .2 shows the influence of p on 1* with the remaining parameters held at their nominal values. Note that 1* = 0 for small p (close to 0) and large p (close to I) as to be expected from Proposition 23.3 . For W = I. T* is zero for all values of p . For W = 2. T* > 0 fo r 0.48 < P < 0.92. Over this range. 1* first increases a nd then decreases as shown in Figure 23.2. The reason for this is as follows. When p = I, no item is no nconforming and. hence, there is no need for testing. As p decreases (for 0.92 s p ~ I) . the fraction of nonconforming is suffi cie ntly small so that testing to weed out nonconforming items is not justified and . as a result . 1* = O. As p decreases still further (for 0.48 :s; p < 0.92), the frac tion of nonconforming items increases and testing is won hwhile. Note that testing for greate r time reduces the warranty cost per item but in-

Table 23.2 per Batch

Expected Number of Nondefective and Defecti ve Items Expected number of items per balch of 100 items

Weeded out during testing W

Nondefect.

I

0.00

2

].] 17 6.085 7.60 1

3 4

Defect.

Released for use

Tolal

Nondefect.

Defect.

Tolal

0.00

0.00

100.00

18 . 185 26.282 29 .65 1

75 .000 7 1.687 68.9 15 67.]99

25.000

14.872 21.397 22.050

10. 128 4.60] 2.950

81.8 15 73.718 70.349

605

Warranty and Manufacturing

....

0.'

....

...

....

..... ..•.

,,

, \ \": ,, \

~

,..--........

\

\ \\

______-cC-~______~,C-____~'~~

! i

/.

u

u

u

u

u

\ \\

11 1:

\ I:

\ l!

u

o. w. ,

figure 23 .2

:

\

\.

I

i

o

'\

u

u w••

T* versus p (fRW policy with minimal repair).

creases the manufacturing cost per item as a greater fraction of items fail during testing and are discarded . The reason that T* increase s as p decreases in the interval 0.69 s p < 0.92 is that the increase in the manufacturing cost per item is less than the decrease in the warranty cost as p decreases. In the interval 0.48 :s:: p < 0.69. T* decreases with p decreasing as the increase in the manufacturing cost per item is more tha n the reduction in the warranty cost per item. Finally. for 0 < p :s:: 0.48. the fraction of nonconforming items is very high. The cost orany testing is not w0l1h the reduction in the warranty cost and . hence. T* = O. As W increases . the interval over which T* > 0 also increases and T* has a shape similar to that for W = 2 except Ihat Ihe values for any given p are increasing with W. This is to be expected for reasons discussed earlier.

606

O. N. P. M urth y

Pro -Rata Warranty (Linear Proration )

Under this policy . the manufacturer agrees to refund a fraction of the original sales price S to the consumer if the item sold fail s within the warrant y period (0. W). (See Chapter 11 .) With linear proration . the asymptotic warranty cost per item with no testing is given by 1(0)

~ P [kS

r (I r (I -

~) { , (I) dl

+ Ii [kS

+ C>F,(W)]

~) { , (I) dl

+ C>F,(W)]

(23.25)

and with testing, it is given b y

1m ~

"[kS

+

(I -

r (1- ~)j,(t)dl + CJ,(W)] P) [kS

foW (, - ~) j2(t)d( + CI, F2(W) ]

(23.26 )

where 1 ;(1 ), I s ; s 2. and p are given by (23 . 18) and (23 . 19), respectively. For the special case w here F t U) and F 2 (t ) are exponential distributions with parameters Xt and A2. respectively, with.\2 > AI > 0, s ufficiency

conditions for optimality , simila r 10 Propositions 23 .3 and 23.4. are given in Ref. 14. Example 23.3 As in Example 23 .2. let F. and F2 be exponential distributions with parameters }\1 and hl. Let hi '= 0.2 per year. ),,2 '= 4.0 per year. C h '= $20. k :: 1.0. ~ :;;:: 1.0. and S "'" $1000 . The remaining parameter values a re tilt: same as in Ex.ample 23 .2. Table 23 .3 shows r· and l ( T·) for the four different warranty periods. For W '= I to 4. T* > O. implying that testing is the optimal strategy in all four cases . Figure 23. 3 shows T* as a fun ction of W . For W < 0.5.

Table 23.3

Optimal Strategy. P and }(P) Under PRW

W

Optimal strategy

1

Testing Testing Testing Testing

2 J 4

T'

}( P)

0. 1539 0. 1871 0. 1565

560.03 650.45 720.22 778.35

0. 1090

607

Warranly and Ma nufacluring

"

"

" 1--1-- - -- - - --\-- ------1

,~

1--+---- - - - -- ---',- ----1

,

,

•

,

,

w

Figure 23.3

P versus W (PRW policy).

r* = O. This is to be expected because small W implies a smaller warranty service cost even with all nonconrorming items released and the cost or testing nOI justified. For 0.5 < W < 6.0, P > O. In contrast to the FRW policy with minimal repair (Example 23 .2) , here T* increases with W ror 0.5 < W < 1.7 and decreases ror 1.7 < W < 6. For 0.5 < W ~ 1.70. the testing period increases with W because the warrant y cost for released nonconforming items increasingly dominates the cost per item released. Thus , by reducing the number of released nonconforming items through increased testing. the warranty cost can be decreased by more than the increase in the manuracturing cost per item released. For 1.70 < W < 6.0, the influence of the warranty cost ror released nonconrorming items decreases while the influence of the other costs increases, resulting in the testing period decreasing as W increases. This cont inues until. ror W > 6.0, testing is again nol justified, so that T* = O. Table 23.4 shows the

608

D. N. P. MUl1hy

Table 23.4 per Batch

Expected Number of Nondefective and Defective Items Expected number of items per batch of 100 items

Weeded out during testing

w I

2 J 4

Released for use

Nondefect.

Defect .

Total

Nondefect.

Defect.

Total

2.273 2.755 2.3 11 1.617

11 .490 13 . 172 11 .633 8.834

13 .763 15.927 13 .944 10.45 1

72 .727 72.245 72.689 73.383

13 .5 10 11.828 13 .367 16. 166

86. 237 84.073 86.056 89.549

expecled number of nonconforming items released and weeded out in each batc h. The effect of p o n T* , with the remaining paramelers held at Iheir nominal values. is similar to Figure 23 .2 for reasons discussed earlier. Finall y. the case in which the items are sold wit h FRW policy and the items are nonrepairable has been studied in Refs. 8 and 14. The results are similar to the two policies discussed in this subsectio n. 23.4.2

Models Based on Improvement

Djamaludin et a l. [15] dea ls with lot sizing as a decision variable to reduce the number of nonco nfo rming items produced . The process deterioratio n is as in Ref. 7 and the model is as foll ows. Model 8 (1 5(

Before a lot production starts, the prO(:css is checked to make sure that it is in-control. This costs C, if the process is in-control and C, + TJ (1l ~ 0) ifit is out-of-control. TJ is the cost of bri nging an o ut-of-control process in.control. This set up cost formu lation is slightly different from Ihat in Section 23 .3.2. The asymptot ic manufacturing (fixed + variable) cost per item is given by

,( LI _ C_ + C, + ( ~ -

(> 0) is the rate of return on investme nt. Let R,( W; $). the

reserves sel aside (and invested). be selected to equal the expected lotal payout over the warranty period. For Model I. the payout (I - xlW) for an item failing at lime x (0 < x < W) can be met by an amount (1 - xl W) exp( - x) of the initial reserves invested for a period x. As a result. it is easily seen that Rt(W ;.0

~

" .Ii

.i ..

0.'

j.1

~

£

0.'

'Ii 0

0.' '---_ _ _ _~----:'_----J

o

" W.....anty Period W

Figure 24 .6

{I-

versus W.

If F is IFR. then it is easily seen that w(O: W) < w( x; W) for all W. To compare w("· ; W) with w(O: W), define the percentage savings. Tl( W). as "

100 ( W) ~ ( "'(0; W) - ",(~' ; W)) Optwdiom' Rt'Jt'tlrch Socit't)', 31 . 1081 - 1088. Mahon. B. H .. and Bailey. R. J . M. (1915). A proposed improved replace· ment policy for army vehicles. Operations Rnetlrt'h QlllIrterly. 26. 417- 494 . Munhy. D. N. P .. and Nguyen. D. G . (1988). An optimal repair cost limit policy for servicing warranty. Mathemaric(ll and Complllt'r Modellin ~. II. 595-599. Issacso n. E .• and Keller, H. B. (1 968). AnClf),sis 0/ N /IIIU:ri('al Mt'lllOds. John Wiley & Sons. New York. Agnihotri. S. R. (1988). A mean value analysis of the travelling repairman problem. liE Tran suctions. 20, 223-229. Brown. R. G. (1961) . Decision Rules f or Im'emory Mlllluj:t'mem. Drysden Press, Hinsdale , IL Hadley. G .. and Whitin . T. M. (1963). Analysi.f o/ lm'ellloT)' Systems. Prentice- Hall . Englewood Cliffs. NJ. Karmarkar. U.S .. and Kubat. P. (1983). Value of loaners in product suppon . liE Transactiom. IS. 5- 11. Palankar, J . G. (1978). Estimation of R esen·es and C(lSi! Flo ws A uoda uJ with Differenl Warranty Policies. Ph .D. Thesis. Clemson University, South Carolina. Chandran . R.. and Lancioni . R. A. (l981 ). Product recall : A challenge for the 1980' s: Intern(ltionai JO/lrnal of Physical Distribution Materials Mallu!!i'melli, 11.46-55. Fisk. G .. and Chandran. R. (1975). How to trace and recall products. H lIrmrd BlIsiness R el'iell's. 53(6), 90-96. Min . H. (1989). A bicriterion reverse distribution model for product recall. Omega. 17. 483-490. Tapiero. C. S .• and Posner, M. J. (1988). Warrant y reserving. Nul'lIl R esf'arch Logislics QUarlerl),. 35 , 473- 479. Dardis. R.. and Zent , C. (1982). The economics of the Pinto recall , jO/mwl of Con:Hllller Affllirs. 16, 26 1-271.

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Part G Warranty and Society

PAIlTG ~~

WAARANl'V ANO ~

“This page left intentionally blank.”

Introduction

Part G contains four chapters, start ing with a treatment of economic and accou nting issues of warranties. An understanding of these two and the relationship between warranty and consumerism sets the scene fOT understanding warranty from a public policy perspective. In Chapter 25. N . A. Lutz looks at warranty from a microeconomic

viewpoint. The author bui ld s on some of the concepts from Chapler 15 to characterize the demand for warranties and the actions of the suppliers of warranties to meet this demand. The resulting outcome determines the market structure. A variety offairly stylized model s are discussed, taking into account issues such as the effect of consumer attitude towa rd risk. moral hazard , and so on. Chapter 26. by R. A. Masc hmeyer a nd K. R. Balachandran. looks at financial accounting and reporting for warranties in the United Stales,

as specified by the Financial Accounting Standards Board (FASB) and the Internal Revenue Service (IRS), Alternative accounting methods for warranty accountillg and some problem areas are di scussed, 657

658

Par! C

In Chapler 27. G. C. Mosier and J . L. Wiener discuss public policy effort s that address warranty issues. They start with a brief hi storical overview of the provisions of the Uniform Commercial Code (Vee) and the Magnuson- Moss Act. Following this . they carry oul a stud y of warranly performance before and afler the Act in the context of automobile and consumer durable warranties. addressing topics such as warranty content, warranties as signals. warranty coverage, and qualit y. Chapter 28. by J . Burto n. looks a l warranties from a consumerist perspective and describes three different consumer movements a nd their impact on the legislati ve ac tions that led 10 the formulation of the public policies related to warranties discussed in Chapter 27 . The author then c ritically evaluates the Magnuson-Moss Act from a consumerist perspec+ live and sugges ls improvements to the Act and possible future legisla tion .

25 The Economic Theory of Warranties Nancy A. Lutz Virg inia Polytechnic Inslitute and Slale University

Blacksburg, Virginia

25.1

INTRODUCTION

The economic theory of warranties is one of many application s of microeconomics, the branch of economics that studies the choices made by individual firm s and consumers and how these choices interact with in a markel . Microeconomics emphasizes rational decisio n making, assuming first that consumers seek to maximize their well-being, or utility, given the information available 10 them at the lime of deci sion ; second . thai firm s act to maximize profit s, again given the available information . Usi ng these assumptions. a wide variety of concl usions can be reached about the behavior of consumers and firms. The great strength of microeconomics when applied to warranties is that it call s for the careful consideration of all the fa ctors affecting 659

N . A Lulz

660

warrant y provi sion.· For example . if we ask what warranty will max imize the profit associated with the sale of some product, we must consider not only the costs of providing the warranty but also the revenue associated with the warranty . The costs depend on the kind of compensation promised in the even! of a failure and the likelihood that the compensation is claimed. The warranty increases revenues if it allows the firm to sell more units or to charge a higher price while selling the same number of unit s. Similarly. if we consider when a consumer will . in fact , be willing to pay more for a product sold with a warranty , we must consider not only the cost of the warranty to the consumer but also the benefit s conferred by the warranty .

Microeconomics, then, answers two quest ions about warrantie s. First, we can a nalyze why consumers demand warranties (in the sense that the y are wi lling to pay more for a product covered by some sort of wammt y) . Second , given this demand on the part of consumers, we can understand when warranties maximize a producer's profit and the kind of warranty that does so. This chapter explains the basic economic answers to these questions. Consumers demand warranties because they provide insurance against the risk of product fa ilure and because a warranty may mean that a product is mo re durable . Firms offer warranties because the cost of in surance is low and because warranties are a tool (0 convince consumers about a product' s durabilit y. 25.2

ATIRIBUTES OF ANY ECONOMIC MODEL OF WARRANTIES

Consider Ihe basic elements of a ny economic model of warranty provision . Suc h a model will begin with the producer of the product being warranted . The first thing to be determined is market structure : Is the producer a mo nopolist. or do competing firm s also produce a more or less equivalent product? Markel structure defines the options available to consumers ; if there are many producers , consumers have a wide range of option s and the producer will face stiff competition for consumers. The next element of the model is technology. This determines the product' s feature s a nd the cost of production . Technology also determines how likely it is that the product will break down in use. Durabilit y is an innate attribute of • For the purpoSC$ of this chapter. I will definc a warran ty ai> II contrac t obligating thc manufacturer of a product to make some specified compensation to the consumer in the eve nt that the product fails to function . A warrant y is th us distinct from a money·back guarantee, under whi, h the consumer can retu rn a produunting and

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