Warranty Cost Analysis [1 ed.] 9780824789114, 9780367810856, 9781000723540, 9781000715729

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\50M1sx 3® Ulo ©DSscMMfe® Departm ent o f Decision Systems School o f Business Administration University o f Southern California Los Angeles, California

Departm ent o f M echanicai Engineering The University o f Queensiand St. Lucia, Australia

Boca Raton London New York

CRC Press is an imprint of the Taylor & Francis Group, an informa business

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Blischke,฀ W.฀ R.฀ Warranty฀cost฀ analysis/฀Wallace฀ R. Blischke,฀ D.฀ N.฀ Prabhakar฀Murthy.฀ p.฀ cm.฀ Includes฀ bibliographical฀ references฀and฀ index.฀ ISBN 0-8247-8911-3฀ 1.฀ Quality฀control-Costs.฀ 2.฀ Warranty-Costs.฀ 3.฀ Quality฀of฀ products-Costs.฀ I.฀ Murthy,฀ D.฀ N.฀ P.฀ II.฀ Title.฀ TS156.B58฀ 1993฀ 658.5'6-dc20฀ 93-25360฀ CIP฀

In memory of B everly, and to Jayashree

Preface

The warranty concept is important to both the seller and the buyer of virtually any consumer or commercial product. Product warranty provides protection for both: The buyer is provided a recourse for dealing with items that fail to fulfill their intended purpose, usually in the form of some stated compensation. The intention is to protect the buyer from shoddy or unreliable goods. The seller, on the other hand, is provided protection because warranty terms explicitly limit responsibility, in terms of both time and type of product failure. The net result of this process is an increase in cost to the seller and a commensurate increase in the price charged for products sold under warranty. There is a very extensive literature, primarily in technical journals, that deals with estimation of these costs and many related issues. Until now, no unified, analytical treatment on the subject of product warranty existed. The aim of this book is to provide such a treatment in a format that will be useful to analysts— engineers, management scientists, statisticians, and so forth— as well as to managers having responsibility for establishing warranty policy, administering the warranty program, or estimating warranty cost. The audience includes both practitioners and researchers. The book is also suitable as a text in a university course at the graduate level. A variety of warranty policies and the mathematical models for analysis of various related engineering and management issues are discussed. The models included cover standard consumer product warranties such as the free replacement and pro rata, as well as certain product warranties sometimes used in large volume or specialized government and commercial transactions. v

vi

Preface

The analytical models deal with cost and optimization problems primarily from the manufacturer’s point of view, although many models that consider the problem from the buyer’s point of view are given as well. Optimal decision rules for design, manufacture, and warranty servicing are discussed. The analytical approaches presented involve stochastic modeling of the warranty process. Methods of collecting and analyzing relevant data are also discussed. Although the treatment is conceptual, real-life examples are used to illustrate the techniques. Applications are stressed throughout. The book assumes an understanding of calculus, basic probability theory, and statistics. In addition, some understanding of stochastic processes would be helpful, but is not essential since the book covers many of the basic concepts needed in this area. The first three chapters deal with warranty in a general framework (Chapter 1), different types of warranties (Chapter 2), and modeling issues (Chapters 2 and 3). Some background in basic statistics and probability is assumed, and an exposure to stochastic point processes is helpful but not essential. Some basic results in this area are proved in Chapter 3; other proofs are given in Appendix B. Proofs of many other results are omitted, but references are furnished so that the interested reader may pursue these topics in more depth. The next five chapters of the book are primarily concerned with warranty cost models and analysis of specific types of warranty policies. Again, results are mainly theoretical, but applications and worked examples are included to illustrate the theory. Chapters 9 and 10 deal with various operational and engineering aspects of product warranties. Chapter 9 addresses a number of issues concerning servicing of warranty claims and some related optimization problems. Chapter 10 deals with design and manufacturing issues and provides some theoretical results for optimization problems in those areas. The last four chapters deal with a variety of topics that are important for application of these results and in the understanding of warranty in a broader context. Chapter 11 covers the simulation approach to solving the analytical problems of Chapters 4-8 and the optimization problems of Chapters 9 and 10. Chapters 12 and 13 apply the models to real-world problems. The former deals with data collection and analysis with emphasis on model estimation, and the latter gives several real-life case studies involving application of the warranty cost models. Finally, Chapter 14 examines product warranty in a comprehensive framework that identifies many more factors and their complex interrelationships. This chapter also discusses additional topics for research in the analysis of warranties.

Preface

vii

As noted previously, this book is intended for use by practitioners as well as researchers. Practitioners need only apply the final results, omitting the mathematical derivations. We hope that researchers will find the mathematical details of sufficient interest to motivate additional analysis. We have included a list of exercises for nearly all chapters. The exercises range from theoretical to applied and vary considerably in degree of complexity and difficulty. We have benefited greatly from interactions with many of our colleagues and students in the preparation of this book. We particularly wish to express our appreciation to Dr. Ernest M. Scheuer (California State University, Northridge) and Dr. Richard Wilson (University of Queensland), with whom we have individually worked closely on many research projects, and our doctoral students, Luis Guin and Vickie Lee Hill at USC, and Dinh Nguyen, Bermawi Iskandar, and Istiana Djamaludin at UQ. We are grateful to these persons as well as to our many other colleagues at our universities and elsewhere for their encouragement and their many helpful comments. We gratefully acknowledge the support of both the University of Southern California and the University of Queensland during a sabbatical visit of Professor Murthy to USC. We would also like to thank the administrative and support staffs of the Department of Decision Systems, USC, and the Department of Mechanical Engineering, UQ, for their help and Patrick Blasa of USC, who prepared most of the illustrations. We wish to thank a number of journals and organizations for permission to reproduce various charts, tables, and other materials. These include the RAND Corp.; the Conference Board, Inc.; John Wiley & Sons, Inc.; Pergamon Press; Gordon & Breach Science Publishers, Inc.; Elsevier Science Pub. Co.; IEE; and numerous other professional publications. Finally, we are grateful to Sierracin/Sylmar Corporation for agreeing to our use of the 747 windshield data and their cooperation in preparing the case presented in Chapter 13, and to all the various organizations who kindly allowed us the use of copyrighted material as examples of warranty policies. The writing of this book has been a true joint venture, with many visits across the Pacific and very extensive use of today’s communications technology. We have followed the tradition of naming the authors in alphabetical order rather than deciding the issue based on the outcome of the toss of an unbiased coin. Wallace R. Blischke D. N. Prabhakar Murthy

Contents

Preface

v

1 An Overview 1.1 Introduction 1.2 The Role of Warranty 1.3 Historical View 1.4 Some Examples of Warranty 1.5 Theories of Warranty 1.6 Warranty and Product Liability 1.7 The Systems Approach to Warranty 1.8 The Study of Warranty 1.9 Aim and Scope of the Book Notes Exercises References

1 1 3 5 9 25 28 30 36 40 41 42 42

2 Warranty Policy and Modeling Issues 2.1 Introduction 2.2 Warranty Policies 2.3 The Analytical Approach 2.4 The Analytical Approach to Warranty Studies 2.5 Modeling First Failure [One-Dimensional Formulation] 2.6 Modeling First Failure [Two-Dimensional Formulation] 2.7 Modeling Rectification Actions 2.8 Modeling Subsequent Failures 2.9 Modeling Item Sales

45 45 45 62 64 65 75 77 81 82 ix

Contents

X

2.10

Cost Analysis Notes Exercises References

3

Stochastic Processes for Warranty Modeling 3.1 Introduction 3.2 Stochastic Processes 3.3 Analysis of Stochastic Processes 3.4 Poisson Processes 3.5 Renewal Processes 3.6 The Renewal Integral Equation 3.7 Additional Topics from Renewal Theory 3.8 Additional One-Dimensional Point Processes 3.9 Two-Dimensional Renewal Processes Notes Exercises References

4

Analysis of the Basic Free-Replacement Warranty 4.1 Introduction 4.2 Modeling the Seller’s Unit Cost for the Nonrenewing FRW 4.3 Modeling the Buyer’s Costs 4.4 Unit Cost Models for the Renewing FRW 4.5 Life Cycle Cost Models for the FRW 4.6 Indifference Prices 4.7 Additional Models for Analysis of the FRW Notes Exercises References

5

Analysis of the Basic Pro-Rata Warranty Introduction 5.1 5.2 Cost Analysis of the Nonrenewing PRW Unit Cost Models for Items Sold Under Renewing PRW 5.3 5.4 Life Cycle Cost Models Indifference Prices for the Pro-Rata Warranty 5.5 5.6 Comparison of the Free-Replacement and Pro-Rata Warranties Notes Exercises References

84 86 88 90 93

93 94 95 96 100 107 115 116 117 127 127 128

131

131 133 142 144 146 155 158 161 163 166 169

169 171 183 198 205 208 213 214 216

Contents

xi

6 Complex One-Dimensional Warranties 6.1 Introduction 6.2 Combination Warranties 6.3 Cumulative Warranties Notes Exercises References

219 219 221 244 263 264 267

7 Reliability Improvement Warranties 7.1 Concepts 7.2 The History of RIW 7.3 Typical RIW Features 7.4 Cost Models for RIW Notes Exercises References

269 269 270 273 274 292 294 296

8 Two-Dimensional Warranty Policies 8.1 Introduction 8.2 Modeling Item Failures 8.3 Free-Replacement Policies [One-Dimensional Approach] 8.4 Life Cycle Costs [One-DimensionalApproach] 8.5 Free-Replacement Policies [Two-Dimensional Approach] 8.6 Life Cycle Costs [Two-DimensionalApproach] 8.7 Pro-Rata Policies [Two-Dimensional Approach] 8.8 Additional Topics Notes Exercises References

301 301 302 306 321 325 344 351 360 362 362 365

9 Warranty Servicing 9.1 Introduction 9.2 Warranty Reserves [PRW Policies] 9.3 Demand for Spares [FRW Policies] 9.4 Demand for Repairs [Repairable Product] 9.5 Repair Versus Replace 9.6 Cost Repair Limit Strategy 9.7 Additional Topics Notes Exercises References

367 367 368 379 382 388 401 406 411 412 413

xii

Contents

10 Warranty and Engineering 10.1 Introduction 10.2 Engineering of Products 10.3 Warranty and Optimal Design 10.4 Manufacturing and Warranty 10.5 Presale Testing 10.6 Additional Topics Notes Exercises References

415 415 416 418 432 462 468 469 470 470

11 The 11.1 11.2 11.3 11.4 11.5 11.6 11.7 11.8 11.9

473 473 475 478 484 487 492 496 499 504 511 512 513

Simulation Approach to WarrantyAnalysis Introduction Simulation Methodology Simulation Modeling Random Number Generation Simulation on a Microcomputer Simulation of One-DimensionalRenewalFunctions Simulation of Two-DimensionalRenewalFunctions A General Purpose Simulator Two Applications Notes Exercises References

12 Statistical Estimation of Warranty Costs 12.1 Introduction 12.2 Parameter Estimation 12.3 Confidence Interval Estimation 12.4 Statistical Estimation of RenewalFunctions 12.5 Estimation of Cost Models for SelectedWarranty Policies Notes Exercises References

515 515 517 537 549 556 567 569 574

13 Case Studies 13.1 Introduction 13.2 Detailed Case Analysis: Sierracin/Sylmar— Prediction of Warranty Costs 13.3 Mini-Cases— Consumer and CommercialProducts 13.4 Case Studies— RIW Notes Exercises References

579 579 580 593 597 600 601 602

Contents

xiii

14 A Comprehensive Framework for the Study of Warranty 14.1 Introduction 14.2 The Systems Approach to the Study of Warranty 14.3 Detailed System Characterization [Consumer/ Manufacturer Perspectives] 14.4 Models Based on the Detailed System Characterization 14.5 System Characterization [Public Policy Perspective] 14.6 Models Based on the Public Policy System Characterization 14.7 Topics for Further Research Notes References

607 607 608

638 642 646 648

Symbols and Notation Appendix A Basic Results from Probability Theory Appendix B Proofs of Results in Chapter 3 Appendix C Calculation of Renewal Functions Author Index Subject Index Index for Policies

653 667 673 679 717 723 731

610 626 635

1

An Overview

1.1 INTRODUCTION

The second half of twentieth century has seen dramatic changes in the role and the importance of warranty in relation to product sales and services. Four main factors responsible for this have been 1.

2. 3. 4.

Activism of the so-called “consumer movement,” which resulted in greater awareness among consumers regarding their rights and the formation of strong and vocal groups to challenge manufacturers and force lawmakers to enact new laws to protect consumer interests; Lawmakers responding to the concerns of consumers and their advocates; Manufacturers acting in a reactive mode to (1) and (2); and Manufacturers initiating proactive actions, using warranty as a powerful marketing tool.

A reason for consumer activism is that modern industrial societies are characterized by two features: (1) products (consumer durables, commercial and industrial) are becoming more and more complex; and (2) new products are appearing on the market at an ever increasing rate. Both of these factors strongly affect the warranty environment and hence the consumer’s concern with warranty as a mechanism for adequate protection of their interests. On the research front, the nature of research on warranties has also undergone a major change. Until 1950, the study of warranty was carried out mainly by the legal profession, and articles on warranty appeared only in the law journals. In contrast, research on warranty since 1950 has been 1

2

Chapter 1

carried out, in addition, by researchers from such diverse disciplines as economics, accounting, management, marketing, engineering, operations research, and statistics. The literature on warranty is very extensive and is scattered across many journals, proceedings, and other publications from different disciplines. It encompasses both empirical and theoretical studies, with a growing trend toward an analytical approach. We begin by attempting to define precisely what is meant by warranty. Many concepts and meanings are associated with the word warranty, as demonstrated by the following nonlegal definitions. “A warranty is the representation of the characteristics or quality of product” (Thorpe and Middendorf [1], pp. 53-54). “A warranty is an expression of the willingness of business to stand behind its products and services. As such it is a badge of business integrity” (NACAA [2], p. 1). Even in the legal sense, the word warranty has many different meanings, since there are many different types of warranties. We shall discuss this issue in a later section. For our purposes, we can view a warranty as a contractual obligation incurred by a manufacturer (vendor or seller) in connection with the sale of a product. In broad terms, the purpose of a warranty is to establish liability in the event of a premature failure of an item or the inability of the item to perform its intended function. The contract specifies the promised product performance and, when it is not met, the redress available to the buyer as compensation for this failure. Note that we shall use the terms manufacturer and seller (and occasionally, vendor) interchangeably. By these we mean the individual or organization that assumes financial responsibility for any future costs incurred as a result of the warranty. Similarly, buyer and consumer will be used interchangeably to designate the recipient of any compensation under warranty. In common usage, the terms warranty and guarantee are often used as synonyms. However, a warranty is a specific type of guarantee: a guarantee concerning goods provided to a purchaser by the vendor. A warranty is often viewed incorrectly as a service contract. A service contract, however, is voluntary, whereas a warranty is a part of a product purchase and is an integral part of the sale. The principal objective of this chapter is to give a broad introduction to the subject of warranty and a brief outline of the book. The organization of the chapter is as follows. In Section 1.2, we discuss the role of warranty. Here the importance of warranty to both manufacturer and consumer are illustrated. Section 1.3 gives a brief historical view and highlights the feature that product warranty is a relatively modern, twentieth century concept.

An Overview

3

Section 1.4 includes a sample of warranties for consumer durables and industrial products and some comments on their terms. These are used to illustrate some of the concepts involved. In Section 1.5, we review the different theories advocated for warranty. Section 1.6 discusses the relationship between warranty and product liability so as to highlight some related topics. We then move on to a simplified systems approach to study warranty in Section 1.7. This approach offers an effective methodology for studying the warranty process, and we indicate the main factors involved and their characterization. In Section 1.8, we briefly discuss the different viewpoints of the manufacturer and consumer. Finally, we outline the scope of the book in Section 1.9. 1.2

THE ROLE OF WARRANTY

The primary role of warranty is to ensure postpurchase remedy for consumers. It offers protection when an item, properly used, fails to perform as intended or as specified by the manufacturer. As such, the critical issues are (1) product performance and (2) the failure of an item to meet the required performance. We discuss these two issues. The performance of a product can vary considerably, from very simple to very complex. In the simplest case it is characterized by a binary variable representing two possible states: working or failed. Such a characterization is appropriate for small electronic or mechanical gadgets. This simple characterization may also be used to model satisfactory or unsatisfactory performance of a complex piece of equipment. An example is noise or vibration in a machine. Either it is below some acceptable level or not. In the latter case, the performance of the machine is not acceptable. Often, however, the product performance is complex, and its characterization involves many variables and a composite performance measure. We illustrate by means of two examples: Example 1.1: Consider a large mainframe computer bought by a banking operation. Acceptable performance measures would include variables such as speed of processing; reliability (measured through mean time between failures); mean time for recovery whenever a failure occurs; operating costs (including personnel); maintenance cost; flexibility (multiple users, compatibility, software issues), etc. Example 1.2: Consider the case where the government issues a contract for a fleet of new aircraft. Here the performance measures might include variables such as load-carrying capacity, fuel efficiency, speed; reliability, maintainability, availability, etc. The performance measure might specify

4

Chapter 1

the acceptable trade-offs between the different variables, with payment to the manufacturer related to the performance of the end product delivered. In the context of consumer goods, actual performance as well as levels of acceptability might vary from consumer to consumer. For example, the efficiency of a car would depend on the driving characteristics of the driver, and various levels of efficiency may be considered acceptable by different owners. Furthermore, product performance can involve varying degrees of subjective evaluation, as would be the case when people buy electronic sound systems. We conclude, then, that there are many dimensions to the concept of product performance. Our concern will be with those aspects of performance that are explicitly or implicity covered under product warranty. Whenever a consumer feels that the performance of a product is below expectation, there is a degree of dissatisfaction. One way of resolving this dissatisfaction is to seek redress through a warranty claim. The purpose of a warranty is to establish liability in the event of the product failing to perform its intended function. Thus, warranties play a critical role in resolving consumer dissatisfaction. The intent is to specify who— consumer, seller, manufacturer— is responsible for what and for how long. The outcome of a claim going in favor of or against the consumer depends on the terms of the warranty and other legal obligations. For example, misuse of product can nullify a claim and deprive the consumer of any compensation. Most often, a warranty claim is settled by the consumer and the manufacturer in a mutually satisfactory manner without any third party intervention. When this fails—for example, the consumer being unreasonable in his expectation or the manufacturer trying to avoid his responsibility (breaching the terms of warranty)—the only option for resolution of the consumer grievance is through a third party such as a government consumer protection agency or through the normal legal channels. For well-established products, the warranty serves a useful role in protecting consumers’ interests. If an item is defective and not performing satisfactorily, warranty ensures that the faulty item is either repaired or replaced by a new and nondefective item, either at a reasonable cost or, in many cases, at no cost to the consumer. This ensures value for money and hence protects the consumer’s interests. This is especially important in the context of complex products (for example, automobiles or expensive electronic goods), where the consumer is unable to evaluate the product performance before purchase either because of lack of knowledge or, more often, because the evaluation requires usage over an extended period of time. In this context warranty serves as a guarantee from the manufacturer to the consumer. At the same time, it protects the manufacturer’s interests

An Overview

5

in terms of requiring certain responsibilities on the part of the consumer (for example, proper use of product, adequate care, and so on) and also by restricting the liability of the manufacturer. Over the last two decades or so, warranty has assumed a new role as an advertising tool for the manufacturer. New and innovative products are often viewed with a degree of uncertainty by consumers at large. The uncertainty is reduced as more people buy the product and information about the product performance is spread by consumer publications or through word of mouth. This can often take a considerable period of time, resulting in slow sales during the early stages of product introduction. Sales may be accelerated by some signaling mechanism that conveys information to reduce the uncertainty or risk perceived by the consumer. Warranty serves as one such signal. Better warranty terms convey the message that the risk is low and hence induce the consumer to buy the product. In addition, warranty has become an instrument, similar to price or product performance, that manufacturers use as an effective advertising tool to compete with other manufacturers.

1.3

HISTORICAL VIEW

It is difficult to pinpoint the origins of warranty exactly, as the law of warranty has evolved from many fields of English law over the last 600 years and from the opinions of American courts during the twentieth century. If warranty is equated to product liability, then one can trace it as far back as 1800 b .c ., when Hammurabi, the King of Babylon, compiled his great code of laws. This code provided stringent penalties for craftsmen found guilty of making defective products. However, the buyer of the product received no compensation, and the law was more in the nature of a deterrent. In the fourteenth century, warranty was a special kind of express representation of fact (Ebright [3], p. 13). In this sense it was similar to the modern concept of fraud. For approximately 400 years thereafter, the action for breach of warranty rested heavily on establishing some form of deceit on the part of the manufacturer. The industrial revolution brought a major change in the method of producing items, and this in turn affected their consumption. As new products were developed, the general rule of caveat emptor (“let the buyer beware”) was generally accepted for both manufacturers and consumers as a fair basis for product transaction. This worked reasonably well as long as the products and their operations were simple and the consumer could understand and evaluate product performance before purchase.

6

Chapter 1

The end of the nineteenth and start of twentieth century saw a dramatic change. Products not only became more complex but became available to larger segments of society in industrialized nations. The courts began to make exceptions to the rule of caveat emptor. Around this time, standardized warranties were first introduced. Initially, these warranties were treated as normal contracts, and the law viewed them as agreements between two informed and competent parties. As such, court decisions were based strictly on the terms of the warranty without any consideration as to whether or not the terms were fair. As time progressed, the courts refused to enforce the terms of the standardized warranty for reasons that will be discussed later. The net consequence of this was the legislation of new codes and acts to protect the consumer. The current law in the United States is laid down by the Uniform Commercial Code (U.C.C.) and the Magnuson-Moss Warranty Act. We briefly outline some of the salient features of these. Although the U.C.C. was promulgated primarily to govern mercantile transactions, Article Two also controls consumer transactions and hence is important for product warranty. The Code includes express warranties (U.C.C. §2-313); implied warranties of “merchantability” (U.C.C. §2-314 (l)-(2 )) and “fitness for a particular purpose” (U.C.C. §2-315); and a few implied warranties. The express warranty provision of U.C.C. §2-313 states that: (1)

(2)

Express warranties by the seller are created as follows: (a) An affirmation of fact or promise made by the seller to the buyer which relates to the goods and becomes part of the basis of the bargain creates an express warranty that the goods shall conform to the affirmation or promise. (b) Any description of the goods which is made part of the basis of the bargain creates an express warranty that the goods shall conform to the description. (c) Any sample or model which is made part of the basis of the bargaining creates an express warranty that the whole of the goods shall conform to the sample or model. It is not necessary to the creation of an express warranty that the seller shall use formal words such as ‘warrant’ or ‘guarantee’ or that he have a specific intention to make a warranty, but an affirmation merely of the value of the goods or a statement purporting to be merely the seller’s opinion or commendation of the goods does not create a warranty.

Note that the emphasis is on the word bargain, and the two parties (manufacturer or supplier and consumer or buyer) can change it by mutual consent.

An Overview

7

The implied warranty of merchantability (U.C.C. §2-314) applies to sales by a merchant and is as follows: (1) (2)

(3)

Unless excluded or modified by (Section 2-316), a warranty that the goods shall be merchantable is implied in a contract for their sale if the seller is a merchant with respect to goods of that kind. . . . Goods merchantable must be at least such as (a) pass without objection in the trade under the contract description; and (b) in the case of fungible goods, are of fair average quality within the description; and (c) are fit for the ordinary purposes for which goods are used; and (d) run, within the variations permitted by the agreement, of even kind, quality and quantity within each unit and among all units involved; and (e) are adequately contained, packaged, labeled as the agreement may require; and (f ) conform to the promises or affirmations of fact made on the container or label if any. Unless excluded or modified (Section 2-316) other implied warranties may arise from course of dealing or usage of trade.

The implied warranty of fitness for a particular purpose (U.C.C. §2-315) is as follows: Where the seller at the time of contracting has reason to know any particular purpose for which the goods are required and that the buyer is relying on the seller’s skills or judgment to select or furnish suitable goods, there is an implied warranty that the goods shall be fit for such a purpose.

The Code is aimed primarily at commercial transactions and hence is essentially suppletory, not regulatory. The rationale for this is that “merchants” are expected to have sufficient knowledge and bargaining power to protect themselves in commercial transactions. Consumers usually do not have the expertise of merchants and almost always lack the bargaining power with respect to the substantive content of warranties. Consumers therefore need the aid of regulatory laws that impose obligations that cannot be disclaimed or varied by the party with greater bargaining power and that compensate for consumer ignorance. The Magnuson-Moss Warranty Act does this. The purpose of the Act is set out in Section 102, which states: In order to improve the accuracy of information available to consumers, prevent deceptions, and improve competition in the marketing of consumer

8

Chapter 1 products, any warrantor warranting a consumer product to a consumer by means of a written warranty shall, to the extent required by rules of the Commission, fully and conspicuously disclose in simple and readily understood language the terms and conditions of such warranty.

Section 110 of the Act sets out a further purpose: Congress hereby declares it to be its policy to encourage warrantors to establish procedures whereby consumer disputes are fairly and expeditiously settled through informal dispute settlement mechanisms.

The Magnuson-Moss Warranty Act defines two types of warranties: full warranty and limited warranty. A full warranty is a warranty that conforms to minimum standards laid down in Section 104(a) of the Act, which reads: In order for a warrantor warranting a consumer product by means of a written warranty to meet the Federal minimum standards for warranty— (1) such warrantor must as a minimum remedy such consumer product within a reasonable time and without charge, in case of a defect, malfunction, or failure to conform with such written warranty; (2) not withstanding Section 108(b), such warrantor may not impose any limitation on the duration of any implied warranty on the product; (3) such warrantor may not exclude or limit consequential damages for breach of any written or implied warranty on such product, unless such exclusion or limitation appears on the face of the warranty; and (4) if a product (or a component thereof) contains a defect or malfunction after a reasonable number of attempts by the warrantor to remedy defects or malfunctions in such product, such warrantor must permit the consumer to elect either a refund for, or replacement without charge of, such product or part (as the case may be). The Commission may by rule specify for purposes of this paragraph, what constitutes a reasonable number of attempts to remedy particular kinds of defects or malfunctions under different circumstances. If the warrantor replaces a component part of a consumer product, such replacement shall include installing the part in the product without charge.

A limited warranty is one that does not meet the requirements of a full warranty. Here again, no supplier may disclaim or modify any implied warranty to a consumer with respect to such consumer product. It is important to note that the law does not require a manufacturer to offer a full warranty. If one is offered, it could result in free replacement for unlimited duration and include incidental damages unless specifically excluded. In the latter case, the warranty is no longer a full warranty,

An Overview

9

but a limited warranty. With the passage of the Magnuson-Moss Act, virtually all manufacturers changed their warranties to read “LIMITED WARRANTY” ! 1.4 SOME EXAMPLES OF WARRANTY

All products sold in the marketplace are covered by implied warranties, and the law does not require the manufacturer to offer an express warranty. Many factors determine the manufacturer’s decision regarding whether or not to offer an express warranty. Once a decision is made to offer an express warranty, important considerations include the following: 1. 2. 3. 4. 5.

Duration of the warranty Parts and labor coverage Consumer obligations under the warranty Warranty servicing mechanism Dispute resolution

What constitutes a fair level of warranty coverage is difficult to define and varies from product to product. Each product must be considered on its own merits and on the basis of reasonable consumer expectations. Exhibits 1-10 are a small sample of warranties offered by manufacturers of consumer durables and industrial products. Exhibits 1 and 2 (Bogart and Fink [4]) are two warranties circa 1930. Exhibit 1 gives the standard warranty for an automobile of this era. The salient features of this warranty are (1) only 90 days coverage, and (2) the need for the consumer to deal directly with component manufacturers to seek redress for certain types of components. Exhibit 2 gives the warranty for one such component, viz., a battery. Here the warranty has a structure that combines two basic consumer warranty concepts, free-replacement and pro-rata warranties, which will receive considerable attention in this text. In both these policies (Exhibits 1 and 2), one can see that the terms favor the manufacturer, with the consumer’s interest being a decidedly secondary issue. Exhibits 3-6 are four warranties dealing with consumer durables. The first belongs to the pre-Magnuson-Moss Act period (i.e., pre-1975) and the remaining three belong to the post-Magnuson-Moss Act period. Exhibit 3 is a copy of the warranty on a small household appliance in the pre-Magnuson-Moss era. A failed item will be replaced free of charge at the retail outlet. For relatively inexpensive items such as this, the Magnuson-Moss Act had little impact (except for possible language clarification in some cases). Exhibits 4 and 5 are examples of warranties on

10

Chapter 1

Exhibit 1: THE STANDARD AUTOMOBILE WARRANTY [Circa 1930] Warrant each new motor vehicle manufactured by us, whether passenger or commercial vehicle, to be free from defects in material and workmanship under normal use and service, our obligation under this warranty being limited to making good at our factory any part or parts thereof which shall, within ninety (90) days after delivery of such vehicle to the original purchaser, be returned to us with transportation charges prepaid, and which our examination shall disclose to our satisfaction to have been thus defective; this warranty being expressly in lieu of all other warranties expressed or implied and of all other obligations or liabilities on our part, and we neither assume nor authorize any persons to assume for us any other liability in connection with the sale of our vehicles. This warranty shall not apply to any vehicle which shall have been repaired or altered outside our factory in any way or so as, in our judgment, to affect its stability, or reliability, nor which has been subjected to misuse, negligence or accident, nor to any commercial vehicle made by us which shall have been operated at a speed exceeding the factory rated speed, or loaded beyond the factory rated load capacity. We make no warranty whatever in respect to tires, rims, ignition apparatus, horns or other signaling devices, generators, batteries, speedometers or other trade accessories in as much as they are usually warranted separately by their respective manufacturers. (From Ref. 4.)

appliances sold soon after the passage of the Magnuson-Moss Act. Both still use the term full warranty. Subsequently, most warranties have been called limited, which is a term used in all but one of the remaining exhibits. Warranties often specify two or more coverages. For example, Exhibit 5 lists separate terms, depending on use. In Exhibit 6, a television warranty, the coverage changes after 90 days and a separate coverage is specified for the picture tube. Exhibit 7 shows a part of the warranty specifications on a 1992 automobile. It is instructive to compare this with the 1930 model. Here separate warranties are offered on certain components with varying coverage and warranty periods, and tires are warranted separately by the supplier. The spare tire warranty is given in Exhibit 8. This is a rather complicated warranty for so seemingly simple an item, based neither on usage nor on calendar time, but on wear!

An Overview

Exhibit 2: THE STANDARD PASSENGER CAR BATTERY WARRANTY [Circa 1930] The manufacturer guarantees to repair or replace at its options f.o.b. factory or any authorized service station, without charge to the user, except transportation, any battery of its manufacture which fails to give a satisfactory service within a period of ninety days from date of sale to the user. ADJUSTMENT WARRANTY: The manufacturer further agrees after expiration of the ninety days guarantee period to replace with a new battery, on a pro-rata basis, any battery which fails in normal passenger service. Normal passenger car service is considered not over 1,000 miles per month. The adjustment period to be established by the manufacturer based on the quality of the battery, but in no case to exceed 18 months. All adjustments are to be based on the current list price plus transportation charges. Example: A battery carrying 12 months adjustment warranty listing at $12.00, fails in service on 9 months from date of purchase. The user receives a new battery of the same type for 9/12 of $12.00 or $9.00 plus transportation charges. (From Ref. 4.)

11

12

Chapter 1

Exhibit 3: Consumer Durable [Pre-M agnuson-M oss Act].

Copyright © Sears, Roebuck and Co. Reproduced with the permission of Sears, Roebuck and Co.

13

An Overview

Exhibit 4: Consumer Durable [Full Warranty] [Pre-M agnuson-M oss Act].

FULL ONE-YEAR WARRANTY* This electric heater is warranted against defects in material and workmanship for one* (1) year from date of original purchase. Titan will repair or replace at its option when the defective unit is delivered prepaid during that period to an authorized service station or to: Titan Sales Corporation 16th and Lamine Streets, Sedalia, Missouri 65301 (Note: Please indicate the defect on the packing slip where possible). This warranty does not apply to commercial use, unreason­ able use, or to damage to the product (not resulting from defect or malfunction) while in the possession of the consumer. The foregoing warranty has been drafted to comply with the new Federal law applicable to products manufactured after July 4, 1975. It replaces the warranties included else* where in this package. This warranty in no material re* spect reduces the coverage provided to you under the warranties it replaces.

This warranty gives you specific legal rights, and you may also have other rights which vary from state to state. Titan Sales Corporation 36th & Bennington, Kansas City, Missouri 64129 *An additional 9 year warranty on the element is extended for m odels 166E, 267B, 267C, 368C, 368E, 469A and 490.

14

Chapter 1

Exhibit 5: Consumer Durable With a Separate Commercial Warranty [Post-M agnuson-M oss Act].

Full One Year Warranty (Dom estic Use)

Full Ninety Day Warranty (Comm ercial Use) (Floor Care €r Sm all Appliances) Your Hoover appliance is warranted in normal household use, in accordance w ith the instruction book, against original defects in material and workmanship for a period of one year from date of purchase. In commercial or rental use, the period of warranty is ninety days. This warranty provides at no cost to you, all labor and parts to place this appliance in correct operating condition during the warranty period. W arranty service can only be obtained by presenting the appliance to one of the following authorized warranty service outlets. 1. Hoover Factory Service Centers. 2. Hoover Authorized W arranty Service Dealers. This warranty does not cover pick up, delivery, or house calls; however, if you mail your appliance to a Hoover Factory Service Center for warranty service, transportation will be paid one w ay under this warranty. W hile this warranty gives you specific legal rights, you may also have other rights which vary from state to state. If there are any questions concerning this warranty, or the availability of warranty service outlets, write or phone the Consumer Affairs Department, The Hoover Company, 101 East Maple Street, North Canton, Ohio (44720). Phone (216) 499-9200. THE HOOVER CO M PAN Y

An Overview

15

Exhibit 6 Consumer Durable [P o st-M ag n u so n - Moss Act]. LIMITED WARRANTY TO ORIGINAL PURCHASER MODEL LKS339C This A S T R A product is warranted by S A M S U N G E L E C T R O N IC S A M E R IC A INC. against manufacturing defects in materials for the period specified. PARTS 1 YEAR

L A B O R (CARRY-IN ) 90 D A Y S

P IC T U R E T U B E 2 YEARS

S A M S U N G will repair or replace (at our option) at no charge, any part(s) found to be defective during the labor warranty period specified at time of sale. This warranty period starts on the date of purchase by the original consumer. The warranty repairs must be performed at S A M S U N G ’S Authorized Service Sta­ tion. A list of S A M S U N G ’S Authorized Service Stations can be obtained from the place of purchase. OBLIGATION OF THE ORIGINAL OWNER 1. 2.

The Dealer’s Original Dated Bill of Sales must be retained, as a proof of purchase and must presented to the Dealer’s Authorized Service Station. Transportation to and from the Service Station is the responsibility of the customer.

EXCLUSION OF THE WARRANTY The warranty does not cover accident, misuse, fire, flood and other Act of God, incorrect line voltage, damage caused by improper installation, improper or unau­ thorized repair, antenna broken or marred cabinet, missing or altered serial numbers and customer adjustments that are not covered in the instruction book. This warranty is valid only on products purchased and used in the United States of America. Som e states do not allow the exclusion of limitation of incidental or consequential damages, or allow limitations on how long an implied warranty lasts, so the above limitations or exclusion may not apply to you. This warranty gives you specific legal rights, and you may also have other rights which vary from state to state.

16

Chapter 1

Exhibit 7 (a) Table of Contents, Automobile Warranty [1992].

An Overview

Exhibit 7 (b) List of Coverages, Automobile Warranty [1992]. Warranty Service________________________________________________________ Warranty (US Cars) The following warranties are provided with every new vehicle: 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

New Car Limited Warranty Emission Control System s Defects Warranty Em issions Performance Warranty Original Equipment Battery Limited Warranty Rust Perforation Limited Warranty Accessory Limited Warranty Replacement Parts Limited Warranty Replacement Muffler Lifetime Limited Warranty Seat Belt Limited Warranty Replacement Battery Limited Warranty

Your car’s original tires are covered by their manufacturer. Tire warranty information is in a separate tire warranty booklet. All warranty details pertaining to your car can be found in the Warranty booklet provided with your new car except the required maintenance schedule which is contained here in the Owner’s Manual. If you are unable to obtain warranty service or are dissatisfied with the warranty decision or service you received at an authorized Honda dealership, you should review the matter with that dealership’s Service Manager. This will normally resolve your problem. If it does not resolve your problem you should appeal the decision with the owner of the dealership. Please bear in mind that your problem will likely be resolved at the dealership, using the dealer’s facilities, equipment, and personnel. So it is very important that your initial contact be with the dealer or his management. After following these steps, if you wish to have the matter reviewed by American Honda you should contact the Zone Office serving your area, as shown on the map on the inside of the back cover. When contacting American Honda, please provide the Zone Office with the following information: • • • • • • •

Vehicle Identification Number Servicing Dealer Name and Address Date of Purchase Mileage on your Car Your Name, Address, and Phone Number Nature of Problem Selling Dealer

After a review of the facts, you will be advised of what can be done. (Canada Cars) Please refer to the 1992 Warranty and Maintenance Guide that came with your car. (Used with permission of American Honda Motor Co., Inc.)

17

C hapter 1

18

Exhibit 8 Tire Warranty [1986]. LIMITED WARRANTY DUNLOP ORIGINAL EQUIPMENT SPACE MISER (T-TYPE SPARE TIRE) ADJUSTMENT POLICY WHAT IS WARRANTED Every Dunlop original equipment “S P A C E M IS E R ” (T-TYPE S P A R E TIRE) that becomes unserviceable for conditions other than those which are listed under “W H AT IS N O T C O V E R E D ” will be replaced at a cost based upon tread wear. REPLACEMENT COST N O C H A R G E — If, during the first two thirty-seconds of an inch (2/32") of tread wear, the tire becomes unserviceable for a condition covered by this warranty, it will be replaced with a new Dunlop “S P A C E M IS E R ” (T-TYPE S P A R E TIRE) of equivalent size. No charge will be made for mounting, balancing or taxes. P R O -R A T A — If, after the first two thirty-seconds of an inch (2/32") of tread wear, the tire becomes unserviceable for a condition covered by this warranty, your Dunlop dealer will replace it at a cost calculated in the following manner: 1.

2.

3.

The total number of thirty-seconds of an inch (/32") tread worn, including the first two thirty-seconds. times ( x ) The adjustment Unit Charge shown in Dunlop’s current dealer price list. The Adjustment Unit charge reasonably reflects the tire prices being charged at retail for the replacement tire, and includes all applicable federal excise taxes. plus ( + ) All applicable local taxes and all charges for dealer services such as mounting and balancing. A C A S H R E F U N D W ILL N O T B E E X T E N D E D IN LIEU O F T H E ABO VE.

DURATION OF WARRANTY A s long as there are at least two thirty-seconds of an inch (2/32") of original tread depth remaining. Beyond this point the tire has delivered its original tread life and there is no warranty regardless of its age or mileage. WHAT IS NOT COVERED • Tires worn beyond the last two thirty-seconds of an inch (2/32") of original tread depth. • Tires on cars normally operated outside the U.S.A. • Tires used in racing or at speeds in excess of 50 M.P.H. • Tires fitted to anything other than the original vehicle. • Tires used in commercial applications. • Tires used for off-the-road service, on trailers, and in any application not rec­ ommended by the vehicle manufacturer. • Claims made by anyone other than the first purchaser of the vehicle. • Tires improperly repaired, injected with a sealant, or whose sidewalls have been modified by the addition or removal of materials, or in which anything other than air has been used as the supporting medium. • Tires rendered unserviceable by road hazard-type damage such as impact breaks, punctures, cuts or snags; or as a result of obstruction on the vehicle, fire, corrosives, running while flat, misalignment, inflation other than 60 P.S.I., overloading, improper mounting or rim fitment; or by spinning, as in mud, snow, sand, or ice or during on-the-vehicle balancing.

19

An Overview Exhibit 8 (cont.) O W N E R O B L IG A T IO N S . MAINTAIN 60 P SI IN Y O U R ‘S P A C E M IS E R ’. . N E V E R E X C E E D 50 M P H ON Y O U R ‘S P A C E M IS E R ’. • F R E Q U E N T L Y C H E C K INFLATIO N P R E S S U R E W ITH A T IR E G A U G E A N D IN S P E C T F O R D A M A G E O R IR R E G U L A R W EAR. • Y O U A R E R E S P O N S IB L E F O R P R O P E R T IR E C A R E A N D P R U D E N T V E ­ H IC LE O P ER A T IO N . Follow the Important Safety Instructions given later in this pamphlet. FOR REPLACEMENT CONSIDERATION • You must present the tire to a participating Dunlop dealer. Consult the yellow pages of your phone book for locations. Should you be unable to contact a dealer, the nearest Dunlop Sales Division Office is listed on the reverse side of this publication. • Except for tires which become unserviceable during the first two thirty-seconds of an inch (2/32") of tread wear, you must pay the adjusted price of a new tire. • Except for tires which become unserviceable during the first two thirty-seconds of an inch (2/32") of tread wear, you must pay all applicable federal excise taxes and local taxes and all charges for dealer services such as mounting and balancing. • You must fill out and sign Dunlop’s Standard Claim Form. LEGAL RIGHTS N O IM P L IED W A R R A N T IE S, E IT H E R O F M E R C H A N T A B IL IT Y O R O T H E R ­ W ISE, A R E E X T E N D E D B E Y O N D T H E T IM E W H EN T H E T IR E H A S D E L IV E R E D ITS O R IG IN A L T R E A D LIFE A S SH O W N B Y W E A R TO O N E O R M O R E T R E A D W E A R IN D IC A T O R B A R S. D U N L O P SH A L L N O T B E R E S P O N S IB L E (1) F O R A N Y C O M M E R C IA L LO SS, (2) F O R A N Y D A M A G E TO, O R L O S S O F P R O P E R T Y O T H E R THAN T H E T IR E ITSELF, O R (3) TO T H E E X T E N T P E R M IT T E D B Y LAW, F O R A N Y O T H E R T Y P E O F IN C ID EN T A L O R C O N S E Q U E N T IA L D A M A G E S. Som e states do not allow limitations on how long an implied warranty lasts, or the exclusion of incidental or consequential damages, so the above limitation or exclusion may not apply to you. This warranty gives you specific legal rights, and you may also have other rights which vary from state to state. MODIFICATIONS NO DEALER, D IST R IB U T O R O R R E P R E SE N T A T IV E H A S AU THO RITY TO M A K E A N Y C O M M IT M EN T , P R O M IS E O R A G R E E M E N T B IN D IN G U P O N DUNLOP, E X ­ C E P T A S S T A T E D H EREIN .

Exhibits 9-12 are a sample of warranties for industrial products. Exhibit 9 indicates different warranties for different types of applications— domestic, commercial, or industrial. Exhibit 10 is an unusual one in that it offers a full warranty as opposed to a limited warranty. Exhibit 11 shows not only the initial warranty but also the extended warranty, which covers only parts for four years. Exhibit 12 is again unusual in the sense that it has 25 years coverage with the buyer paying an increasing fraction of the cost with passage of time.

C hapter 1

20

Exhibit 9 Commercial and Industrial Product.

Onon

MANUFACTURER S LIMITED WARRANTY

Onan extends to the original purchaser of goods for use. the following warranty covering goods manufactured or supplied by Onan. subject to the qualifications indicated. T H ER E IS NO O T H ER E X P R E S S W ARRANTY. IM PLIE D W A R R A N T IE S IN C LU D IN G M E R C H A N T A B ILIT Y A N D F IT N E S S FOR A PA R T IC U LA R PU RPO SE. A R E LIM IT ED TO P E R IO D S OF W A R R A N T Y SE T FORTH BELO W A ND TO TH E EXTENT PER M IT T E D BY LAW. A NY A N D ALL IM PLIED W A R R A N T IE S A R E E XC LU DED . IN NO EVEN T IS O N A N L IA B L E FOR IN C ID E N T A L OR C O N S E Q U E N T IA L DA M A G ES. Note: Some states do not allow limitations on how long an implied warranty lasts, so the above limitations may not apply in every instance. (1)

Onan warrants to original purchaser for the periods set forth below that goods manufactured or supplied by it will be free from defects in workmanship and material, provided such goods ‘ire installed, operated, and maintained in accordance with Onan's written ihstructions. and further provided, that installation inspection and initial start-up on commercial-industrial generator set or power system installations are conducted by an Onan Authorized Distributor or its designated service representative.

PRODUCT APPLICATION

PERIOD OF WARRANTY

| Goods used in personal, family and household applications.

One (1) year from date of purchase.

|

| Goods used in commercial-industrial applications.

One (1) year from date of purchase.

(

| Commercial-industrial stationary generator sets.

I

One (1) year from date of initial start-up.

| | Commercial-industrial, standby power systems with nominal 1— operating speeds of 1800 rpms or less which are installed in the U.S. or Canada (must include Onan supplied generator sets, automatic transfer switch, exerciser and running time meter).

• Five (5) years or 1500 hours, whichever occurs first from the date of initial start-up Labor allowance for the first two (2) years or 1500 hours, whichever occurs first from the date of initial start-up.

|__ | Commercial-industrial, standby power systems with nominai operating speeds of 1800 rpms or less which are installed outside the U.S. or Canada (must include Onan supplied generator set. automatic transfer switch, exerciser and running time meter).

• Two (2) years or 1500 hours, whichever occurs first from the date of initial start-up.

| | Repair or replacement parts.

Ninety (90) days from date of purchase, excludes labor.

* Must be registered within thirty (30) days of initial start-up on Form No. 23C065. to be provided and completed by seller ’ Optional engine coolant heaters are warranted for one (1) year only. (2)

Onan's sole liability and Purchaser's sole remedy for a failure of goods under this warranty and for any and all other claims arising out of the purchase and use of the goods, including negligence on the part of the manufacturer, shall be limited to the repair of the product by the repair or replacement, at Onan's option, of parts that do not conform to this warranty, provided that the product or parts are returned to Onan's factory at 1400 73rd Avenue N.E., Minneapolis. Minnesota 55432, or to an Onan Authorized Distributor or its designated service representative, transportation prepaid

21

An Overview

Exhibit 9 (cont.) Except as indicated below, this warranty does not include travel time, mileage, or labor for removal of Onan product from its application and reinstallation a)

Removal and Reinstallation Onan will pay the following stated labor at straight time only for warranty work requiring removal and reinstallation of Onan Products in the following applications, provided, such warranty labor is performed by an Onan Authorized Distributor or its designated service representative: i- On-Highway Recreational and Commercial Vahicla Applications—Up to a maximum of one (1) hour for Onan engineered Power Drawer* or slide-out type generator set installations. — Up to a maximum of two (2) hours for all other permanent type generator set installations. ii. Marina Product Installations—Up to a maximum of four (4) hours for all single and two cylinder engine powered Marine Generator Sets installed below-deck. — Up to a maximum of eight (8) hours for all four and six cylinder engine powered Marine Generator Sets installed below-deck.

iii. All single and two cylinder gasoline, and single cylinder diesel industrial engines— Up to a maximum of two (2) hours. iiii. All four cylinder gasoline, two and four cylinder diesel industrial engines— Up to a maximum of four (4) hours. b)

Travel Time and Mileage i. Marina Ganarator Sat Installations— Onan will lo r six (6) months afterdate of purchase, pay travel time up to two and one half (2'/i)) hours and mileage cost up to one hundred (100) miles on generator sets with a kilowatt (kW) rating of fifteen (15) or less, and up to six and one half (6V4) hours and mileage cost up to two hundred fifty (2S0) miles on generator sets with a kilowatt (kW) rating above fifteen (15) for related warranty repairs, provided, such travel and repairs are performed by an Onan Authorized Distributor or its designated service representative. ii. Industrial Floodlightar Generator Sets—Onan will for six (6) months after date of purchase, pay travel time up to two and one half (2%) hours and mileage cost up to one hundred (100) miles for related warranty repairs, provided, such travel and repairs are performed by an Authorized Distributor or its designated Service Representative. iii. Commercial-Industrial Standby Generator Set and System Installations—Provided the generator set or system is permanently wired in a stationary installation. Onan will, tor six (6) months after initial start-up. pay travel time up to two and one half (2'4) hours and mileage cost up to one hundred (100) miles on generator sets with a kilowatt (kW) rating of seventeen and one half (17.5) or less, and up to six and one half (6V£) hours and mileage cost up to two hundred fifty (250) miles on generator sets with a kilowatt (kW) rating above seventeen and one half (17.5) and for transfer switches used with industrial standby generator set and system installations, for warranty repairs performed by an Onan Authorized Distributor or its designated service representative.

(3) All claims must be brought to the attention of Onan or an Authorized Distributor or its designated service representative within thirty (30) days after discovery that goods or parts fail to meet this warranty. (4) T H IS W AR R A N T Y SH A LL NOT A PP LY TO: a) Cost of maintenance, adjustments, installation and start-up b) Failures due to normal wear, accident, misuse, abuse, negligence or improper installation. c) Products which are altered or modified in manner not authorized by manufacturer in writing. d) Failure of goods caused by defects in the system or application in which the goods are installed. e) Telephone, telegraph, teletype or other communication expenses. f) Living and travel expenses of persons performing service, except as specifically included in Section 2. g) Rental equipment used while warranty repairs are being performed. h) Overtime labor requested by purchaser. i) Starting batteries. No person is authorized to give any other warranties or to assume any other liabilities on Onan's behalf, unless made or assumed in writing by an officer of Onan, and no person is authorized to give any warranties or assume any other liability on behalf of Seller unless made or assumed in writing by Seller. (5) This warranty gives the user specific legal rights, and the user may also have other rights which vary from state to state

(From Ref. 12.)

22

C hapter 1

Exhibit 10 Industrial Product. Rollei of Am erica, Inc.

(From Ref. 12.)

< R o ilei

(From Ref. 12.)

Exhibit 11 Industrial Product. An Overview

23

24 (From Ref 12.)

H& %h

Exhibit 12 Industrial Product.

An Overview

25

1.5 THEORIES OF WARRANTY

The theories of warranty aim to either answer or explain one or more of the following issues: 1. What determines the content of a warranty? 2. How do the contents relate to reliability or durability of the goods? 3. When is there an equilibrium in the warranty market? 4. What is the effect of Government intervention on the warranty market in the context of public policy? 5. What are the social welfare implications of warranties? Three different theories which attempt to deal with these issues have been proposed. Priest [5] calls these: 1. The exploitative theory 2. The signal theory 3. The investment theory Theories (1) and (3) are also dealt with by Schwartz and Wilde [6], who refer to (1) as the market power theory and (3) as the comparative advantage theory. In this section we give a brief outline of each of these three theories. For more details and references to original sources, see the notes at the end of the chapter. 1.5.1

The Exploitative Theory

The exploitative theory evolved in a legal framework and was developed mainly by scholars in the legal profession. Its origins can be traced to the initial standardized warranties offered in the last decades of the ninteenth century and the prevailing view of warranties as contracts. A normal contract rests on the assumption that the two parties to the contract are well informed and competent. The starting point of the exploitative theory is that standardized warranty, viewed as a contract, is unique because its terms are drafted by the seller unilaterally and only involuntarily “adhered to” by the consumer. The seller, being in a superior position, has the advantage of incorporating terms that serve his interests rather than those of the consumer. There was a certain element of truth to this, and this formed the basis for decisions of the courts not to uphold and enforce the

26

Chapter 1

terms of warranties and to make exceptions. Some scholars argued that warranties were devices to build up industrial empires, leading to monopoly and collusion between sellers in industries with multiple sellers. In early years, the main thrust of the theory was that without judicial intervention to imply warranty of quality in sales transactions, many manufacturers would provide consumers with “worthless junk.” The exploitative theory predicts that manufacturers will limit their legal obligations to consumers as much as possible, and they will especially attempt to exclude coverage of risks that are difficult to calculate. Later on, the theory focused on the market power gained from coordinating advertising that makes extravagant promises to consumers with warranties that disclaim responsibility for these promises. Some scholars have argued that standardized contracts are instruments of this type of fraud. Although there is some support for the contention that the consumer’s interests are not fully protected, the theory has serious drawbacks. It fails to explain why firms would choose to exploit market power by shifting risks to buyers rather than raising prices if buyers prefer to pay higher prices. Also, the theory is inconsistent with the fact that warranty coverage and market power are uncorrelated. The exploitative theory found wide acceptance, because it was the only coherent theory of warranty until 1970 and formed the basis for many court decisions. 1.5.2

The Signal Theory

The signal theory has its origins in the economic literature on “markets” for information and maintains that warranty terms provide information to consumers about product reliability. According to the theory, a consumer is often unable to determine product reliability at the time of purchase by direct inspection, due either to high cost or lack of expertise. In such a situation, a consumer may view warranty as a “signal” of product reliability, because reliability is correlated negatively with the cost of warranty coverage. Schwartz and Wilde [7] list the five basic assumptions on which the theory rests. They are as follows: 1. 2. 3.

Consumers cannot distinguish among competing products based on their likelihood of failure. Consumers believe that product quality correlates positively with the extent and duration of warranty coverage. The cost to firms of offering warranties varies inversely with product quality.

An Overview

4. 5.

27

If firms do not signal their level of product quality, then consumers will assume that the average quality in the market is low. Consumers can costlessly observe the prices and terms of every firm in the market, i.e, zero search cost.

Based on these assumptions, the theory predicts that firms with better quality can offer warranty as signals, and firms with poor quality cannot duplicate them because of the high cost of complying. Empirical studies of 108 warranties by Gerner and Bryant [8] show some mild support for the theory, and the theory has exerted substantial influence on consumer product warranty policy. One of the aims of the MagnusonMoss Act, described in Section 1.3, is to make warranties more efficient signals. The theory has serious shortcomings. It is inconsistent with the real world for two reasons. Firstly, the theory predicts that warranty coverage and product quality have a strong positive correlation, and this does not seem to be the case in the real world (see Priest [5]). Secondly, the theory predicts that better products would be sold with longer warranties, whereas in actuality the warranties are often identical irrespective of product quality. Also, the assumption of zero search cost is not valid (see Schwartz and Wilde [7]). 1.5.3

The Investment Theory

The investment theory, first proposed by Priest [5], is based solely on the relative costs to consumers and manufacturers to insure against product defects and to prolong productive capacity or duration of the product. In this theory, a warranty is viewed both as an insurance policy and a repair contract. As an insurance policy, if the product becomes defective within a certain period, the manufacturer compensates the buyer by repair, replacement, or refund. As a repair contract, it imposes an obligation on the manufacturer to provide services to repair defects (often without charge) in order to prolong the productive capacity of the item. A warranty acts as insurance to the extent that it views the occurrence of product defects as being probabilistic. As a repair contract, it reflects the respective costs to consumer and manufacturer of repair services, with repair by consumers and manufacturers being viewed as substitutes. This implies that some repairs are best done by the manufacturer and others by the consumer. In this framework, a warranty allocates responsibility to the manufacturer for those types of repair that are difficult and burdensome for consumers themselves to provide. This, in turn, forces on the manufacturers certain obligations to invest in good design and quality control

28

Chapter 1

and requires more effective care on the part of consumers. Thus, the warranty as a contract divides responsibility for allocative investment and insurance between consumers and manufacturers. Schwartz and Wilde [7] list the seven assumptions on which the theory rests. They are as follows: 1. 2. 3. 4. 5. 6. 7.

Firms can reduce the costs of defects because of their expertise (e.g., fixing a defective motor in a refrigerator). Consumers can better ensure the duration of service by suitable actions (e.g., the life of a refrigerator door can be extended by careful use). Consumers are perfectly informed as to the risks of product defects and know what steps are needed to reduce this risk. Search costs (for both price and warranty terms) are zero. Consumers minimize net purchase costs. Manufacturers maximize profits. Some consumers demand broader warranty coverage than others.

Based on this, the theory predicts that firms will offer optimal warranties. As in the case of the first two theories, this theory also has serious shortcomings. The main difficulties are assumptions (3) and (4). In the real world these are not valid. When these two assumptions are dropped, Schwartz and Wilde [7] show that the theory cannot provide unambiguous explanations of warranty behavior. They develop a modification that takes into account nonzero cost for obtaining information. Comment: None of the preceding theories offers a complete or satisfactory explanation for the issues listed in the beginning of the section. It is doubtful that any comprehensive theory for warranty behavior can be developed, since the warranty process is a very complicated one. Some further observations on this topic will be offered in the last chapter of the text. 1.6 WARRANTY A ND PRODUCT LIABILITY

“Product liability can be explained as the legal obligation of a manufacturer or vendor to indemnify persons who have suffered bodily injury or property damage from a defective product or representation about the product by the manufacturer or vendor” (Chandran and Linneman [9]). A manufacturer or vendor can be held liable under the following: 1. Negligence 2. Breach of warranty 3. Strict liability 4. Misrepresentation

An Overview

29

We have discussed warranty in earlier sections. In this section we discuss briefly the remaining three bases for liability. 1.6.1

N eg lig en ce

Negligence can be defined as a breach of the duty of reasonable care on the part of the manufacturer or vendor. The duty to exercise reasonable care extends to all parts of the production and distribution process. Redress can be sought for the following: 1. 2. 3. 4. 5.

Improper design Improper manufacture Failure to inspect Failure to warn Failure to foresee possible damage

As an example, if a manufacturer relies on retailers to inspect the product before sale, the manufacturer can be held negligently liable if the retailers are lax. Similarly, retailers (vendors) are held negligently liable if they fail to warn whenever they have knowledge about a product’s dangerous condition and it appears that the consumer will not discover the danger. 1.6.2

Strict Liability

Strict liability developed as a response to a complex, consumer-oriented society in which buyers and users acquired products from manufacturers through a series of intermediate sellers (Morgan [10]). Strict liability is a tort action, and the manufacturer is held liable when it can be established that the product was defective when it left the manufacturer and the injury or damage occurred when the product was being used in a reasonable manner. As such, strict liability represents the melding of warranty and negligence principles, the quality of the product being under question and not the breach of reasonable care. The basic statement of strict liability is set forth in the Restatement (1965, Section 402 A): (1) One who sells any product in a defective condition unreasonably dangerous to the user or consumer or to his property is subject to liability for physical harm thereby caused to the ultimate user or consumer, or to his property, if (a) The seller is engaged in the business of selling such a product, and (b) It is expected to and does reach the user or consumer without substantial change in the condition in which it is sold.

Chapter 1

30

(2) The rule stated in Subsection (1) applies although (a) the seller has exercised all possible care in the preparation and sale of his product, and (b) the user and consumer has not bought the product from or entered into any contractual relation with the seller. 1.6.3

M isrepresentation

The common theme behind action under negligence, breach of warranty, and strict liability is the defective product. Without a defective product, recovery is not allowed. The exception is tort recovery under misrepresentation. It is set forth in the Restatement (1965, Section 402 B): One engaged in the business of selling chattels who, by advertising, labels, or otherwise, makes to the public a misrepresentation of material fact concerning the character or quality of a chattel sold by him is subject to liability for physical harm to a consumer of the chattel caused by justifiable reliance upon the misrepresentation, even though (a) it is not made fraudulently or negligently, and (b) the consumer has not bought the chattel from or entered into any contractual relation with the seller.

The manufacturer is held liable if someone is injured who relied on false representation, even if these were made honestly based on laboratory and consumer research, or even if the product itself is not defective but unrealistic performance claims were made about it. 1.6.4

Econom ic Im p a c t of Product Liability

Figure 1.1 (from Dungworth [11], p. 34) shows the number of suits filed in U.S. Federal Courts over the period 1974-1986 for a select grouping of industries. It shows an increasing trend. The cost figures indicate a similar trend. The following statistics (from Morgan [10], p. 69) illustrate the point: Manufacturers’ and retailers’ liability insurance jumped from $1.3 billion in 1975 to $2.75 billion in 1978, and the average bodily injury settlement rose from $6800 in 1972 to $19,500 in 1974. This is a matter of serious concern to both manufacturers and lawmakers and hence will be a topic of considerable interest for years to come. 1.7

THE SYSTEMS APPROACH TO WARRANTY

The systems approach to warranty offers a unified approach to the study of warranty. The first step in the systems approach is system characteri-

An Overview

Figure 1.1 Ref 11.)

31

Federal product liability filings for different industry groups. (From

zation. This is a process of simplification and involves identifying the important factors of interest and their interrelationships. Each factor in turn might involve one or more variables as well as interactions among the variables. In this section, we discuss a conceptual system characterization that will serve to highlight the scope of the book. (A more comprehensive system characterization is given in the last chapter.) The important factors are as follows: 1. Manufacturer 2. Consumer 3. Product 4. Product performance 5. Warranty policy The manufacturer produces products and sells them to consumers with a warranty policy attached. Product performance is determined by the interaction between product characteristics (determined by manufacturer) and product usage (determined by consumer). When the consumer is not

32

Chapter 1

satisfied with the performance of a product, a claim under warranty results. The cost incurred by the manufacturer of servicing a claim under warranty is called warranty cost, and this depends on the terms of the warranty policy. Figure 1.2 shows a schematic representation of the different factors and their interrelationships in its most simple form. We discuss each of the factors in some detail. 1.7.1

Manufacturers

Many variables characterize a manufacturer. Some of these are as follows: 1. 2. 3. 4. 5. 6. 7.

Size Organizational structure Management objectives Marketing orientation Reputation Range of products produced Quality control philosophy

For the study of warranty, all of these are important. However, from the warranty point of view, we need to differentiate two cases. The first corresponds to a monopolistic, or near monopolistic, situation characterized by a single manufacturer or a relatively small number of manufacturers. Manufacturers of very specialized equipment with limited sales would fall into this category. Typical examples are manufacturers of (1) heavy earth-

Figure 1.2

A simplified system characterization of the warranty process.

An Overview

33

moving equipment used in road construction; (2) large electrical machinery such as generators; (3) weapons systems or components; and so on. The second case corresponds to a more competitive situation characterized by a large number of manufacturers with very little scope for collusion. Typical examples are manufacturers of consumer durables such as television sets, dishwashers, small appliances, tools, etc. In the context of warranty, the latter case would force manufacturers to offer warranties that are fairer to the consumer due to competition, while in the former case there are no such pressures. A good example is the automobile industry, where in the late sixties and early seventies the warranties being offered were becoming less favorable to the consumer. This trend was altered with the penetration of the market by Japanese cars, which offered terms more favorable to the consumer. As a result, terms such as two or three years and increased mileage are fairly common. 1.7.2

Consumers

For our purposes, consumers may be grouped into three different categories. The first corresponds to the case of a single consumer A typical example of such a consumer is the federal government as a consumer of certain types of defense related products such as rockets, navy ships, etc. Here the consumer is more powerful than the manufacturer, has a strong input into product development, and may dictate the product support subsequent to product sale. In fact, until fairly recently, these items were procured without warranty. Over the last fifteen years or so, Congress has become progressively more insistent that warranties be part of the procurement process. The second case corresponds to a relatively small number of consumers for the product. Typically, the consumers are industrial or commercial organizations or government agencies buying very specific industrial or commercial products. Typical examples are (1) airline companies buying aircraft; (2) manufacturing firms purchasing large machine tools; and (3) cities and towns buying fire fighting equipment, police cars, trash trucks, etc. The U.C.C. also covers transactions between manufacturer and consumers belonging to this category, including government agencies. Finally, the last category corresponds to the case where there are a large number of consumers, who, in general, are not well informed about the technical aspects of the product and have no direct influence on the design of the product or the warranty terms offered. An average citizen buying consumer durables would be the typical example of this category. The Magnuson-Moss Warranty Act was aimed primarily at educating these consumers and looking after their interests.

34

Chapter 1

1.7.3

Products

Products can be categorized into four groups as follows: 1. 2. 3. 4.

Specialized defense related products Industrial and commercial products Consumer durables Consumer nondurables

Specialized defense-type products (for example, military aircraft and ships, tanks, rockets, etc.) are characterized in the domestic market by a single consumer, the federal government (or by a small number of consumers, if international sales are included), and a relatively small number of manufacturers. The products are usually complex and expensive, and they involve state of the art technology with considerable research and development effort from the manufacturers. As a result, the warranties for such items are typically very complex and involved and are bid and negotiated as a part of the government procurement process. Industrial and commercial products are characterized by a relatively small number of consumers and manufacturers for such products. The technical complexity of such products and the mode of usage by the user can vary considerably. In general, the consumers for goods of this type are well informed about the technical aspects of the product and hence are in a better position to negotiate fairer terms of warranty. The products can be either complete units such as cars, trucks, pumps, and so forth, or parts such as, for example, batteries, tires, and light bulbs needed by an auto manufacturer. The quantities involved can vary from small (orders of hundreds) to very large (orders of millions). Similarly, the prices of such products can vary over several orders of magnitude. Consumer durables (typical examples: stereo systems, television sets, automobiles, kitchen and laundry appliances) and nondurables (typical examples: food items, cosmetics, paper goods) are consumed by society at large and hence characterized by a large number of consumers for each product. The complexity of the product can vary considerably, and in general the typical consumer often is not well informed about the product. Consumer durables vary considerably in their functional use and their prices. Small domestic appliances, such as coffee grinders or can openers, cost a few tens of dollars, while items such as cars, furniture, and expensive stereo systems can cost several thousand dollars. It is this latter category for which the Magnuson-Moss Act ensures that consumers’ interests are adequately protected. Many nondurables are relatively cheap (costing less than $5.00) and as such are not sold with warranty. However, manufacturers can be held liable

An Overview

35

under negligence, strict liability, or misrepresentation. Typical examples are actions resulting from (1) food poisoning due to negligence in preparation; or (2) failure to warn about possible adverse reactions to a suntan lotion, to name a couple. 1.7.4

Warranty Policies

Warranty policies can be either simple or complex, depending on the type of product being covered by the warranty and the coverage offered. Many different types of policies exist, and we shall list them in the next chapter. In the context of consumer durables, two types of warranty policies have been used extensively. These are the free-replacement policy and the prorata policy. In the free-replacement policy, the manufacturer agrees to either repair or provide a replacement at no cost to the consumer up to a specified time from the time of initial purchase. Such a policy is typically offered with products that are repairable— for example, television sets, automobiles, and appliances. In the pro-rata warranty policy, replacements are provided at pro-rata cost. Such a policy is usually offered with nonrepayable items such as tires, batteries, tools, etc. 1.7.5

Product Performance

The performance of a product depends on product characteristics and the mode of usage by the consumer. Product characteristics in turn depend on the design and manufacturing decisions made by the manufacturer. The mode of usage, in its simplest characterization, can be either normal or abnormal. The latter represents the case where the user uses the product in a manner for which it has not been designed—for example, operating a machine at speeds above the safe limit; using a flashlight to drive a nail; using a microwave to heat food in a metal container. The degree of misuse (or abuse) of a product can vary. Failures occurring due to abnormal use are not usually covered by warranty. However, in certain instances, the manufacturer can be held liable for either not having foreseen a usage that caused the damage or failure to warn of the dangers of certain types of usage. Product performance can be assessed objectively in some cases, whereas in many others a fair degree of subjective evaluation by the consumer plays an important role. A typical example involving a high degree of subjective evaluation is the sound quality of a musical instrument. In certain instances, the environment can have a significant impact. An example is poor picture quality in a television set due to a geographical location where a hill blocks the signal reaching the set. Note that in this case, the consumer can take action against a vendor under the implied warranty of fitness for a particular

36

Chapter 1

use if the vendor was informed of the problem of reception before the set was bought and the buyer depended on the vendor’s judgment. When a consumer feels dissatisfied with the performance of the item purchased, the warranty offers an avenue for redress. When a claim is made under warranty, the manufacturer is legally obliged to respond to it. Failure to do so would constitute a breach of warranty, and the consumer then can initiate legal action against the manufacturer. 1.7.6

Warranty Costs

Whenever there is a claim under warranty, the manufacturer incurs a cost. If the claim is not valid, then the only cost is the administrative cost of handling the complaint. A claim is not valid if it is not covered by the warranty, if the warranty has expired, if the claim is bogus (i.e., the item has, in fact, not failed as claimed), or if the warranty ceases to apply due to consumer misuse of the product. If the claim is valid, there are additional costs. These include the cost of labor and parts for repairable items; replacement by a new item or part for nonrepairable items; incidental costs such as shipping costs, and, in some instances, additional costs to compensate the consumer for having been deprived of the use of the item while it is undergoing repair. A typical example of such an additional cost is the cost of providing a replacement for the duration of repair. When a dispute arises between a consumer and a manufacturer, additional costs are incurred by both parties— e.g., time and effort spent in resolving the dispute and, in extreme cases, legal fees and court costs. In general, the total warranty cost for each unit sale is unpredictable, because product defects are random and consumer usage varies significantly. Table 1.1 (McGuire [12], Table 5, p. 13) shows warranty costs, as a percentage of net sales, in various product sectors. The results are based on a survey of 369 U.S. manufacturing companies of industrial products. Warranty servicing costs vary from less than 1% to 10% or more, with the majority of firms having these costs less than 5%. Table 1.2 (McGuire [12], Table 7, p. 15) shows warranty servicing costs, as a percentage of net sales, as a function of warranty duration, based on the same survey. Again, the majority of firms have costs less than 5% for all warranty durations from three months to five years. 1.8

THE STUDY OF WARRANTY

The study of product warranty is of importance to both manufacturers and consumers, although the reasons and the motivation for such study can be different for a number of reasons.

23%

26

26

24%

37

20

11

Less than 1.0%

1.0-1.9% 2.0-2.9%

3.0-4.9%

100%

100%

aD eta ils do not add to total because o f rounding. Source: R ef. 12.

Total3

1

10.0% or more

0

3

3

4

4

5.0-6.9%

7.0-9.9%

20

Business equipment

100%

0

2

0

20

38 20

20%

Electrical equipment

100%

0

0

3

10

24

41

21%

Heating and refrigeration

100%

0

2

0

13

22

33

33%

Machinery products

100%

0

12

12

16

16

21

23%

Scientific equipment and controls

Type of product manufactured

Warranty Servicing Costs in Different Product Sectors.

All companies

Warranty servicing costs as a percentage of net sales

T a b le 1-1

100%

0

4

2

2

17

54

21%

Mobile equipment and components

100%

3

1

4

7

16

38

30%

All other products

An Overview 37

0

5

3

100%

aD eta ils do not add to total because o f rounding. Source: R ef. 12.

100%

1

10.0% or more

Total3

5

7.0-9.9%

18

11

44

3.0-4.9%

5.0-6.9%

13

20

45

37

1.0-1.9%

2.0-2.9%

18%

24%

0 -3

Below 1.0%

All warranties

100%

0

3

3

17

14

40

23%

4 -9

100%

1

4

5

10

22

34

25%

100%

0

0

0

0

22

67

11%

13-18

100%

0

6

0

6

24

29

35%

19-24

Length of warranty period (in months) 10-12

Warranty Servicing Costs Versus Warranty Duration.

Warranty servicing costs as a percent of net sales

T a b le 1.2

100%

0

0

0

14

14

43

29%

25-36

100%

0

0

0

33

0

33

33%

49-60

38

Chapter 1

An Overview

1.8.1

39

The M anufacturer’s View point

A strong motivating factor for any manufacturer is the desire to maximize profits. Offering a warranty results in additional cost due to servicing of the warranty but at the same time, if used properly as a marketing tool, increases sales and hence revenue generation. Warranty servicing costs depend on product characteristics and the usage patterns of consumers. If the extra revenue generated exceeds the warranty servicing costs, then it is more sensible to sell the product with a warranty. As a result, manufacturers are interested in the study of warranty in order to seek answers to some or all of the following questions: 1. 2. 3. 4. 5. 6. 7.

What is the cost of offering a specific warranty policy? How does this compare with other warranty policies? How does the warranty cost change with the parameters of the policy (for example, duration and form of a pro-rata warranty)? How does one optimize the choice of warranty when multiple business objectives are involved? What is the optimal strategy for servicing a warranty? (This would involve, for example, establishing a policy with regard to repair/replace decisions in the case of repairable items.) What kinds of data (laboratory, field, etc.) are needed to decide warranty policy, and how should the data be analyzed? What are the optimal decisions with regard to product design and manufacture, given that the product must be sold with a specific type of warranty policy as dictated by the marketplace?

Any study of warranty that aims to answer one or more of the preceding questions would necessarily be based on an analytical approach involving mathematical models. Without such a study, the consequences can be costly or catastrophic. The costs involved with product recall is one such case. See, for example, Dardis and Zent [13] on the economics of the Pinto recall. 1.8.2

The Consumer’s View point

For a consumer, the difficult problem is the choice between products with different characteristics and warranty policies. In addition, in situations where the warranty is optional, a buyer would like to know if the warranty is worth the additional cost. This is becoming more and more important, since there is a growing trend among manufacturers to offer extendedterm warranties. These involve additional costs, and the terms can vary considerably. For example, a warranty or extended warranty might cover

Chapter 1

40

both labor and parts initially and only cover parts later in the warranty period. The consumer has to decide, often at the time of purchase and based on very limited information, whether to opt for an extended warranty or not and to determine the best extended terms for his situation when there are multiple options. For consumers of industrial and commercial products, these costs can have a significant impact on profits. As mentioned earlier, consumers of such products often have the skills and expertise, and the bargaining power, to demand relevant data from manufacturers to carry out such analyses. Here again, the most effective and appropriate approach is the analytical approach involving mathematical models. In the case of consumer durables, it is beyond the scope of a typical consumer to carry out such analyses, because the consumer neither has the skills and expertise for analysis nor the bargaining power to obtain relevant data from the manufacturer. However, consumer bureaus and regulatory agencies can carry out such analyses and inform the consuming public. Here too, data collection can be a problem in many instances. 1.9

AIM AND SCOPE OF THE BOOK

The objective of this book is to develop a framework for the analytical study of various issues related to warranty and of importance to manufacturers and consumers. The study will deal with the modeling and analysis of a variety of warranty policies and some related optimization problems. Some of the warranty policies studied are currently offered by manufacturers of both consumer durables and industrial products. Others are new but should be of interest to both manufacturers and consumers. Our study focuses on warranty based on the system characterization indicated in Figure 1.2. We confine ourselves to product performance that is described in simple terms. The performance is satisfactory if the item is in a working state and unsatisfactory if it is in a failed state. This is a meaningful criterion for a large number of consumer durables and industrial products. We ignore many features such as misuse of the product, disputes leading to legal actions, behavioral aspects of consumers, marketing, and informational aspects. The modeling and analysis for the simplified approach taken in this book involves complex mathematical formulations and challenging data collection and parameter estimation problems. Finally, our study deals mainly with warranty policies for consumer durables, but we do discuss some policies appropriate only for industrial and commercial products. Our objective in this book is to present a unified treatment of the technical literature on warranty, with emphasis on analytical models of warranties and their associated costs.

An Overview

41

A comprehensive framework to study warranty in greater detail, with all aspects included, is discussed in the last chapter. Such a study will need an interdisciplinary approach, and the modeling and analysis would be considerably more complex than that in this book.

NOTES

Section 1.1 1. Consumer activism on a national level can be identified with Ralph Nader’s battles with the automobile manufacturers in the early sixties. Some interesting survey data on consumer opinion on warranties, done in 1976-1977, can be found in NACAA ([2], p. 1). A 1977 poll (Harris [14]) reported that only 7% of the American public felt that the auto industry was doing a good job in servicing consumers. For more on liabilities associated with servicing, see Morgan [15] and the references cited therein. Section 1.3 1. For an interesting historical discussion of warranties, see Ebright [3]. Section 1.4 1. McGuire [12] contains a large number of warranties for industrial products, with interesting comments on each. Section 1.5 1. Our presentation of the different theories has been very brief. The literature relevant for a better understanding is vast, and most of it is cited in Priest [5] and Schwartz and Wilde [6,7]. Section 1.6 1. There are many books on product liability— see, for example, Sherman [16]; Owles [17]; and Thorpe and Middendorf [1]. Most of the books are written for people in the legal profession. 2. The issue of product liability in the context of marketing and planning has become very important. Further details on this can be found in Morgan [10] and Chandran and Linneman [9]. Section 1.7 1. Product design, manufacturing, modeling of item failures, etc., belong to the “hard” physical sciences, while consumer behavior, marketing, etc., belong to the “soft” social sciences. The study of warranty requires the effective blending of the hard and the soft sciences. 2. There are many books on the systems approach— see, for example, Checkland [18] and Wilson [19]. 3. The warranty literature on each factor indicated in Figure 1.2 is very extensive. Some of this literature is cited in Chapter 14.

42

Chapter 1

EXERCISES

1.1.

1.2. 1.3. 1.4. 1.5. 1.6. 1.7. 1.8.

Find at least five warranties, and determine their terms exactly. Include in your description of the warranties at least the following: (a) buyer’s and seller’s responsibilities; (b) separate warranty coverages (e.g., for certain components), if any; (c) the form of compensation to the buyer should the item fail in the warranty period; (d) the length of the warranty period(s). Develop a taxonomy to group the warranties in Exhibits 1-12 and in Exercise 1.1. Devise an appropriate warranty for electronic equipment used in a medical application. Formulate a warranty policy for consumer durables that reflects the incentive theory point of view. Indicate how the policy incorporates incentives for consumer and manufacturer. How might the flow chart of Figure 1.2 be modified to incorporate dispute resolution involving an independent third party? This chapter has been concerned with product warranties. How would you define warranty for service (e.g., repair)? What additional issues are relevant in this context? Suppose that you are Warranty Manager for a large automobile dealership. Discuss the organizational structure that you would require to manage this activity in an efficient and effective manner. Discuss the collection and organization of relevant claims data in situations such as that of Exercise 1.7.

REFERENCES

1. 2.

3. 4.

Thorpe, J. F., and Middendorf, W. H. (1979). Product Liability, Marcel Dekker, Inc., New York. NACAA Product Warranties and Servicing (1980). National Association of Consumer Agency Administration and Society of Consumer Professionals in Business, U.S. Govt. Printing Office, Washington, D.C. Ebright, A. H. (1961). A Survey o f the Historical Setting— Past and Present— o f the Law o f Warranty, Master of Law Thesis, University of Southern California, Los Angeles, CA Bogart, G. G., and Fink, E. E. (1930). Business practice regarding warranties in the sale of goods, Illinois Law Rev., 25, 400-417.

An Overview

5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19.

43

Priest, G. L. (1981). A theory of the consumer product warranty, Yale Law J., 90, 1297-1352. Schwartz, A., and Wilde, L. L. (1983). Warranty market and public policy, Information Economics and Policy, 1, 55-67. Schwartz, A., and Wilde, L. L. (1983). Imperfect information in market for contract terms: The example of warranties and security interests, Virginia Law Rev., 69, 1387-1485. Gerner, J. L., and Byrant, K. W. (1981). Appliance warranties as market signals, J. Consumer Affairs, 15, 75-86. Chandran, R., and Linneman, R. (1978). Planning to minimize product liability, Sloan Management Rev., Fall 1978, 33-45. Morgan, F. W. (1982). Marketing and product liability: A review and update, J. Marketing, 46, 69-78. Dungworth, T. (1988). Product Liability in the Private Sector, Rand Corp., Santa Monica, CA. McGuire, E. P. (1980). Industrial Product Warranties: Policies and Practices, The Conference Board Inc., New York. Dardis, R., and Zent, C. (1982). The economics of the Pinto recall, J. Consumer Affairs, 16, 261-277. Harris, L., and Associates (1977). Consumerism at the Crossroads, Marketing Science Institute of Harvard University and Lou Harris Associates, Boston, MA. Morgan, F. W. (1987). Strict liability and the marketing of services and goods: A judicial review, J. Public Policy and Marketing, 7, 4357. Sherman, P. (1981). Product Liability, Shepards/McGraw-Hill, Colorado Springs, CO. Owles, D. (1978). The Development o f Product Liability in the USA, Lloyd’s of London Press, Ltd., London. Checkland, P. (1981). Systems Thinking, System Practice, John Wiley and Sons, Inc., New York. Wilson, B. (1984). Systems; Concepts, Methodologies and Applications, John Wiley and Sons, Inc., New York.

2 Warranty Policy and Modeling Issues

2.1

INTRODUCTION

In Chapter 1, product warranty was discussed in a general context, and the need to study warranty from the points of view of both manufacturer and consumer was determined. In this chapter, we define the different types of warranty policies that are in common use or have appeared in the literature on warranty analysis. Since our study is analytical in nature, the first step is the building of mathematical models of the warranty process, which is the main focus of this chapter. The outline of the chapter is as follows. In Section 2.2, we propose a taxonomy to categorize the many different warranty policies that have been formulated, and we give a precise statement of each policy to be studied later. Section 2.3 gives a brief discussion of the analytical approach in general, while Section 2.4 focuses on the analytical approach in the context of the study of warranty. The remaining sections (Sections 2.5-2.10) deal with specific modeling techniques that are relevant for our study. These will form the building blocks for models to be developed in later chapters.

2.2

WARRANTY POLICIES

We begin with the formulation of a taxonomy of warranty structures. The purpose of this taxonomy is to provide a framework for organization of the many different policies that have been used for consumer and commercial warranties. Precise statements of each specific policy will be given in later subsections. 45

Chapter 2

46 2.2.1

Taxonomy

A taxonomy to classify the different types of warranty policies is given in Figure 2.1. Initially, the policies can be divided into two groups based on whether or not a policy involves product development after sale. Policies that do not involve product development can be further divided into two subgroups: Group A, comprising policies applicable for single item sales, and Group B, comprising policies applicable only for the sale of groups of items (also called block sales). Policies in Group A can be subdivided into two further subgroups based on whether the policy is renewing or nonrenewing. In a renewing policy,

Figure 2.1

Taxonomy for warranty policies.

Warranty Policy and Modeling Issues

47

whenever an item fails under warranty, it is replaced by a new item with a new warranty replacing the old one. In contrast, in the case of a nonrenewing policy, replacement of a failed item does not alter the original warranty. Thus, for renewing policies, the warranty period begins anew with each replacement, while for nonrenewing policies, the replacement item assumes the remaining time of the item it replaced. A further subdivision comes about in that warranties may be classified as simple or combination. The free replacement and pro-rata policies defined in Chapter 1 are simple policies. A combination policy is a simple policy combined with some additional features or a policy that combines the terms of two or more simple policies. As a result, we have four different types of policies under category A, which we have labelled as A 1-A 4 as shown in Figure 2.1. Each of the preceding four groupings can be further subdivided into two subgroups based on whether the policy is one dimensional or two (or more) dimensional. The dimension of a policy is the number of variables specified in defining the warranty limits. A one-dimensional policy is almost always based on either time or age of the item but could instead be based only on usage. In contrast, a typical two-dimensional policy is based on time or age as well as usage. As an example, in the case of automobile warranties, a warranty involving only a time limitation (two years) would be a one-dimensional warranty, while a warranty involving time and usage limitations— for example, a policy specifying “two years or 30,000 miles, whichever comes first” — would be a two-dimensional warranty policy. Policies in Group B can be subdivided into two categories based on whether the policy is simple or combination. These are labelled B1 and B2 in Figure 2.1. As in Group A, B1 and B2 can be further subdivided based on whether the policy is one dimensional or two dimensional. Finally, policies involving product development are labelled Group C. Warranties of this type are typically part of a maintenance contract and are used principally in government acquisition of large, complex items— for example, military equipment. We now give the precise details of a variety of warranty policies. For each we indicate, wherever possible, typical products sold with such a warranty.

2.2.2

One-Dimensional Group A Policies

SUBGROUP A1 [Nonrenewing Simple Policies] There are two policies belonging to this grouping. The first is as follows:

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48

Policy 1 F R E E - R E P L A C E M E N T P O L I C Y : Under this policy the manufacturer agrees to repair or provide replacements for failed items free of charge up to a time W from the time of the initial purchase. The warranty expires after time W from the time of initial purchase.

The time W is called the warranty period. The free-replacement warranty will be abbreviated FRW. The warranty is nonrenewing. Thus, in the case of nonrepair able items, should a failure occur at age X 1 (< W), then the replaced item has a warranty for a period (W - X x)— the remaining duration of the original warranty. Should additional failures occur, this process is repeated until the total service time of the original item and its replacements is at least W. In the case of repairable items, repairs are made free of charge until the total service time of the item is at least W. This type of policy is the most common of all consumer warranties and probably of commercial warranties as well. It is offered on a wide range of consumer products, ranging from inexpensive items such as photographic film to relatively expensive repairable items such as automobiles, refrigerators, large screen televisions, etc., and on expensive nonrepairable items such as microchips and other electronic components as well. Of the examples given in Chapter 1, Exhibits 1, 3, 4, 5, and 10 are straightforward FRWs and many of the warranties in Exhibits 7 and 9 are either FRWs or FRWs subject to special conditions. All of the warranties of the remaining exhibits of Chapter 1 include some FRW features. The second warranty in this grouping is as follows: Policy 2

P R O -R A T A P O L I C Y :

Under this policy the manufacturer agrees to refund a fraction of the purchase price should the item fail before time W from the time of the initial purchase. The buyer is not constrained to buy a replacement item.

This policy will be abbreviated PRW. The refund depends on the age of the item at failure (A^), and, for X 1 < W, it can be either linear or a nonlinear function of W - X l9 the remaining time in the warranty period. This defines a family of pro-rata policies, each of which is characterized by the form of the refund function. Three members of this family are as follows: Policy 2a: The refund is a linear function given by [(W - X^)IW\cb, where cb is the buyer’s purchase price.

49

Warranty Policy and Modeling Issues Policy 2b: The refund is a linear function given by 0 < a < 1.

[a (W

-

X x) / W ] c b,

where

Policy 2c: The refund is a nonlinear function given by [(W - X x)/W]2cb.

Policies such as 2a and 2b are sometimes offered on relatively cheap nonrepairable products such as batteries, tires, ceramics, etc. In application, however, this type of warranty is usually renewing and most often it is offered as a part of a combination warranty. SUBGROUP A2 [Nonrenewing Combined Policies] These are policies that are combinations of the free-replacement and pro-rata policies previously defined. Warranties of this type typically feature terms that change one or more times during the warranty period (e.g., full to limited warranty, or FRW to PRW). One such policy is the following: Policy 3

C O M B I N E D F R W A N D P R W P O L I C Y : Under this policy the manufacturer agrees to provide a replacement or repair free of charge up to time W x from the initial purchase; any failure in the interval W l to W (> W x) results in a prorated refund. The warranty does not renew.

The proration can be either linear or nonlinear. Again, depending on the form of the proration cost function, we have a family of combined freereplacement and pro-rata policies similar to that in Policy 2. Warranties of this type are sometimes used to cover replacement parts or components where the original warranty covers an entire system. Note that the general nature of the proration function allows for a good deal of flexibility in the structuring of Policy 3 warranties. For example, multistage warranties such as the following are included in this family: Policy 3a: Replacements or repairs are provided free of charge up to time W x after initial purchase, at cost C x if the failure time X is in the interval (Wj, W 2], and at cost C2 if X is in the interval (W2, W], where C x < C2 and 0 < W x < W2 < W.

Here Q and C2 may be random quantities that depend on the nature of the failure. Policy 3a is basically the warranty of Exhibit 6 of the previous chapter. The television is given full FRW for the first 90 days, parts only for the next nine months, and coverage on only the picture tube for the remaining year of the warranty. (The policy is considered nonrenewing, since labor coverage is not extended beyond 90 days in the event of early failure nor are parts covered under a new warranty. Presumably, a replacement picture tube would be covered under separate warranty.)

50

Chapter 2

The second policy of this type is modeled on that of Exhibit 12 of Chapter 1. The item is a stand-alone fireplace. An FRW covers the first twoyear period, after which the buyer pays only freight for an additional eight years. For the next five years, a replacement is provided at 50% of the then current price. This is increased to 75% for the final 10 years of a 25year warranty. (Even though a replacement would undoubtedly be covered under a new warranty, we have chosen to treat this as nonrenewing. It seems highly likely that warranty terms would differ to some extent after a couple of decades!) The general form of this type of policy is as follows: Policy 3b: Replacements or repairs are provided free of charge up to time Wl after initial purchase, at cost c^ C ^ ) if the failure time X is in the interval (Wl5 W2], at cost a 2C(X) if X is in the interval (W2, W3], and so forth, up to cost a kC(X) if X is in the interval ( W k, W], where a x < a 2 < • • • < a* are the proportions of the current price C(X) at time of failure for failures in each of k time intervals.

In the fireplace example, k is 3, Wx is 2 years, axC(X) may be interpreted as shipping cost in the interval from 2 to W2 = 10 years; a 2 = .5, W3 = 15, a 3 = .75, and W = 25. Another version of this is a combination lump sum rebate warranty, under which a refund which is a declining proportion of the original purchase price is given rather than a replacement item or a repair. This policy is given as follows: Policy 3c: A rebate in the amount is given for any item that fails prior to time Wx from the time of purchase; the rebate is a2cb for items that fail in the interval (Wx, W2], a 3cb for items that fail in the interval (W2, W3\, and so forth, up to a final interval (Wk_ x, W], in which the rebate is with 1 > a x > a 2 > • * • > a* > 0.

(Note that the ordering of the proportionality constants a, is reversed in Policies 3b and 3c. Here the a, indicate the decreasing magnitude of the rebate.) Policies such as 3b and 3c are widely used in warranties covering television sets or picture tubes, automobile tires, small and large appliances, and other consumer durables. Warranties in which parts and labor are covered initially and parts only later in the warranty period also fall in this general category. The warranty on a stand-alone fireplace in Exhibit 12 of Chapter 1 could also be interpreted as an example of a declining rebate policy. Note, incidentally, that if k = 1 and a x = 1, Policy 3b reduces to the ordinary FRW, while Policy 3c reduces to the rebate from of the FRW.

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If k > 1 and a 2 = 1, Policy 3b includes an initial free-replacement period, while Policy 3c provides a money back guarantee in the initial period. Still another version of Policy 3 is a combination lump sum and pro rata refund, given in the following: Policy 3d: Under this policy, the manufacturer agrees to provide a full refund of the original purchase price up to time W x from the time of initial purchase; any failure in the interval from W l to W (> W x) results in a pro-rata refund. The warranty does not renew.

To distinguish between this and other forms of the combined FRW/PRW, we shall call this the rebate combination warranty. The following combination policy is a modification of the free-replacement policy (Policy 1). It is particularly appropriate for items that are sold as spares and hence not used immediately after purchase. Policy 4

W A R R A N T Y W I T H S T O R A G E L I M I T A T I O N : The policy is characterized by two parameters w and 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 operation. Let X l denote the service life of the item. The item, upon failure, is covered under warranty only if and

If this condition is met, the coverage may be under FRW, PRW, or a combination warranty such as Policy 3.

This implies that a failed item may not be covered by warranty, even though X 1 is smaller than W, if S is sufficiently large. This policy is offered by a manufacturer of aircraft windshields and will be discussed in one of the case studies of Chapter 13. It is also a feature of the warranty of Exhibit 11 of Chapter 1. SUB GROUP A3 [Renewing Simple Policies] Two types of policies, analogous to those of Subgroup A l, are included in this subgroup. An example of the first of these is the following: Policy 5

F R E E - R E P L A C E M E N T P O L I C Y : Under

this policy the manufacturer agrees to either repair or provide a replacement free of charge up to time W from the initial purchase. Whenever there is a replacement, the failed item is replaced by a new one with a new warranty whose terms are identical to those of the original warranty.

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Under this policy, the buyer is assured of one item that operates for a period W without a failure. This type of policy is usually offered with inexpensive electrical, electronic, and mechanical products such as coffee grinders, alarm clocks, tools, and so forth, for which the warranty is contained inside the item package. Upon failure, the item is returned to the seller, who merely replaces it with an identical package. If the buyer simply returns the new warranty card, he or she gets a new warranty with each replacement. (Thus, the warranty is effectively renewed whether or not the manufacturer intended to do so!) The second simple renewing policy is the following: Policy 6

P R O - R A T A P O L I C Y : Under this policy the manufacturer agrees to provide a replacement item, at prorated cost, for any item (including the item originally purchased and any replacements made under warranty) that fails to achieve a lifetime of at least W .

Proration can be either a linear or a nonlinear function of W - X x, where X x is the age at failure and is less than W. Depending on the proration function, this again defines a family of pro-rata policies. Note: The difference between this and the pro-rata warranty of Policy 1 is that under Policy 1 the refund is unconditional, whereas here it is only provided as a discount on the purchase of a replacement item. Many nonrepairable items are sold with this type of policy or a combination having this following an initial FRW period. Most auto tires and batteries are sold under renewing PRW, the buyer being offered a replacement for a failed item at a reduced price, without a cash rebate option. SUBGROUP A4 [Renewing Combination Policies] These are policies based on combinations of the terms of two or more simple renewing policies. We describe two such policies. Policy 7

C O M B I N E D F R E E - R E P L A C E M E N T A N D P R O - R A T A P O L I C Y : Under this policy the manufacturer agrees to provide a replacement free of charge up to time from the initial purchase; any failure in the interval W 1 to W (> W J 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 identical to the original one.

This is the most common combination warranty. It is featured in Exhibits 2, 8, and several of the items of Exhibit 7 of Chapter 1.

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A great many additional renewing combination warranties could be defined. For example, for the relatively simple (and widely used) twoperiod version of Policy 7, two additional policies may be defined by having the warranty renew only in the first time period or only in the second. These are called partially renewing combination policies and will be discussed in somewhat more detail and analyzed to some extent in Chapter 6. In multiperiod policies, many additional versions (most of primarily academic interest) would be possible. The following is an example of a partially renewing warranty: Policy 7a: P A R T I A L L Y R E N E W I N G C O M B I N A T I O N F R W / P R W : Under this policy the manufacturer agrees to provide a replacement free of charge up to time W 1 from the time of the initial purchase. Replacement items in this time period assume the remaining warranty coverage of the original item. Failures in the interval to W (> W J are replaced at pro-rata cost. Replacement items in this interval are provided warranty coverage identical to that of the original item.

Under this policy, the warranty renews only in the PRW period. This means that upon failure of an item at, say, x < Wu a free replacement is provided, and this replacement item is covered under FRW until time Wl — 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 Wx < x f < W, the replacement is provided at prorated cost to the buyer and the warranty begins anew. In theory, the proration during the second period of this warranty may again be either linear or nonlinear. 2.2.3

Two-Dimensional Group A Policies

In the one-dimensional case, discussed in the previous subsection, a policy is characterized by an interval, called the warranty period, which is defined in terms of a single variable— e.g., time, age, usage. In the case of twodimensional warranties, a warranty is characterized by a region in a twodimensional plane with one axis representing time or age and the other representing item usage. As a result, many different types of warranties, based on the shape of the warranty coverage region, may be defined. We shall confine ourselves to two-dimensional versions of the policies in Subgroup A l. Policies for Subgroups A 2-A 4 are similar. SUBGROUP A1 [Nonrenewing Simple Two-Dimensional Policies] We shall define five different two-dimensional policies of type A l. For these policies, the warranty coverage regions in the two-dimensional plane

54

Chapter 2

are indicated as shaded regions in Figure 2.2. The first such policy is the following: Policy 8

F R E E - R E P L A C E M E N T P O L I C Y : Under this policy the manufacturer agrees to repair or provide a replacement for failed items free of charge up to a time W or up to a usage U , whichever occurs first, from the time of the initial purchase. The time W is called the warranty period, and U the usage limit. The warranty region is the rectangle shown in Figure 2.2(a).

Note that under this policy, the buyer is provided warranty coverage for a maximum time period W and a maximum usage U. If the usage is heavy, the warranty can expire well before W, and if the usage is very light, then the warranty can expire well before the limit U is reached. Should a failure occur at age X 1with usage Yx, it is covered by warranty only if X x is less than W and Y 1 is less than U. If the item is replaced by a new one, the replacement item is a warranted for a time period W X 1 and for usage U — Y v This type of policy is offered by nearly all auto manufacturers, with usage corresponding to distance driven. In the illustration of Exhibit 7, Chapter 1, there are several two-dimensional FRW’s: 36 months or 36,000 miles on the car; 12 months or 12,000 miles on the accessories; 5 years or 50,000 miles on the emission equipment, etc. (Note that this warranty package is actually more complicated than those under discussion in that it is only partially renewing— replacement parts are warranted separately with less coverage than that given the originals. This is a complication that will be ignored in our analysis.) Aircraft manufacturers also offer this type of policy, with usage corresponding to number of flying hours. The second two-dimensional Group A policy is the following Policy 9

F R E E - R E P L A C E M E N T P O L I C Y : Under this policy the manufacturer agrees to repair or provide a replacement for failed items free of charge up to a minimum time W from the time of the initial purchase and up to a minimum total usage U . The warranty region is given by two strips, as shown in Figure 2.2(b).

Note that under this policy the buyer is provided warranty coverage for a minimum time period W and for a minimum usage U. If the usage is heavy, the warranty will expire at time W, and if the usage is very light, the warranty will expire only when the total usage reaches the limit U. This type of policy would have some obvious advantages for the consumer. As far as we know, however, it is not offered by any manufac-

Warranty Policy and Modeling Issues

F igu re 2.2

Warranty regions for two-dimensional policies.

55

56

Chapter 2

turer at present. The reason is that a manufacturer would much prefer Policy 8. Policy 8 tends to favor the manufacturer, because it limits the maximum time and usage coverage for the buyer. For a buyer who is a heavy user, the warranty expires before time W due to the usage reaching U. Similarly, for a buyer who is a light user, the warranty expires at time W with the total usage below U. In contrast, Policy 9 favors the buyer. Here a heavy user is covered for a time period W, by which time the usage would have well exceeded the limit t/, and a light user is covered well beyond W, for the policy ceases only when the total usage reaches U. This implies that in the latter case the manufacturer has to carry spares or replacement units for a time period well beyond W. We now describe two policies that achieve a compromise between these two extremes. The first is as follows: Policy 10

F R E E - R E P L A C E M E N T P O L I C Y : Under this policy the manufacturer agrees to repair or provide a replacement for failed items free of charge up to a time W x from the time of the initial purchase, provided the total usage at failure is below U 2, and up to a time W 2, provided the total usage at failure does not exceed The warranty region is given by the region shown in Figure 2.2(c).

Note that under this policy the buyer is provided warranty coverage for a minimum time period Wx and for a minimum usage Ux. At the same time, the manufacturer is obliged to cover the item for a maximum time period W2 and for a maximum usage i/2. We are not aware of any product being sold with this type of warranty. However, such a policy may be of interest to manufacturers of automobiles, for example, since they can use the upper limits as promotional features in marketing their products. Note that some care must be exercised in choosing the values of and t/j. By proper choice of warranty parameters, the manufacturer’s warranty servicing costs can be made the same for both light and heavy users, so that the manufacturer would not have to differentiate between the two types of users in determining warranty costs. The second policy that is a compromise between Policies 8 and 9 is the following: Policy 11

Under this policy the manufacturer agrees to repair or provide replacements for failed items free of charge up to a maximum time W from the time of the initial purchase and for a maximum total usage of U . Let X = time since purchase and Y = total usage at failure. The item is

F R E E -R E P L A C E M E N T P O L IC Y .

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57

covered under warranty if [ Y + ( U I W ) X \ < U . If [ Y + ( U / W ) X ] > U , then the item is not covered by the warranty. The warranty region is given by the triangle shown in Figure 2.2(d).

Note that under this policy, warranty coverage extends for a maximum time period W and a maximum usage U. The time instant at which warranty ceases depends on the usage rate. We are not aware of any products being sold with this type of warranty. The policy would appeal to car buyers, since it gives better coverage for a wider spectrum of usage than the typical new car warranty. The final two-dimensional Group A policy that we shall consider is the following: Policy 12

P R O -R A T A R E P L A C E M E N T P O L IC Y :

Under this policy the manufacturer agrees to refund to the buyer a fraction of the original purchase price should the item fail before time W from the time of the initial purchase and the total usage at failure is below U. The fraction refunded depends on W — X 1 and U -

Y x.

As in the case of Policy 2, this leads to a family of policies, based on the form of the function that determines the refund. This function can be either linear or nonlinear. Two examples are Policy 12a:

and Policy 12b:

where

cb

is the purchase price per unit.

Note: The warranty coverage region for Policy 12 is given by the rectangular shown in Figure 2.2(a). Pro-rata policies for warranty regions shown in Figures 2.2(b)-(d) are defined similarly. SUBGROUP A2: [Nonrenewing Combination Two-Dimensional Policies] Many two-dimensional combined versions of the simple warranty policies are possible. The following is one example of this type: Policy 13

C O M B I N E D R E P L A C E M E N T P O L I C Y : Under this policy the manufacturer agrees to provide replacements for failed items free of charge up to a time W x

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from the time of the initial purchase provided the total usage at failure is below U 1. Any failure with time at failure greater than W l but less than W2, and/or usage at failure greater than U l but less than U 2, is replaced at a prorated cost.

The proration can be either a linear or nonlinear function of the warranty parameters and the age and usage at failure. The coverage area is identified in Figure 2.2(e).

2.2.4

One-Dimensional Group B Policies

The policies in this group are called cumulative warranties and are applicable only when items are sold as a single lot of n items and the warranty refers to the lot as a whole. The policies are conceptually straightforward extensions of the nonrenewing free replacement and pro-rata warranties discussed in Section 2.2.1. Under a cumulative warranty, the lot of n items is warranted for a total time of nW , with no specific service time guarantee for any individual item. Cumulative warranties would quite clearly be appropriate only for commercial and governmental transactions, since individual consumers rarely purchase items by lot. In fact, warranties of this type have been proposed in the United States for use in acquisition of military equipment. The rationale for such a policy is as follows. The advantage to the 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 contract. The advantage to the seller is that fewer warranty claims may be expected, because longer-lived items can offset early failures. Conceptually, the cumulative warranty seems a reasonable compromise between buyers’ and sellers’ interests. It provides warranty protection without unduly penalizing the seller for a few early failures. There are, however, some problems in implementing such a policy, and, to our knowledge, cumulative warranties are not in use at this time. Nonetheless, the approach is an interesting one, and we shall devote some attention to it. In our analysis, it will be assumed that the policies are nonrenewing. One of the initial problems one encounters in attempting to deal with cumulative warranties is formal specification of the warranty terms. Some of the many approaches that might be considered are discussed by Guin [1] (who provides the only analysis of this problem area of which we are aware). In this section we discuss a variety of cumulative versions of the simple FRW and PRW policies, based on the ideas of Guin, as well as one cumulative version of a combination policy. The following notation will be

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59

useful for expressing the terms of these policies:

SUBGROUP B1 [Cumulative Simple Policies] The following two policies are cumulative versions of the FRW of Policy 1: Policy 14

C U M U L A T I V E F R W : A lot of n items is warranted for a total (aggregate) period of n W . The n items in the lot are used one at a time. If S n < n W , 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 n W .

This type of policy is applicable to components of industrial and commercial equipment bought in lots as spares and used one at a time as items fail. Examples of possible applications are mechanical components such as bearings and drill bits. The policy would also be appropriate for military or commercial airline equipment such as mechanical or electronic modules in airborne units. In contrast to Policy 14, in the following it is assumed that more than one item is in use at a given time. Policy 15

C U M U L A T I V E F R W : A lot of n items is warranted under cumulative warranty for a total period of n W . Of these, k (< n ) are put into use simultaneously, with the remaining n - k items being retained as spares. Spares are used one at a time as failures occur. Upon failure of the nth item, free replacements are supplied as necessary until a total service time of n W is achieved.

The following are two policies, also due to Guin [1], are cumulative pro-rata warranties: Policy 16

C U M U L A T I V E P R W : A lot of n items is purchased at cost n c h and warranted for a total period n W . The n items may be used either individually or in batches. The total service time S n is calculated after failure of the last item in the lot. If S n < n W , the buyer is given a refund in the amount of c b( n - S J W ) , where cb is the unit purchase price of the item.

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Chapter 2

Rather than waiting for the last failure in the lot before a settlement is made (which could occasionally take a very long time), in the following policy a settlement is made after every ATth item failure. (The quantity K is an integer agreed upon by the buyer and the manufacturer and preferably is an integral divisor of n.) Policy 17

C U M U L A T I V E P R W : A lot of n items is warranted for a total time n W . The lot is divided into subsets of size K . Under a cumulative pro rata warranty, the refund to the buyer at the instant of the Afh failure in each subset is given by max{0, c b( K - 5^/W)}, where S K is the sum of the service times of the K failed items in the subset and cb is the unit purchase price of the item.

SUBGROUP B2 [Cumulative Combination Policies] Again, many combinations could be devised. We illustrate the possibilities by the following: Policy 18

C U M U L A T I V E C O M B I N E D F R W A N D P R W : A set of n items is warranted for a total time of n W . Upon failure of the final item in the group, the total service time S n is calculated. If S n < n W u where W x < W is a prespecified age, free replacements are provided until a total service time of n W x is achieved, say with Item n + J . Upon failure of this item, the buyer receives a rebate in the amount of cb[max{0, (n - 5„+//W)}]. If n W l < S n < n W , the buyer receives a rebate of c b(n - S J W ) .

2.2.5

Reliability Im provement Warranties

The basic idea of a reliability improvement warranty (RIW) is to extend the notion of a basic consumer warranty (usually the FRW) to include guarantees on the reliability of the item and not just on its immediate or short-term performance. This is particularly appropriate in the purchase of complex, repairable equipment that is intended for relatively long use. The intent of reliability improvement warranties is to negotiate warranty terms that will motivate a manufacturer to continue improvements in reliability after a product is delivered. This intent is explicitly stated in the following definition of RIW (Trimble [2]): A Reliability Improvement Warranty is defined as a provision in either fixedprice acquisition or fixed price overhaul in which: (a) the contractor is provided with a monetary incentive, throughout the period of the warranty, to improve the production design and engineering of the equipment so as to enhance the field/operational reliability and maintainability of the system/equipment; and

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61

(b) the contractor agrees that, during a specified or measured period of use, he will repair or replace (within a specified turnaround time) all equipment that fails (subject to specific exclusions, if applicable).

This approach to “buying” reliability was first used by the airlines in purchasing commercial aircraft. The idea was later adopted by the military, where it was considerably expanded in concept and much more widely applied. From these applications, four main components of an RIW have evolved: 1. 2. 3. 4.

A guaranteed mean time between failure (MTBF) Manufacturer supported engineering changes (including modification of existing units, if necessary) A guaranteed turnaround time for repaired or replaced units A supply of consignment spares for use by the buyer at no cost until the guaranteed MTBF is demonstrated

In applications of RIW, it is essential that all terms be defined precisely and that methods for computing performance measures be negotiated and specified in the contract. For example, MTBF could be computed as a simple point estimate or as the lower confidence limit at some specified level of confidence, and so forth. Under RIW, the contractor’s fee is based on his ability to meet the warranty reliability requirements. Not all of these features are used in all procurements, of course. Furthermore, the terms of an RIW are quite specific to the individual transaction, so it is quite difficult to formulate a general RIW policy. For purposes of illustration, we list the following two somewhat simplified examples: Policy 19

R E L IA B IL IT Y IM P R O V E M E N T W A R R A N T Y :

Under this policy the manufacturer agrees to repair or provide replacements free of charge for any failed 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 buyer, (1) engineering 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 engineering changes, and (4) consignment spares for buyer use until such time as it is shown that the MTBF is at least M (Gandara and Rich [3]).

The following RIW (Balaban [4]) provides for an initial period during which no MTBF guarantee is in effect followed by successive periods in which specific improvements in MTBF are required:

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Policy 20

R E L IA B IL IT Y IM P R O V E M E N T W A R R A N T Y :

Under this policy the manufacturer agrees to repair or provide replacements for any failed parts or units until time W after purchase. In addition, the manufacturer guarantees the MTBF of the purchased equipment to be as follows: no MTBF is guaranteed until time W l after date of first production delivery; during the period from W x to W 2 after first delivery, the MTBF is guaranteed to be at least M x\ from W 2 to W 3 the MTBF is guaranteed to be at least M 2\ and from W 3 to W the MTBF is guaranteed to be at least M 3 (with 0 < W x < W 2 < W 3 < W and 0 < M x < M 2 < M 3). If during any period the MTBF guarantee is not met, the manufacturer will provide, at no cost to the buyer, engineering changes and product modifications as necessary to achieve the MTBF requirements.

A variation of this (Kruvand [5]) allows the manufacturer some “free” failures at the outset: Policy 20a: A lot of n items is purchased with individual warranty periods W . Items that fail prior to W are repaired or replaced at the buyer’s expense until k such failures occur, after which the manufacturer will repair or replace failed items until each of the n items in the lot and its replacements achieve a total service time of W .

2.3

THE ANALYTICAL APPROACH

In this section, we look at several aspects of the analytical approach in general. This approach to problem solving typically involves three stages: Stage 1 Mathematical modeling Stage 2 Parameter estimation and model validation Stage 3 Analysis and optimization 2.3.1

M a th e m a tica l M odeling

In Chapter 1, we introduced the term system characterization as a simplified representation of the real world relevant to the study of interest. It involves identifying the variables and relationships among variables that are of importance. Mathematical modeling refers to the translation of a system characterization into a mathematical description suitable for analysis and optimization. In a sense, mathematical modeling can be viewed as linking the physical variables of a system characterization and their relationships to the variables and relationships of an abstract mathematical formulation. Without such a linking, the abstract mathematical formulation makes little sense outside mathematics.

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63

The type of mathematical formulation appropriate for model building depends on the system characterization. It time plays no part, then a static formulation suffices. On the other hand, if the variables of the system characterization change with time, then dynamic formulations are needed. If the variables of the system characterization change only at set time instants, then the independent variable of the formulation (representing time) assumes only discrete values, and hence the modeling would involve discrete time formulations (e.g., time series). In contrast, if changes can occur over a time continuum, then one would need continuous time formulations (e.g., differential equations) for modeling. Finally, depending on whether the variables of the system characterization change in a deterministic or uncertain manner, one would need either a deterministic framework or a stochastic framework for modeling. The complexity of the model depends on the framework (deterministic or stochastic) and the degree of detail incorporated into the system characterization. In general, deterministic models are simpler than their stochastic counterparts. 2.3.2

Param eter Estimation/Model V alidation

For a mathematical model to be of use in solving real-world problems, it is essential to establish that it represents the real world relevant to the problem in an adequate manner. An adequate model is defined as a model that provides a satisfactory representation of the phenomenon under study in the problem at hand. Not every model built is an adequate model. A model may be inadequate because (1) the system characterization is inadequate due to oversimplification, or (2) the mathematical formulation used in the modeling is inappropriate. It is necessary to validate a model before one can meaningfully apply the results to a specific real-life application. Validation requires a testing procedure where model behavior is compared with the real world in a meaningful way. In carrying out a model validation, it is necessary to assign specific numerical values to model parameters. The model parameters are the coefficients of the mathematical formulation used in modeling, and they correspond to certain physical parameters of the real world as it relates to the problem under study. The values to be assigned are determined by estimation methods and are based on data from the real world. Thus, we see that for both parameter estimation and for validation, we need real data. Often the data available are not the most suitable for the problem at hand (for example, having been collected for reasons very different from the current objective). Furthermore, data can be corrupted in various ways: improper sampling, missing data, recording inaccuracies,

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and so on. A variety of methods have been developed for both parameter estimation and model validation for different types of models and data structures. Some of these will be discussed in Chapter 12. 2.3.3

Analysis an d Optimization

There are two approaches to analysis and optimization: (1) mathematical and (2) computational. The techniques for analysis depend on the approach used. Only for very simple mathematical models can one obtain solutions as closed-form analytical expressions. For the majority of models, the computational approach is necessary. In the context of stochastic models, the computational approach involves simulating the time histories of changes in variables on a computer and drawing inferences based on the statistics obtained from a large number of repetitions of such simulations. 2.4 2.4.1

THE ANALYTICAL APPROACH TO WARRANTY STUDIES M odeling

In analyzing warranties, the appropriate framework for system characterization is a stochastic framework with time treated as a continuum. Since we confine ourselves to warranty claims resulting from random item failures, changes to variables occur at points along the time continuum in a random manner. We need to differentiate the first failure from subsequent failures, since the latter may depend on the type of rectification action used after the first failure. Modeling of the first failure can be done using basic probability theory, whereas the modeling of subsequent failures requires formulations belonging to the theory of stochastic processes—in particular, the theory of reliability and the theory of point processes. Reliability theory will be discussed in Section 2.5, and we shall discuss point processes and their use in modeling item failures in the next chapter. In order to carry out a thorough cost analysis, it is necessary to model many other issues as well. Some of the more important of these are alternative rectification actions, sales of items over time, product design and development, inspection, and burn-in. Several of these issues are discussed in the remainder of this chapter. Others will be dealt with in later chapters, as appropriate. 2.4.2

Param eter Estimation and M odel V alidation

A variety of methods have been developed for estimating model parameters in the context of point process formulations and different data structures. We postpone our discussion of this topic to Chapter 12.

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65

Analysis an d Optimization

Most of our analysis will deal with the expected warranty cost per unit sale and the expected cost over either item life or product life. Since our models are stochastic, a variety of analytical tools developed for the analysis of stochastic point processes are relevant. We shall discuss these in Chapters 3-8. As mentioned previously, it is necessary to resort to computational schemes for most models, and we discuss both numerical and simulation approaches in Chapters 4-8, with simulation methodology per se covered in Chapter 11. We shall deal with various optimization problems in the context of product design and manufacturing. Again, a variety of optimization techniques are available, and some of these will be discussed in Chapters 9 and 10. 2.5

MODELING FIRST FAILURE [ONE-DIMENSIONAL FORMULATION]

Let X x denote the age of an item at its first failure. (This is also called the time to first failure.) One can model X x in two different ways. The first, called black box modeling, models X 1 directly as a random variable with a distribution function based on the modeler’s intuitive judgment or on historical data. The second, called physically based modeling, models the physical mechanism of the item failure and then derives the distribution function for X x from that model. Thus, the black box approach is based on a simple system characterization where an item is characterized through two states—working or failed— and the physically based model involves a more detailed system characterization of the physics of the failure. 2.5.1

The Black Box A pproach

Let F(x) (or more completely, F(x\ 0), with 0 the parameter or vector of parameters) denote the distribution function for the first time to failure, i.e.,

The probability that the first failure does not occur prior to x is given by

We will assume that F(jc) is differentiable almost everywhere, so that/(x) = dF(x)/dx exists almost everywhere. Here f(x) is the density function

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associated with F(x). The conditional probability of item failure in the interval [.x , x + i), given that it has not failed prior to x , is given by

For probability functions admitting of a density, we define the following important function: Definition: The failure rate r(x) [associated with a distribution function F(jc)] is defined as

The quantity r{x) bx is interpreted as the probability that the item will fail in [jc , x + 8jc ) given that it has not failed prior to x. In other words, it characterizes the effect of age on item failure more explicitly than does the failure distribution or the density function. The failure rate r(jt), density function f(x ), and distribution F(x) are related as follows:

and

One can classify the distribution function F(x) into many categories based on the form of the failure rate r(jt). The three most important such categories are the following: Definition: F(jc) is said to have an increasing failure rate (IFR) if r(x) is increasing in x > 0. Definition: F(x) is said to have a decreasing failure rate (DFR) if r(jt) is decreasing in x > 0. Definition: Fix) is said to have a constant failure rate if r(x) is constant for all x > 0. The form of the failure rate depends not only on the distibution function but also on the values of the parameter(s) of the distribution function. We illustrate this by means of three examples.

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Example 2.1 [Exponential Distribution] The density function and the failure rate are given by

Example 2.2 [Gamma Distribution] The density function and the failure rate are given by

Figure 2.3 shows r(x) for different values of p, with X = 1. The value p = 1 corresponds to a constant failure rate, p < 1 implies that the distribution is DFR, and p > 1 implies that it is IFR. Example 2.3 [Weibull Distribution] The distribution function and the failure rate are

for 0 < x < oo, p > 0, and X > 0, and

Figure 2.4 shows r{x) for different values of p, with X = 1. Again, p = 1 corresponds to a constant failure rate, P < 1 implies that the distribution is DFR, and p > 1 implies that it is IFR. The Gamma and Weibull distributions both reduce to the exponential distribution when P = 1. They are the most commonly used distributions for characterizing time to failure in the black box approach. In addition, many physical failure mechanisms lead to these distributions. Many electrical and mechanical products exhibit a failure rate that has a “bathtub” shape. It is characterized by a decreasing failure rate from 0 to some point x ly a nearly constant failure rate over a range x x to x2 and an increasing failure rate beyond x2, as shown in Figure 2.5. The failures during the initial period are mainly due to defective material or poor

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Figure 2.3

Failure rate for Gamma distribution as a function of p.

manufacturing processes. In the case of repairable items, such failures are called teething problems and may often be fixed through some form of testing program. Failures over the middle period are due purely to chance and hence are not influenced by age. Finally, failures over the last period reflect a true aging process, which results in the failure rate increasing with age. Note that in some instances x 1 can be zero and/or equal to jc2. Many distributions have failure rate functions that exhibit a bathtub shape. The notes at the end of the chapter indicate some references where further details may be found. In practice, one often starts with a specific form for r{x) and then obtains F(x) using the relations indicated earlier. 2.5.2

Bounds on Distribution Functions

In warranty analysis (as will be indicated in Section 2.5.3) numerical values of F(jc) for different values of x are needed. For some common distribution functions, F(x) is available in tabulated form. For other, more complex distributions functions, no such tables are available, and F(;t) must be

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Figure 2.4

69

Failure rate for Weibull distribution as a function of p.

evaluated by some numerical integration scheme. This can be difficult and time consuming. As an alternative approach, it is often possible to obtain lower and upper bounds that can be evaluated more easily and used as approximations for F(jc). In this section we present some results for obtaining such bounds. The following additional definitions are needed: Definition: The distribution function F(x) is said to have increasing failure rate average (IFRA) if - ( 1 /*) logF(jt) is increasing in x > 0. This implies that (Jg r(t) dt)lx, the average of the failure rate function r(t) over the interval [0, *], is increasing in x. Definition: The distibution function F(x) is said to have decreasing failure rate average (DFRA) if -(1/*) log F(x) is decreasing in x for x > 0. Definition: The distribution function F(jc) is said to be new better than used (NBU) if

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Figure 2.5

Bathtub failure rate.

This implies that the probability of a item surviving for y units, conditioned on its age being x, is less than the corresponding probability for a new item. Definition: The distribution function F(x) is said to be new worse than used (NWU) if

Definition: The distribution function F(jc) is said to be new better than used in expectation (NBUE) if 1. 2.

F(x) has finite mean |x; J* F(t) dt < |jlF(jc) for x > 0.

This implies that an item of age x has a smaller mean remaining life than a new item. Definition: The distribution function F(x) is said to be new worse than used in expectation (NWUE) if 1. 2.

F(x) has finite mean (jl ; F(t) dt > |jl F(x) for jc > 0.

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The chain of implications that links these different concepts is given in the following proposition: Proposition 2.1 (Barlow and Proschan [6]): IFR DFR

IFRA =» NBU => NBUE DFRA 4> NWU => NWUE

We now state various bounds without proof. The proofs can be found in the references at the end of each result. In the following, (xy, j > 1, denotes the yth moment of X. Proposition 2.2: If F(jt) is IFR, then

(Proof: See Barlow and Proschan [6], p. 113.) Proposition 2.3: If F(x) is IFRA, then

where w > 0 is a function of x , and is gotten by solving

(Proof: See Barlow and Proschan [6], p. 115.) Proposition 2.4: If F(x) is DFR, then

(Proof: See Barlow and Proschan [6], p. 116.) Proposition 2.5: If F(x) is IFRA, then

where bs is determined by

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and (Proof: See Barlow and Marshall [7].) 2.5.3

An A pplication in the Analysis of Warranties

To illustrate the potential usefulness of the results of Section 2.5.2, we consider a few simple examples. Consider an item being sold with either a free-replacement warranty policy (Policy 1) or a pro-rata policy (Policy 2) with warranty period W. The probability that the manufacturer does not incur any warranty cost as a result of the sale, due either to repair or replacement under Policy 1 or refund under Policy 2, is given by F(W). It follows from Propositions 2.1 and 2.2 (and the fact that IFR implies IFRA) that if the item failure distibution is IFR A, then, for W < (xl9

and, for W > (xx,

where w is given by Proposition 2.3. The IFRA assumption says that, on the average, failure rates are increasing during the lifetime of the item, which seems a very reasonable characterization of the aging process. From the preceding results, we conclude that for items of this type if the warranty period is, say, 2/3 of the average lifetime of the item, (W = 2p,j/3), the probability of incurring no warranty cost is at least 1 - .51 = .49. On the other hand, if, say, W = 1 year and |xx = 1.5 years (in which case the w of Proposition 2.3 is about .59), then the probability of not incurring any warranty cost is less than .41. 2.5.4

Physically Based Models

As mentioned previously, in the physically based modeling approach, one models explicitly the mechanism that causes item failure. In this section we confine our attention to this approach for modeling single component items. The next section deals with multicomponent items. A variety of models have been developed to describe different mechanisms of failure. We briefly discuss two such models. Model 1 [Shock Damage Model] Suppose that an item is subjected to shocks that occur randomly over time. Suppose further that the magnitude of the shock is also a random

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variable and results in a short duration stress on the item with the magnitude of the stress related to the intensity of the shock. 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 (to be discussed in Chapter 3), where the point process characterizes occurrence of shocks and the associated mark characterizes the magnitude of the shock. A typical example of a failure phenomenon of this type is an electronic component failing due to current surges in a network. Model 2 [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 fails at the first time 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 (to be discussed in Chapter 3). Typical examples of item failure due to cumulative damage are crack growth in metals and tears in a conveyor belt. Many other models involving different types of stochastic formulations have been developed for item failure. These are beyond the scope of our analysis and are discussed in the notes at the end of the chapter.

2.5.5

Multicom ponent Items

Most items and systems are made of more than one component. One approach to modeling multicomponent item failure is to model each component failure separately either as a black box or based on the physical mechanism causing the failure and then to relate component failure to item failure. The distribution for item failure would depend on how the components are interconnected and the effect 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 distribution function and from this obtain the time to item failure. The analysis of such formulations is, in general, fairly involved and difficult. Often when a component of a complex item or system fails, it can either induce failure of one or more components or cause damage so as to weaken them and hence accelerate their failure. Such types of failures are termed

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failure interactions and involve interacting point processes (to be discussed in Chapter 3) for modeling component failures. Nearly all complex items and systems are built with a modular structure, a module being a collection of components. For such items, failure of a component results in a module failing. Thus, one needs to relate component failure to module failure first and then module failure to item failure in order to obtain the distribution function for item failure. Such models are even more complex and usually require computer analysis or simulation to obtain even approximate results. 2.5.6

M odeling Failure of Items Used Intermittently

Some products are used continuously— clocks, lights in certain applications, pumps in industrial operations, air conditioning in large, closed buildings, and so forth. Many other products are used intermittently or only occasionally— e.g., a refrigerator, television, or dishwasher in a home, an elevator in a building, 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 twostate continuous time Markov chain (a special type of stochastic process). When the duration of each usage is very small in relation to the time interval between usages, one can model usages as points along a time continuum. For this situation, usages may be modeled as a stochastic point process (to be discussed in Chapter 3). 2.5.7

Justification for the Black Box A pproach

Although modeling items in terms of components, and component failure in terms of the mechanisms of failure results in models that may be more realistic, this approach is not appropriate for the study of warranty, for two reasons: 1. 2.

The resulting models become extremely complex and very difficult to analyze. The validation of such detailed models requires a large amount of data, and in general it is very difficult, if not impossible, to obtain sufficient data of the types needed for model validation.

Consequently, we shall confine ourselves to modeling via the black box approach. Our starting point is the characterization of the first time to

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75

failure through a distribution function selected either on an intuitive basis or on the basis of an analysis of available failure data. This is an approximation to the real world and is in the spirit of modeling, for model selection must be based on a sensible trade-off between the sometimes conflicting factors of realism, complexity, solvability, and verifiability. 2.6

MODELING FIRST FAILURE [TWO-DIMENSIONAL FORMULATION]

As in the previous section, one can either take a black box or a physically based approach to modeling the first failure. We shall confine ourselves to the former case for the reasons discussed in the previous section. 2.6.1

The Black Box A pproach

Let (X 1, Yj) denote the age of an item and its usage at first failure. Since failure occurs in an uncertain manner and the item usage is also uncertain, X 1 and Y l are (nonnegative) random variables. We can model (Xu Yx) by a two-dimensional distribution F(x, y), defined by

Analogous to the one-dimensional case, under appropriate assumptions on F(x, y), one can describe the age and usage at first failure by a density function/( jc, y), which is related to F(x, y) by the relation

or through a instantaneous failure rate r(x, y) given by

where

The quantity r(x, y) 8jc 8y is essentially the probability that the first failure will occur with (X t, Yx) E [x, x + 8jc) x [y, y + 8y), given that X l > x and Yx > y.

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Alternately, one can model the time to first failure, X l9 by a onedimensional distribution function Ft(x) as discussed in Section 2.5. The item usage at first failure is modeled by a conditional distribution function F2(y\x), with

The product of F{{x) and F2{y\x) is the two-dimensional distribution function F(jt, y). 2.6.2

Some Specific Distributions

Since Y 1 represents the item usage at failure, it is reasonable to assume that E ( Y 1\ X i ) is an increasing function of X x. As a result, one must choose distribution functions F{x, y) that have this property. We indicate three such distribution functions: Example 2.4 [Beta Stacey Distribution] The density function/(x, y ) for ( X x, Yx) is given by

where x > 0, 0 < y < c[>jc, and a, b, a, , 01? 02 > 0. This is a slightly modified version of the Beta Stacey distribution proposed by Mihram and Hultquist (see Johnson and Kotz [8]). The conditional expectation E ( Y 1\ X l ) is given by

Example 2.5 [Multivariate Pareto Distribution] The density function/(x, y) for (X u Yx) is given by

with a > 1, x > > 0, and y > 02 > 0. The conditional expectation E (Yl\X1) is given by

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Example 2.6 [Multivariate Pareto Distribution of the Second Kind] The density function/(*, y) for (X u Yx) is given by

with x > 0l9 y > 02, ax > 2, a2 > 2, p2 < 1, and /0(*) a modified Bessel function of order zero (see Abramowitz and Stegun [9]). The expected usage conditioned on the time to failure, £(Y1|2f1), is given by

Note that in contrast to the earlier two models, this conditional expectation is no longer a linear function of x. By suitable choice of parameters, the exponent of x in this expression can be controlled. Alternate approaches to modeling the age and usage at first failure will be discussed in Chapter 9. 2.7

MODELING RECTIFICATION ACTIONS

Whenever a repairable item fails under warranty, the manufacturer has the option of either repairing the failed item or replacing it by a new item. For nonrepairable items, the only option is to replace a failed item by a new one. In this section, we discuss various modeling issues related to rectification actions. We shall confine our discussion to one-dimensional warranties. The extension to the two-dimensional case is straightforward. 2.7.1 Types of Repair

In the case of repairable items, a failed item can be made operational by subjecting it to repair. The behavior of the item after repair depends on the nature of repair carried out. In this context, one can define five different types of repair actions: 1. Repaired Good As New: Here, after each repair, the condition of the repaired item is assumed to be as good as that of a new item. In other

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words, the failure distribution of repaired items is the same as that of a new item. In real life, this is seldom the case. 2. Minimal Repair: When a failed item is subjected to a minimal repair, the failure rate of the item after repair is the same as the failure rate of the item just before item failure. This type of rectification model is appropriate for repair of multicomponent items where item failure occurs due to a component failure. When the failed component is replaced by a new working one, the item becomes operational. Since all other components are of age X l9 the repaired item as a whole is effectively a working item of age X l9 and hence the failure rate after repair is the same as that just before failure. 3. Repaired Items Are Different from New (I): Often when an item fails, not only are all the failed components replaced but also components that have deteriorated sufficiently. In other words, the item is subjected to a major overhaul, which results in the failure distribution of all repaired items being F^x), say, which is different from the failure distribution F(jt) for new items. Since repaired items are assumed to be inferior to new ones, the mean time to failure for a repaired item is taken to be smaller than that for a new item. 4. Repaired Items Are Different from New (II): In (2), the failure distribution for a repaired item is different from that of a new item but is independent of the number of times the item has been subjected to repair. In some instances, the failure distribution of a repaired item is a function of the number of times the item has been repaired. This can be modeled by assuming that the failure distribution of an item after yth repair (/ > 1) is given by F;(x) with mean (jl;. The sequence of |xy is assumed to be a decreasing sequence in /, implying that an item repaired j times is inferior to an item repaired j - 1 times. 5. Imperfect Repair: Minimal repair implies no change to the failure rate, whereas a repair action under (2) results in a predictable failure rate associated with the distribution function F^*). Often, however, the failure rate of a repaired item after repair is uncertain. This is called imperfect repair and can be modeled in many different ways. Figure 2.6 shows two different imperfect repair actions: (a) corresponds to the failure rate after repair being lower than that before failure, and (b) corresponds to the reverse situation. The change in the failure rate is a random variable in both cases. Another form of imperfect repair is one where the item becomes operational with probability p after it is subjected to a repair action and continues to be in a failed state with probability 1 - p. This implies that the item may have to be subjected to repair more than once before it becomes operational or perhaps would have to be replaced.

Warranty Policy and Modeling Issues

Figure 2.6

2.7.2

79

Different types of repair.

Rectification Duration

The time duration needed to carry out a rectification action is important in the context of warranty. When warranty terms include a penalty for down time, it is in the manufacturer’s interest to reduce this duration to the minimum possible. The duration is also of interest to buyers, since an item that is out of action deprives the buyer of its use and, in some cases, of the revenue that may be generated by the item. The total time involved in a rectification action consists of the following: 1. Processing time of the warranty claim 2. Investigation time 3. Repair/replace time 4. Testing time 5. Time to return the item to the buyer Processing time consists of the time needed (at the retail outlet level) for handling the claim for rectification under warranty, the time involved in transporting the failed item to the manufacturer (or workshop), and the waiting time at the workshop. Investigation time is the time needed to locate the fault and decide on an appropriate action. Repair time includes the time needed to carry out the actual repair and the waiting times that can result due to lack of spares or because of other failed items awaiting

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rectification actions. This time is dependent on the inventory of spares and the manning of the repair facility. For the case where the failed item is replaced by a new one, the replacement time is nearly zero as long as there is a new unit in storage. Testing time is important where the success of a rectification action demands that the item be subjected to considerable testing before it is returned to the buyer. Some of these times can be predicted precisely, while others (e.g., repair time) can be highly variable, depending on the type of product. The easiest approach is to aggregate all of the just mentioned times into a single time, called service time, X , modeled as a random variable with a distribution function G(*) = P{X < x}. We assume that G(jc) is differentiable and let g(jc) = dG(x)/dx denote the density function and G(jc) the probability that the service time will exceed x, i.e.,

Analogous to the concept of a failure rate function, we can define a service rate function p(;c), given by

The quantity p(x) bx is interpreted as the probability that the service activity will be completed in [x, x + 8x), given that it has not been completed in [0, x). In general, p(x) will be a decreasing function of x, indicating that the probability of the service being completed in a short time increases with the duration that the service has been going on. In other words, p(x) has a decreasing service rate, a concept similar to decreasing failure rate. If the variability in the service time is small in relation to the mean time for service, then one can approximate the service time as being deterministic. If this mean value is very small in comparison to time between failures, then we can view service time as being nearly zero. This point of view allows a much simpler characterization of failures over time, as will be demonstrated in the next chapter. 2.7.3

Repair Cost

When an item is returned under warranty, the manufacturer incurs a variety of costs. These are as follows: 1. 2.

Administrative costs Transportation cost to the repair facility

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3. 4. 5. 6.

81

Repair/replacement cost, comprising material cost and labor cost Transportation cost to return the item to the consumer Handling costs of the retailer Spare parts inventory costs

One can aggregate all of these costs into a single cost, termed service cost, for each warranty claim. Since some of the costs are uncertain (e.g., repair cost may depend on the type of repair), the service cost is a random variable and needs to be modeled by a suitable distribution function. If the variability in the service cost is small, it can be treated approximately as a deterministic quantity.

2.7.4

R ep lace Versus Repair Decisions

Often when a failed item is returned under warranty, the manufacturer has to make a decision whether to repair the failed item or replace it with a new one. The aim of the manufacturer is to minimize the cost of servicing the warranty, and hence the choice must be made on the basis of economic considerations. We will discuss this topic in greater detail in Chapter 9.

2.8 MODELING SUBSEQUENT FAILURES

Thus far we have discussed modeling of the first failure and the different rectification actions available to the manufacturer. Subsequent failures of an item often may depend on the nature of the rectification action after the first failure. In the case of one-dimensional warranties, the time instants of item failures and of rectified items being returned to the buyer can be viewed as events occurring randomly along a time continuum. As such, they can be modeled by a stochastic point process, the form of which would depend on the nature of the rectification action. In the next chapter, we indicate a variety of point process formulations and their use in modeling item failures over the warranty period. In the case of two-dimensional warranties, the failures and the return of rectified items to the buyer can be viewed as points on a two-dimensional plane, with the horizontal axis representing time or age and the vertical axis representing item usage. Appropriate models would involve twodimensional point processes, and these will also be discussed in the next chapter.

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2.9 MODELING ITEM SALES

For the manufacturer, item sales are important in the context of warranty for purposes of planning repair facilities and for deciding on warranty reserve levels. Item sales can be divided into the following three categories: 1. 2. 3.

First purchases Replacement purchases under warranty Repeat purchases outside warranty

Repeat purchases are important in the context of evaluating total cost over the product life cycle. Modeling of first and repeat purchases of a product has received a good deal of attention, and the literature on the subject is extensive. In this section, we discuss a fairly simple model formulation that has been generalized in many ways. 2.9.1

First Purchase Sales

For certain types of products (e.g., defense products) a sale can be treated as taking place within a very short time interval. In these cases, the transaction can be approximated as a point sale in which a lot of items is sold to a buyer. The more interesting, and more general, case, however, is that in which sales occur over time, with a changing sales rate. One approach to this problem is to model aggregate sales, so that the time domain is of no importance. The advantage of such an approach is that a static formulation can be used to model aggregate quantities. When modeled explicitly, sales over time can be modeled in two ways. The first approach involves a discrete time formulation in which sales aggregated over small time intervals (e.g., monthly, quarterly, or yearly) are the variables being modeled. In the second approach, sales over time are modeled as continuous functions of time. Static Models: A fairly simple static model for aggregate sales is the CobbDouglas Model (see Henderson and Quandt [10]) given by

where Sa represents total sales, W the warranty duration, and P the price per unit. The quantities k , a, and p are constants with a > 0 and p < 0, implying that total sales increase with increasing warranty duration and decrease with increasing price. A slightly more complex model is one where

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a similar model is assumed for the competitor and sales are a function of the relative ratios of prices and warranty duration. Dynamic Models: We shall confine our discussion to sales occurring continuously over time. Let S(t) denote the total sales until time t, and s(i), the time derivative of S(t), i.e., the sales rate at time t. Sales occur as a result of diffusion of information about the product through the population of potential buyers. The spread of information occurs in two modes: external (e.g., advertising) and word of mouth (e.g., contact between a person who has bought the product and a person yet to buy the product). For new product sales, a simple model of this type, using an ordinary differential equation, is given (see Bass [11]) by

where L represents the total population that will buy the product, a and b represent, respectively, the external and word-of-mouth effects, and 5(0) = 0. Here the sales rate increases initially and then decreases to zero as t becomes large. A variety of extensions of this model to incorporate the effect of price, advertising, and other variables can be found in Mahajan and Peterson [12]. Since a warranty can be viewed as a signal, it can be incorporated into the model in a manner similar to advertising. Thus, we can model L, a, and b as functions of W with 1. 2.

L increasing with W (implying greater total sales with longer warranty terms) da(W)/dW > 0 and db(W)/dW > 0 (implying a faster rate of adoption with better warranty terms)

For very expensive products (e.g., large mainframe computers), the total sales are few and occur as points along a time continuum. In this case, a model based on a point process formulation (to be discussed in the next chapter) is more appropriate for modeling sales. The probability of a new sale occurring will depend on the total sales that have taken place earlier and the product’s performance over time. 2.9.2

R epeat Purchase Sales

We define a repeat purchase as any purchase other than the first purchase where the buyer pays the full price. Thus, replacements bought at reduced price and under warranty are not considered to be repeat purchases.

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The time intervals between the first purchase and the first repeat purchase, and between successive repeat purchases, are random variables. The distribution functions for these random variables are dependent on many factors—product reliability, type of warranty policy, and consumers’ reactions, both to the product and to the warranty services offered. In the case of nonrepair able products, a potential repeat purchase situation occurs at the first time instant outside the warranty period that the item fails. Whether a purchase occurs or not depends on factors such as advertising, competition, the general economy, buyer satisfaction with the product and its performance, and the warranty service provided by the manufacturer. This uncertainty can be modeled by a binary-valued random variable assuming the value 1 (implying a repeat purchase) with probability p and 0 (implying no repeat-purchase sale) with probability 1 - p. In the case of a repairable product, when an items fails outside warranty the buyer has the option of either repairing it or replacing it with a new item. In this case, the probability of a repeat purchase is a function of the buyer’s economic evaluation of the choice between repair and replacement. In addition, for both repairable and nonrepairable products, a buyer may sometimes decide to replace a working item by a new one. In this case, we have an additional feature, namely, the market for secondhand items. A typical example is the market for automobiles. When a change of ownership occurs before the warranty has expired, the original warranty terms are, in many cases, invalidated. Comment: We shall confine ourselves to fairly simple static and dynamic models for both first and repeat purchases. In addition, we will assume that repeat purchases occur only when an item in use fails outside warranty. Thus, we do not incorporate the effect of a secondhand market or the analysis of warranty for secondhand items. 2.10

COST ANALYSIS

In the context of the study of warranty, the following costs are of importance: 1. 2. 3.

Expected warranty cost per unit sale Expected cost of operating a unit over its lifetime Expected cost of operation over the product life cycle

The last two costs are often called life cycle costs (LCC). As such, we shall denote them as LCC-I and LCC-II, respectively. In this section we discuss some issues related to obtaining these cost measures.

Warranty Policy and Modeling Issues 2.10.1

85

Warranty Cost per Unit Sale

Whenever an item is returned for rectification action under warranty, the manufacturer incurs various costs as outlined in Section 2.7.3. These costs, too, can be random variables. The total warranty cost (i.e., the cost of servicing all warranty claims for an item over the warranty period) is thus a random sum of such individual costs, since the number of claims over the warranty period is also a random variable. In general, it is very difficult to obtain the distribution function for the warranty cost per unit sale, even for the simplest model formulations, because of mathematical intractability. Hence, we shall confine ourselves to obtaining expressions for the expected value of this cost. We do this for a variety of warranty policies in Chapters 4-8. 2.10.2

Life C y c le Cost LCC-I

The total cost of operating a single item over its life consists of the following cost elements: 1. Acquisition (or purchase) cost [CA] 2. Maintenance and repair cost of operating the item beyond the warranty period [CM] 3. Operating cost (energy, labor, etc.) [CQ\ 4. Incidental ownership costs [CJ 5. Disposal cost (if necessary) [Cs] The cost CA is a fixed cost. The cost Cm is a function of the time period beyond warranty for which the item is in use. This time period usually depends on the warranty period and will be denoted g(W). The remaining costs are functions of W + g(W), the life of the unit. 2.10.3

Life C y c le Cost LCC-II

This cost depends on the life cycle of the product, that is, the time interval over which buyers buy the product. After this time period, sales of the product cease, often because of the introduction of a new and better replacement. Let L denote the product life cycle. We assume that the buyer continues repeat purchases over this period. The number of repeat purchases is a random variable. The total life cycle cost is given by the product of this random variable and the life cycle cost per item, LCC-I. As a result, the total cost over the product life cycle is also random. In general, the distribution function for warranty cost over the product life cycle is also mathematically intractable. Hence, we shall again confine

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ourselves to obtaining expressions for the expected value of this cost. This is done for a variety of warranty policies in Chapters 4-8. NOTES

Section 2.2 1. One can define many additional policies for each of the subgroupings of Section 2.2. The bulk of the warranty literature deals with onedimensional warranty policies belonging to Groups A and C. As mentioned in the text, although policies belonging to Group B are extensions of Group A policies, they have received very little attention. 2. Extending the idea of a two-dimensional warranty, one can formulate warranties with three or more dimensions as well. An example of a three-dimensional warranty is as follows: For an airplane engine, usage and age would define the first two dimensions as in the usual twodimensional warranty policy. A possible third dimension is the fuel efficiency of the engine, with the warranty requiring some compensation to the user if the efficiency falls below some specified value. In this case, the servicing of warranty would involve periodic maintenance by the manufacturer (or operator) to reduce the penalty for drops in the engine efficiency. 3. Although a great deal has been written on reliability improvement warranties (see Dhillon [13] and Blischke [14] for a partial list of references), the subject has been dealt with primarily on a conceptual basis, with few attempts at formulating operational definitions such as those given in this chapter as Policies 19 and 20. Part of the problem is that the warranty terms are very much application specific, so that it is difficult to devise general warranty models. Further discussion of these notions will be given in Chapter 7. Section 2.3 1. Mathematical modeling and the use of mathematical models in problem solving has received a good deal of attention, and there are many books on the topic. Murthy et al. [15], gives an integrated treatment of the subject and highlights the different facets of modeling. An annotated guide to many other texts on this subject, where interested readers can find additional material on specific aspects of modeling, is also given. Section 2.4 1. The literature on the analytical approach to the study of warranty is extensive. For a survey of mathematical models for cost analysis of one-dimensional warranties belonging to Groups A -C , see Blischke

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87

[14]. In the remainder of the book, we examine various types of policies and many related aspects. Notes at the end of each chapter contain additional relevant references. Section 2.5 1. Most books on reliability theory (e.g., Barlow and Proschan [6], Kapur and Lamberson [16], Dhillon [13]) discuss the modeling of item failures. We have indicated only a few distributions for modeling item failure using the black box approach. Johnson and Kotz [8] list a large number of distribution functions, many of which are suitable for characterizing item failures. 2. Many alternate formulations to model the bathtub failure rate have been proposed— see, for example, Kao [17], Hjorth [18], and Glaser [19] 3. For more on physically based models, see Gertsbakh and Kordonsky [ 20] . 4. Murthy and Nguyen [21,22] deal with models for failure interactions in multicomponent systems. Murthy [23] deals with an unreliable multicomponent system with nested modular structure. 5. Murthy [24] considers an intermittent usage model based on a twostate, continuous time Markov chain formulation. Murthy [25] deals with a model where usage is modeled as a point process. Section 2.6 1. Johnson and Kotz [26] list a large number of multivariate distributions appropriate for modeling item failures in two dimensions. Section 2.7 1. The concept of minimal repair was first proposed by Barlow and Hunter [27]. Many other types of repair action have been proposed and can be found in various journals on reliability theory. Section 2.8 1. As indicated in the text, the modeling of subsequent failures involves stochastic process formulations. These are discussed in the next chapter. Section 2.9 1. The literature on mathematical modeling of product sales is vast. Most books on marketing contain a chapter on this topic— see, for example, Kotler [28] and Parsons and Schultz [29]. For more on diffusion-type models for product sales, see Mahajan and Peterson [12]. Section 2.10 1. The concept of life cycle cost is used extensively in evaluating expensive products. Again, the literature on this subject is vast— see, for example, Dhillon [13], Chapter 9. (This source alone includes over 150 additional references on LCC.)

Chapter 2

88 EXERCISES

2.1.

2.2. 2.3. 2.4.

2.5. 2.6. 2.7. 2.8. 2.9. 2.10. 2.11. 2.12. 2.13.

2.14.

Classify the warranties found in Exercise 1.1 according to the taxonomy of this chapter, expressing them in the terminology of the taxonomy. Include in your description of the warranties whether the warranty is simple or compound; the value(s) of W (Wu etc.); whether or not the warranty is renewing; and so forth. Give explicit definitions of at least two additional pro-rata warranties. Give explicit definitions of at least two additional combination warranties. Suppose that a policy has three terms: (1) free replacement up to time Wx after purchase; (2) replacement at 50% of the selling price from W1 to W2; and (3) replacement at pro-rata cost to the buyer from W2 to W, where W1 < W2 ^ 0.5W. Describe all of the possible partially renewing warranties of this type that could be defined. Which of these might make sense in practical applications? Extend the warranty of Exercise 2.4 to a four-period policy in at least two ways. How many partially renewing policies are now possible? Are any of these of practical interest? Think of at least three applications other than automobiles in which two-dimensional warranties are used or might be appropriate. Think of at least two applications for three-dimensional warranties. How about four-dimensional warranties? Prove Proposition 2.1. Prove Proposition 2.2. Prove Proposition 2.3. Prove Proposition 2.4. Prove Proposition 2.5. Show that a mixture of exponential distributions, with distribution function

where X1, \ 2 > 0 and 0 < p < i, has the DFR property. Consider a system consisting of n components connected in series. Assume that the components operate independently and that component lifetimes are identically distributed with distribution function F(t). Let G(t) denote the failure distribution for the system. Show that

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89

2.15.

Consider a system consisting of n components in parallel. Assume that the components operate independently and that component lifetimes are identically distributed with distribution function F(t). Let G(t) denote the failure distribution for the system. Show that

2.16.

Suppose that T1 and T2 are independent random variables having Gamma distributions with the same scale parameter, i.e.,

2.17.

with t; > 0, 'J\ > 0, and ~; > 0, for i = 1, 2. Show that S = T1 + T2 has a Gamma distribution with parameters 'J\ and~ = ~1 + ~2 . Let T1 , T2 denote the failure times of two dependent components. Suppose that the joint failure distribution of T1 , T2 is given by the bivariate exponential distribution (BVE) F(t 1 , t2 ) with joint survival probability

for t 1 , t2 , 'J\ 1 , 'J\ 2 > 0. Show that the marginal distributions of T1 and T2 are given by

2.18.

for i = 1, 2. Using the results of Exercise 2.17, show that

and

for i = 1, 2.

Chapter 2

90 2.19.

Show that the BVE distribution leads to the following conditional distribution function:

2.20.

Using the result of Exercise 2.19, show that

REFERENCES

1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.

Guin, L. (1984). Cumulative Warranties: Conceptualization and Analysis, Doctoral Dissertation, University of Southern California, Los Angeles, CA. Trimble, R. F. (1974). Interim Reliability Improvements (RIW) Guidelines, USAF Memorandum. Gandara, A., and Rich, M. D. (1977). Reliability Improvement Warranties for Military Procurement, Report No. R-2264-AF, Rand Corp., Santa Monica, CA. Balaban, H. S. (1975). “Guaranteed MTBF for Military Procurement,” Proc. 10th Int. Logistic Symp., SOLE. Kruvand, D. H. (1987). Army aviation warranty concepts, 1987 Proc. Annual Reliab. and Maint. Symp., 392-394. Barlow, R. E., and Proschan, F. (1965). Mathematical Theory o f Reliability, John Wiley and Sons, Inc., New York. Barlow, R. E., and Marshall, A. W. (1964). Tables of bounds for distributions with monotone hazard rate, J. Amer. Stat. Assoc., 60, 872-890. Johnson, N. L., and Kotz, S. (1970). Continuous Univariate Distributions I and II, Houghton Mifflin, Boston, MA. Abramowitz, M., and Stegun, I. A. (1964). Handbook o f Mathematical Functions, Applied Mathematics Series No. 55, National Bureau of Standards, Washington, D.C. Henderson, J. M., and Quandt, R. E. (1958). Microeconomic Theory, McGraw-Hill, Inc., New York. Bass, F. W. (1969). A new product growth model for consumer durables, Man. Sci., 15, 215-227.

Warranty Policy and Modeling Issues

12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29.

91

Mahajan, V., and Peterson, R. A. (1985). Models for Innovation Diffusion, Sage Pub. Co., Inc., Beverly Hills, CA. Dhillon, B. S. (1983). Reliability Engineering in System Design and Operation, Van Nostrand Reinhold Co., New York. Blischke, W. R. (1990). Mathematical models for analysis of warranty policies, Math, and Computer Modelling, 13, 1-16. Murthy, D. N. P., Page, N. W., and Rodin, E. Y. (1990). Mathematical Modelling, Pergamon Press, Oxford. Kapur, K. C ., and Lamberson, L. R. (1977). Reliability in Engineering Design, John Wiley, New York. Kao, J. H. K. (1959). A graphic estimation of mixed Weibull parameters in life testing of electron tubes, Technometrics, 1, 389-407. Hjorth, U. (1980). A reliability distribution with increasing, decreasing, constant and bathtub-shaped failure rates, Technometrics, 22, 99-107. Glaser, R. E. (1980). Bathtub and related failure rate characterizations, J. Amer. Stat. Assoc., 75, 667-672. Gertsbakh, I. B., and Kordonsky, Kh. B. (1969). Models o f Failure, Springer-Verlag, Berlin. Murthy, D. N. P., and Nguyen, D. G. (1985). Study of two component system with failure interaction, Nav. Res. Log. Quart., 32, 239-248. Murthy, D. N. P., and Nguyen, D. G. (1985). Study of multicomponent system with failure interaction, Euro. J. Oper. Res., 21, 330338. Murthy, D. N. P. (1983). Analysis and design of unreliable multicomponent system with modular structure, Large Scale Systems, 5, 245-254. Murthy, D. N. P. (1992). A usage dependent model for warranty costing, Euro. J. Oper. Res., 57, 89-99. Murthy, D. N. P. (1991). A new warranty costing model, Math, and Computer Modelling, 13, 59-69. Johnson, N. L., and Kotz, S. (1972). Distributions in Statistics: Continuous Multivariate Distributions, John Wiley, New York. Barlow, R. E., and Hunter, L. (1961). Optimum preventive maintenance policies, Oper. Res., 8, 90-100. Kotler, P. (1971). Marketing Deision Making: A Model Building A p proach, Holt, Rinehart, and Winston, New York. Parsons, L. J., and Schultz, R. L. (1976). Marketing Models and Econometric Research, North-Holland, New York.

3

Stochastic Processes for Warranty Modeling

3.1

INTRODUCTION

In this chapter, we give a brief introduction to stochastic point processes and their role in modeling item failures for warranty study. The outline of the chapter is as follows. In Section 3.2, we define a stochastic process and introduce the oneand two-dimensional counting processes. In Section 3.3, we discuss different approaches to the analysis of stochastic processes. Sections 3.4 and 3.5 deal with two specific counting processes, the one-dimensional Poisson process and the one-dimensional renewal process. For each, we give the precise details of the formulations, indicate some techniques used in the analysis, and present some results that will be used in later chapters. Where possible, we illustrate through an application in warranty modeling. Of particular significance to warranty study, as will be illustrated in later chapters, is the renewal integral equation associated with a renewal process. This is discussed in detail in Section 3.6. In Section 3.7, we discuss some further topics from renewal theory that are useful in warranty study. In Section 3.8, we describe some additional one-dimensional point processes, useful for modeling item failures using the physically based approach. Finally, in Section 3.9, we introduce two-dimensional renewal processes, discuss the analysis, and present some results in a manner similar to that in Section 3.4 for the one-dimensional case. Our motivation in this chapter is to introduce the mathematics needed to model item failures for warranty study. Readers whose main interests are in the application of the models discussed in later chapters can skip the mathematical analysis presented in this chapter. Researchers and analysts interested in a deeper understanding of the underlying theory will 93

C hapter 3

94

find this chapter to be a starting point to further study of point processes. Notes at the end of the chapter give references to books on stochastic and point processes. 3.2 STOCHASTIC PROCESSES 3.2.1

One-Dimensional Stochastic Processes

A one-dimensional stochastic process Z(i), t E T, can be viewed as a collection of random variables— that is, for each t E T, Z(t) is a random variable. The independent variable is interpreted as time, and we call Z(t) the state of the process at time t. If T is countable, then Z(i) is called a discrete time stochastic process; if T is a continuum, then it is called a continuous time stochastic process. 3.2.2

One-Dimensional Point Processes

A one-dimensional point process is a continuous time stochastic process characterized by events that occur randomly along the time continuum. Examples of an event, in the context of product warranty, are an item being put into operation or an item failing. The theory of point processes is very rich, as a variety of such processes have been formulated and studied. Of particular interest to warranty modeling is a counting process. 3.2.3

One-Dimensional Counting Processes

A point process {N(t), t > 0} is a counting process if it represents the number of events that have occurred until time t. It must satisfy the following: 1. N(t) > 0. 2. N(t) is integer valued. 3. If 5 < r, then N(s) < N(t). 4. For s < t, {N(t) - N(s)} is the number of events in the interval (,s, t\. We shall confine ourselves to t > 0. The behavior of N(t), t > 0, depends on whether t = 0 “ corresponds to the occurrence of an event or not. The analysis of the case with t = 0~ corresponding to the occurrence of an event is simpler than the alternate case. Also, we assume that N(0) = 0. Two special counting processes of particular importance to warranty modeling are the following: 1. 2.

Poisson processes Renewal processes

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95

The precise details of the mathematical formulations of these and a few other one-dimensional point processes will be given in later sections. Definition: A counting process {N(t), t > 0} is said to have independent increments if, for all choices 0 < ij < t2 < ¿3 • • • < tn, the n — 1 random variables

are independent. Definition: A counting process {N(t), t > 0} is said to have stationary independent increments if, for each s > 0, {A ^ 4- s) — N{tf)} and {N{t2 + s) - N(t2)} have the same distribution function. In other words, the distribution function of {N(t + s) - N(t)} does not depend on t. 3.2.4 Two-Dimensional Counting Processes

A two-dimensional counting process N(x, y ) is a natural extension of the one-dimensional case. Here events occur randomly over a two-dimensional continuum given by X x Y and the random points (jc , y) E X x Y. In the warranty context, the variables x and y are interpreted as time and usage, respectively. As such, the process can be viewed as generating random points on a two-dimensional plane with the horizontal axis representing time and the vertical axis representing usage. The quantity N(x, y) is the number of events occurring over the rectangle [0, x) x [0, y) with both x and y > 0. In contrast to the one-dimensional case, the theory of general twodimensional point processes is not yet very well developed. However, the theory of two-dimensional renewal process has been well developed and is of particular interest in warranty modeling. We will discuss this special process in a later section. 3.3 3.3.1

ANALYSIS OF STOCHASTIC PROCESSES Methods of Analysis

The analysis of stochastic processes can be done either analytically or computationally. In the analytical approach, one seeks solutions to specific problems in closed-form analytical expressions. Unfortunately, this approach works only for very simple formulations. In most cases, one needs to use the computational approach. For example, as will be illustrated later, in the analysis of renewal processes, the analytical approach often leads to integral equations that cannot be solved except through computational methods.

Chapter 3

96 3.3.2

Approximations and Bounds

When exact analytical approaches become difficult, it is sometimes possible to use an approximation that makes the analysis more manageable. An example is the asymptotic approximation. In some instances, one can obtain analytical expressions for either upper or lower bounds more easily. The usefulness of such results depends on how tight the bounds are. 3.3.3

Simulation

In the simulation approach, sample paths (or time histories) of the process are generated on the computer using random number generators. In other words, the computer simulates the process. The simulation is repeated a large number of times, and the statistics obtained from the simulation runs are used to obtain estimates of various quantities of interest. For the method to be effective, the simulation technique must be efficient and the number of simulation runs sufficiently large to ensure a high degree of confidence. Simulation of stochastic processes is sometimes also called Monte Carlo simulation. 3.4 POISSON PROCESSES

We start with the most simple case, viz., a stationary Poisson process. We then discuss some extensions. 3.4.1

Stationary Poisson Processes

Definition (1): A counting process N(t), t > 0, is a stationary Poisson process if the following hold: 1. 2. 3.

N( 0) = 0. The process has independent increments. The number of events in any interval of length s is distributed according to Poisson distribution with parameter Xs, i.e.,

for n = 0, 1, 2, . . . , and all s > 0 and t > 0. It can be shown through simple analysis (see e.g., Ross [1]) that for a stationary Poisson process, the times between events (also called interevent times) are independent and identically distributed exponential random variables with mean l/X. This gives us a second definition for a stationary Poisson process:

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97

Definition (2): Consider a counting process. Let X x denote the time instant of the first event occurrence, and for j > 2, let Xj denote the time interval between the (j - l)st and yth events. The counting process is a stationary Poisson process with parameter Xif the sequence Xjyj > 1, are independent and identically distributed exponential random variables with mean 1/X. For a stationary Poisson process it can be shown (see, Ross [1]) that the following hold: 1. The probability of an event occurring in [t, t + 81) is X bt -f o(bt). 2. The probability of two or more events occurring in [t, t + 81)is o(bt). 3. The occurrence of an event in [t, t + 81) is independent of the number of events in [0, t). As a result, we have a third definition for a stationary Poisson process: Definition (3): A counting process {N(t), t > 0} is a stationary Poisson process if the following hold: 1. The probability of an event occurring in [t, t + 81) is X 8i + o(bt). 2. The probability of two or more events occuring in [f, t + 81)is o(bt). 3. The occurrence of an event in [i, t + bt) is independent of the number of events in [0, t). The quantity X is called the intensity of the process. Comment: The preceding discussion illustrates the point that there is more than one way of characterizing a counting process. In the context of warranty modeling, a particular characterization may be more appropriate than the other equivalent characterizations. For example, in the warranty analysis of nonrepairable items, Definition 2 is more appropriate; in the case of repairable items with the item being subjected to minimal repair after each failure, Definition 3 is more appropriate, as will be illustrated later. Expected Number of Events in [0, t)

Let M(t) denote the expected number of events in [0, t). Since N(t) is distributed according to Poisson distribution with parameter Xi, we have (3.1) 3.4.2

Nonstationary Poisson Processes

In a stationary Poisson process, the probability of an event occurring in [t, t + bt) is X bt + o(8i), with X a constant. A nonstationary Poisson process is a natural extension in which X changes with time.

98

Chapter 3

Definition: A counting process {N(t), t s 0} is a nonstationary Poisson process if the following hold: 1. N ( 0 ) = 0 . 2. / > 0} has independent increments. 3. P{N(t + 8/) - N(t) = 1} = X(r) 8f + o(8/). 4. P{N(t + 8/) - N(t) > 2} = o(8/).

The function k(t) is called the intensity function. Let (3.2) Then it can be shown (see Ross [1]) that (3.3) for j > 0. This statement can be used to define a nonstationary Poisson process in a manner similar to Definition 1 for a stationary Poisson process. Expected Number of Events in [0, t]

Since the probability of j events (j > 0) in [0, t) is given by (3.4) the expected number of events in [0, i), Af(r), is given by (3.5) An Application in Warranty Modeling

Consider the free-replacement policy where failed items are repaired minimally (see Section 2.7.1). If the time to repair is small, then it can be approximated as being zero. Since the failure rate is unaffected, failures over time occur according to a nonstationary Poisson process with intensity function \{t) equal to the failure rate r(t).

Stochastic Processes for Warranty Modeling

3.4.3

99

Conditional Poisson Processes

For a stationary Poisson process, the intensity Xis a deterministic quantity. For a conditional Poisson process, which we now define, the intensity is a random variable A with a distribution G(-). Definition: A counting process {N(t), t > 0} is a conditional Poisson process, if, conditional on the event A = X, {N(t), i > 0} is a stationary Poisson process with intensity X. In this case, the probability of the event {N(t + s) - N(t) = n} is given by (3.6) Note that {N(t), t > 0} is not a Poisson process except for trivial measures dG. An Application in Warranty Modeling

Most electronics components do not deteriorate with age, and their failure is due to pure chance. Under controlled laboratory conditions, the failures over time can be modeled by a stationary Poisson process with intensity X0. When the components are used in the field, the environment is often different from that in a controlled laboratory. Although the failure is still due purely to chance, its occurrence is influenced by the randomness of the environment. In this case, component failures over time can be modeled by a conditional Poisson process with A = 7X0, where 7 is a random variable assuming values in [1, 0°). Higher values of 7 imply a more hostile environment and hence an increased intensity of failures. 3.4.4

Doubly Stochastic Poisson Processes

This is an extension of the conditional Poisson process formulation discussed in Section 3.4.3 to the nonstationary case. In other words, the intensity functions are themselves stochastic processes. Conditioned on the intensity function, the process is a nonstationary Poisson process. The analysis of such processes is done by first conditioning on the intensity function and then removing the conditioning. In general, the analysis is involved and often intractable, even for simple stochastic characterizations of the intensity function. An Application in Warranty Modeling

Consider an item that at any given time can be in one of two states, “in use” or “idle.” It switches from one state to another in a random manner.

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100

When the item is in the in use state, the failure rate of the item is when it is in the idle state, the fialure rate is \ 0>with X0 < K- Suppose that a failed item is immediately replaced by a new one. In this case, item failures over time can be modeled by a doubly stochastic Poisson process. 3.5

RENEWAL PROCESSES

3.5.1 Ordinary Renewal Processes

In Section 3.4 we mentioned that a counting process can be characterized in terms of interevent times and that for a stationary Poisson process these times are independent and identically distributed exponential random variables. A natural generalization of this is a counting process where the interevent times are independent and identically distributed with an arbitrary distribution. Definition: A counting process {N(t), / > 0} is an ordinary renewal process if the following hold: 1. N{ 0) = 0. 2. X l9 the time to occurrence of the first event (from t = 0) and Xj9 j > 2, the time between (; — l)st and yth events, are a sequence of independent and identically distributed random variables with distribution function F(x). 3. N(t) = sup{n: Sn < t}, where (3.7) Note: The stationary Poisson process is a special case of the ordinary renewal process with F ( jc ) an exponential distribution function. An Application in Warranty Modeling

Consider the free-replacement policy where items that fail need to be replaced by new ones because the items are nonrepairable. If the time to replace is negligibly small, then it can be treated as being zero. In this case, times between failures are independent and identically distributed with a failure distribution the same as that of the original item. As a result, failures over time occur according to an ordinary renewal process. Note: Since in the preceding situation the event corresponds to item failure and its replacement by a new item, the event can be viewed as a renewal point for the system. Hence, one tends to use the term renewal as opposed to event in the context of renewal processes. We shall use these terms interchangeably.

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3.5.2

101

Analysis of Ordinary Renewal Processes

Note that Sn is the time instant for the nth renewal (or event) and is the sum of n independent and identically distributed random variables. Since the X /s are distributed with distribution function F(x), from the result of Appendix A, the distribution of Sn is given by the «-fold convolution of F with itself, i.e.,

Distribution of N[t)

It is easily seen that N(t) > n if and only if Sn < r. As a result,

Since

this implies that (3.8) From this, expressions for the moments of N(t) can be obtained. Of particular interest in warranty analysis is the first moment, the expected number of renewals in [0, t). Expected Number of Renewals in [0, t)

The expected number of renewals in [0, r), M(i), is given by

Using (3.8), this can be written as (3.9)

102

C hapter 3

Let M(s) denote the Laplace transform (see Appendix A) of M{t). Then we have (3.10) This follows from the result of Appendix A for the Laplace transform of F*n\t). As a result, M(s) can be written as

or (3.11) since F(s) = ?(s)/s. On taking the inverse Laplace transform, we have

(where * is the convolution operation), or (3.12) An Alternate Derivation

An alternate derivation for M(t), based on conditional expectation, is as follows. Conditioned on X x, the time to first failure M(t) can be written as (3.13) But (3.14) Note that we have used the renewal property in deriving the preceding expression. If the first failure occurs at x < t, then the number of renewals over t - x occur according to an identical renewal process, and hence the

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103

expected number of renewals over this period is M(t — x). Using (3.14) in (3.13), we have

which the same as (3.12). Equation (3.12) is called the renewal integral equation, and M(t) is called the renewal function associated with the distribution function F(t). The function M(t) plays an important role in warranty analysis, and we shall discuss it further in the next section. Renewal Density Function

The renewal density function, m(t), is given by (3.15) and satisfies the equation (3.16) w here/(i) is the density function associated with F(t). Variance of the Number of Renewals in [0, t)

From (3.8) it can be shown that the variance of N(t) is given by (3.17) Excess [or Residual] Life at t

Let B(t) denote the time from t until the next renewal, that is, B(t) = SN^ +1 — t

(3.18)

The function B(t) is called the excess or residual life at t. It represents the remaining life of the item in use at time t. The distribution function for B(t) is given, for x > 0, by (3.19)

104

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where M(t) is the renewal function associated with the distribution F(x), given by (3.12). In general, it is difficult to solve (3.19) analytically. In the limits as t —> oo, this becomes

(3.20) where |x = E[X]. The derivations of (3.19) and (3.20) are given in Section 2 of Appendix B. An Application in Warranty Modeling

Consider the case where nonrepairable items are sold with a nonrenewing free replacement warranty with warranty period W. All failures under warranty are replaced by new items at no cost to the buyer. If we assume that the time required to replace a failed item is zero, the failure replacements occur according to a renewal process. The first repeat purchase occurs when an item fails outside the warranty period. This occurs at time B{W) from the instant the warranty ceases. As a result, the time intervals between repeat purchases occur according to another renewal process, with the interval a random variable given by W + B(W). The distribution function for this random variable is easily obtained once the distribution function for B{t) is obtained. Age at Tim e t

Let A(t) be the time from t since the last renewal. That is, (3.21) The function A(t) is the age of the item in use at time t and hence is called the age at t. The distribution function for A(t) is given by (3.22)

where M(t) is the renewal function associated with F(t). The derivation of (3.22) is given in Section 3 of Appendix B.

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105

Delayed Renewal Processes

Definition: A counting process {N(t), t > 0} is a delayed renewal process if the following hold: 1. N( 0) = 0. 2. X 1, the time to first event, is a nonnegative random variable with distribution function F(x). 3. Xj, j > 2, the time intervals between yth and (y - l)st events, are independent and identically distributed random variables with distribution function G(jc), which is different from F(x). 4. N(t) = sup{n: Sn < i}, where S0 = 0 and, for n > 1.

Note that when G(jk) equals F(x), the delayed renewal process reduces to an ordinary renewal process. An Application in Warranty Modeling

Consider repairable items sold with a nonrenewing free replacement warranty. All failed items are repaired and the failure distribution of repaired items is G(jc), which is different from F(jc), the failure distribution of new items. If the time to repair is negligible, then failures over the warranty period occur according to a delayed renewal process. Expected Number of Renewals in [0, t)

Let Md(t) denote the expected number of renewals over [0, t) for the delayed renewal process, that is, (3.23) An expression for this can be easily obtained using the conditional expectation approach used for obtaining M{t) for the ordinary renewal process. Conditioning on X x, the time to first renewal, we have (3.24) where Mg(t) is the renewal function associated with the distribution function G(i). This follows from the fact that if the first event occurs at x < i, then

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106

over the interval (x, t) the events occur according to a renewal process with distribution G. As a result, Md(t) is given by (3.25) Excess [or Residual] Life at Tim e t

Let Bd(t) denote the time from t until the next renewal. The distribution function for Bd{t) is given by (3.26) for x > 0. For a proof of this, see Cinlar [2]. 3.5.4

Alternating Renewal Processes

In an ordinary renewal process, the interevent times are independent and identically distributed. In an alternating renewal process, the interevent times are all independent but are not identically distributed. More specifically, the odd-numbered interevent times (X 1, X 3, X 5,. . .) are identically distributed with distribution F(x) and the even-numbered ones (X2, X 4, X 6, . . .) are identically distributed with distribution G(x). An Application in Warranty Modeling

Consider a repairable item sold with a free-replacement policy. Whenever an item fails, it is subjected to repair and is assumed to be restored to good as new after repair. Suppose that the repair time is nonzero and is a random variable with distribution function G(x). The failure times are distributed according to a distribution function F(x). In this case, the failures over the warranty period occur according to an alternating renewal process as previously described. (Here it is assumed that the warranty clock continues to run during repair.) State of an Item at a Given Tim e

Suppose that at any given time an item can be either in working state or in a failed state and undergoing repair. Of interest is the probability P(t) that the item is in working condition at time t. The function P(t) is given by (3.27)

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where H{x) is the convolution of F(x) and G(x), i.e.,

The derivation of (3.27) is given in Section 4 of Appendix B. 3.6 THE RENEWAL INTEGRAL EQUATION

As will be seen in later chapters, the renewal integral equation plays an important role in the analysis of many warranty policies. In this section, we study the renewal integral equation in more detail and focus our attention on alternate approaches to solving it. We discuss five different approaches: 1. The analytical approach 2. Approximations 3. Bounds 4. Numerical integration 5. The simulation approach The advantage of the analytical method is that one can carry out parametric studies of the renewal function, i.e., the behavior of M(t) as a function of the parameters of the distribution. Only for a small class of distribution functions can one obtain analytical expressions for M(t). The advantage of approximations is that sometimes the approximation can be obtained analytically, and in this case one can carry out the parametric study easily. Bounds on M(t) are useful in obtaining bounds on warranty costs for many warranty models. The usefulness of this approach depends on how tight the bounds are; loose bounds are of limited value. Numerical integration and the simulation approach involve computer analysis. This implies that estimates of M{t) are obtained for specific parameter values of the distribution function. The accuracy of the numerical integration depends on the integration mehod used and the nature of the distribution function. The numerical efficiency of the simulation approach depends critically on the efficiency with which the X /s can be generated. For complex distribution functions, this can be very time consuming, making the method inefficient. In this section we discuss each of the preceding methods of obtaining M(t) for given F(t).

Chapter 3

108 3.6.1

The Analytical Approach

It is possible to obtain M{t) analytically only for a small class of distribution functions F{t). If F(t) is such that its Laplace transform F(s) is a rational polynomial (i.e., a ratio of two polynomials in 5), then from (3.11) it follows that M{s) is also a rational polynomial. By carrying out the partial fraction expansion of this polynomial, one can obtain M(t) by use of the Laplace inverse transformation. We illustrate this approach by means of a few examples. Example 3.1 [Exponential Distribution] Let F(t) be given by

Then

On taking the Laplace transform, we have

and

Using (3.11), we have

which, on taking the inverse transformation, yields

and

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This is the expected result, for which F(t) is exponential the renewal process reduces to a stationary Poisson process with intensity \. The expected number of events in [0, t) for a Poisson process is given by (3.1) as Kt. Example 3.2 [Erlang Distribution with k Stages] An Erlang distribution with k stages is a special case of the Gamma distribution defined in Section 2.5.1, for which the distribution function is given by

Using the Laplace transformation approach in a manner similar to that of Example 3.1, we have

where 0 = exp(2Ttilk) with i = V - 1. (For details of the derivation, see Barlow and Proschan [3].) For a two-stage Erlang distribution (that is, k - 2), we have and

Example 3.3 [Shifted Exponential Distribution] The shifted exponential models the case where the item cannot fail until it reaches an age L, and beyond L the failure is due to pure chance. The distribution F(t) is given by

The renewal function is given by

where [t/L\ is the greatest integer less than or equal to t!L. (For details of the derivation of this result, see Cinlar [2].)

Chapter 3

110 3.6.2

Approximations

The difficulty in obtaining M(t) from (3.12) is that it appears on both sides of the equation. If Equation (3.12) can be approximated so that M{T) appears only on the left hand side of the equation and the right hand side contains only known or prescribed functions, then we have an approximate solution that may be obtained either analytically or computationally. Bartholomew [4] proposed one such approximation, Mb(t), given by (3.28) where

with K = l/|x, jjl being the expected value of interevent times. Ozbaykal [5] proposed a different approximation, M0{t), given by (3.29) Yet another approximation, Mde(r), is due to Deligoniil [6] and is given by (3.30) Deligoniil [6] compares the different approximations with the exact M(t) using a numerical method. Example 3.4 [Gamma Distribution] Let the failure distribution function be given by a Gamma distribution with density

Table 3.1 (from Deligoniil [6]) shows MQ(t), Mb(t), Mdc{t) for (3 = 2 and \ = 0.5. Also included is M(t) obtained using the analytical approach and

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Table 3-1 Comparison of Various Approximate Methods for M(t), Gamma Distribution, p = 2, X = 0.5.

t

Ma(t)

KU)

Mde(i)

M(t)

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 3.0 4.0 5.0 8.0

0.00000 0.09562 0.18461 0.26952 0.35520 0.43394 0.51566 0.59796 0.68123 0.76571 0.85150 1.29979 1.77289 2.26011 3.75075

0.00000 0.01760 0.06262 0.12640 0.20313 0.28880 0.38067 0.47678 0.57575 0.67663 0.77871 1.29458 1.80740 2.31475 3.82099

0.00000 0.01760 0.06261 0.12631 0.20285 0.28814 0.37937 0.47454 0.57225 0.67151 0.77166 1.27411 1.77143 2.26546 3.74982

0.00000 0.01758 0.06233 0.12530 0.20047 0.28383 0.37268 0.46520 0.56019 0.65683 0.75458 1.25062 1.75008 2.25001 3.75000

Source: Ref. [6].

given by

Series Approximation

For the Weibull distribution, Leadbetter [7] proposed a series expansion for M(t) in terms of where p is the shape parameter of the distribution. Here F(t) is expanded in terms of an infinite series, namely

where the coefficients Cr are given by

The infinite series representation of M(t) is given by (3.31)

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112

where the A /s are given by

and, for j > 1,

For t < 1, by truncating the series at some finite value K , one obtains an approximation Me(t) given by (3.32) The accuracy of the approximation depends on K. Asymptotic Approximation

For large t, one can approximate M{t) by MJJ) given by (3.33) where |x and a 2 are the mean and variance of the interevent times. (See Ross [1] and Cinlar [2].) The error in the approximation is 0(1) and is small if t is sufficiently large. As t —> oo, we have the asymptotic result (3.34) This is known as the Elementary Renewal Theorem and provides the approximation M(t) ~ tl|x for large t. 3.6.3

Bounds

In this section we will present upper and lower bounds for M{t) that can be obtained relatively easily. These are useful in warranty analysis, as they yield upper and lower bounds for warranty costs. We present the results without proofs. These are based on certain properties of F(t) and can be found in Barlow and Proschan [8].

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Proposition 3.1: If F(t) is NBU, then

and

Proposition 3.2: For all F(t),

If F(t) is NBUE, then

Proposition 3.3: If F(t) is NBUE, then

Proposition 3.4: If F(t) is DFR, then

3.6.4

Numerical Integration

One can obtain M(t) numerically by using (3.9) with the infinite summation replaced by the first K terms of the series. By suitably choosing K , any desired numerical accuracy can be achieved. This method, however, requires the numerical evaluation of the convolutions that yield F (;)(jc) for 2 < j < K. Soland [9] uses this approach to obtain M(t) for Weibull and Gamma distributions. Baxter et al. [10], using the cubic spline method developed by Cleroux and McConalogue [11], computed tables for M(t) for five different distributions: Weibull, Gamma, lognormal, truncated normal, and inverse Gaussian. These are the most comprehensive tables for M(t).

114

Chapter 3

Deligoniil and Bilgen [12] propose a method using cubic splines and the Galerkin technique to obtain M(i), and present limited computational results for the Gamma distribution. Finally, a numerical method for obtained M(t) for phase-type distributions is discussed by Kao [13]. Example 3.5 [Weibull Distribution] The Weibull distribution is given by

for t > 0, with two parameters p > 0 and X > 0. Plots of M(t), obtained from the tables in Baxter et al. [14], for X = 1 and several values of p, ranging from p = 0.5 (decreasing failure rate) to p > 1 (increasing failure rate), are shown in Figure 3.1. A small portion of the Baxter et al. tables is reproduced in Appendix C.

Figure 3.1 M{t) for the Weibull distribution, X = 1.

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3.6.5

115

The Simulation Approach

In the simulation approach, the renewal process is simulated on a computer using random number generators to produce a sample path, or a time history, of the process. Using the statistics of the simulation, one obtains an estimate M(t). This approach will be discussed in detail in Chapter 11. 3.7

ADDITIONAL TOPICS FROM RENEWAL THEORY

In this section, we present some additional topics from renewal theory that will be useful later in the analysis of warranty policies. 3.7.1

Renewal Type Equations

Definition: A renewal type equation is an equation of the form (3.35) where h and F are known and g is the unknown function to be obtained as a solution to the integral equation. The result given in the following proposition is often used in the analysis of warranty policies involving renewal processes. Proposition 3.5. The function g(t) given by (3.36) where M{x) is the renewal function associated with F(x), is a solution of (3.35). The proof of this is given in Section 1 of Appendix B. 3.7.2

Wald’s Equation

This result is useful in evaluating sums of random numbers of random variables. We need the concept of stopping time. Definition: An integer-valued positive random variable N is said to be a stopping time for the sequence X l9 X 29 . . . if the event {N = n} is independent of X n+1, X n+2i . . . for all az = 1, 2, . . . . Wald’s Equation: If X 1,X 2, . . . are independent and identically distributed random variables with finite mean, and if A is a stopping time for the

Chapter 3

116 sequence, such that E[V] < oo, then

(3.37) We omit the proof. Interested readers can find a proof in Ross [1]. 3.8 ADDITIONAL ONE-DIMENSIONAL POINT PROCESSES

In this section we indicate some additional one-dimensional point processes of interest in warranty modeling. 3.8.1

Marked Point Processes

A marked point process is a point process with an auxiliary variable, called a mark, associated with each event. Let Yh / > 1, denote the mark attached to the ith event. For example, in the case of a multicomponent item, failure of a component can cause induced failures of one or more of the remaining components. If the number of components that need to be replaced at the ith failure of the item is a random variable, then it can be viewed as a mark attached to an underlying point process characterizing item failures. A simple marked point process is characterized by the following: 1. 2.

{N(t), t> 0}, a stationary Poisson process with intensity X A sequence of independent and identically distributed random variables {y,}, called marks, that are independent of the Poisson process

The foregoing simple marked point process is also called as a compound Poisson process. Various extensions (e.g., nonstationary point process, and marks constituting a dependent sequence) yield more complex marked point processes. 3.8.2 Cumulative Processes

A cumulative process involves a marked point process with {N(t), t > 0} and {y,} as indicated above and an additional variable X(t) given by

The variable X(t) is called a cumulative process or, occasionally, a mark accumulator process.

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In Section 2.5.4, we discussed the physically based approach to modeling item failure and discussed two models. These are examples of marked and cumulative point processes, respectively. In Model 1 of Section 2.5.4, the item is subjected to shocks modeled by a marked point process with the mark representing the magnitude of the shock. The failure occurs at the first time instant the shock magnitude exceeds some critical value. In Model 2 of that section, each shock does a random amount of damage to the item. Here the mark represents the damage caused by the shock. The damage is cumulative, and the item fails at the first time instant the cumulative damage exceeds some critical value. 3.8.3

Interacting Point Processes

As the name implies, here we have two or more point processes that interact with each other in some sense. If there is no interaction, then each process can be treated separately as a simple point process. Many different types of interactions can be defined. We illustrate one such interaction by considering the modeling of a two-component system with failure interaction. Let the two components be denoted Jx and J2. They fail randomly. When ever Jx [J2\ fails, it can cause an instantaneous failure of J2 [/J with probability p 1 [p2] or have no effect with probability 1 — [1 —p 2]. The failures of Jx and J2 define two point processes, and there is interaction between the two in the sense that an event occurring in one can induce an event in the other with nonzero probability. 3.9 TWO-DIMENSIONAL RENEWAL PROCESSES 3.9.1

Formulation

A two-dimensional renewal process is a natural extension of a one-dimensional renewal process. Here events occur on a two-dimensional plane as opposed to a line in the one-dimensional case. Definition: A counting process {A(jc, y), (x, y) G R2+} (where is the positive quadrant in the two-dimensional plane) is an ordinary two-dimensional renewal process if the following hold: 1. 2.

N{0, 0) = 0. {{Xh y f), i > 1} is a sequence of independent and identically distributed nonnegative bivariate random variables with a common joint distribution function F(x, y), where

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118

3.

N(x, y) = max{n: S(nl) < x , S 1} is a sequence of independent and identically distributed random variables with distibution function F^x) = F(jc, °°), the marginal distribution function of X. As such, it defines a univariate renewal process. Similarly, the sequence {Yh i > 1} defines another renewal process with distribution function F2(y) = F(°°, y), the marginal distribution function of Y. We can modify (2) in the preceding definition as follows: 2'.

{X±, Yj) is a nonnegative bivariate random variable with joint distribution function F(x, y), and {(Xh Yz), i > 2} is a sequence of independent and identically distributed nonnegative bivariate random variables with a common joint distribution function Fx(jt, y), which is different from F(*, y).

Then we have a “modified” two-dimensional renewal process. This is the extension of the “delayed” renewal process for the one-dimensional case. An Application in Warranty Modeling

Consider a two-dimensional free-replacement policy defined by a rectangle [0, W) x [0, U) in a two-dimensional plane with the horizontal axis representing time and the vertical axis representing usage. The item is nonrepayable, and all failed items under warranty are replaced by new ones. Let {(A"7, Y,), i > 1} denote the age and usage of the ith item at failure. We assume that the time to replace a failed item is negligible, and that the failures are independent. This implies that {{Xh Y,), i > 1} is a sequence of independent and identically distributed nonnegative bivariate random variables with a common joint distribution function F(jt, y), and the number of failures over the rectangle [0, x) x [0, y) is given by N(x, y), a twodimensional renewal process. 3.9.2

Analysis of Renewal Processes

The analysis of the two-dimensional renewal process can be carried out in terms of the two univariate renewal processes generated by the sequences

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{XÙ and {Y,}. Define (3.39) (3.40) Then it is easily seen that (3.41) and, as a result, we have

(3.42) where F(/i)(jc, y) is the joint distribution of {S ^ \ S*,2)}. Since {Sjj1*, S^2)} involves sums of n bivariate, independent, and identically distributed random variables with common joint density function F(x, y), F(n)(x, y) is given by the n-fold convolution of F(x, y) with itself. That is, (3.43) for n > 2, where ** is the convolution operation given by (3.44) As a result, from (3.42) we have (3.45) From this, one can obtain means, variances, covariances, and so forth. Expected Number of Renewals in [0, x) x [0, /)

Let M (x, y) denote the expected number of renewals over the rectangle [0, x) x [0, y). This is given by (3.46)

120

Chapter 3

One can obtain M(x, v) directly using (3.45), that is, (3.47) However, we will use the conditional expectation approach. Note that conditional on X x = x 1 and Y x = y l9 we can write M (x, y) as (3.48) where the expectation is done over the conditional variables. We have

(3.49) This follows from the fact that if the first event occurs with (xl9 y x) E [0, x) x [0, y), then, since (*, y) is a renewal point, the expected number of events in [xl9 x) x [y1? y) is equal to the expected number of events in [0, x - *i) x [0, y - yx). As a result, using (3.49) in (3.48) yields (3.50) This equation is similar to Equation (3.12), which gives M{t) for the onedimensional case, and is called the two-dimensional renewal integral equation. Similarly, M (x, y) is called the two-dimensional renewal function associated with the two-dimensional distribution function F(x, y). The two-dimensional renewal integral equation plays an important role in the analysis of two-dimensional warranty policies, as will be indicated in Chapter 9. Unfortunately, it is impossible to obtain M (x, y) analytically even for the simplest forms of F(jt, y). Thus, one needs to use computational methods for obtaining it. Example 3.6 [Beta Stacy Distribution] The density function /( x, y) for (Xh Y,), i > 1, is given in Example 2.4. Various moments of interest are the following (Johnson and Kotz

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[15]):

Note that the expected usage between failures, conditioned on the failure time Xi = x, increases with x. Figure 3.2 shows a two-dimensional plot of M(x, y ) as a function of x and y for the following set of parameter values: a = 0.2, = 2.6, a = 1.9, c = 2.5, 0! = 1.1, and 02 = 1.1. Figure 3.3 is a contour plot, with contours of constant M(x, y) plotted as functions of x and y. Example 3.7 [Multivariate Pareto Distribution] The density function/(x, y) for (Xh Y,), i > 1, is given in Example 2.5. The various moments of interest are as follows:

Again, the expected usage between failures, conditioned on the failure time Xi■= x, increases with x. Example 3.8 [Multivariate Pareto Distribution (Type 2)] The density function/(x, y) for (Xh Y,), i> 1, is given in Example 2.6. The expected usage between failures, conditioned on the time to failure, is given by

Note that, in contrast to the earlier two models, E[Yt\Xi = x\ is no longer a linear function of x. Figure 3.4 shows a two-dimensional plot of M(x, y) as a function of x and y for the following parameter values: ax = 4.0, a2 = 2.1, 0! = 0.3, 02 = 0.2, and p = 0.4472.

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122

Figure 3.2 M(x, y ) for the Beta Stacy distribution.

Figure 3.5 is a contour plot with contours of constant M{x, y) plotted as functions of x and y. 3.9.3

Bounds on M(x, y)

In this section we give some lower and upper bounds for M(jc, y) that can either be evaluated analytically or obtained easily by some computational

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123

Figure 3.3 Contour plots of constant M(x, y) for the Beta Stacy distribution.

method. Such results are useful in obtaining lower and upper bounds for cost analysis of two-dimensional warranty policies. We state the results without proofs. Interested readers can find the proofs in Hunter [16]. Define (3.51) (3.52) (3.53)

124

Chapter 3

Figure 3-4 M(x, y) for the multivariate Pareto distribution. and (3.54) where F\n\x ) (i = 1, 2) is the «-fold convolution of F^x) with itself. Let M a (x , y) and MB(x, y) be the two-dimensional renewal functions associated with the two-dimensional joint distribution functions A (x , y) and B(x, y), respectively.

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125

Figure 3.5 Contour plots of constant M(x, y ) for the multivariate Pareto

distribution.

Proposition 3.6: (3.55) where

(3.56)

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126

The next result gives bounds for M(jc, y) in terms of a one-dimensional renewal function. This is appealing, since one-dimensional renewal functions can be numerically evaluated more easily than two-dimensional renewal functions. Define Z, = aXt + bYh i > 1, with a and b > 0. Let (3.57) Then Nz defines a univariate renewal counting process for {ZJ. Let M(z) denote the renewal function associated with the process {Az}. Proposition 3.7: For all x, y > 0 and a, b > 0, a + b > 0, (3.58) 3.9.4

A Related Counting Process

The two-dimensional renewal process N(x, y) is related to the two univariate renewal processes A^1} and A^2) by the relation given in (3.41), namely, (3.59) A related counting process is N(x, y), given by (3.60) An Application in Warranty Modeling

In Chapter 9 we shall show that the number of failures for a two-dimensional warranty policy characterized by two strips as shown in Figure 2.4(b) can be modeled by Sl(x, y). A nalysis of N[x, y)

It is easily seen that

(3.61) From this, one can obtain expressions for the various moments of N(x, y). Of particular interest is the first moment, which, from (3.61), is given by M (x, y) and the renewal functions associated with the two univariate renewal processes.

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NOTES

Section 3.2 1. There are a large number of books on stochastic processes, varying from introductory to advanced and from intuitive to abstract and rigorous. Ross [1] and Bhat [17] are two introductory level books that do not require the measure theory background needed for the rigorous approach. A slightly more advanced book is Cinlar [2]. Books on point processes are also numerous— Thompson [18] and Cox and Isham [19] are two. Section 3.3 1. The books listed in the preceding note give a good treatment of the analytical approach to the analysis of stochastic processes. Sections 3.4 and 3.5 1. Most books on stochastic processes discuss the Poisson process and renewal processes in depth. Most books on reliability theory also deal with renewal processes—for example, see Barlow and Proschan [3,8], Ross [1], and Cox [20]. Section 3.9 1. For more on two-dimensional renewal processes, see the three papers by Hunter [16,21,22]. EXERCISES

3.1. 3.2. 3.3. 3.4.

3.5. 3.6.

Prove that the three characterizations given in Section 3.4.1 for the stationary Poisson process are equivalent. Prove Equations (3.3) and (3.5). Suppose that N(t) is a conditional Poisson process with intensity function given by an exponential distribution with parameter X. Derive an expression for P{N(t + s) - N(t) = n). A complex system (e.g., an engine) cannot operate without a critical component (e.g., a carburetor) being operational. Suppose that the critical component is unreliable, with distribution of time to failure F(t). Suppose that n - 1 spares for the critical component are available, and that as soon as one of these components fails, it is replaced by one of the spares. The system fails when the last spare fails. Let G(t) denote the distribution of time to failure for the system. Show that if F(t) is the exponential distribution, then G(t) is a Gamma distribution. Prove Equation (3.17). Using Equation (3.19), derive the distribution of the residual life of the item at time t, assuming that item failure times are expo-

128

3.7. 3.8. 3.9.

3.10. 3.11.

3.12. 3.13.

Chapter 3

nentially distributed. Indicate how the result could have been derived in a straightforward, intuitive manner. Derive the expression for M{t) in Example 3.2. Derive the expression for M(t) in Example 3.3. Obtain an analytical expression for the renewal function M{t) associated with the mixed exponential distribution, given for t > 0 by

where \ 2 > 0 and 0 < p < 1. Prove Propositions 3.1-3.4. A device is subject to shocks occurring randomly in time according to a stationary Poisson process with intensity X. The ith shock causes a random amount Y, of damage, where X l9 X 2, . . . are independent and identically distributed with distribution F(x). The device fails when the total accumulated damage exceeds a specified level d. Let G(jc) denote the distribution of time to failure of the device. Show that

where F 0)(-) is the /-fold convolution of F(-) with itself. How would the expression for G(x) in Exercise 3.11 change if successive shocks caused greater damage? [Hint: Assume that the distribution function Ffx) for is such that E{X^) is increasing in i.] Prove Propositions 3.6 and 3.7.

REFERENCES

1. 2. 3. 4. 5.

Ross, S. M. (1970). Applied Probability Models with Optimization Applications, Holden-Day, San Francisco. Cinlar, E. (1975). Introduction to Stochastic Processes, Prentice-Hall, Englewood Cliffs, NJ. Barlow, R. E., and Proschan, F. (1965). Mathematical Theory o f Reliability, John Wiley and Sons, Inc., New York. Bartholomew, D. J. (1963). An approximate solution of the integral equation of renewal theory, /. Royal Statist. Soc., 25B, 432-441. Ozbaykal, T. (1971). Bounds and Approximations for the Renewal Function, unpublished M.S. Thesis, Dept, of Operations Research, Naval Postgraduate School, Monterrey, CA.

Stochastic Processes for Warranty Modeling

6. 7. 8. 9. 10. 11. 12. 13. 14.

15. 16. 17. 18. 19. 20. 21. 22.

129

Delingoniil, Z. S. (1985). An approximate solution of the integral equation of the renewal theory, J. Appl. Prob., 22, 926-931. Leadbetter, M. R. (1963). On series expansion for the renewal moments, Biometrika, 50, 75-80. Barlow, R. E., and Proschan, F. (1981). Statistical Theory o f Reliability and Life Testing, To Begin With, Silver Spring, MD. Soland, R. M. (1968). A renewal theoretic approach to estimation of future demand for replacement parts, Oper. Res., 16, 36-51. Baxter, L. A., Scheuer, E. M., McConalogue, D. J., and Blischke, W. R. (1982). On the tabulation of the renewal function, Technometrics, 24, 151-156. Cleroux, R., and McConalogue, D. J. (1976). A numerical algorithm for recursively defined convolution integrals involving distribution functions, Management Sci., 22, 1138-1148. Deligoniil, Z. S., and Bilgen, S. (1984). Solution of the Volterra equation of renewal theory with Galerkin technique using cubic splines, J. Stat. Comp. Simul., 20, 37-45. Kao, E. P. C. (1988). Computing the phase-type renewal and related functions, Technometrics, 30, 87-93. Baxter, L. A., Scheuer, E.M., McConalogue, D. J., and Blischke, W. R. (1981). Renewal Tables: Tables of Functions Arising in Renewal Theory, Tech. Rept., Decision Systems Dept., Univ. of Southern California, Los Angeles, CA. Johnson, N. L., and Kotz, S. (1972). Distributions in Statistics: Continuous Multivariate Distributions, John Wiley and Sons, Inc., New York. Hunter, J. J. (1974). Renewal theory in two dimensions: Basic results, Adv. Appl. Prob., 6, 376-391. Bhat, N. U. (1984). Elements o f Applied Stochastic Processes, John Wiley and Sons, Inc., New York. Thompson, W. A ., Jr. (1988). Point Process Models with Applications to Safety and Reliability, Chapman and Hall, New York. Cox, D. R., and Isham, V. (1980). Point Processes, Chapman and Hall, London. Cox, D. R. (1960). Renewal Theory, Methuen, London. Hunter, J. J. (1974). Renewal theory in two dimensions: Asymptotic results, Adv. Appl. Prob., 6, 546-562. Hunter, J. J. (1977). Renewal theory in two dimensions: Bounds on the renewal function, Adv. Appl. Prob., 9, 527-541.

4

Analysis of the Basic Free-Replacement Warranty

4.1

INTRODUCTION

The free-replacement warranty is the most commonly used consumer product warranty and is frequently used in commercial sales as well. Under a free-replacement warranty (which, as before, will be abbreviated FRW), a failed item is replaced at no cost to the buyer if the failure occurs prior to a specified time W. If the replacement item fails before time W from the original purchase, it too is replaced at no cost to the buyer. If additional failures occur, this process is repeated until a total service time of W is attained. Ordinarily, any necessary replacements are new items of the same type as the original. Most mathematical models for warranty costs and other aspects of warranty assume this to be the case. Note that the notion of “no cost to the buyer” is seldom literally true— there are always costs associated with returning a failed item and obtaining its replacement; there may also be costs associated with loss of use of the item until the replacement is put into service. These costs are almost never covered by warranty and are usually ignored in modeling the process. In fact, most models assume that replacements are put into service instantaneously. The free-replacement warranty has also been called the failure free warranty, the standard warranty, full warranty, and the lump-sum or lumpsum rebate warranty. “Failure free,” obviously not meant to be taken literally, is used to indicate that the seller assumes all responsibility for any failures, “standard” to indicate that this is the most common policy, and “full” to imply complete coverage or that any other policy provides less coverage. The remaining terms convey the notion that the FRW is equivalent to a cash rebate (“full refund”), which, in fact, is occasionally the case. This idea will be developed more fully in the next chapter. 131

132

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The FRW is ordinarily not renewing, that is, the warranty period for a replacement item (say item n) is ordinarily not W, but W - Sn, provided Sn < W, where Sn = 2"=1 X h with X t the life or service time of the ith item. Thus, warranty coverage ends at time W, and the item then in service is no longer covered by warranty. Occasionally, items are sold under renewing FRW. Under this policy, all replacement items would be warranted anew for (ordinarily) the same period W. Thus, the warranty process associated with a single sale ends only when a single item with a lifetime X > W is encountered. Although this chapter will be concerned primarily with the nonrenewing FRW, we will look briefly at renewing FRW as well. In practice, the FRW may effectively be renewing under special circumstances, even though this is not the intent of the manufacturer. For example, inexpensive items (e.g., alarm clocks, toys, inexpensive small appliances, and so forth) may simply be replaced free of charge by the seller (or by some sellers). The buyer renews the warranty simply by indicating the replacement date as the initial date of service on the warranty registration card. Whether this is a cost incurred by the seller or the manufacturer depends on the contractual agreements between the two. In any case, existing warranty models do not account for these somewhat unusual situations, and we shall assume that the FRW is either strictly renewing or strictly nonrenewing. When used explicitly, the renewing FRW is often called an unlimited warranty or unlimited free-replacement warranty. As indicated in the preceding and in Chapter 2, the FRW is a very widely used warranty policy. Applications range from small, quite inexpensive items such as single rolls of film, basic household goods, and cheap tools, to very expensive items such as luxury watches and automobiles. Specific illustrations may be found in Exhibits 1, 3, 4, 5, and 10 of Chapter 1. The FRW is also used to provide separate coverage on individual components of complex items, for example, color picture tubes, computer CPUs, and so forth. In this chapter, we will be concerned with cost models and related quantities for the basic free-replacement warranty. Here, as in most of the book, the emphasis will be on the manufacturer’s or seller’s costs. (Although it is recognized that these costs are not necessarily the same, the two terms will be used interchangeably.) There are several reasons for emphasizing the seller’s point of view. First, the seller is ordinarily in possession of vastly more information about his product than an individual buyer can ever hope to be. This information is essential for application of even the simplest models of the warranty process. A second and related reason is that the majority of the modeling effort reported in the literature is devoted to analysis from this point of view. This is due, in great part,

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to the fact that the seller usually has a much greater stake in the outcome, being responsible for sales, under warranty, of many items. Finally, many of the models can be used to analyze buyers’ costs as well by suitable modification of the cost elements and the random quantities involved. Warranty costs for the seller and benefits for the buyer depend fundamentally on two elements: the structure of the warranty policy and the life distribution of the items (both the items sold under warranty and the replacement items in the event of failure). These jointly determine the magnitude of the costs or benefits that will be incurred, the instances of their occurrence, and the time frame or usage period over which the process is in operation. In this chapter, we consider both unit and life cycle cost models for the nonrenewing FRW defined in Policy 1 as well as the renewing FRW defined in Policy 5. In determining unit cost, we are concerned with the costs associated with the sale or purchase of a single item, including warranty costs. In determining life cycle cost, repeat purchases after warranty expiration are assumed, and costs are determined over this larger time frame rather than on a per-unit basis. The organization of the chapter is as follows: Section 4.2 will consider unit cost models for the nonrenewing FRW from the point of view of the manufacturer. Both nonrepairable and repairable items will be considered. Both repair and replacement are assumed to be instantaneous. Models for discounted and undiscounted long-run costs will be given. Section 4.3 will deal with these issues for the nonrenewing FRW from the consumer’s point of view. Section 4.4 will deal with the renewing FRW from the manufacturer’s and consumer’s points of view. Life cycle cost models for manufacturer and consumer, with and without discounting, will be discussed in Section 4.5. Section 4.6 will deal with buyer’s and seller’s indifference prices, that is, with models for determining a pricing structure such that the buyer (seller) incurs the same long-run costs whether an unwarrantied item is purchased (sold) repeatedly at a specified price or a warrantied item is purchased (sold) at a predetermined higher price. Finally, some models incorporating additional features of the warranty process, including intermittent use and invalid warranty claims, are discussed briefly in Section 4.7.

4.2

MODELING THE SELLER’S UNIT COST FOR THE NONRENEWING FRW

The terms of the nonrenewing free-replacement warranty are stated in Chapter 2 as follows:

Chapter 4

134

Policy 1 The manufacturer agrees to repair or provide replacements for failed items free of charge up to time W from the time of initial purchase. In practice, most FRWs are expressed in simple terms such as these, particularly since the Magnuson-Moss Act. Usually, however, some restrictions such as “when used properly” or “for the use intended,” and so forth, are appended. The fact that the warranty is not renewing, i.e., that the repaired or replacement item is covered under warranty only for the time remaining in the original warranty period, is implicit.

4.2.1

Modeling Expected Warranty Cost

The cost to the seller of a warranted item will be modeled as the total expected cost of the sale, say E[C(W)\, including the cost of supplying the original item and the costs of all replacements necessary under the warranty. On a per-item basis, these costs are taken to include manufacturing costs, distribution costs, and all other costs associated with providing the item to the consumer, as well as marketing and all other costs associated with the sale, amortized over all items provided (whether by sale or under warranty). Thus, the average per-item cost to the seller of providing an item, which will be denoted cs, is taken to be the same for the original item sold as for items supplied under warranty. For later purposes, cs may also be thought of as the seller’s cost of an item sold without warranty. Both it and the buyer’s cost (i.e., the selling price), say c5, are assumed to be constant throughout the period of interest. Correspondingly, CS(W) and Cb(W) will be used to denote the seller’s and buyer’s costs, respectively, of a single item sold under warranty with warranty period W. Both of these are random variables whose distributions depend on the distribution(s) of the X t and the structure of the warranty and may depend on any of many other factors as well. Cost models almost universally deal with average or expected cost, e.g., F[CS(W)]. Initially, a number of additional simplifying assumptions will be made. Unless otherwise stated, we shall assume that all failures before time W result in a warranty claim (i.e., that all legitimate warranty claims are, in fact, made), that no false claims are made, that replacement of a failed item by a repaired or new item is instantaneous, and that the lifetimes X u X 2, . . . of the initial and all replacements items are independent and identically distributed with distribution function F( ). We consider repairable and nonrepairable items separately, beginning with the latter.

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4.2.2

135

Basic Models for Nonrepairable Items

The earliest models of seller’s expected cost for an item under FRW were simple single-failure models, that is, models that accounted only for the first failure in the warranty period. In this case, the cost of the warranty is modeled as CSF(W), so that the expected total unit cost to the seller is (4.1) This approximation is adequate when it is reasonable to assume that the probability of a second failure within the warranty period is negligible, which is often not the case. When multiple failures may occur within the warranty period with nonnegligible probability, this model may significantly underestimate the true warranty cost. To model the process allowing for the possibility of multiple failures within the warranty period, it is necessary to consider the random variable N = N(W) = number of replacements in the interval [0, W]. The random variable N(W) may also be thought of as the first N such that = 2£LV Xi > W. This random variable was discussed at length in Chapter 3. Under the assumption that the X t are independent and identically distributed, the expected value of N(W) is given by F[N(VF)] = M(W), where M( ) is the ordinary renewal function associated with the distribution function F(*). Since N(W) + 1 is a stopping time for the renewal process, it follows from Wald’s Theorem that the expected total cost per item under FRW is

(4.2) Example 4.1 [Exponential Distribution] Suppose that color picture tubes are sold under free-replacement warranty with warranty period W = 1 year, and that the lifetimes of the tubes are independent and identically exponentially distributed, i.e.,

for each X t. The renewal function for the exponential distribution is given in Example 3.1 of Chapter 3 as M(t) = \t. Suppose that X = 0.5. This corresponds to an average lifetime of (x = 1/X = 2.0 years, or twice the warranty period. Under the simplifying assumptions stated previously, the average cost to the seller of an item

136

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sold under FRW is found from Equation (4.2) to be

or 50% more than the cost of an item sold without warranty. (Note, incidentally, that if the single-failure model of Equation (4.1) were used to approximate this cost, the result would be £[CS(1)] = cs[l + (i e 5(i))j _ 1393Cs? which significantly underestimates the true cost of the warranty.) If, instead, X = .4, so that the average life of a picture tube is 2.5 years, the expected cost of a warrantied item is 1.4cs, still a substantial warranty cost. The exponential distribution is appropriate when dealing with items having constant failure rate. For many items, it is more realistic to assume that the failure rate is increasing. We next look at a distribution that has this property for appropriate choices of the values of its parameters. Example 4.2 [Weibull Distribution] The distribution function for the Weibull distribution is

This reduces to the exponential distribution if (3 = 1, and it has an increasing failure rate if 0 > 1. The Weibull distribution has mean

and variance

The renewal function for the Weibull distribution cannot be expressed analytically in closed form, but it has been tabulated extensively (Baxter et al. [1,2] and Giblin [3]). Table 1 in Appendix C is a reproduction of a portion of the Baxter et al. tables, giving values of M{t) = M(t; X, 0) for t = 0(.05)5.00, 0 = .5(.5)3.0(1)7.0, and X = 1. For X * 1, the value of the renewal function is obtained as M{t\ X, 0) = M(Xt; 1 ,0 ). For larger

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values of t, the approximation

may be used (see Section 3.6.2), with |x and a 2 as given in the preceding. We consider the Weibull distribution with p = 2 and 4. To make the results comparable with those for the exponential distribution, we select values of X for each case so that (x = 2.0 and 2.5. From the previous expression for |jt, we obtain

The required values of T(1 4- 1/p) may be obtained from any of a number of standard tables of special functions (e.g., Abramowitz and Stegun [4]). The values are T(1.5) = .88623 and T(1.25) = .90640. The corresponding values of X are given in Table 4.1. The required values of Af(l; X, p) are obtained by interpolation in Table C .l, using the relationship M (x; X, p) = M (\x\ 1, p). (See [1,2].) Finally, Table 4.1 gives values of 1 4- M( 1; X, p) = £[Cs(l)]/cs for the parameter combinations considered. Thus, for example, for p, = 2 and p = 2, the expected cost of supplying an item under FRW is 18.4% higher than the cost (cs) of supplying an item without warranty. Note from these results that far fewer warranty replacements will be needed and far lower costs incurred if failure times follow Weibull distributions with the same mean as the exponential but increasing failure rates. The reason for this is that the Weibull distributions used in this example have far smaller standard deviations than the exponential, and correspondingly fewer early failures. (With early failures relatively unlikely, it can be

Table 4.1 Factors for Calculating Cost of Nonrenewing FRW, Weibull Distribution X

ß 2 4

£[c.(i)]/c,

2.0

.443

.354

1.121

2.0

.453

1.041

2.5

2.5

.363

1.184

1.017

Chapter 4

138

expected that the single-failure model of Equation (4.1) will provide a much better approximation. This is easily seen to be the case in this example.) 4.2.3

Discounted Expected Costs for Nonrepayable Items

If the seller’s costs are not constant, but vary with time, so that cs = cs(i), this can be accounted for by replacing M(W) in Equation (4.2) by the expression

A particular case of interest is discounting of future warranty costs to present value. A standard approach is to express discounted costs as cs(0 = cse~h\ where 8 is the discount rate. If we write CS(W; 8) = discounted seller’s cost of supplying an item with FRW, Equation (4.2) becomes (4.3)

4.2.4

Basic Cost Models for Repairable Items

In the case of repairable items, there are a number of additional factors that may affect the supplier’s total cost of warranty, and, correspondingly, a number of additional options that may be available to the supplier. The principal consideration is, of course, the cost of repair. An important new option that is generally available to the supplier is whether to replace a failed item by a new one rather than repairing the failed item. The relative cost of repair versus replacement is then a key consideration, and models of the process are used to devise optimal repair/replace strategies. Analyses of these and related issues will be considered in Chapter 9. The second important element in modeling the warranty process for repairable items is the failure distribution of the items after repair. If it can be assumed that items are returned to their original state after each repair (i.e., are “good as new”), the models of the previous section are applicable with only minor modifications. Since good-as-new repair implies that new and repaired items have the same life distribution F, we again have an ordinary renewal process, and the models need only be changed to reflect the differential costs of replacement and repair.

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Many other assumptions concerning the life distributions of repaired items have been considered. A common alternative is minimal or “badas-old” repair, where it is assumed that the life distribution after repair is the same as the conditional life distribution given that the item had survived up to the time of the observed failure. Other models are based on the assumption that the failure distribution is modified after each repair, for example, with each repair effecting an increase in the failure rate. In this section, we shall confine attention to the good-as-new repair model. Other repair models will be considered in Chapter 9. Under this assumption, the cost to the supplier of replacing a single repairable item under FRW is

where cr is the expected total cost of supplying a repaired item, including parts, labor, shipping, administrative costs, and so forth, and Nr(t) is the number of repairs required (or repaired items supplied) in the interval [0, t). Note that the use of average repair cost is again a simplification. In reality, the cost associated with each repair is a random variable Cr, and its distribution may change (for example, shifting toward higher costs) through time and with each repair. The quantity cr is the expected value of Cr, averaged over all of these various conditions. Another random quantity that is ignored in most models for repairable items is repair time. In the context of warranty, this is usually not a problem, for several reasons. Firstly, repair time is usually short enough relative to the warranty period and relative to the mean time between failures that it can be safely ignored for most items. Secondly, the warranty period is almost never extended to compensate for lost time while repairs are being performed. (Thus, the repair time effectively counts as service time!) Thirdly, in some cases it may be possible to employ a policy that in effect removes this factor from consideration in the warranty process. Two methods of doing this are (1) provision of a “loaner,” and (2) replacement of a failed item by a repaired, previously failed item from a stock of such items kept on hand for this purpose. Under the assumptions of good-as-new repair, constant repair cost, and the additional simplifying assumptions discussed in the preceding, it follows from a derivation similar to that given previously that the supplier’s expected cost under FRW is given by (4.4) Realistically, in most situations, repair is less than perfect and repaired

140

Chapter 4

items are not good as new. An alternative model in which this is recognized is based on the assumption that the original item has a lifetime X x with distribution F(-), and repaired items are assumed to have lifetimes X 2, X 3, . . . , identically distributed with distribution G(-). The expected number of failures over [0, W] is given by the delayed renewal function given in Equation (2.25) as (4.5) where MG( ) is the ordinary renewal function corresponding to G, rather than by the ordinary renewal function M( ). As a result, the expected cost of warranty is given by (4.6) 4.2.5

Discounted Expected Costs for Repairable Items

As in the case of nonrepairable items, if cTis a function of r, the expected cost function can be modified accordingly. For discounting costs to present value, this modification results in the expression (4.7) where, as before, CS(W; 8) denotes the seller’s costs for warranty period W, discounted at rate 8. Example 4.3 [Exponential Distribution] Suppose that new and repaired items have exponentially distributed lifetimes with respective parameters \ 1 and with < ^2 (so that > 1x2)- Then

Note, incidentally, that in this case Md(t) = MG(t) - [(X2 _ ^i)^i]^(0* i.e., the delayed renewal function is the renewal function for repaired items reduced by an amount proportional to the relative difference between the failure rates for new and repaired items.

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141

For illustration, suppose \ 1 = .4 and \ 2 = .5 (corresponding respectively to mean times to failure (MTTFs) of 2.5 years for new items and 2.0 for repaired items). For W = 1 year, we find Md( 1) = .5 - (.1/.4) (1 - e~ 4) = .4176, so that

If failed items are replaced rather than repaired, this expected cost (Example 4.1) is given by 1.4cs. It follows that it is better (less costly) to repair than to replace an item as long as cs + .4176cr < 1.4cs, i.e., as long as cs > 1.044cr. Thus, it is virtually always better to repair an item. On the other hand, if \ 2 = 1 (so that repaired items have an MTTF of 1 year), then £[CS(1)] = cs + .5055cr, and it is better to repair an item only if cs > 1.263cr. With discounting, the analysis for the exponential distribution proceeds as follows: The delayed renewal density is

giving expected cost discounted at rate 8 as

For W = 1, X2 = .4, and X2 = .5, a discount rate of 10% yields

For X2 = 1, this expected cost is cs + .4795cr. Both of these are only slightly lower than the undiscounted values. For most other common life distributions, Equations (4.5) and (4.7) must be solved by numerical methods or approximated to determine warranty costs for repairable items. 4.2.6

Modeling Aggregate Sales and Profit

In order to estimate the total cost of warranty (and, ultimately, total profit) for a product, it is necessary to model sales as well. Again, many factors

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are involved, and many models expressing sales through time as a function of these factors have been developed. A demand model that explicitly considers warranty as a factor has been proposed by Glickman and Berger [5]. Demand is assumed to be a decreasing function of price and an increasing function of W, and it is assumed to be of the form (4.8) where q{•, •) is the demand function, and f l > l , 0 < 6 < l , f c 1 > 0 , and k2 ^ 0 are parameters whose values must be provided or estimated, for example, on the basis of some relevant marketing data. The interpretation of the parameters of this model is as follows: k x is a scale (“amplitude”) factor; k2 is a displacement parameter to account for the fact that there will be sales even if no warranty is offered; a represents price elasticity; and b is a measure of (displaced) warranty period elasticity. Given the cost structure and analysis of the previous sections, it is easy to model seller’s expected profit, say t t (W), as well, since this is simply the difference between selling price and seller’s cost. Thus, the expected per-unit profit is given by (4.9)

(4.10) for repairable items having lifetimes distributed differently from those of new items. (Similar results obtain for other types of repair.) Total expected profit may now be modeled simply as ir(W)q(ch, W). In principle, this quantity may now be maximized with respect to c5 and W to obtain optimal choices for price and warranty period. Differentiation will lead to expressions involving renewal densities. Computer methods are required for solution. Glickman and Berger [5] present a few results for repairable items with lifetimes following an Erlang distribution. Additional details are given in Chapter 10. 4.3

MODELING THE BUYER’S COSTS

We consider now the cost to the buyer, Cb(W), of an individual item purchased under nonrenewing FRW. Cost models for the buyer’s point of view have been considered under basically the same simplifying assump-

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tions used in analyzing the seller’s costs. Thus, we again assume that failed items are replaced or repaired instantaneously, that certain key elements are constant rather than random, and so forth. If we consider only the cost of acquisition of the item, then, under these assumptions, the buyer’s total cost during the warranty period is simply cb, since any needed replacement items are supplied free of charge up to time W after purchase. (Here we assume that incidental costs of acquisition—information gathering, travel, etc.— are either ignored or are included in cb.) If we consider instead the total cost of ownership (either during the warranty period or over the lifetime of the item), many other cost factors must be recognized. Suppose, for example, that we model the buyer’s total lifetime cost of a single item covered under nonrenewing FRW. This cost includes not only the original purchase price, but also costs of installation, energy, maintenance, and repair (except as covered by warranty) and many potential incidental costs, including costs due to loss of service of the item while it is undergoing maintenance or repair, possible legal costs, cost of invoking the warranty, unrecoverable incidental damages caused by failures, and, ultimately, cost of disposal. We shall include installation cost as part of the purchase price and ignore disposal cost; all other costs will be included under incidental costs. Note that, for nonrenewing FRW, the total time period involved may be written as W + B(W), where B{W) is the residual life of the item in service at time W after purchase (see Section 3.5). This may be, in fact, the lifetime of a single item. Otherwise, it is W plus the residual life of the item then in service. Note also that under FRW the buyer is responsible for maintenance and repair only after W, but is responsible for energy and most incidental costs for the entire life of the item. Based on these factors, the total cost of ownership of a single item under FRW may be expressed as

(4.11) where cb is the original purchase price, CG(W + Z?(W)) is the total operating cost over the lifetime of the item and any replacements (usually primarily energy costs), Cm(W, W -f B(W)) is the cost of maintaining the item from time W to time W + B(W ), including parts, service, shipping, possible overhaul, and so forth (costs are assumed to be covered by the warranty until time W), and CY(W + B(W )) denotes incidental costs of ownership for the lifetime of the item.

144

Chapter 4

Other than c5, the cost elements in this model are random variables, each of which may be a function of many other random variables. At least in the context of warranty processes, analyses of these variables under various distributional assumptions have not been pursued. 4.4

UNIT COST MODELS FOR THE RENEWING FRW

We now look at unit cost, that is, cost per unit sale, for items sold with a renewing FRW, that is, an FRW where the warranty period begins anew after each replacement under warranty as well as after each purchase. The renewing FRW, sometimes termed the unlimited free-replacement warranty, was stated in Chapter 2 as follows: Policy 5 FREE-REPLACEMENT POLICY: The manufacturer agrees to repair a failed item or replace it free of charge up to time W from the time of initial purchase. The repaired or replacement item is covered by a new warranty whose terms are identical with those of the original warranty. This warranty is not often used in practice and, when it is, its use is basically limited to small, nonrepairable consumer products. As a result, very little attention has been paid to modeling costs of renewing FRWs. We briefly discuss two results: per-unit expected cost to the seller of items warranted under Policy 5, and discounted per-unit net profit. 4.4.1 Modeling the Seller’s Costs

Consider the case of independent and identically distributed items with distribution function F( ). Let K(W) be the number of free replacements (and/or repairs, if repair is good as new) required under renewing FRW. Thus, X K(W) +1 is the first item lifetime in the sequence of replacements that is at least of length W. K{W) is a random variable with P{K(W) = 0} = 1 - F(W) = F(W), and, in general, (4.12) This is a geometric distribution and has mean E[K(W)] = F(W)/F(W). From this it follows that the expected per-item total cost to the seller under renewing FRW is (4.13)

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This result is easily extended to the case of repairable items having identical life distributions G(-) different from that of new items. The distribution of K{W) now becomes (4.14) From this it follows that E[K(W)] = F(W)/G(W), so that, for repairable items, (4.15) These results are further modified in an obvious way if the item lifetimes X t, X 2, . . . , are independent with respective distributions Fi(-), F2(-), . . . . Example 4.4 [Exponential Distribution] As in the previous examples, take X = .5 and W = 1. For nonrepairable items,

The renewing warranty is thus seen to be somewhat more costly than the nonrenewing version (Example 4.1), for which the expected seller’s cost was found to be 1.50cs. For X = .4, the corresponding values are 1.49cs for the renewing versus 1.39cs for the nonrenewing FRW. Similar results are obtained for repairable items. Taking Xx = .4 for new items and X2 = .5 and 1.0 for repaired items, we find the expected costs to be cs + .5436cr and cs + .8962cr, respectively, for the renewing FRW versus cs + .4176cr and cs + .5055cr for nonrenewing. Example 4.5 [Weibull Distribution] For the Weibull distribution with parameters as in Example 4.2, the differences between renewing and nonrenewing FRWs are not nearly as striking. Here F(t) = 1 (f> 0). For W = 1 and the combinations of parameter values used previously (P = 2 with X = .443 and .354, and 0 = 4 with X = .453 and .363, giving, in each case, |x = 2.0 and 2.5), relative expected costs are given in Table 4.2. These costs are only slightly higher than those of the nonrenewing FRW. The reason for this is that

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Table 4.2 Factors for Calculating Cost of Renewing FRW, Weibull Distribution

£[Cs(l)]/cs

p 2 4

2.0 2.5 2.0 2.5

1.217 1.134 1.043 1.017

relatively few failures are expected to occur during the warranty period, particularly for p = 4. 4.4.2

Modeling the Buyer’s Costs

With appropriate modifications, Equation (4.11) can, in principle, be used to express the buyer’s total cost of ownership for the renewing FRW as well. In practice, however, the renewing feature considerably complicates the analysis because the time elements on the right-hand side of the equation are not, in fact, the constant VF, but involve the random total time until an item with lifetime W is found. A detailed analysis of this cost estimation problem has not been carried out. 4.5

LIFE CYCLE COST MODELS FOR THE FRW

We turn now to the concept of life cycle costs to seller and buyer. In the analysis to follow, the life cycle of concern is not the time horizon based on the useful life of a single item purchased under warranty as in the previous two sections; rather it is taken to be a longer period L over which identical replacements will continue to be purchased or provided under warranty. Examples are replacement tires, batteries, and other components over the lifetime of an automobile (for which L may be, say, 10 years); replacement engines, windshields, tires, and a large number of other components on military or commercial jet aircraft (for which L may in some cases be as much as 30 years or more). Note that L is typically again a random variable. This, however, considerably complicates the analysis, and we shall assume in our models that L is a known, deterministic quantity. In dealing with life cycle costs in this sense, it is necessary to analyze repeated realizations of the warranty cycle, beginning with the initial purchase and accounting for replacements under warranty in each cycle as

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well as repeat purchases up to time L from the initial purchase. In this sequence, the seller will incur a cost each time an item fails, while the buyer will incur a cost (and hence, the seller will realize income) at the beginning of each cycle, that is, at the first failure outside of warranty coverage, which ends the preceding cycle. Thus, in looking at the time horizon L, we have two renewal processes involved, that associated with the sequence X u X 2, . . .o f lifetimes within each cycle, and that associated with the sequence of purchase times, say, Y u Y2, . . . . Since purchases occur upon failure of the item in service at the end of warranty coverage, the Y/s are of the form Y = W + B(W) for the nonrenewing FRW, where B(-) is the excess life, a random variable defined previously. (We assume, as before, that repairs or replacements occur instantaneously. Here this means, in addition, that repurchases are instantaneous. Again, this will be a reasonable approximation to reality as long as the actual times until replacement and repurchase are small relative to W and L.) A schematic representation of the life cycle warranty process, including cost elements based on our previous analyses, is given in Figure 4.1 for the nonrenewing FRW and nonrepayable items. (For repairable items, the seller’s costs are altered as in Section 4.4.2.) For the renewing FRW, the process is somewhat more complicated. Since in this case a cycle is completed only after a single item achieves a lifetime of at least W, the length of the warranty coverage period is itself a random variable, and the seller realizes income only after failure of the item in service at the end of that period. In this case, the random variable Y, which represents cycle length, is not expressed simply in terms of W and the excess random variable. This is illustrated in Figure 4.2, in which a few typical cycles of a renewing FRW are represented, with X t and S, representing lifetimes and partial sums, in the usual notation. Note in this illustration that since Sx < W and S2 < Sx + W, but S3 > S2 + W, the

Figure 4.1 Cost cycles for the nonrenewing free-replacement warranty.

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Figure 4.2 Some typical cost cycles for the renewing free-replacement warranty. first cycle involves two free replacements. In general, Y is given by

Expected life cycle costs in this case will involve the number of renewals of this process over the life cycle L. It is clear from this initial analysis that, as before, life cycle costs will depend on the life distributions of the items—whether or not the warranty is renewing, whether or not the items are repairable— the warranty period W, and the various cost elements, in addition to the life cycle L. In the remainder of this section, life cycle costs (LCC), including discounting to present value, will be analyzed for a number of these warranty structures from the seller’s and buyer’s points of view. In addition, an indifference price structure— that is, differential prices such that the long-run costs are the same with and without warranty— will be derived for both seller and buyer. It is convenient to begin with the buyer’s cost since that involves only the renewal function of the variable Y. The seller’s profit can then be expressed as the difference between this and the seller’s cost. 4.5.1

Modeling the Buyer’s Life Cycle Cost

Basic Model

We consider the nonrenewing FRW, assuming nonrepairable items. Most of the results to be given can easily be modified to deal with repairable items as well, as long as repair costs are constant and repair is good as new. Thus, we assume that lifetimes X u X 2, . . . are independent and identically distributed with distribution function F(*). It follows that the buyer’s purchase intervals Y1? Y2, . . . are also independent and identically

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distributed. Let My(*) be the renewal function associated with the Yf. The expected number of purchases in the interval [0, L], i.e., during the life cycle of the item, is 1 + Aiy(L). Let Cb(L, W) denote the buyer’s LCC. Then by Wald’s Theorem, the buyer’s undiscounted expected total cost of purchasing the item over its life cycle is (4.16) In practice, considerable difficulty is encountered in evaluation of My(*). We begin with the distribution of Y. Since Y = W + B(W), its distribution is simply a translation of the distribution of the excess random variable, B(W). This was given in Equation (3.19) as (4.17) From this it follows that the distribution function of Y is (4.18) In concept, one could now derive the renewal function either by solving the renewal integral equation (Equation (3.12)) or by determining the A>fold convolutions of Fy(*) with itself (k = 1, 2, . . .) and using Equation (3.9), which expresses the renewal function in terms of these convolutions. Alternatively, My(*) can be expressed in terms of FB(*), given in Equation (4.17). The result (Nguyen and Murthy [6]) is

(4.19) In practice, all of these approaches lead to substantial mathematical difficulties in all but the simplest cases. In addition, tables for My(-) are not available for any distributions. As a result, it is generally necessary to resort to numerical computation, analytical approximations, bounds, or simulation in evaluating this renewal function, except for the few distributions for which exact results are possible.

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One approach, appropriate if L is large relative to W and E (Y), is to use asymptotic results for renewal functions. The simplest of these is the Elementary Renewal Theorem (given in Equation (3.34)), which states that M Y(t)/t —» 1IE(Y) as t —> oc. it can be shown that E(Y) = |x[l + M(W)], where p, = E(X) and M () is the ordinary renewal function of X. For large L, this provides the approximation (4.20) If F(-) is NBU, then the bounds of Proposition 3.1 of Section 3.6.3 can be used to obtain the improved approximation (4.21) The result of Equation (4.20) can be further improved, for any F(*) having finite higher moments, by use of a higher order asymptotic result. The second-order approximation (Cox [7]) of this type is (4.22) where V{Y) is the variance of the random variable Y. Note that V(Y) is equal to V[B(W)], since Y is simply a translation of £(•). An expression for V(Y) is given (Nguyen and Murthy [6]) by (4.23) where a 2 = V(X). Evaluation of the right-hand side of Equation (4.23) is usually not possible analytically. Note, however, that the final term can be expressed as

The integral of the renewal function has been tabulated for several important distributions (e.g., by Baxter et al. [1]), and this may be used to provide numerical results.

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Note, incidentally, that Equation (4.20) overestimates the expected number of renewals if F(-) is NBU. On the other hand, if the item is not NBU, then, compared with Equation (4.22), (4.21) would tend to underestimate the value. Example 4.6 [Exponential Distribution] For the exponential distribution, the conditional distribution of X , given that the item has survived to time r, is again exponential with the same parameter X. It follows that the distribution of the excess random variable is also exponential with parameter X. Thus, the distribution of Y is simply the translated exponential:

The renewal function in this case can also be expressed analytically. The result (Cinlar [8]) is

where [x] denotes the largest integer less than x. We consider again the previous examples with X = .4 and .5 (jx = 2.0 and 2.5 years) and W = 1 year. Suppose we look at life cycles of 7.5 and 15 years. For X = .4 and L = 7.5, we find

Thus, the total number of expected purchases is 2.898, with a total expected cost to the buyer over a life cycle of 7.5 years of 2.898cb. For L = 15 years, this value is found to be 5.041cb. For X = .5, these values are 3.222cb for L = 7.5 years and 5.722cb for L = 15 years. For the exponential distribution, E(Y) = jx[l + M(W)] = X-1[l + KW]. Thus, for X = .5, E(Y) = 3 in this example. Further, since translation does not affect the variance, V(Y) - V(X) = X-2 = 4. The approximation of Equation (4.20) yields M( 1) ~ 7.5/2[l + .5] = 2.5 versus an exact value of 2.222. Equation (4.22), however, gives the exact value for the exponential distribution; in this case, M(l) ~ 2.5 + 2/9 - .5 = 2.222, which is exactly correct. Example 4.7 [Weibull Distribution] For the Weibull distribution, Ai(-) cannot be evaluated analytically. Thus, it is also not possible to determine exact analytical expressions for

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Fb{•) from Equation (4.17) or Fy(*) from Equation (4.18), and the analysis will be based on the approximations. In previous examples, parameters were chosen so that the same means as the exponential resulted. Pursuing this illustration, we calculate the expected number of purchases for L = 7.5 and 15 years for the Weibull with 3 = 2 and 4. Since these choices of 3 give increasing failure rates, so that the distributions are NBU, we use the approximation of Equation (4.21) . The values for 1 + M(W) that are required are given in Example 4.2. The resulting approximate values for expected number of purchases are given in Table 4.3. Note that these values are substantially higher than the corresponding values for the exponential distribution. To improve on this approximation, we require V(Y) for use in Equation (4.22) . Note that the improvement involves the addition of the term V(Y)I 2E 2(Y), which is always an upward adjustment. The value of V(Y) may be obtained from Equation (4.23), using the alternate form of the last term in this expression and the tables cited. We also require the result that, for the Weibull,

The required tabulated values of the gamma function are T(1.25) = .90640, T(1.5) = .88623, and T(2) = 1.0. The values of the integral of the renewal function for the four cases in the table of the previous paragraph are found, by interpolation in the Baxter et al. [1] tables, to be .0632, .0423, .0088, and .0055. We now obtain, for 3 = 2 and (jl = 2 (for which combination X = .443),

Table 4.3 Factors for Determining Life Cycle Costs, Nonrenewing FRW, Weibull Distribution 1 + My (L)

p 2 4

2.0 2.5 2.0 2.5

L = 7.5

L = 15

3.667 3.176 4.102 3.450

6.834 5.852 7.705 6.400

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and

The resulting correction to the preceding approximation is .9053/2(2.3742) = .080. We thus have estimated average purchases of 3.659 + .080 = 3.739 for L = 7.5 and 6.818 + .080 = 6.898 for L = 15. For the remaining three lines of the table, the adjustments are .093, .033, and .034. In the preceding example, the refinement in the approximation leads to very small changes, and certainly to no significant difference in interpretation of the results. The distributional assumptions, however, are important, and they can lead to substantial differences in estimated costs. In the preceding illustration, the buyer’s costs are nearly 50% higher if item lifetimes are Weibull distributed with (3 = 4 than if they are exponentially distributed with the same mean. Discounted Buyer's Cost

Discounting to present value can be accomplished as before by use of the discount factor e~bt, where 8 is the discount rate, in the defining integral equation expressing expected cost. This leads to Equation (4.3) with My( ) replacing M( ). The result is an expression involving the Laplace transform of the renewal function My(*), which in practice again leads to substantial analytical difficulties. A simpler approach is to base the discounting on expected purchase intervals. As we have seen, under the FRW, purchases are made on the average at time intervals of length E(T) = |x[l + M(W)\. The expected number of purchases until time L is approximately L/E(Y). Suppose, for ease of computation, that this is an integer (say k). Then the expected present value of all payments over the life cycle L, say Cb(L, W; 8), is approximated by

(4.24)

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Table 4.4 Factors for Calculating Discounted LCC, Nonrenewing FRW, Weibull Distribution

L = 7.5

4

II bL /l

2

8 = .10

8 = .05

8 = .10

3.481 3.079 3.853 3.307

2.969 2.631 3.282 2.823

5.186 4.508 5.814 4.892

3.899 3.402 4.360 3.684

QO

p 2.0 2.5 2.0 2.5

L = 15.0

If L is large, the exponential term in the numerator of (4.24) vanishes, and the expected present value may be approximated simply as (4.25) Example 4.8 [Weibull Distribution] Discounted costs of warranty, calculated from Equation (4.24), for the same sets of Weibull parameters used in the previous examples, with discount rates of 8 = .05 and .10, may be obtained from the factors given in Table 4.4. In each case, the long-run expected cost is calculated as cb times the tabulated factor. The effect of discounting can be seen by comparison of these results with those of Example 4.7 (in which 8 = 0).

4.5.2

M od eling the S e lle r’s Life C y c le Profit

Based on the results regarding the buyer’s costs given in the previous section, it is easy, in principle, to model the seller’s profit. If we assume that the seller’s expected cost per item cs includes not only cost of manufacture, but also all other costs of sale, including marketing, distribution, dealer’s profit, and so forth, then, since cost to the buyer, cb, is income to the seller, the seller’s profit is simply a function of these two costs. Over the life cycle L of the item, the expected number of sales is 1 + Afy(L). The expected cost to the seller of each sale is cs[l + M{W)]. If Pf r w is taken to be the profit on a stream of items sold under FRW to a single buyer (that is, an initial purchase plus all purchased replacements

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over the life cycle L), we have (4.26)

4.6

INDIFFERENCE PRICES

Another approach to determining the value of a warranty is to consider the proportional price differential that a buyer should be willing to pay or that a seller would have to charge for a warrantied item as opposed to an unwarrantied item. From the buyer’s point of view, this is equivalent to finding an indifference price c£, say, below which a warrantied item is less expensive in the long run and above which an unwarrantied item, purchased at price cu, say, is cheaper. Thus, cb is such that if the buyer had the option of purchasing an unwarrantied item at price cu or a warrantied item at price c5, he would prefer to purchase the warrantied item if cb < c£, would prefer the unwarrantied item if c5 > cb, and would be indifferent between the two if cb = cb. From the seller’s point of view, the equivalent result is an indifference price, say cb*, below which the seller would prefer to sell the item without warranty at price cu and above which he would prefer to sell the item with warranty at price cb. We again consider the two points of view separately, beginning with the buyer. We shall analyze only the nonrenewing FRW and, as before, we will consider only basic purchase and replacement costs over a life cycle of length L. Indifference prices with and without discounting will be given.

4.6.1 Buyer’s Ind iffe re n c e Price

The basis of the analysis is simply to equate the buyer’s long-run expected costs of items with and without warranty. For unwarrantied items, sold at cost cu, say, the total expected cost to the buyer over the life cycle L is cu[l + M(L)], since the buyer must purchase a replacement at each failure in [0, L). For warrantied items, the buyer’s total expected cost is cb[l + My(L)], as we have previously seen. Equating these two long-run costs leads to the indifference price:

(4.27)

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As before, because of its complexity, it is often useful to approximate M Y(L ) when L is large. The approximation used previously leads to (4.28) The adequacy of this approximation is discussed by Blischke and Scheuer [9]. The approximation is good for large values of L, but it appears to lead consistently to overestimates of cl if L is not large. As L —» oo, M(L) is approximately L/|x, and (4.28) becomes (4.29) Discounting of future payments may also be considered. If future payments are discounted to their present value at rate 8, the indifference price becomes (as L —» o°) (4.30) 4.6.2

S e lle r’s In d iffe re n c e Price

In our analysis of the seller’s point of view, we consider the differential pricing structure such that the seller’s long-run profit would be the same if items were sold with warranty at price ch or without warranty at price cu. Without warranty, each transaction results in a profit of cu — cs, so that the seller’s total expected profit is simply (cu - cs)[l + M(L)]. Equating this to the seller’s expected profit under warranty and solving will yield the seller’s indifference price, say cl*. With warranty, if the administrative cost of servicing the warranty is ignored, the expected long-run profit to the seller is his long-run income, cb[l + A/y(L)], minus expected expenses of cs[l + Af(L)], the cost of supplying all of the required items over the period to time L. This is equivalent to assuming that cs includes all of the seller’s costs, including administrative costs of servicing the warranty, amortized over all items. In this case, it is easily seen that cl* = cb, where cb is given by (4.27); i.e., the seller and buyer indifference prices are the same (but note, of course, that the buyer prefers the warrantied item if the actual selling price c5 is less than c l, whereas the seller prefers to offer the warranty if cb > cl). If the per-item administrative cost, say ca, of servicing the warranty is included, then the indifference price to the seller is increased. Considering, again, expected costs in each instance, the result is as follows: As before,

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the net profit without warranty is (cu - cs)[l + M(L)]. Total expected income is cb[l + M Y{L)\. The total cost of supplying all items over the entire life cycle, exclusive of administrative costs, is cs[l + M(L)]. The expected number of items supplied under warranty (each of which has an additional cost ca) is M(L) - M Y(L), the remaining items being repurchases. Equating total profits for the two situations, we obtain

giving as the indifference price, including administrative costs, (4.31) Note that, in this situation, the buyer prefers a warrantied item at cost cb to an unwarrantied item at cost cu if cb < cb, the seller prefers to supply an item with warranty only if cb > c£*, and both would prefer an unwarrantied item at price cu to a warrantied item at price cb if cb < cb < cb*. Example 4.9 [Weibull Distribution] We again consider the Weibull distributions with (3 = 2 and 4 and p = 2.0 and 2.5, with W = 1 and L = 7.5 and 15. The values of M(L) required for calculation of the indifference prices, obtained by interpolation in the Baxter et al. [1] tables, are given in Table 4.5. The multipliers of cu for calculation of cb are obtained by division of these values by the corresponding quantities 1 + M Y{L). For this purpose, the approximations given in Example 4.7 will be used. The resulting ratios, along with the limiting values obtained from (4.2), are given in Table 4.6. Thus, for p = 2, p = 2, and L = 7.5, the buyer prefers a warrantied item only if its cost does

Table 4.5 Factors for Calculating

Indifference Prices, Weibull Distribution 1 + M(L) p 2 4

2.0 2.5 2.0 2.5

L = 7.5

L = 15

4.405 3.650 4.306 3.543

8.173 6.683 8.072 6.567

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Table 4.6 Factors for Calculating Discounted Indifference Prices, Weibull Distribution

cb*/cu p 2 4

2.0 2.5 2.0 2.5

L = 7.5

L = 15

L = oc

1.204 1.149 1.049 1.027

1.199 1.142 1.047 1.026

1.184 1.121 1.041 1.017

not exceed 20.4% more than that of an unwarranted item. Equivalently, the seller’s long-run profits (ignoring the administrative costs of the warranty) would be the same if items were sold without warranty at price cu or with warranty at price cb = 1.204cu. 4.7

ADDITIONAL MODELS FOR ANALYSIS OF THE FRW

The free-replacement warranty has been the subject of rather extensive modeling efforts. In this chapter, we have been concerned mainly with cost models for manufacturers and, to a lesser extent, buyers, under relatively simple, straightforward conditions. We turn now to a few of the models that represent attempts to relax these conditions, generalize the results, and broaden their applicability. Some additional cost models are discussed briefly in the notes at the end of the chapter. Other aspects of the FRW are modeled and analyzed in later chapters of the book. In the remainder of this section, we shall briefly consider models that incorporate certain special features, including intermittent use and discrete usage patterns, and that take into account the fact that not all valid warranty claims are made nor are all claims that are made valid. 4.7.1

Intermittent Usage

Most of the models given previously in this chapter are based on ordinary or other renewal processes. Models of this type are appropriate if (among other things) use is continuous once a product is put into service and replacements or repairs of failed items are instantaneous. Suppose, instead, that usage is intermittent, i.e., the item alternates between being in an idle state (State 0, say) and being in use (State 1). This is realistic for many consumer goods, for example, appliances, automobiles, electrical items such as television sets, mixers, power tools, lawn mowers, and many other

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durable goods. Murthy [10] considers two approaches to modeling this usage structure for both repairable and nonrepairable items. Both models are based on a Markov chain formulation; i.e., it is assumed that transitions between States 0 and 1 in a given time period depend only on the current state of the item and not on its past history of usage. The two models are as follows: Model 1: In this model, the failure rate t in a linear function of 1. Y(i), the age of the current unit 2. t (Y(0), the total duration for which the current unit is in State 1 during the period [t - Y(t), t] 3. N(Y(t)), the number of times the current unit has been switched from State 0 to State 1 during the period [t - Y(t), t\ Under this model, the failure rate increases with age, the number of times the item is used, and the total duration of usage. Model 2: In Model 2, the failure rate at time t depends on whether the item is in State 1 or State 0 at time t. The model formulation involves an additional binary random variable X(t), which assumes the value 1 when the item is in State 1 and 0 when the item is in State 0. Conditional on X(t) = 1, the failure rate at time t is a linear function of Y(t), t (Y(>)), and N(Y(t)) as in Model 1; conditional on X(t) = 0, the failure rate is a constant. The latter condition implies that when the item is in State 0, failures are due to pure chance. 4.7.2

Point Usage

When the item usage is intermittent and the duration of each usage is very small in relation to the time between usages, then each usage can be viewed as a point along the time continuum. Since usages are random, the usage is best modeled as a one-dimensional point process. This is a good description for many products, for example, coffee grinders, handyman tools, hair dryers, and many other such goods. If the failure rate depends only on the number of times the item is used, then the failure rate can be modeled by a discrete formulation (an indexed sequence with the index increasing with each usage). Murthy [11] develops a model for item failure based on this characterization and carries out an analysis to obtain the expected warranty costs for both repairable and nonrepairable products. 4.7.3

Modeling Invalid Claims and Unclaimed Benefits

Implicit in the models of the preceding sections of this chapter are the assumptions that all warranty claims are valid and that all covered failures

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within the warranty period lead to claims under warranty. In practice, neither assumption holds. From the manufacturer’s point of view, claims not actually made are a difficulty only in that they complicate the cost estimation problem, but invalid claims actually increase warranty costs, and many companies devote some effort to validating claims in order to reduce these costs. An approach to modeling the frequency of warranty claims, following Patankar and Mitra [12], is to define a warranty execution function, e(t) = ^{Warranty claim is made|Product failure at time t} and use this, in essence, as a discounting function in calculating expected warranty cost. A general model of this type is given by (4.32) where, as in previous models, cs( ) expresses replacement cost as a function of time, including discounting, if desired. Patankar and Mitra consider the cost function cs(i) = cse-'

(4.33)

where 8 is the discount rate and is the rate of inflation. They propose a warranty execution scheme under which all possible claims are made during an initial period, say [0, w), where 0 < w < W, and the claim frequency decreases linearly to a final proportion tt during the remainder of the warranty period. The resulting expression for e(t) is (4.34) where b = (tt - 1)/(W - w). Models along these lines that account for invalid claims as well have not been developed. One approach would be to generalize the function e{-) to include probabilities of claim for failures not covered by warranty as well as for items that have, in fact, not failed, both of which are situations encountered in real-life applications. 4.7.4

A General First-Failure Model

General models of warranty costs, incorporating as many as possible of the features discussed in this chapter, have been formulated by Hill [13].

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We shall consider a first-failure model for the nonrenewing FRW that is based on these results. The model includes a choice of two methods of discounting to present value, but it is easily modified to include a general cost function as in Equation (4.32). It also includes constant probabilities of false claims and unexecuted claims. The result is

(4.35) where a and a* are the proportions of valid and invalid claims made, 0 and 0* are the proportions of valid and invalid claims rejected, /(•) and /*(•) are the life distributions of items leading to valid and invalid claims, and 8( ) is the discounting function. An expression for b(t) that provides for both the usual forms of discounting is (4.36) where 8 is the discount rate, p is the optimal rate of return on invested funds, and ax and a2 are indicator (0-1) variables used to select the form of discounting desired. In principle, this model could be generalized to include multiple failures by the introduction of appropriate renewal functions. Models of this type would again involve complex renewal processes, and mathematical analysis of these has not been pursued. Some work has been done, however, on simulation of Hill’s generalized models. The results are discussed in Chapter 11. NOTES

Section 4.2 1. Historically, the first cost analyses based on probabilistic models were apparently those of Lowerre [14] and Menke [15]. Lowerre developed a binomial model for nonrepairable items. Menke was concerned with cash reserves required to cover future costs for repairable products sold under warranty, assuming exponential failure distributions. Both used, in effect, the first-failure model given in Equation (4.1). These results were extended to include discounting to present value by Amato and Anderson [16]. 2. Many additional results regarding repairable items, sold in lots of fixed size, are given by Nguyen and Murthy [17]. The various types of repair

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are considered and prediction intervals (based on the normal approximation) are given for the total expected cost of warranty. Additional important results in this area are given by Karmarkar [18]. 3. A very different approach to modeling and estimating warranty costs, with emphasis on accounting aspects, is presented by Balachandran et al. [19] and others. The analysis is based on a Markov chain representation of states of items and components and transitions between states over accounting periods. A major difficulty with this approach is that the number of states that must be defined becomes very large in any realistic application, leading to significant conceptual and computational difficulties. An analysis of this approach and some further observations on it are provided by Nguyen [20]. Section 4.3 1. User cost of repairs after the warranty period are modeled by Park [21], with repair cost over the life of the product discounted to the present. Section 4.4 1. As mentioned, few treatments of the renewing FRW appear in the literature. In addition to Nguyen and Murthy [6], Mamer [22] provides an analysis of some depth (discussed in what follows). Elsewhere, this warranty is given only cursory mention. Section 4.5 1. Mamer [23] analyzed “short-run” (life cycle L ) and “long-run” (L —> °°) average total cost of ownership for the buyer and average profit for the seller for the nonrenewing FRW. The results involve difficult integral equations for which the results of this section are an approximate solution. 2. The most comprehensive model of buyer and seller expected unit costs as well as discounted long-run costs for both the nonrenewing and renewing FRW is that developed by Mamer [22]. Two modes of failure are identified, one occurring under “ideal” conditions and the other due to random damage. It is assumed that the warranty covers both types of failure (or that the cause is not known) and that damage occurs according to a Poisson process. The results involve only the renewal function M () (ordinarily much more easily obtained than Afy(-)), but they also involve Laplace transforms of considerable complexity. 3. Means and variances of total LCC to the buyer can be obtained from the results of Balcer and Sahin [24]. In practice, the usual computational problems are encountered. 4. Additional results on discounted future costs for repairable items are given by Karmarkar [18] and Park and Yee [25].

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163

Section 4.6 1. The results of this section are based on long-run averages. A sharper result regarding the buyer’s indifference price, based on embedded renewal processes, is given by Baxter [26]. 2. Many articles deal implicitly with the notion of an indifference price in that the authors compare costs of warrantied and unwarrantied items. Two additional sources that deal explicitly with the concept are Park [27] and Ip-Tamayo and Deuermeyer [28]. Section 4.7 1. Patankar and Mitra [12] use the warranty execution function in analyzing the monetary reserves required for covering anticipated future warranty costs on a lot of N items. 2. Amato et al. [29] consider accounting aspects of warranty, including administrative costs, for general warranty structures, including the FRW, and arbitrary failure distributions. Their model includes constants representing the probabilities of false claims and of unexecuted claims. Time-dependent probabilities are not considered. 3. Several of the models developed by Hill [13] are discussed in Hill and Blischke [30]. EXERCISES

4.1.

4.2.

4.3. 4.4.

Suppose that, for a particular product, the distribution of time to failure, F(i), is a mixture of two exponential distributions, i.e.,

where \ u X2 > 0, and 0 < p < 1. Suppose further that items are nonrepairable and are sold with a nonrenewing FRW policy. Obtain an expression for the seller’s expected cost per unit. Suppose that the item failure distribution is an Erlang distribution with shape parameter p = 2, i.e.,

X, t > 0, and that items are nonrepairable and sold with nonrenewing FRW. Obtain an expression for the seller’s expected cost per unit. Determine expressions for the means and variances of the mixed exponential and Erlang distributions of Exercises 4.1 and 4.2. Suppose that p = 0.05 in the mixed exponential distribution of Exercise 4.1. Select three sets of values for and p,2 so that p, = 2.0. Calculate the seller’s expected cost per unit for the nonrenewing

164

4.5.

4.6.

4.7. 4.8.

4.9.

Chapter 4

FRW with W = 1. Repeat this for |x = 2.5. Compare the results with those of Examples 4.1 and 4.2. Determine values of X so that jx = 2.0 and 2.5 for the Erlang distribution of Exercise 4.2. Calculate the seller’s expected cost per unit for the nonrenewing FRW in each case, and compare the results with those of Exercise 4.4 and Examples 4.1 and 4.2. [Note: Recall that the Erlang distribution is a special case of the gamma distribution (see Example 2.2). Values for M(t) are given in Table C.2.] Calculate the seller’s expected cost assuming the Weibull distribution as in Example 4.2, with p = 1.5, 3, and 5, and |x = 2.0 and 2.5. [Note: r(1.667) = 0.9027, T(1.333) = 0.8930, T(1.40) = 0.8873, and T(1.20) = 0.9182.] Compare the results with those for the p-values used in Example 4.2. Calculate expected costs in Examples 4.1 and 4.2 for W = 2 and 3. Repeat Exercises 4.4-4.6 with W = 2 and 3, and compare all results. Note that the mixed exponential distribution is DFR. Suppose that parameter values were chosen so that the remaining distributions of time to failure also were DFR. How would the results of the previous four exercises be affected? To get some idea of this, try the Weibull distribution with p = 0.50. Determine appropriate corresponding values of the parameters of the remaining distributions. Compare the results for these distributions with each other and with the results of the previous exercises. Extend the results of the previous exercises to the lognormal distribution, with density

Show that, for this distribution, and

4.10.

Determine values of j \ and 0 corresponding to the values of |x and a 2 for the Weibull distributions of Example 4.2 and the previous exercises. Use Table C.3 to obtain values of the renewal function, and calculate expected costs. Compare the results with those of the previous exercises. What do the results of Exercises 4.1-4.9 tell you concerning the importance of distributional assumptions, particularly in the context of calculating expected warranty costs?

The Basic Free-Replacement Warranty

4.11.

4.12.

4.13.

4.14. 4.15. 4.16. 4.17.

165

Assume that the distribution of time to failure is exponential with parameter X and that items are nonrepairable and are sold with nonrenewing FRW. Obtain an exact analytical expression for the discounted seller’s cost per unit with discount rate 8. Suppose that the failure rate of an item is given by

with 0, > 0, i = 0, 1, 2. Suppose further that the item is sold with nonrenewing FRW, that the seller services the warranty by always minimally repairing a failed item, and that the cost of each repair is Cr. Obtain an expression for the discounted seller’s cost per unit with discount rate 8. Suppose that items are nonrepairable and are sold with a nonrenewing FRW with warranty period W. Time to failure follows an arbitrary distribution F(t). Let p denote the probability that the buyer is satisfied with the first item purchased. Assume that buyers who are satisfied obtain free replacements for all items that fail within the warranty period and continue to purchase the product each time the item owned fails outside the warranty period. Assume that buyers who are not satisfied with the first item purchased do not make repeat purchases (but do exercise warranty claims). This happens with probability 1 - p. Obtain an expression for the expected LCC to the seller. How would the expression for LCC obtained in Exercise 4.13 change if the seller’s costs were discounted at rate 8? Indifference prices are discussed in Section 4.6. Calculate the buyer’s indifference price when the item failure distribution is Erlang with 0 = 2. Calculate the seller’s indifference price when the time to failure is Erlang with 0 = 2. Suppose that a repairable item with distribution of time to failure F(t) is sold with nonrenewing FRW. The seller uses the following strategy for servicing failed items: 1. An estimate of the cost to repair is obtained. 2. If the estimate is less than some specified amount CL, the failed item is minimally repaired. 3. If the estimate is greater than CL, the failed item is replaced by a new item. Assume that the cost of repair is a random variable with distribution function G(-). Obtain an expression for the seller’s cost per unit.

166

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4.18.

Suppose that items are repairable and that the failure distribution of a repaired item is a function of the number of times the item has been subjected to repairs. Let F;(t) denote the failure distribution of an item subjected to j repairs, with F0(t) the failure distribution of a new item. Let the cost to the seller of a new item be C0 and the cost of the yth repair be Cy, with C7, j > 1, an increasing sequence in ;. Obtain an expression for the seller’s cost per unit when the item is sold with nonrenewing FRW and the manufacturer always repairs items that fail under warranty. 4.19. Many products are used intermittently, i.e., an item alternates between being idle (State 0) and being in use (State 1). Assume that the changes between States 0 and 1 occur according to a continuous time Markov chain with parameters \ 0 and (see [8]). Assume that when the item is in State 0, the failure rate is zero (i.e., the item cannot fail when idle), and when it is in State 1, failures occur at a constant rate |x. Obtain an expression for the time to first failure. 4.20. Suppose that the item in Exercise 4.19 is nonrepayable and is sold with nonrenewing FRW with warranty period W. Determine the seller’s cost per unit. 4.21. It is very common in certain types of applications that not all warranty claims are exercised. Equation (4.34) characterizes this situation and defines the probability that a warranty claim is made, conditional on the age of the item at failure. Using this function, calculate the seller’s cost per unit. How does this cost change as a function of the parameter t t ?

REFERENCES

1.

2. 3. 4.

Baxter, L. A., Scheuer, E. M., Blischke, W. R., and McConalogue, D. J. (1981). Renewal Tables: Tables o f Functions Arising in Renewal Theory, Tech. Rept., Decision Systems Dept., Univ. of Southern California, Los Angeles, CA. Baxter, L. A., Scheuer, E. M., Blischke, W. R., and McConalogue, D. J. (1982). On the tabulation of the renewal function, Technometrics, 24, 151-156. Giblin, M. T. (1983). Tables o f Renewal Functions Using a Generating Function Algorithm, Tech. Rept., Postgrad. School of Studies in Industrial Technology, Univ. of Bradford, West Yorkshire, England. Abramowitz, M., and Stegun, I. (eds.) (1964). Handbook o f Mathematical Functions with Formulas, Graphs, and Mathematical Tables,

The Basic Free-Replacement Warranty

5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19.

20.

167

National Bureau of Standards Applied Mathematics Series, No. 55, U.S. Government Printing Office, Washington, D.C. Glickman, T. S., and Berger, P. D. (1976). Optimal price and protection period decisions for a product under warranty, Management Science, 22, 1381-1390. Nguyen, D. G., and Murthy, D. N. P. (1988). Failure free warranty policies for non-repairable products: A review and some extensions, RAIRO Operational Research, 22, 205-220. Cox, D. R. (1962). Renewal Theory, Methuen and Co., Ltd., London. Cinlar, E. (1975). Introduction to Stochastic Processes, Prentice-Hall, Inc., Englewood Cliffs, NJ. Blischke, W. R., and Scheuer, E. M. (1981). Applications of renewal theory in analysis of the free-replacement warranty, Naval Research Logistics Q., 28, 193-205. Murthy, D. N. P. (1992). A usage dependent model for warranty costing, Euro. J. Oper. Res., 13, 89-99. Murthy, D. N. P. (1991). A new warranty costing model, Math, and Comput. Modelling, 13, 59-69. Patankar, J. G ., and Mitra, A. (1989). “Effects of warranty execution under various warranty rebate plans,” TIMS XXIXth International Meeting, Osaka, Japan. Hill, V. L. (1983). A Quantitative Model for the Analysis o f Warranty Policies, Doctoral Dissertation, Univ. of Southern California, Los Angeles, CA. Lowerre, J. M. (1968). On warranties, J. Industrial Engineering, 19, 359-360. Menke, W. W. (1969). Determination of warranty reserves, Management Science, 15, 542-549. Amato, H. N., and Anderson, E. E. (1976). Determination of warranty reserves: An extension, Management Science, 22, 1391-1394. Nguyen, D. G., and Murthy, D. N. P. (1984). A general model for estimating warranty costs for repairable products. HE Transactions, 16, 379-386. Karmarkar, U. S. (1978). Future cost of service contracts for consumer durable goods, A IEE Transactions, 14, 380-387. Balachandran, K. R., Maschmeyer, R. A., and Livingston, J. L. (1981). Product warranty period: A Markovian approach to estimation and analysis of repair and replacement costs, Accounting Rev., 56, 115-124. Nguyen, D. G. (1984). Studies in Warranty Policies and Product Reliability, Doctoral Dissertation, Dept, of Mechanical Engineering, Univ. of Queensland, Brisbane, Australia.

168

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21.

Park, K. S. (1985). Optimal use of product warranties, IEEE Transactions on Reliability, R-34, 519-521. Marner, J. W. (1982). Cost analysis of pro rata and free-replacement warranties, Naval Research Logistics Q., 29, 345-356. Marner, J. W. (1987). Discounted and per unit costs of product warranty, Management Science, 33, 916-930. Balcer, Y., and Sahin, I. (1986). Replacement costs under warranty: Cost moments and time variability, Operations Research, 34, 554559. Park, K. S., and Yee, S. R. (1984). Present worth of service cost for consumer product warranty, IEEE Transactions on Reliability, R-33, 424-426. Baxter, L. A. (1982). Reliability applications of the relevation transform, Naval Research Logistics Q. , 29, 323-330. Park, K. S. (1985). Development o f a Price Indifference Function with Parameters o f Reliability, Maintenance and Warranty, Doctoral Dissertation, Texas A&M Univ., Lubbock, TX. Ip-Tamayo, T. C., and Deuermeyer, B. L. (1985). Establishing an Indifference Function in Quantity Purchases with Optional Warranty, Tech. Rept., Univ. of Houston, TX. Amato, H. N., Anderson, E. E., and Harvey, D. W. (1976). A general model of future period warranty costs, The Accounting Review, 51, 854-862. Hill, V. L., and Blischke, W. R. (1987). An assessment of alternative models in warranty analysis, J. Information and Optimization Sciences, 8, 33-55.

22. 23. 24. 25. 26. 27. 28. 29. 30.

5

Analysis of the Basic Pro-Rata Warranty

5.1

INTRODUCTION

In the preceding chapter, we studied warranty structures under which replacement items were provided free of charge on failure of an item covered under warranty. The basic notion of a pro-rata warranty, on the other hand, is that replacements are not provided free of charge, but are provided at a prorated cost, with the proration depending on the amount of usage or service time provided by the item prior to its failure. Service is ordinarily defined in terms of calendar time (e.g., “two-year limited warranty” on an automobile battery) or usage (40,000 miles on high-quality radial tires), but it may also be defined in other terms (e.g., “cycles,” for example, takeoffs and landings of aircraft). The rationale is that the customer has received some service from the item purchased, and hence should be willing to pay for the benefits obtained. This form of warranty is also very commonly used for basic consumer durables. Most of the common applications involve nonrepair able goods such as automobile batteries and tires (for which ordinary punctures are usually excluded from coverage), television picture tubes, and so forth. The warranty is also in common use in commercial transactions involving items ranging from consumer goods such as those just mentioned to complex systems and equipment. Warranty Policies 2, 2a-c, and 6 of Chapter 2 are some of the many forms of pro-rata warranties. A number of additional applications are also discussed in Chapter 2. The pro-rata warranty, which we shall denote PRW, has also been called a partial warranty, since only a part of the initial cost is covered. Another term sometimes used for the PRW is rebate warranty. This term reflects 169

170

Chapter 5

the fact that, in practice, failures under warranty are rectified by some manufacturers by means of an actual cash rebate rather than a discounted replacement. It also reflects the fact that cost analyses of the PRW have often been based on the use of rebate functions to express warranty costs and payoffs (though this could, of course, be done for the FRW as well, and it sometimes has been). We shall, in fact, pursue this approach in some of our analyses in this chapter. Note that while proration is calculated as a function of service, it need not be a linear function. This leads to a rich class of possible warranty structures. Several of the policies referred to in Chapter 2, in fact, feature nonlinear proration. Many other nonlinear forms may be reasonable in certain applications and should be considered as possible candidates when choosing a warranty policy. In addition, the pro-rata feature is also a part of many combination warranties. For example, many items are sold with a warranty that provides for a free replacement for failures up to a particular time and replacement at pro-rata cost during a succeeding time interval. Warranties of this type will be discussed in the next chapter. The pro-rata approach to warranty provides a compromise between the no-warranty situation, in which the consumer assumes the entire financial responsibility for replacement of failed items, and the free-replacement warranty, under which the seller assumes this responsibility entirely for a specified period. This cost sharing feature of the PRW is especially important to the seller. In fact, as opposed to the free-replacement warranty, where large numbers of failures near the end of the warranty period (which is, after all, where they would be expected to occur) could be disastrous, under PRW this may be quite advantageous, since the consumer may be induced by the warranty to repurchase an identical item (feeling that something is being gotten for nothing) rather than switching brands. In fact, this type of warranty has been used as a marketing tool in just this way, selecting warranty periods and designing products with mean time to failure so that many items can be expected to fail shortly before the end of the warranty period, and determining a pricing structure that will cover the relatively small expected warranty costs. Note, incidentally, that since the PRW is suggested as a compromise between the FRW and no warranty, the expected costs for the PRW may be expected to lie somewhere between those of the other two structures. This is, in fact, true for the buyer and will usually, but not necessarily, be the case for the seller as well. As we shall see, exceptions for the seller may occur in the case of the nonrenewing PRW, where the amount of the rebate is ordinarily calculated as a portion of the selling price and not of the cost of producing an item.

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171

As noted previously, the PRW is ordinarily offered primarily on nonrepayable items (or as a separate warranty on nonrepairable components of more complex items such as automobiles). The concept does not lend itself as well to repairable items, since cost sharing would then involve the cost of repair, which can vary significantly and is much less predictable and controllable. In principle, however, proration can be applied on this basis, and a few results regarding repairable items will be included in our analysis. In practice, the PRW is almost always renewing, that is, the replacement item, provided at prorated cost, is warranted anew for a period W from the time of replacement. A nonrenewing version of the PRW results when a rebate is given instead of a replacement. This is occasionally the case, and the nonrenewing PRW will also be included in our analysis. In discussing the pro-rata warranty, we shall use much of the same notation used in Chapter 4 (e.g., cs for seller’s cost, cb for buyer’s cost, W for the warranty period, and so forth), with these now being interpreted as costs, etc., under PRW rather than under FRW. Except for certain elements unique to one or the other of the warranty structures, our coverage of the pro-rata warranty parallels that of the FRW in the previous chapter and is as follows: In Section 5.2, we discuss the nonrenewing PRW from the points of view of both buyer and seller. Various rebate functions will be considered, and undiscounted and discounted costs will be analyzed. Policies 2, 2a, 2b, and 2c, given in Chapter 2, are of this type. Section 5.3 is devoted to cost models for the renewing PRW from the buyer’s and seller’s points of view. The basic PRW of this type is Policy 6 of Chapter 2. Again, repairable and nonrepairable items will be considered, and discounting will be included in the analysis. Because of the importance of the renewing aspect of the standard PRW in many applications (e.g., in encouraging repeat sales), it is particularly meaningful to look at long-term costs of this warranty. This is done in Sections 5.4 and 5.5, which deal with life cycle costs and indifference prices, respectively. In Section 5.6, we provide some comparisons of the costs of items sold under PRW and FRW, both with each other and with the cost of an unwarranted item. To facilitate these and other comparisons, examples throughout the chapter will follow the pattern set in Chapter 4 in analyzing the FRW, employing the exponential and Weibull distributions. Finally, a few additional models relevant to analysis of the PRW will be discussed briefly. 5.2

COST ANALYSIS OF THE NONRENEWING PRW

The nonrenewing pro-rata warranties defined previously are as follows:

Chapter 5

172

Policy 2 PRO-RATA POLICY: The manufacturer agrees to refund a fraction of the purchase price should the item fail before time W from the time of the initial purchase. The buyer receives a cash rebate and is not constrained to buy a replacement item. The refund may be either a linear or nonlinear function of the lifetime X x of the purchased item and the length W of the warranty period. This function is called the rebate function and denoted q(-). The three forms of q(-) specified previously are as follows: P olicy 2 a:

The refund is a linear function given by (5.1)

where cb is the purchase price of the item. P olicy 2 b:

The refund is a proportional linear function given by (5.2)

where 0 < a < 1. P o licy 2c:

The refund is a quadratic function given by (5.3)

Policies 2a and 2b are fairly widely used. In 2b, a is a proportionality constant used to attempt to control warranty cost for the seller while still providing reasonable protection to the buyer. This is needed particularly for items with a large markup or a large differential between the seller’s basic production cost and the selling price of the item, since the rebate is calculated on the latter basis. Another approach to controlling this cost is Policy 2c, which has the feature that the rebate, though still relatively simple, will decrease even more rapidly as a function of the time remaining in the warranty period. This is, of course, only one of many such functions, and we know of no actual instances of its use. Functions of this type are suggested for consideration, however, in situations of the type discussed. We proceed to a cost analysis of these three policies, concentrating on nonrepairable items. Since the warranties are nonrenewing, only single-

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173

sale costs are relevant. We look at both the buyer’s and seller’s cost. Since in both cases the rebate amount is the principle uncertain cost determinant, the form of the rebate function is of essential importance to the analysis. As before, we analyze expected cost in each case. 5.2.1

Modeling the Seller’s Per-Unit Expected Cost for Nonrepayable Items

The cost to the manufacturer of a single item sold under PRW may be written simply as (5.4) where, as usual, cs denotes the average total cost of supplying a single item, and X x is the lifetime of the item supplied. Thus, we express the seller’s cost as a “fixed” cost plus the variable cost of the rebate. Since the rebate function will always be zero outside the interval [0, W], the seller’s expected cost of an item sold under PRW is given by (5.5) For the rebate functions of Policies 2a, 2b, and 2c, we obtain the following: Policy 2a:

(5.6) where, as before, (5.7) is the partial expectation of X. Policy 2b: (5.8)

174

Chapter 5

Policy 2c:

(5.9) where (x^(2) is the partial expectation of X 2, given by (5.10) Expected Profit per Unit Sale

Note that since the seller’s per-unit income is simply the buyer’s cost cb, the per-unit average profit, t t (VF), may be modeled simply as the difference between this value and £[CS(VF)]. For example, for Policy 2a, we obtain (5.11) To illustrate the calculation of pro-rata warranty costs, we shall pursue the same examples used in Chapter 4, namely a warranty period of W = 1 year and items having exponential distributions or Weibull distributions with increasing failure rates. In each case, we consider mean lifetimes of |i = 2.0 and |x = 2.5. Expected costs will be calculated for each of the three policies discussed in the preceding. Example 5.1 [Exponential Distribution; Policies 2a and 2b] For the exponential distribution, we have

and

175

The Basic Pro-Rata Warranty

We begin with Policy 2a. From the preceding results and Equation (5.6), we obtain the seller’s expected cost under this PRW policy as

From Equation (5.11), the expected profit for this policy is given by

For Policy 2b, these results are easily modified to obtain

and

We consider again k = .5 and .4 (giving mean times to failure (MTTFs) of |x = 2.0 and 2.5, respectively), and a warranty period of W = 1. For these choices of \ , the corresponding values of F(l) are 0.3935 and 0.3297, and the corresponding values of |xw are 0.1804 and 0.1539. Expected warranty costs under Policies 2a and 2b are given in Table 5.1. These results may be used to explore costs and profit levels for the two policies and for various choices of seller’s cost, selling price, and a.

Table 5.1 Expected Cost to Seller, Nonrenewing PRW, Exponential Distribution, (x = 2.0 and 2.5 £[Cs(W0]

k

Policy 2a

Policy 2b

.5 .4

cs + ,2131cb cs + .1758cb

cs + .2131acb cs + .1758acb

176

Chapter 5

Example 5.2 [Weibull Distribution; Policies 2a and 2b] For the Weibull distribution, given by

analytical results are considerably more difficult. The partial expectation |xw requires evaluation of the incomplete gamma function (see Abramowitz and Stegun [1]). The result is

where y(a, jc)/T(«) is the incomplete gamma function evaluated at a, x, and T(-) is the (complete) gamma function. Abramowitz and Stegun (Section 6.5) give an expression for y(a, x) in terms of another special function, the confluent hypergeometric M(-, -, •), namely

which, in turn, can be evaluated ([1], Section 13.1) by use of the infinite series expression

We again consider |x = 2.0 and 2.5 and 0 = 2 and 4. For comparability with the results of the previous example, we set X = T(1 + l/0)/|x. (See Example 4.2.) The values of 0, p,, X, the y-function (evaluated by use of the series expression just given), F(W), |xw, and expected cost for Policy 2a are given for this example in Table 5.2. Costs for Policy 2b are easily obtained from these results as well, simply by replacing cb by acb. Note that, as in the case of the free-replacement warranty (and for the same reason), the expected costs here are much less than they would be for the corresponding exponential distributions.

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177

Table 5.2 Expected Cost to Seller, Nonrenewing PRW, Weibull Distribution, |x = 2.0 and 2.5 ß 2.0 4.0

2.0 2.5 2.0 2.5

X

7(1 + 1/ß, \ p)

F(l)

.44312 .35449 .45320 .36256

.05163 .02756 .01494 .004964

.17828 .11809 .04131 .01713

E[CS(1)] .11651 .07773 .03297 .01369

cs + cs + cs + cs -1-

.0618cb .0404cb .0083cb .0034cb

Example 5.3 [Exponential Distribution; Policy 2c] Here we require the additional term

Expected warranty costs are now obtained from Equation (5.9) (after some simplification) as

Expected costs for the exponential with X = .5 and .4 (|x = 2.0 and 2.5) are given in Table 5.3. In these examples, the cost element associated with the selling price of the item is about one-third less with quadratic proration than it would be with the ordinary linear pro-rata warranty.

Table 5.3 Expected Cost to Seller, Nonrenewing PRW, Exponential Distribution, (x = 2.0 and 2.5, Quadratic Proration F(l) - 2^w +

X .5 .4

.1151 .0991

.1478 .1210

£IC.( l)] cs + .1478cb cs + .1210cb

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Chapter 5

Example 5.4 [Weibull Distribution; Policy 2c] For the Weibull distribution, the second-order partial expectation also involves the incomplete gamma function. The result is

This may be evaluated using the series expression given in Example 5.2. The additional quantities needed for calculation of expected costs may be found there as well. Table 5.4 gives values of the second-order partial expectation and the resulting expected costs under PRW for the Weibull distributions considered previously. Here the influence of selling price on warranty cost is about one-half as much with quadratic prorating as with linear when p = 2, and about one-third as much when p = 4. 5.2.2

Discounted Seller’s Expected Unit Cost for Nonrepayable Items

The seller’s discounted costs, denoted CS(W; 8) as previously, are calculated by discounting costs incurred during the warranty period, excluding the initial cost (cs) of supplying the item. If we again use the usual exponential discount function, the resulting discounted expected cost to the seller is

Table 5.4 Expected Cost to Seller, Nonrenewing PRW, Weibull Distribution, jx = 2.0 and 2.5, Quadratic Proration

2.0 4.0

E[CX1)]

y(l + 2/p, \P)

0 2.0 2.5 2.0 2.5

.44312 .35449 .45320 .36256

.016931 .007264 .003755 .001499

.08623 .05781 .02742 .01140

cs + cs -1cs + cs -1-

.0315cb .0204c 5 .00279cb .00115cb

179

The Basic Pro-Rata Warranty

given by (5.12) for the usual linear proration function (Policy 2a), and by the corresponding integral with the function qx(t) = 1 - t/W replaced by the appropriate alternative proration functions for other pro-rata policies. Discounted expected profit is now determined simply as the difference between cb, the seller’s income, and the seller’s expected cost, given in Equation (5.12). Note that the seller’s income is also obtained as a result of the initial transaction at the outset of the warranty period and hence is not discounted. Example 5.5 [Exponential Distribution] For the exponential distribution, Equation (5.12) gives the result

Discounted costs for discount rates of 8 = .05, .10, and .15 are given in Table 5.5 for the exponential distribution with \ = .5 and .4. Note that discounting does not have much of an effect on costs in this instance because of the short length of the warranty period relative to the expected life of the items. Discounting would have an even smaller effect in the case of Policies 2b and 2c. The Weibull distribution will lead to even more severe computational difficulties than those encountered in Examples 5.2 and 5.4 if exponential

Table 5.5 Discounted Expected Costs to Seller, Pro-Rata Warranty, Exponential Distribution 8

|x = 2.0

|x = 2.5

.05

cs + .1731cb cs + .1705cb cs -I- .1679cb

cs + .2098cb cs + .2067cb cs + .2036cb

.10

.15

Chapter 5

180

discounting is employed. In this case, other forms of discounting, for example, power functions or polynomials, will lead to more tractable results. 5.2.3

Modeling the Buyer’s Per-Unit Expected Cost for Nonrepayable Items

Since we consider only purchase price in our analysis of buyer’s costs, the expected per-unit cost to the buyer for a PRW with period W, C5(W), is equivalent to the per-unit income to the seller. This may be calculated from the relationship (5.13) where q(-) is the rebate function defined in the beginning of Section 5.2. The buyer’s expected per unit cost under PRW for the three rebate functions considered previously (Equations (5.1)—(5.3)) are easily obtained. The results are as follows: Policy 2a: The buyer’s cost of a single unit purchased under PRW is given by (5.14) Thus, under linear proration given by qi(-), the buyer’s expected unit cost is

(5.15)

Policy 2b: Similarly, under proportional linear proration given by q2(•), the buyer’s expected unit cost is (5.16) Policy 2c: From our previous results regarding quadratic proration, as defined by q3{-) in Equation (5.3), we obtain the buyer’s per unit expected

The Basic Pro-Rata Warranty

181

cost for this policy to be

(5.17) The buyer’s costs can be discounted to present value in the usual way by suitable modification of the integral expressions in Equations (5.15)— (5.17). As observed in the case of the seller’s costs, discounting will ordinarily have relatively little impact in the case of single-item purchases. Example 5.6 [Exponential Distribution; Policies 2a and 2b] We consider Policies 2a and 2b with, as usual, W = 1. The buyer’s costs are given in Equations (5.15) and (5.16). For the exponential distribution with |x = 2.0 and 2.5 (X = .5 and .4), the values of F(W) and |xw are given in Example 5.1. The results of the cost analysis from the buyer’s point of view are given in Table 5.6 for Policies 2a and 2b. For Policy 2b, the results are given for 10% and 20% discounted rebates (a = .9 and .8, respectively). We see that these policies are equivalent to about a 15-20% discount to the buyer under the assumption of exponential lifetimes with MTTF 2.0-2.5 times the length of the warranty period. Example 5.7 [Weibull Distribution; Policies 2a and 2b] The Weibull distributions with 0 = 2.0 and 4.0 and the same MTTFs provide a comparison of the results of Example 5.6 with those obtained for increasing failure rate distributions. The required values of F(W) and |xw are given in Table 5.2. With increasing failure rates and the same MTTF, fewer early failures (and hence smaller rebates) are expected, and the buyer’s cost should be T a b le 5.6 Expected Cost to Buyer, Nonrenewing PRW Policies 2a and 2b, Exponential Distribution, jjl = 2.0 and 2.5

mmi

Policy 2b

X

F(W)

.5 .4

.3935 .3297

.1804 .1539

Policy 2a

a = .9

a = .8

General

.7869cb .8242cb

.8082cb .8418cb

.8295cb .8594cb

(1 - .2131a)c, (1 - .1758a)c,

182

Chapter 5

Table 5-7 Expected Cost to Buyer, Nonrenewing PRW Policies 2a and 2b, Weibull Distribution, |x = 2.0 and 2.5

E[Ch(W)] Policy 2b p

P'

F(w ) - m.

Policy 2a

a = .9

a = .8

2.0

2.0 2.5 2.0 2.5

.06177 .04036 .00834 .00344

.9382cb .9596cb .9917cb .9966cb

.9444cb .9637cb .9925cb .9969cb

.9506cb .9677cb .9933cb .9972cb

4.0

General (1 (1 (1 (1

-

.0618a)c, .0404a)c, .0083a)C| .0034ot)c,

expected to increase (so, of course, will his “benefit,” i.e., his usage time). The results, given in Table 5.7, confirm this. Example 5.8 [Exponential and Weibull Distributions; Policy 2c] Policy 2c represents another approach to discounting the rebate to the buyer in the event of a failure. The additional quantities required for calculating costs using Equation (5.17) may be found in Tables 5.3 and 5.4. Table 5.8 gives the buyer’s expected cost per unit for the Weibull with P = 2.0 and 4.0 and for the exponential, which is equivalent to the Weibull with 0 = 1.0. Again, the buyer’s costs are significantly higher for the Weibull with increasing failure rate than for the exponential. 5.2.4

Cost Models for Repairable Items Sold Under PRW

Although repairable items are not often sold under pro-rata warranty protection, particularly with a nonrenewing warranty, the situation is concep-

Table 5.8 Expected Cost to Buyer,

Nonrenewing PRW Policy 2c, Exponential and Weibull Distributions, |x = 2.0 and 2.5 £ [cb(W)]

P 1.0 2.0 4.0

2.0 2.5 2.0 2.5 2.0 2.5

.8522cb .8571cb .9685cb .9795cb .9973cb .9989cb

The Basic Pro-Rata Warranty

183

tually possible. We shall assume that implementation of such a warranty would be through repairs offered at a prorated discount cost to the buyer. The buyer would then have a working item that, since we are concerned with the nonrenewing case, would have no further warranty coverage. Under these conditions, the cost models are easily modified for both buyer and seller to account for repair costs. Suppose that cr is the average cost (to the buyer or seller) for repair of an item. Then under linear proration (Policy 2a), for example, the cost to the buyer of a unit purchased under PRW is (5.18) This is, of course, equivalent to a rebate that is calculated on the basis of repair rather than replacement cost, and, as such, is much less advantageous to the buyer, unless repair is good as new and the buyer’s preference is to replace the item rather than simply pocket the rebate. Under these same conditions, the buyer’s expected per unit cost now becomes (5.19) Similar results may be obtained for Policies 2b and 2c by simply replacing ch by cr in all but the first term on the right-hand sides of Equations (5.16) and (5.17). The seller’s expected per-unit costs and expected profits for repairable goods sold under the various nonrenewing PRWs may similarly be obtained by replacing cb by cr in Equations (5.6), (5.8), and (5.9). Modifications along these lines will also lead to expressions for the discounted seller’s and buyer’s expected costs for repairable items. Numerical illustrations of the effects of repair on unit costs can be obtained by replacing cb by cr as appropriate in the examples of the previous sections. In addition, all previous numerical results regarding Policy 2b are directly applicable to repairable items under Policy 2a, simply by reinterpreting a as the proportional cost of repair relative to replacement, i.e., setting a = cjch. 5.3

UNIT C O ST M ODELS FOR ITEM S SO LD UNDER RENEWING PRW

We turn now to the renewing PRW. This is the more usual form of the pro-rata warranty. In practice, it is offered on both repairable and non-

184

Chapter 5

repairable goods, but is ordinarily especially preferred in the latter case. The general form of the renewing PRW was stated in Chapter 2 as follows: Policy 6 PRO-RATA POLICY: Under this policy the manufacturer agrees to provide a replacement item, at prorated cost, for any item that fails to achieve a lifetime of at least W, including the item originally purchased and any replacements made under warranty. Implicit in this definition is that, under policies of this type, replacement items are warrantied anew. Under this policy, failed items are replaced at some prorated cost to the buyer, which again may be linear or nonlinear. We assume that coverage under the new warranty is identical to that of the original warranty. In analyzing costs for the renewing PRW, renewal-type equations are again encountered. Under the assumptions just stated, the renewal functions themselves are ordinary renewal functions, but, as we shall see, some of the additional renewal-type equations encountered are considerably more complex. We again consider both repairable and nonrepairable items, though, for reasons previously mentioned, the emphasis is on the latter case. We again look at both the seller’s and buyer’s points of view. Here, however, it is convenient to begin with the expected cost to the buyer. This presents the most difficulty and is also the starting point in dealing with the seller’s point of view, since the buyer’s costs represent income to the seller in our models. 5.3.1

M od eling the Buyer’s Short-Term C osts fo r N o nre p a ira b le Items Purchased Under Renew ing PRW

We begin the cost analysis of the renewing PRW by noting its relationship with the nonrenewing case. For a nonrenewing warranty, costs to the buyer were modeled by defining a rebate function, which expressed the penalty paid by the buyer upon failure of an item during the warranty period. In the renewing case, this rebate may be considered to be a discount on the purchased replacement item, with the constraints that no rebate will be forthcoming unless the purchase is made and that each new item so purchased carries the same warranty as that of the original purchase. It follows that the initial purchase of an item under PRW leads to a whole sequence of purchases, which, in practice, only stops either when the buyer chooses not to repurchase the item (thereby foregoing the rebate/ discount) or when an item achieves a lifetime of at least W. As in our previous analyses, we shall again make the simplifying assumption that the

The Basic Pro-Rata Warranty

185

process does not stop for either of these reasons, i.e., that on failure of an item, a new item of the identical type is purchased and that the item is purchased and put into service instantaneously. Thus, we consider a constant stream of items that have independent and identically distributed lifetimes, are manufactured at the same cost, bought at identical prices, and covered by identical warranty terms. We are interested in modeling the net cost per unit of items sold under these conditions. Under the stated assumptions, Equation (5.13) may be used to model cost to the buyer of each purchase. Thus, if we use X {to denote the lifetime of the ith item as usual, and /, to denote the item itself, the cost to the buyer of item Ii+1 is given by, say, (5.20) where q(-) is the rebate function. In the remainder of this section, we shall consider only linear proration, for which the buyer’s cost of item Ii+1 is given by Equation (5.14) with X t replacing X x. The extension to include a proportionality constant (as in Equation (5.2)) is immediate. Other proration functions (e.g., that of Equation (5.3)) could be handled by the methods given in what follows as well. At this point in the analysis, the renewing nature of the warranty provides an additional complication. The point is that, with renewing, it may not be appropriate to consider either only a single-item sale or the sequence of purchases generated by a single sale. In fact, in this context there are at least three ways in which one can define per-unit cost: (1) the average price of each item purchased, whether under warranty or not; (2) the total cost generated by a single item purchase, including all repeat purchases under warranty; and (3) the average cost per item purchased in some finite time period T. We consider each of these in turn. Average Purchase Price of Items Under Renewing PRW

The first item purchased under warranty is purchased at full price, cb. The cost of each subsequent item is given by Equation (5.20). A simple conditional argument gives the expected cost of the ith subsequent item purchased under linear PRW as

(5.21)

186

Chapter 5

Note that this result is independent of i, and is precisely the result given in Equation (5.15) for the nonrenewing case. Thus, we have the very logical conclusion that under renewing PRW the expected cost of each item, except for the initial purchase, is the same as that under nonrenewing PRW with otherwise identical warranty terms and assumptions. This will also hold for other PRW policies. Thus, the results for Policies 2b and 2c given in Equations (5.16) and (5.17) are applicable here as well. Numerical results are given in Examples 5.6-5.8. Expected Total Cost of Item and Replacements Purchased Under PRW

We next consider the total cost of an item purchased under renewing PRW and all subsequent purchases generated by this initial purchase. Denote this total cost Cb(*). We assume that, with probability one, any item that fails before age W will be instantaneously replaced by an identical item and that the process will continue until an item achieves a lifetime of as least W. Again, we consider only linear proration. Under these conditions, the expected total cost incurred by the buyer is the cost of the initial item, c5, plus the expected cost of all replacements. By Wald’s Theorem, this is calculated as the expected cost of each replacement, given in the previous section as cb|xw/W, times the expected number of replacements. The number of replacements until an item with lifetime at least W is encountered has a geometric distribution with parameter F(W). This result is given in Equation (4.12). From this, it follows that the expected number of replacements is given by F(W)/F(W). Thus, (5.22) It is easy to see that the expected cost given in Equation (5.22) is an increasing function of W. This is due to the fact that as W increases relative to |x, failures become more likely and additional purchases will be required. Note that cost to the buyer is equivalent to income to the seller. Thus, from the seller’s point of view, this says that the marketing strategy of choosing W relatively large increases sales, albeit at discounted prices. Realistically, of course, purchase of a replacement is not mandatory. If repurchases are made with some probability less than one, the actual buyer’s cost, and hence, seller’s income, will be less than the values calculated by Equation (5.22). If the items are priced so that the discounted replacement sales are profitable, it may be worthwhile to increase the coverage

187

The Basic Pro-Rata Warranty

period for items sold under renewing PRW. We illustrate some of these notions with the exponential and Weibull distributions used previously. Example 5.9 [Exponential and Weibull Distributions] As in Example 5.8, we look at the Weibull distribution with p = 1, 2, and 4. In all three cases, we consider |x = 2.0 and 2.5. The values of the CDF and the partial expectation required for evaluating Equation (5.22) are given in Examples 5.1 and 5.2 for W = 1. Suppose we also consider W = 2, i.e., a warranty period at or a little below the mean lifetime of the items. The additional terms required may be determined by the methods of Examples 5.1 and 5.2. The values of y(l + 1/p, kpWp) for W = 2 and the p,\-combinations in Table 5.2 are found to be (in the sequence of the table) .29598, .17877, .34231, and .13794. Expected costs, calculated by Equation (5.22), are given in Table 5.9. The increase in total expected costs with W is apparent from these results. Note also that the relative increase in additional cost beyond the initial purchase (i.e., in E[Cb(W)\ - cb) as W increases from 1 to 2 becomes far more substantial as p increases. As expected, these costs are, of course, always higher than those derived in the previous section. The costs are not really comparable, however, because here we are dealing with (possibly) multiple items and an extended, indeterminate period of time. In the next section, we look, instead, at a fixed time period. Average Short-Term Per-Unit Cost to Buyer

Here we look at the average purchase price of all items that the buyer may expect to purchase (whether or not covered by warranty) during some fixed time period of length T from the time of initial purchase (which, for sim-

Table 5-9 Expected Total Cost to Buyer, Renewing PRW, Exponential and Weibull Distributions, |x = 2.0 and 2.5

W=1 ß

P'

1.0

2.0 2.5 2.0 2.5 2.0 2.5

2.0 4.0

.1804 .1539 .1165 .0777 .0330 .0137

W=2

F(W)

E[Cb(W)]

.3935 .3297 .1783 .1181 .0413 .0171

1.1170cb 1.0757cb 1.0253cb 1.0104cb 1.0014cb 1.0002cb

.5285 .4780 .6680 .5043 .7553 .3805

F(W)

E[Cb(W)]

.6321 .5507 .5441 .3951 .4908 .2415

1.4540cb 1.2929cb 1.3985cb 1.1647cb 1.2801cb 1.0606cb

188

Chapter 5

plicity, is taken to be time 0). The time period T may be thought of as a relatively short period with one or a few purchases, rather than as a life cycle, which we take to be a relatively long period with, typically, many repeat purchases. An example of a short term might be tire replacement on an automobile owned for, say, a two- or three-year period. Denote this cost Cb(7\ W). (Note that we include in this cost the cost of the initial item purchased. In much of the literature— e.g., Mamer [2]—the quantity corresponding to Cb( v ) refers to the total cost exclusive of the initial purchase.) We wish to determine the expected value of Ch(T, W). It is convenient to express this as a conditional cost, given the first item lifetime X x. We denote this conditional cost Cb(T, W\Xx = x). The form of this cost function depends on the rebate function, which here will be taken to be linear proration, and on whether T is chosen so that T < W or so that T > W . We denote these Case (a) and Case (b), respectively. The cost models for these two cases are as follows: Case (a) For the linear PRW and T < W, the conditional cost function is given by

(5.23) where Cb(y, W) is the cost to the buyer in a period of length y for the initial and all replacement items under pro-rata warranty with warranty period W. To complete the analysis, we now determine conditional expected replacement costs given that X x = *, and uncondition on X x. This approach, due to Mamer [2], results in renewal-type equations, to which the methods of Chapter 3 can be applied to obtain a formal solution. Here we obtain, on unconditioning,

(5.24) If we write h(t) = cb|x,/W, where, as usual, |x, is the partial expectation of X , we see that Equation (5.24) is a standard renewal equation. The so-

The Basic Pro-Rata Warranty

189

lution, by Equation (3.39), is

(5.25) where A/( ) is the ordinary renewal function associated with the CDF F (). From Equation (5.25) and the definition of partial expectation, we now obtain, for T < W, (5.26) Case (b) If T the analysis is identical to that of life cycle costs, and we defer the derivations to the next section, which is devoted to that topic. In Section 5.4, it will be shown that for the PRW with linear proration, the expected cost is given by

(5.27) Finally, a first-order approximation of the average short-term unit cost to the buyer may be obtained as the ratio of this result to the expected total number of purchases in [0, T). As before, the latter is given by 1 + M(T), the initial purchase plus the expected number of replacements in [0, T). We now have that the buyer’s approximate expected unit cost under renewing PRW, say UCB, is given by (5.28) where the expected cost in the numerator is given by Equation (5.26) if T < W, and by Equation (5.27) if T s W. (Note: It is easily seen that

190

Chapter 5

(5.27) reduces to (5.26) when T = W, so that the simpler result may be used in that case.) Example 5.10 [Exponential Distribution] For the exponential distribution, M(t) = Xt, and we find, from the partial expectation formula previously determined,

We consider Cases (a) and (b) and the parameter values used in previous examples. For T < W, the remaining cost element is the integral expression in Equation (5.28). For the exponential distribution, this is given by

From Equation (5.28), we now obtain

For T > W, the required integrals in Equation (5.29) are (after some algebra)

and

191

The Basic Pro-Rata Warranty

Some further algebra now yields

for T > W. Since M(t) = kt for the exponential distribution, we have expected unit cost to the buyer given by

Table 5.10 gives values of UCB/cb for selected values of T for the exponential with W = 1 and |x = l / \ = 2.0 ancf2.5. We see that, unless items are in use for only a relatively short time, the effect of the warranty is equivalent to about a 15-20% reduction in the average price of the item. Equivalent results for the Weibull and most other failure distributions of interest cannot be obtained explicitly because of the complexity of the

Table 5-10 Average Unit Cost to Buyer of Items Sold Under Renewing PRW, Exponential Distribution, |x = 2.0 and 2.5

.1 .2 .3 .4 .5 .6 .7 .8 .9 1.0

T

|x = 2.0

[L = 2.5

.9547 .9179 .8882 .8646 .8461 .8320 .8218 .8147 .8105 .8087

.9634 .9331 .9083 .8882 .8724 .8601 .8511 .8450 .8413 .8399

1.1 1.2 1.3 1.4 1.5 2.0 2.5 3.0 4.0 5.0

.8080 .8073 .8067 .8061 .8056 .8032 .8014 .8000 .7978 .7963

.8394 .8390 .8386 .8383 .8379 .8364 .8352 .8342 .8326 .8315

O

|x = 2.5

*P

T

UCB!ch

il N>

UCB/cb

Chapter 5

192

integrals of Equations (5.26) and (5.27). Simulation is suggested as an alternate approach. 5.3.2

M od eling the S e lle r’s Unit C ost and Pro fit Under Renew ing PRW

We again consider the three interpretations of unit cost. The seller’s expected cost and expected profit under renewing PRW will be derived for each. We consider linear proration; other proration functions can be analyzed as well. Average Profit of Items Sold Under Renewing PRW

If we look at a single item, the cost to the seller is the same whether the warranty is renewing or not. In either case, the seller’s expected total cost is simply the average cost cs of producing and providing the item to the buyer plus the expected cost of the rebate. The seller’s expected profit is the difference between the buyer’s expected cost, which is taken to be the seller’s expected income, and this expected total cost. Thus, we have seller’s expected cost as (5.29) and the seller’s expected profit t t (W) given by either of the following two expressions: (5.30) For the linear PRW, the expected cost to the buyer is given in Equation (5.21). From this and Equation (5.30) we now obtain (5.31) This result provides a means of determining the minimum markup required for profitability in terms of the ratio of the two costs. For t t (W) to be nonnegative, we require (5.32) It should be noted that this result ignores future sales and costs that may be generated by the sale of an item.

193

The Basic Pro-Rata Warranty

Example 5.11 [Exponential and Weibull Distributions] For the exponential and Weibull distributions previously considered, the expected profit to the seller may easily be calculated using Equation (5.31) and the results for |xw and F(W) given in Table 5.9. Here we consider the maximum cost ratio required for profitability. These are given in Table 5.11 for the same distributions and for W = 1 and 2. Tabulated values are the break-even values of cs/c5; values less than those tabulated indicate a profit. We see that if items are exponentially distributed with |x = 2.0, the total cost of sales must be less than 78.7% of the selling price for a warranty of 1.0 time units, and less than 63.2% for a warranty of 2.0. For the Weibull distribution, these numbers are significantly higher, again because far fewer early failures are expected. Expected Total Cost and Profit to Seller of Item and Replacements Sold Under Renewing PRW

The total cost to the seller associated with a single-item sale under renewing PRW is the cost per item cs times the total number of items supplied, say 1 + K(W), where K(W) is the number of replacements until an item with lifetime at least W is obtained. As before, K{W) is a random variable having a geometric distribution with mean F(W)/F(W). The total cost to the seller of an item and the resulting replacements under renewing PRW will be denoted CS(W). We have

(5.33)

Table 5.11 Minimum Cost Ratio for

Profitability, Renewing PRW, Exponential and Weibull Distributions, p = 2.0 and 2.5 ß

P'

W=1

W=2

1.0

2.0 2.5 2.0 2.5 2.0 2.5

.7869 .8242 .9382 .9596 .9917 .9966

.6322 .6883 .7899 .8570 .8868 .9488

2.0 4.0

Chapter 5

194

The expected profit to the seller, t t (W), may now be calculated as the difference between expected income, which is the expected cost to the buyer given by Equation (5.22), and the seller’s expected cost. The result is

(5.34) For this cost basis, the item is profitable as long as (5.35) which is always an amount less than that for a single sale without considering replacements (given on the right-hand side of (5.32)). Example 5.12 [Exponential and Weibull Distributions] For the exponential and Weibull distributions of the previous examples, we again calculate the break-even cost ratios. These are obtained from the right-hand side of the inequality (5.35) and are given in Table 5.12 for the parameter combinations considered. Note that, as expected, the breakeven ratios are smaller when costs of replacements are considered as well. It is instructive to look at profit itself as a function of the cost ratio. Let i\i = cjch. The ratio t t (W)/cb is given in Table 5.13 for these same distri-

Table 5.12 Minimum Cost Ratio for

Profitability, Total Cost, Renewing PRW, Exponential and Weibull Distributions, |x = 2.0 and 2.5 p

P'

W=1

W=2

1.0

2.0 2.5 2.0 2.5 2.0 2.5

.6774 .7210 .8425 .8911 .9601 .9831

.5349 .5809 .6376 .7045 .6946 .8044

2.0 4.0

195

The Basic Pro-Rata Warranty

butions, with W = 1 and 2, and for values of i|j ranging from 0.4 to 0.9. The tabulated values express profit as a proportion of selling price. We see from Table 5.13 that for W = 1, expected profit invariably increases as either p or |x increases. As 3 increases, fewer early failures occur, so costs to both buyer and seller decrease, but the cost to the seller evidently decreases more rapidly, so profits increase. For W = 2, the picture is somewhat more complicated. For the most part, the patterns are similar to those for W = 1, but there are some unexpected results (e.g., for 3 = 2). For the values of p tabulated, profits always decrease as W increases from 1 to 2, but again in a somewhat complicated pattern. This complexity apparently reflects the varied nature of the Weibull distribution as p changes—for p = 1, the distribution reduces to the exponential and is highly skewed to the right; by the time 3 reaches 4, the distribution is more nearly symmetrical, and, in fact, it is very nearly normal. In practice, it is necessary to carefully study the distributional aspects and to vary many parameters in investigating competing warranty policies. The Se lle r's Short-Term Average Profit per Unit

Here we consider the average per-unit income and profit to the seller for items and replacements, whether or not purchased under warranty, over a fixed time period T. Average per-unit income to the seller is given for the renewing linear pro-rata warranty by substituting Equations (5.26) and

Table 5.13 Profit as a Proportion of cb, Total Cost Model, Renewing PRW, Exponential and Weibull Distributions, \x, = 2.0 and 2.5 * = cjcb

w

p

P'

.4

.5

.6

.7

.8

1.0

1.0

2.0 2.5 2.0 2.5 2.0 2.5 2.0 2.5 2.0 2.5 2.0 2.5

.4575 .4790 .5386 .5568 .5842 .5932 .3667 .4026 .5211 .5034 .4946 .5332

.2863 .3298 .4168 .4434 .4799 .4915 .0949 .1801 .3018 .3381 .2982 .4014

.1214 .1806 .2951 .3301 .3755 .3898 -.1769 -.0425 .0824 .1728 .1018 .2696

- .0435 .0314 .1734 .2167 .2712 .2880 -.4487 - .2652 -.1369 .0075 -.0946 .1377

- .2083 -.1178 .0517 .1033 .1669 .1863 -.7205 - .4876 -.3563 -.1578 - .2910 .0059

2.0 4.0 2.0

1.0 2.0 4.0

.9 -

.3731 .2670 .0700 .0101 .0626 .0846 - .9923 - .7100 - .5757 - .3231 - .4874 -.1259

Chapter 5

196

(5.27), which give E[Cb(T, W)] for T < W and T > W, respectively, into Equation (5.28). Since by assumption the average per-unit cost to the seller is cs, the expected profit per unit over [0, 7], denoted t t (T, W), is given by (5.36) Application of this result requires the evaluation of the integral expressions in Equations (5.26) and (5.27). This is possible analytically only for a few life distributions. Example 5.13 [Exponential Distribution] Evaluation of the various terms in Equations (5.26) and (5.27) is done in Example 5.10 for the exponential distribution. For W = 1 and |x = 2.0 and 2.5, Table 5.10 gives the bracketed term on the right-hand side of (5.36) in units of cb, for T ranging from 0.1 to 5.0. Thus, multiplication of the entries of Table 5.10 by ch gives the seller’s expected income, and the seller’s expected profit is gotten as the difference between this quantity and cs. We have, for |x = 2.0, for example,

and

Note that the expected profit is a decreasing function of T, approaching a minimum as T —» W, however, discounting may be a very important factor. This situation is essentially equivalent to determining life cycle cost and will be dealt with in Section 5.4. 5.3 .4

C o st M od els fo r Re p a ira b le Items

As we have seen in previous models involving repairable items, the repairability of an item changes the cost structure in that repairs are always less costly than replacements. As noted, however, the PRW is not usually used in such cases because of the difficulty of determining proration of repair costs. Formally, however, results similar to those of Equations (5.21), (5.22), (5.26), (5.27), and so forth, can be obtained by replacing cb by cr for all but the initial item purchased in the derivation of the models. The analysis then parallels that leading to Equation (5.19) in the nonrenewing case.

198

Chapter 5

As an example, we apply this approach to the buyer’s total cost of an item and its expected replacements under warranty given for nonrepayable items by Equation (5.22). For repairable items, the corresponding expected total cost is (5.38) Similarly, (5.33) becomes (5.39) from which the result corresponding to (5.34) becomes (5.40) Of somewhat more interest from a practical standpoint would be the more realistic situation in which repairs are not good as new. Because of the limited use of the pro-rata warranty for repairable items, however, we shall not pursue the analysis of cost models of this type here. Some results along these lines will be given in the discussion of warranty servicing and warranty reserves in Chapter 10. 5.4

LIFE C YC LE C O ST M ODELS

Here we look at the total cost to both buyer and seller in a period [0, L). In this analysis, L is assumed to be a relatively long period of time, so L < W is not of interest, and attention will be restricted to L ^ W, which is Case (b) in the short-term cost analysis of Section 5.3.1. We begin with the buyer and denote this long-run cost, called the life cycle cost or LCC, C5 (L, W). We consider only the renewing PRW with linear proration, and nonrepairable goods. 5.4.1 The Buyer’s Life C ycle C ost

As in the previous analysis along these lines in Section 5.3.1, it is convenient to express Cb(L, W) as a conditional cost, given the first item lifetime X x. We denote this conditional cost Cb(L, W\Xx = x). As before, we take Cb(L, W) to be the cost to the buyer of all items purchased under pro-rata warranty of length W, in a period of length L, including the initial purchase.

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199

For L > W, we have

(5.41)

To complete the analysis, we determine conditional expected replacement costs given that X x = x, and uncondition on X x. We obtain

(5.42) If we now write H (L) for the first integral on the right-hand side of (5.42), we have a standard renewal equation with solution, from Equation (3.39), given by (5.43) where M () is the ordinary renewal function associated with F(-). Evaluation of the integrals yields

(5.44) and, since

(5.45)

200

Chapter 5

(5.46) from which it follows that

(5.47) This provides a formal solution to the problem. In practice, the integral expressions in (5.47) cannot be evaluated analytically except for a few simple life distributions. Example 5.14 [Exponential Distribution] The expected cost to the buyer for the exponential distribution was derived in Example 5.10. The result for a life cycle L is

Note that this is a linear function of L. For the \-values previously considered and W = 1, the buyer’s expected life cycle costs are given in Table 5.14.

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201

Table 5.14 Buyer’s

Expected Life Cycle Cost, Renewing PRW, Exponential Distribution X

E[Cb(L, W)}

.5 .4

(.8195 + ,3935L)cb (.8461 + .3297L)cb

5.4.2

The S e lle r’s Life C yc le C o st and Profit

The seller’s cost of supplying items over the lifetime of the item is calculated, as usual, as the average cost cs per item supplied times the expected number supplied. Thus, the total expected cost to the seller is simply (5.48) The seller’s expected profit is again calculated as the difference between expected income and expected cost. The result is (5.49) Example 5.15 [Exponential Distribution] The buyer’s expected cost for the exponential distribution is given in Example 5.14. The renewal function is M(L) = XL. From (5.51) and Example 5.10, we obtain,

For W = 1 and X = .4 and .5, the seller’s profit is given in Table 5.15. 5.4.3

D iscounted Life C ycle C osts

Discounting to present value is of particular interest as L becomes large. We again consider L > W and linear proration. Additionally, we look only at the exponential discount function, for which the discounted value of x is given by xe~bx. Extensions to nonlinear pro-rata warranties and alter-

202

Chapter 5

Table 5-15 Expected Life Cycle Profit, Renewing PRW, Exponential Distribution X .5 .4

1) (.8196 + .3935L)cb - (1 + .5L)cs (.8461 + .3297L)cb - (1 + AL)cs

native discount functions can be obtained by obvious modifications of the results that follow. As before, we begin with the buyer’s point of view. The seller’s profit can then be determined by the approach used previously. As will be seen, the analysis again leads to complex renewal-type equations that are very difficult to evaluate in practice. Finally, we look at the limiting forms of the cost expressions as L —> «>. These are found to be more amenable to analytical solution. The Buyer's Discounted Cost

The basis of the analysis is a reexpression of Equation (5.41) as a discounted value for the remaining cost to the buyer, denoted Cb(L; W, 8). For L > W and the exponential discount function, we obtain

(5.50) The conditional analysis now gives, in place of (5.42),

(5.51)

If we write H(L, 8) for the first integral on the right-hand side of (5.51), this is again seen to be a renewal-type equation, with solution given by Equation (3.37). Evaluation of the integrals corresponding to (5.44) and

The Basic Pro-Rata Warranty

203

(5.45) yields (5.52) and

(5.53)

We now obtain (5.54) Explicit expressions for discounted LCC can be obtained only for the exponential and a few other life distributions. The Se lle r's Discounted Cost and Profit

A similar approach can be taken to analyzing the seller’s cost and profit. Write it (L; W, 8) = seller’s total profit over [0, L) under linear PRW with period W and exponential discounting at rate 8. Note that the profit on the first item is simply cb - cs. If we now look at the additional profit after the initial sale, given that the failure time of the first item is jc, we have the relationship

(5.55)

An analysis along the lines of that of the preceding section will lead to

204

Chapter 5

similar renewal-type equations, which will again be found to be intractable. As an alternative, Mamer [3] considered discounted total profit over an infinite sequence of purchases. In this case, since

the relationship given in (5.55) simplifies somewhat in the limit and the conditional argument does not lead to a renewal-type equation as in the previous analyses. In fact, if we replace 7T(y; W, 8) by its limiting value on both sides of (5.55), the conditional analysis leads directly to the result

(5.56) where F(*) is the Laplace transform of F(*). Example 5.16 [Exponential Distribution] For the exponential distribution with parameter X, the Laplace transform is given by

The remaining integral expressions in (5.58) are

205

The Basic Pro-Rata Warranty

and

Thus, the asymptotic values of profit for the exponential distribution are given by

Values of the limiting discounted life cycle profit, calculated from this expression, are given in Table 5.16 for W = 1, X = .5 and .4 (|x = 2.0 and 2.5), and several values of 8. 5.5

INDIFFERENCE PRICES FOR THE PRO-RATA W ARRANTY

In Chapter 4, we defined the indifference prices for the buyer and seller and derived expressions for these for the free-replacement warranty. Our approach to this is to hypothesize a pricing structure whereby an item may be purchased without warranty at price cu or with warranty at price cb. Indifference prices are those choices of cb for which the long-run cost to the buyer or the long-run profit to the seller are the same whether the warrantied or the unwarrantied item is purchased. Here we apply this concept to nonrepairable items sold under pro-rata warranty with linear proration. We consider first the buyer’s point of view. The long-run average cost to the buyer under pro-rata warranty is given in Equation (5.49). Note

Table 5.16 Asymptotic Discounted Profit Under Linear Pro-Rata Warranty, Exponential Distribution, |x = 2.0 and 2.5

Limit of t t (L; W, 8) 8 .05 .10 .15 .20

= 2.0 8.692cb 4.760cb 3.451cb 2.798cb -

11.000cs 6.000cs 4.333cs 3.500cs

^ := 2.5 7.442cb 3.148cb 2.051cb 1.504cb -

9.000cs 5.000cs 3.667cs 3.000cs

206

Chapter 5

that this is a multiple of cb, and write it as E[Cb(L, W)] = K (L , W)ch. The long-run expected cost to the buyer of an unwarrantied item is simply cu[l + M(L)]. Equating these two long-run costs and solving gives the indifference price for the buyer, say cb = cb(L, W), as (5.57) In analyzing the seller’s point of view, we shall ignore administrative costs. To determine the seller’s indifference price, cb*, we equate seller’s profit with and without PRW. Long-run average profit on items sold under PRW is given in Equation (5.49). For unwarrantied items, the long-run average profit is given by [1 H- M(L)](cu - cs). Equating these two quantities yields

(5.58) Thus, (5.59) which is the same as the buyer’s indifference price. Here cb = cb* in a sense measures the true value of a warranty. If the actual selling price cb exceeds cb, the buyer would prefer an unwarrantied item at price cu. On the other hand, if cb were to be less than cb, the seller would prefer to sell the item without warranty. Thus, cb is the unique price at which buyer and seller would be in agreement, and at that point both are indifferent as to whether the item is sold (purchased) with warranty at cb = cb = cb* or without warranty at cu. Thus, the cost (value) of the warranty is the same to both, namely cb - cu. The indifference price analysis can also be done with discounting to present value. The necessary expressions are given in Section 4.3. In analyzing indifference prices, it is sometimes instructive to consider the limiting value as L —>°c. This analysis may be pursued using the results of this section or the indifference prices resulting when discounting is introduced into the models. As is often the case in warranty analysis, many of the integral expressions that result are analytically intractable except for a few simple life distri-

207

The Basic Pro-Rata Warranty

butions. We consider only the exponential distribution in the example to follow. In other cases, the limiting forms may be less intractable, and this approach is suggested. Example 5.17 [Exponential Distribution] For the exponential distribution, the buyer’s expected cost was determined in Example 5.10. From this we find

so that

Note that

For the parameter values used in previous examples, K (L , 1) = .81959 + .39347L for X = .5 and K (L , 1) = .84612 + .32968L for X = .4. Indifference prices for selected values of L are given in Table 5.17.

Table 5.17 Indifference Prices c£ = cj*,

L

>* II in

Renewing PRW, Exponential Distribution, X = .5 and .4 c,b* II

2 5 10 50 00

1.2449cu 1.2559cu 1.2620cu 1.2687cu 1.2707cu

1.1956cu 1.2026cu 1.2069cu 1.2118cu 1.2133cu

Chapter 5

208

5.6

CO M PARISON OF THE FREE-REPLACEMENT AND PRO-RATA W ARRANTIES

In this section, we attempt to summarize some of the many results given in this chapter and the previous chapter for a number of the important forms of the free-replacement and pro-rata warranties. The objective is to provide a framework for choosing between various types of warranties. This is of particular interest to the seller, since the selling organization is ordinarily the only one to which the option of determining warranty terms is available. Thus, we shall compare various warranties from the seller’s point of view. Many of the comparisons could also be made on the basis of the buyer’s costs as well by reference to the appropriate sections of Chapters 4 and 5. One of the difficulties in attempting comparisons of this type is finding a common basis for comparison. The choice here is expected cost to the seller on a per-item basis for nonrepairable items. (Note that expected cost means long-run average in the statistical sense, that is, the expected value over many items rather than over a long period of time such as a life cycle). In this comparative study, costs will not be discounted to present value. In interpreting the results to be given in the following, it should be noted that many other important factors have been omitted (and were, in fact, omitted in the analyses of these two chapters). For example, it is assumed that all items that fail within the warranty period are returned for rectification. This is not true in practice and will affect the seller’s expected cost. Furthermore, the return rate will almost surely be different under the different warranty policies, so the effect on cost will depend on the policy considered, further complicating the comparison. Administrative costs and other costs associated with the warranty are also ignored in the comparisons. These are also important factors that may vary from policy to policy. Another complication that is ignored is the fact that many of the policies involve indirect future costs or benefits, to both seller and buyer. For example, the seller may reap ill or good will as a result of his warranty policy. Most items will have a significant residual life after the end of the warranty period. This benefits the buyer but affects the seller’s future income, since it delays the time of future purchases. It may also affect the likelihood of brand switching. Recognizing that these and other difficulties should be taken into consideration in interpreting the results, we proceed to a comparison of warranty policies. In the comparison of alternative policies, we wish to include renewing and nonrenewing versions of both the FRW and PRW. The basic difference is that under a renewing warranty, a failed item is replaced by a new item with the warranty period beginning anew (i.e., the warranty clock is set

The Basic Pro-Rata Warranty

209

to zero), whereas under a nonrenewing warranty, rectification is either in the form of a cash rebate or in the form of a new item that is warrantied only for the time remaining in the original warranty period. The distinction between these two approaches to rectification for the nonrenewing warranty has not been emphasized for the PRW, because in practice only the rebate form is used. Thus, for the PRW, the customer is given either a cash rebate or (in the renewing case) an equivalent discount on the purchase price of a new item and the new item includes a new warranty rather than assuming the remainder of the previous warranty period. In the case of a cash rebate, the amount of the rebate depends on the form of the rebate function. In our comparison, we shall include linear and quadratic rebate functions and the proportional linear rebate with a = .80 as well as the linear renewing PRW. In the analysis of the FRW of Chapter 4, models for determining the cost of the renewing and nonrenewing versions were given. In both cases, it was assumed that a failed item was replaced by an identical new item. Both will be included in the comparison study. Finally, for completeness, a version of thc first-failure model of Chapter 4 will also be included. (This model was discussed as a first approach to analysis of the free-replacement warranty, in which the possibility of multiple failures within the warranty period was ignored.) In this context, the model represents a rebate-type warranty, in which a full refund is given on failure of any item within in the warranty period. To distinguish between these types of warranties, we shall call the nonrenewing version of the FRW the standard FRW (since that is the most commonly used version) and the latter warranty simply a rebate warranty. The rebate concept was briefly mentioned in the Introduction to Chapter 4 in the context of the FRW. In the setup just described, the expected cost to the seller is the cost of providing the item plus the warranty cost expressed as a proportion of the selling price, namely (5.60) To summarize, the following versions of the free-replacement and prorata warranties will be compared: Free-Replacement Warranties Rebate warranty—full purchase price is refunded if the item fails before time W.

210

Chapter 5

Standard (nonrenewing) FRW—free replacements are provided until a total service time of W is attained (equivalently, replacement items are warrantied for the balance of the time remaining in the original warranty). Renewing FRW—free replacements are provided for all items that fail before achieving a lifetime at least W time units in length (i.e., items are provided free of charge until an item having a lifetime of at least W is obtained). Pro-Rata Warranties Nonrenewing (rebate) PRW—on failure of any item prior to time W, the buyer is refunded a portion of the original purchase price. Three versions will be included: Linear proration—the percentage of the purchase price refunded is equal to the percentage of time remaining in the warranty period. Proportional linear proration—the percentage refunded is .80 times the percent of time remaining. Quadratic proration—the proportion refunded is calculated as the square of the proportion of remaining time. Renewing PRW—items that fail prior to time W are replaced at a cost determined by linear proration with full linear pro-rata warranty coverage provided on each replacement item. A final difficulty that is encountered in attempting to compare these various policies is the fact, discussed in some detail in Section 5.3, that there are a number of ways of defining per-unit costs for the renewing PRW. For comparability, we look at the total cost generated by a single sale under renewing PRW, adjusted so that the income is cb. (The point is that extra income is generated as well by the repeat sales under warranty, and this must be accounted for in formulating a common basis for comparison.) The adjustment is made by looking at the total expected cost to the buyer, £[Cb(W)], and crediting the excess amount above cb as an effective reduction in the seller’s cost. (See Table 5.9.) The net result is that the seller’s expected cost for the renewing PRW is calculated as the same cost for the renewing FRW less this excess amount. Thus, in every case, we calculate the cost to the seller equivalent to a buyer’s cost of cb, i.e., we determine the cost that the seller will incur in generating income in the amount of cb. It follows that the seller’s expected profit can be calculated as the difference between the selling price cb of the item and the amount tabulated in what follows.

The Basic Pro-Rata Warranty

211

Numerical results will be given for the examples used to illustrate the cost models in Chapters 4 and 5. The distributions used in these analyses were the exponential and the Weibull with shape parameters p = 2 and 4. (In our tabulation, the exponential will be listed as a Weibull with p = 1.) We used a warranty period of W = 1 and selected the remaining parameters of the distributions so that |x = 2.0 and 2.5. The following is a summary of the results obtained in relation to the warranty policies listed 1. 2. 3.

Rebate FRW. Model: Equation (5.60); values of F(W): Table 5.9 Standard FRW. Model: Equation (4.2); costs: Table 4.1 Renewing FRW. Model: Equation (4.13); costs: Table 4.2

Nonrenewing PRW: 4. Linear rebate. Model: Equation (5.6); costs: Tables 5.1, 5.2 5. Proportional rebate. Model: Equation (5.8); costs: Tables 5.1, 5.2 6. Quadratic rebate. Model: Equation (5.9); costs: Tables 5.3, 5.4 Renewing PRW: 7. Model for £[Cb(W)]: Equation (5.22); values: Table 5.9 The numerical results are summarized in Table 5.18. In the table, policies are numbered 1 through 7 in accordance with the preceding list. From the summary presented in Table 5.18, one can get a reasonable idea of the effect of changing warranty policies as well as of changing the distributional assumptions. For example, if an item selling for cb = $100 costs a total of cs = $60 to produce and sell, then profit on this item net of warranty costs may range from nearly $40 (Policy 5 with P = 4) down to a few cents (Policy 1 for the exponential distribution), even though items have a mean time to failure of 2.0 and are warranted for 1.0 time units in each case. In addition, many of the observations made previously concerning these two basic warranty policies are apparent from the results of Table 5.18. For example, the PRW is usually thought to be less costly to the seller than the FRW, but this is not always the case. One cannot, however, determine the effect of varying the warranty period, which will obviously also have a significant effect on costs, since this parameter was held constant in the comparison. The many other factors mentioned previously are also not apparent in these results. An overview of the effects of possible alternative policies can be obtained from studies such as this, however, and it is suggested that the practitioner pursue this approach, using simulation, if necessary, in attempting to formulate a warranty policy.

P = 1

cs + .394cb 1.500cs 1.649cs cs + .213cb cs + .171cb cs + . 148cb 1.649cs - .117cb

Policy

1 2 3 4 5 6 7

P = 2

|x = 2.0

cs + .178cb 1.187c, 1.217c, cs + .062cb cs + .049cb cs + .032cb 1.217c, - .025cb

Distributions, |x = 2.0 and 2.5

cs + .041cb 1.040c, 1.043c, cs + .008cb cs + .007cb cs + .020cb 1.043c, - .001cb

P = 4

= i

cs + .330cb 1.400c, 1.492c, cs -f .176cb cs + .141cb cs + .121cb 1.492c, - .076cb

p

cs + .118cb 1.121c, 1.134c, cs + ,040cb cs + .032cb cs + .003cb 1.134cs - ,010cb

p = 2

M. = 2.5 P = 4

cs + .017cb 1.017c, 1.017c, cs + .003cb cs + .003cb cs + .001cb 1.017c, - .0002cb

Table 5.18 Per-Item Cost to Seller for Free-Replacement and Pro-Rata Warranties, Exponential and Weibull

212 Chapter 5

The Basic Pro-Rata Warranty

213

NOTES

Section 5.2 1. The first important results on the pro-rata warranty were reported by Menke [4], who calculated warranty reserve funds required for servicing a lot of items of fixed size. The results were extended by Amato and Anderson [5] to include discounting. Patankar and Worm [6] further extended these results, including derivation of an asymptotic confidence interval for predicted, discounted warranty reserves. 2. Some results on the effects of claim frequency and the proportion of invalid claims on estimated warranty costs are given by Amato et al. [7], Section 5.3 1. Summaries of some of the approaches discussed in this section and the previous section are given in Hill and Blischke [8] and Blischke [9] as well as in many of the other articles cited. 2. We consider unit costs on a per-item basis, and look at three bases for calculation. Still another approach is to define unit cost as cost per unit of time. This is the basis of Mamer’s [2] unit cost analysis. Unit costs are obtained in this analysis as the limiting value of the ratio of total cost of purchases to total length of the time period. 3. Balcer and Sahin [10] derive expressions for the variance and higher moments of replacement costs to the buyer under PRW. Examples include the exponential, gamma, and mixed exponential distributions. 4. Optimal levels of warranty reserves in the context of multiple management goals are discussed by Mitra and Patankar [11]. Section 5.4 1. The initial attempt at determining life cycle costs for the pro-rata warranty was by Blischke and Scheuer [12]. In that analysis, the conditional expectation given was incorrect because the condition assumed changed the distribution in question. This was pointed out by Mamer [2], who gave an alternate derivation on which the approach used in this chapter is based. The Mamer result, however, is also incorrect in that a term was omitted in evaluating one of the integrals, resulting in omission of the quantity [(xw/W - F(W)]M(L - W) in the final expression (Mamer’s Equation [2]). The incorrect result was quoted in Blischke [9]. This cost model was also derived by Biedenweg [13]. Except for a few typographical errors, Biedenweg’s basic results agree with those given here. Finally, the correct result is also given by Nguyen and Murthy [14]. 2. Many of the integral expressions that result in deriving the cost models in this section and elsewhere are found to be more than a little in-

Chapter 5

214

tractable. Some important results regarding approximations are given by Frees and Nam [15]. Section 5.5 1. The notion of indifference prices was introduced in the warranty context by Blischke and Scheuer [12]. Discounting was also included in the analysis. The results given must be modified along the lines indicated in Note 1 on Section 5.4. 2. Models for obtaining indifference prices were derived by Park [16] in the context of repairable items under a preventive maintenance program. It was concluded that the indifference price increases with product quality but at a decreasing rate, and that the price becomes more sensitive to quality as the period of preventive maintenance increases. A number of additional interesting findings are given for this special application. EXERCISES

5.1.

5.2.

5.3.

5.4.

Suppose that an item is sold with nonrenewing PRW with refund q(t) given by the following piecewise linear function:

where 0 < y < 1. Determine the seller’s expected cost per unit. Compare the three cases: 7 < WJW, 7 = WJW, and 7 > WXIW. Find an expression for for the gamma distribution, with density function/(t) given, for t > 0, by

Express the general result in terms of the incomplete gamma function. Obtain explicit, closed-form expressions for |xw for the Erlang distribution with two stages, i.e., for the distribution function

Find an expression for |xw for the mixed exponential distribution with distribution function, for t > 0, given by

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215

5.5.

Find expressions for |x, a 2, and jjl w for the lognormal distribution, with density function, for i > 0, given by

5.6.

Obtain expressions for the seller’s cost per unit when an item is sold with nonrenewing PRW Policies 2a and 2b and the failure distribution is the gamma distribution. Obtain an expression for the seller’s cost as in Exercise 5.6 for Policy 2c. Obtain expressions for the seller’s costs as in Exercises 5.6 and 5.7 with exponential discounting to present value with discount factor 8. Obtain expressions for the seller’s cost per unit when an item is sold with nonrenewing PRW Policies 2a and 2b and the failure distribution is the mixed exponential distribution. Repeat Exercise 5.9 for Policy 2c. Obtain expressions for the cost per unit in Exercises 5.9 and 5.10 with exponential discounting with discount factor 8. Determine the seller’s expected cost per unit for the nonrenewing PRW Policies 2a and 2b assuming that time to failure follows a lognormal distribution. Obtain expressions for the seller’s cost per unit when an item is sold with renewing PRW Policies 2a and 2b and the failure distribution is the mixed exponential distribution. Repeat Exercise 5.12 for Policy 2c. Determine parameter values for the gamma, lognormal, and mixed exponential distributions so that the mean in each case is 2.0 and the variance is 1.0929 (equivalent to a Weibull with |x = 2 and p = 2). Calculate expected costs in the preceding examples for W = 1 and 2 and (where appropriate) 8 = .05 and .10. Compare the results with the corresponding results for the Weibull given in the examples of this chapter. Repeat Exercise 5.15 with |x = 2 and a 2 = 20. Include in the analysis the equivalent Weibull distribution, for which p = .50 and X = 1.0. Suppose that not all failures under warranty result in a warranty claim. We can model this as follows: If the age of the item at failure is t (0 < t < W), then the probability that a claim results is given by e~pt. Obtain an expression for the seller’s expected cost per unit under nonrenewing PRW.

5.7. 5.8. 5.9. 5.10. 5.11. 5.12. 5.13. 5.14. 5.15.

5.16. 5.17.

Chapter 5

216 5.18.

CLASS PROJECT: Repeat the analysis of Section 5.6, comparing policies as in Table 5.18, for the gamma, mixed exponential, Weibull, exponential, and lognormal distributions. Investigate a range of values for |x and a sufficient range of choices of W so that the resulting costs can be plotted as a function of W. Extend the results of the preceding exercises and the examples of this chapter to investigate in detail the relationship between warranty costs, failure distributions, and both policy and distribution parameters.

REFERENCES

1.

2. 3. 4. 5. 6. 7. 8.

9. 10. 11. 12.

Abramowitz, M., and Stegun, I. (eds.) (1964). Handbook o f Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Nat. Bureau of Standards Applied Math. Series, No. 55, U.S. Government Printing Office, Washington, D.C. Mamer, J. W. (1982). Cost analysis of pro rata and free-replacement warranties, Naval Research Logistics Q., 29, 345-356. Mamer, J. W. (1987). Discounted and per unit costs of product warranty, Management Science, 33, 916-930. Menke, W. W. (1969). Determination of warranty reserves, Management Science, 15, 542-549. Amato, H. N., and Anderson, E. E. (1976). Determination of warranty reserves: An extension, Management Science, 22, 1391-1394. Patankar, J. G., and Worm, G. H. (1981). Prediction intervals for warranty reserves and cash flows, Management Science, 27, 237-241. Amato, H. N., Anderson, E. E., and Harvey, D. W. (1976). A general model of future period warranty costs, The Accounting Review, 51, 854-862. Hill, V. L., and Blischke, W. R. (1987). An assessment of alternative models in warranty analysis, J. Information and Optimization Sciences, 8, 33-55. Blischke, W. R. (1990). Mathematical models for analysis of warranty policies, Math, and Computer Modelling, 13, 1-16. Balcer, Y., and Sahin, I. (1986). Replacement costs under warranty: Cost moments and time variability, Operations Research, 34, 554559. Mitra, A., and Patankar, J. G. (1988). Warranty cost estimation: A goal programing approach, Decision Sciences, 19, 409-423. Blischke, W. R. and Scheuer, E. M. (1975). Calculation of the cost of warranty policies as a function of estimated life distributions, Naval Research Logistics Q., 22, 681-696.

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13. 14. 15. 16.

217

Biedenweg, F. M. (1981). Warranty Policies: Consumer Values vs. Manufacturer Cost, Tech. Rept. No. 198, Dept, of Operations Research, Stanford Univ., Stanford, CA. Nguyen, D. G., and Murthy, D. N. P. (1984). Cost analysis of warranty policies, Naval Research Logistics 0 ., 31, 525-541. Frees, E. W., and Nam, S. H. (1988). Approximating expected warranty costs, Management Science, 34, 1441-1449. Park, S. K. (1985). Development o f a Price Indifference Function with Parameters o f Reliability, Maintenance and Warranty, Doctoral Dissertation, Texas A&M Univ., College Station, TX.

6

Complex One-Dimensional Warranties

6.1

INTRODUCTION

In Chapters 4 and 5, we discussed the two most common basic warranty structures, the free-replacement (FRW) and pro-rata warranties (PRW). These, in fact, form the basis of virtually all consumer warranties and most commercial warranties as well, in the sense that more complex warranty structures are typically devised as generalizations or combinations of the FRW and/or the PRW. Two basic extensions of this type are the subjects of this chapter. These are the combination warranty and the cumulative warranty. The combination warranty results when warranty terms change at one or more points in time during the course of the warranty period, which we again take to be the time interval [0, W]. Warranty coverage in each subinterval may be free-replacement, rebate, proportional rebate, or prorata (with any reasonable proration function), subject only to the constraint that coverage be a nondecreasing function of time throughout the warranty period. The most common combination in nearly all types of applications is an initial period of FRW coverage, followed by a (usually longer) period under PRW. In fact, this is usually what is meant by combination warranty. Note, however, that we use the term in a more general sense than is usual in the literature, to mean a multiperiod warranty with rebate, free-replacement, or pro-rata terms, renewing or nonrenewing, in any combination. Warranty Policies 3, 3a, 3b, 4, and 7 of Chapter 2 are combination warranties of various types. In Section 6.2, most of these and a few similar variations or generalizations will be covered. In each case, we again consider cost models for buyer and cost and/or profit models for the seller. 219

220

Chapter 6

Nonrenewing and renewing versions of the combination warranty will be considered. Note, however, that this differentiation is also more complex for this type of warranty, since this feature can also change from period to period over the course of the warranty coverage. We see that a great many combination warranties are possible. Our analysis in this chapter is not intended to be exhaustive of all of these possibilities. Rather, we have selected a few of the more widely used combination warranties for analysis, in order to illustrate the methodology, provide cost models for some important applications, and note some of the special difficulties encountered in analyzing warranties of this type. The cumulative warranty is a different type of combination, one in which items are combined under joint warranty terms. The cumulative warranty is also a generalization of the basic FRW or PRW, in this case covering a group of n items. Rather than covering each item separately for a period W, however, the cumulative warranty covers the group of items for a total service time of nW. Thus, individual items may have lifetimes less than W without leading to a warranty claim; whether or not the buyer is compensated under warranty is determined only on the basis of whether or not the total service time of all n items exceeds nW. If it does not, compensation may be in the form of a rebate, an agreed upon number of free or prorata replacements, or, conceptually, any of a number of other possibilities. By their very nature, cumulative warranties are not appropriate in the context of consumer transactions, since these typically involve only one or, at most, a few items. They may be appropriate in some commercial and government purchases, however, and, in fact, have been proposed for use in the United States in military acquisition. (Whether or not they have actually been implemented in this context and what the resulting outcomes may have been are not known to the authors at present.) The appeal of such a warranty to the seller is that he is not penalized for a few early failures if most of the items provided, in fact, have lifetimes exceeding W or if some have quite long lifetimes. Thus, the seller’s warranty cost should be decreased under cumulative warranty terms. Advantages to the buyer may be a lower purchase price (made possible because of this decreased cost to the seller) and a reduction in administrative cost due to settling warranty claims on a group rather than an individual basis. Although conceptually such a warranty could be either renewing or nonrenewing, we shall consider only the nonrenewing (i.e., rebate) version. The cumulative forms of both the nonrenewing FRW and PRW will be analyzed. This warranty has received very little attention in the literature. Our discussion of cumulative warranty structures and some associated cost models is given in Section 6.3.

Complex One-Dimensional Warranties 6.2

221

COM BINATION W ARRANTIES

The most common combination warranties include features of both the FRW and PRW. The usual combination warranty begins with a freereplacement term, followed by a period in which replacements will be supplied at pro-rata cost. This is Policy 3 of Chapter 2. Other combinations, with different terms in more than two periods (and decreasing compensation through time), are also used. For example, Policies 3a and 3b begin with free-replacement periods followed by periods of fixed payments in decreasing amounts rather than prorated rebates. These fixed payments are, in fact, partial rebates, making these policies combination FRW/rebate warranties. In the next section, we analyze a slight modification of these in which failures in the first period are also compensated by a rebate (in practice, usually a full refund) rather than a free replacement or repair. Policies of this structure are fairly commonly used in practice for some types of items, and will be designated combination lump-sum rebate warranties. The rationale for combination warranties is that they are a kind of compromise between the free-replacement warranty, which tends to favor the buyer, particularly in the latter portion of the warranty (where failures are perhaps more likely to occur), and the pro-rata warranty, which tends to favor the seller, since failures late in the warranty period entail very little compensation to the buyer and hence very little cost to the seller. Thus, the combination warranty gives the buyer full protection against potentially costly early failures, while also protecting the seller against full liability for later failures, where the buyer has received nearly the full amount of service that was guaranteed under the warranty. In short, this type of warranty is very common, because it has a significant promotional value to the seller while at the same time providing adequate control over costs for both buyer and seller in most applications. Exhibits 2, 6, 7, 8, and 12 of Chapter 1 include examples of various types of combination warranties. In the following subsections we shall be concerned with cost models that may be used to assess the value to both buyer and seller of various versions of these policies. We shall assume in each case that the items are nonrepayable, though most models are easily modified to include simple repair regimes such as good-as-new and minimal repair. As noted, the concept of renewability introduces still further latitude in specifying warranty terms, since this, too, may vary from period to period. We shall consider renewable warranties only for the standard two-period warranty, that is, FRW followed by PRW. In this case, the warranty may renew only in the PRW period or it may be set up so that it is always renewing.

Chapter 6

222 6.2.1

C o m b in a tio n Lump-Sum Re b a te W a rra nties

Policy Definition

A combination warranty that begins with a free-replacement period followed by decreasing lump-sum payments was given as Policy 3b in Chapter 2 as follows: Replacements or repairs are provided free of charge up to time W1 after initial purchase, at cost a xC(X) if the failure time X is in the interval (Wu W2], at cost a2C(X) if X is in the interval (W2, W3], and so forth, up to cost akC(X) if X is in the interval ( Wk, W], where < a2 < • • • < otk are the proportions of the current price C(X) at time of failure for failures in each of the k + 1 time intervals.

P o licy 3 b :

Here we consider only rebate forms. Thus, we are interested in a modification of Policy 3b in which a rebate rather than a free replacement is given if an item fails in the period [0, Wx). We shall refer to a warranty of this type as a combination lump-sum rebate warranty. The notion of a combination lump-sum rebate warranty is essentially that of the FRW, except that the full rebate is given only for an initial, relatively short period (if at all), with successively decreasing proportions of the initial sale price being refunded in later periods. Note that since we are considering the rebate form of the warranty, we have implicitly assumed that the warranty is nonrenewing. Since this type of warranty would seldom be renewing in practice, this constraint is not particularly restrictive. We shall make the additional assumption that the selling price of the item is constant throughout the warranty period, so that C(X) in Policy 3b becomes simply cb. This results in the following policy: P o licy 3c: COMBINATION LUMP-SUM REBATE WARRANTY : A rebate in the amount is given for any item that fails prior to time W1 from the time of purchase; the rebate is a2cb for items that fail in the interval (Wlf W2], a3cbfor items that fail in the interval (W2, W3\, and so forth, up to a final interval {Wk_x, W], in which the rebate is a*cb, with 1 > a x > a2 > • • • > a* > 0.

(Note that the ordering of the proportionality constants a x is reversed in Policies 3b and 3c. Here the a f indicate the decreasing magnitude of the rebate.) Policies such as 3b and 3c are widely used in warranties covering television sets or picture tubes, automobile tires, small and large appliances, and other consumer durables. Warranties in which parts and labor are covered initially and parts only later in the warranty period also fall into this general category. The warranty on a stand-alone fireplace in Exhibit 12 of Chapter 2 is an example of a declining rebate policy.

Complex One-Dimensional Warranties

223

Note, incidentally, that if k = 1 and a x = 1, Policy 3b reduces to the ordinary FRW, while Policy 3c reduces to the rebate from of the FRW. If k > 1 and a 2 = 1, Policy 3b includes an initial free-replacement period, while Policy 3c provides a money-back guarantee in the initial period. Cost Analysis

In this section, we analyze Policy 3c. Cost analyses of this policy are most easily approached by means of a rebate function q(-), as in the case of the pro-rata policy. This function is given explicitly in the second statement of the policy as follows:

(6 .1) where W0 = 0, Wk = W, and cb is the fixed cost to the buyer (the selling price) of a single item warranted under Policy 3c. Expected costs to both buyer and seller can easily be determined as functions of the expected rebate. As in previous chapters, we take X to be the lifetime of a randomly chosen item and use F(-) to denote the CDF of X. The expected rebate is given by

(6 .2) We use the vector notation a = (a1? . . . , a*)' and W = (W1, . . . , Wky. The buyer’s and seller’s cost under Policy 3c are given, respectively, by Cb(a, W) = cb - q(X) and Cs(a, W) = cs + q(X). Thus, the average cost per unit to the buyer of an item covered under combination lumpsum warranty is (6.3) The expected per-unit cost to the seller is (6.4)

Chapter 6

224

We next look at a numerical example. In this section, we will, where possible, continue the analysis of the exponential and Weibull distributions with means of 2.0 and 2.5 and a total warranty period of W = 1. This will permit comparison of the costs of the various warranty policies analyzed here and in Chapters 4 and 5, at least in that limited context. Example 6.1 [Weibull and Exponential Distributions; Policy 3c] Consider a three-stage rebate policy under which an item is warrantied for one year, with the buyer being given a full refund if a failure occurs in the first three months, a rebate of 75% of the initial purchase price if a failure occurs in the second three months, and a 50% rebate if a failure occurs in the remaining six months of the warranty period. This is Policy 3c with k = 3, c*! = 1.0, a 2 = 0.75, a 3 = 0.50, = 0.25, W2 = 0.50, and W = W3 = 1.0. The CDF for the Weibull distribution is given by

This reduces to the exponential distribution when p = 1. In Chapters 4 and 5, we considered the cases p = 1, 2, and 4. In each case, X was chosen so that the mean time to failure was |x = 2.0 or 2.5. The required values of X corresponding to these choices of |x may be found in Example 4.1 for p = 2 and 4; for the exponential, |x = 1/X. Simple substitution into Equations (6.3) and (6.4) will provide the buyer’s and seller’s expected costs for this warranty. The results are given in Table 6.1. As in previous examples, cs is the average cost to the seller of producing an item, and ch is the fixed per-item cost to the buyer. For purposes of comparison, costs for the standard FRW (which results when k = 1, a 2 = 1, and Wx = W) are also given in Table 6.1. Relative costs of other policies may be determined by comparison of these results with those given in Table 5.16. 6.2.2

C o m b in a tio n N onrenew ing Lump-Sum and PRW W a rra nties

As noted previously, the most common combination warranty is freereplacement or full refund up to time, say Wu from purchase, followed by linear pro-rata coverage from W1 until some later time W2 = W. In this section, we consider the rebate form of this warranty; renewing FRW forms will be analyzed in the two sections that follow.

Complex One-Dimensional Warranties

225

Table 6-1 Expected Cost to the Buyer and Seller Under FRW and

Combination Lump-Sum Warranties, Exponential and Weibull Distributions Combination lump-sum

FRW

0 1 2 4

2.0 2.5 2.0 2.5 2.0 2.5

£[Cs(i)]

E[Cb(l)]

£[CS(«, W)]

cs + .3935cb .3297cb .1783cb .1181cb .0413cb .0171cb

.6065cb .6703cb .8217cb .8819cb .9587cb .9829cb

cs + cs + cs + cs + cs + cs +

cs + cs + cs + cs + cs +

.2814cb .2340cb .1042cb .0687cb .0214cb .0089cb

E[Cb(a, W)] .7286cb .7660cb .8958cb .9313cb .9786cb .991lcb

As noted previously, the basic nonrenewing combination FRW/PRW was given in Chapter 2 as Policy 3. The rebate version of this is P olicy 3 d: COMBINED LUMP-SUM REBATE AND PRW POLICY: Under this policy, the manufacturer agrees to provide a full refund of the original purchase price up to time Wl from the time of initial purchase; any failure in the interval from Wx to W (> W^) results in a pro-rata refund. The warranty does not renew.

To distinguish between this and other forms of the combined FRW/PRW, we shall call this the rebate combination warranty. For the rebate as well as the renewing versions, this type of warranty can also be analyzed most easily by means of a rebate function. For Policy 3d with linear proration, the rebate function is

(6.5)

Note, incidentally, that this combination reduces to the FRW if Wx = W, and to the linear PRW if Wx = 0. The expected cost to the seller, E[Cs(Wly W)], is obtained, as in the previous section, as the fixed cost of supplying the item plus the expected

226

Chapter 6

rebate. This gives

(6 .6) where |xw and |xWlare partial expectations of X , defined in Equation (5.7). The buyer’s expected cost of the rebate combination warranty is easily calculated as well, as cb minus the expected rebate. The result is (6.7) Other combination warranties can be analyzed similarly. One possibility that may be useful in certain applications is a three-period warranty with a full-refund period followed by a period in which a smaller lump-sum rebate is given, and concluding with a pro-rata rebate period. A modification of this would be a proportional prorated rebate in the final period. Many other such combinations are possible. Analyses of the nonrenewing, rebate versions of such warranties, along the lines of this and the previous sections, are straightforward. Example 6.2 [Weibull and Exponential Distributions; Policy 3d] Suppose that a one-year warranty is offered, with full refund if a failure occurs within the first three months and pro-rata refund for the remainder of the warranty period. Thus, Wx = .25 and W = 1. Expected costs per unit to the seller and buyer are

and

For the exponential distribution, the partial expectation is given in Example 5.1 as

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227

For the Weibull distribution, this partial expectation requires evaluation of the incomplete gamma function. An expression for |xw and a series expression for calculating its values are given in Example 5.2. The CDF of the Weibull distribution is given in Example 6.1. Expected costs for the buyer and seller are given in Table 6.2. For comparison, the corresponding expected costs for the straight PRW are included as well. Additional cost comparisons may be accomplished by comparing these results with Tables 5.16 and 6.1. Note that, as expected, both buyers’ and sellers’ costs associated with the combination warranties lie between those of the FRW and PRW, all other conditions being equal. 6.2.3

Renew ing C o m b in a tio n W a rra nties

Introduction of the concept of renewing leads to many more possible combination warranties. It also leads to many additional aspects for analysis, since long-run costs are now relevant as is discounting to present value. In addition, a seemingly endless number of permutations of warranty conditions would be possible in multiperiod warranties. To limit the analysis, attention will be restricted in this section to the two-period combination FRW/PRW, that is, an initial period in which a free replacement is given on failure of an item, and a second period in which a replacement is provided at pro-rata cost. In this case, two reasonable forms of renewing may be considered. These are as follows: (1) Renew only in the PRW portion (i.e., in the FRW period of the warranty, replacement items assume only the remaining time in the warranty period); and (2) Always renew. The latter case is Policy 7 of Chapter 2. We shall refer to this as the fully renewing combination FRW/PRW. We first con-

Table 6.2 Expected Cost to the Buyer and Seller Under PRW and

Combination FRW/PRW Warranties, Exponential and Weibull Distributions

1 2 4

2.0 2.5 2.0 2.5 2.0 2.5

cs + cs + cs + cs + cs + cs +

.2131cb .1758cb .0618cb .0404cb .0083cb .0034cb

E[Cb(1)]

£[CS(.25, 1)]

.7869cb .8242cb .9382cb .9596cb .9917cb .9966cb

cs + cs -1cs + cs + cs -1cs +

.2643cb .2183cb .0810cb .0530cb .0111cb .0046cb

E[Cb(.25, 1)] .7357cb Wx) are replaced at pro-rata cost. Replacement items in this interval are provided warranty coverage identical to that of the original item. P o licy 7a:

Under this policy, the warranty renews only in the PRW period. This means that upon failure of an item at, say, x < Wl9 a free replacement is provided, and this replacement item is covered under FRW until time Wx - 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 Wt < x' < W, the replacement is provided at prorated cost to the buyer and the warranty begins anew. In theory, the proration during the second period of this warranty may again be either linear or nonlinear. We consider only the linear case. The cost analysis proceeds as follows:

Complex One-Dimensional Warranties

229

Buyer's Average Cost of Replacement

We begin with the long-run average cost to the buyer of replacing a failed item that was purchased with the warranty coverage of Policy 7a. This is determined as a function of the random variable Y = interval between purchases. Since replacements are made at no cost to the buyer (i.e., no purchase is necessary) in the interval [0, WJ, we have that Y > W1. After this initial interval, the buyer’s replacement cost, say R( ), is given by ( 6 . 8)

To determine the expected replacement cost, we require Fy(*), the distribution of Y. This is simply (6.9) where Fy(-) is the distribution of the excess random variable. (See Equation (3.19).) We now obtain

( 6 . 10)

The average cost per unit time, say A (W U W), is now calculated as the ratio of the expected cost per replacement to the expected time between replacements,! i.e.,

(6 . 11) tThis is an asymptotic result, used as a first-order approximation. The exact result is the expected value of the ratio rather than the ratio of expected values and is usually much more difficult to evaluate.

230

Chapter 6

where £(7?(Y)] is given in Equation (6.10) and E(Y) is given by ( 6 . 12)

with Af() being the ordinary renewal function corresponding to F ( ) , the life distribution of the items. (See Chapter 3 and Section 4.5.) The difficulty in applying these results is that closed-form expressions exist for only a few distributions, the exponential, as usual, being one of these. In other cases, approximations or simulation are required for evaluation of expected costs. In the following examples, we look at the exact result for the exponential distributions and some numerical results obtained for the Weibull distributions used in our previous examples. A simple simulation, generating Weibull variates, is used in this numerical study. Some general simulation methods and approaches for use in more complex warranty situations are discussed in Chapter 11. Example 6.3 [Exponential Distribution; Policy 7a] Because of the “memoryless” feature of the exponential distribution, the distribution of the excess random variable is simply Fy(t) = F(t). Thus, from Equation (6.9),

From this, one easily obtains

Since M{t) = \ t for the exponential distribution, we have

The average cost per unit time is obtained by substitution of these results into Equation (6.11). For the cases considered previously (|x = 2.0 and 2.5), expected costs are given in Table 6.3 for a total warranty period of 1.0 time units and a free-replacement period of 0.25.

Complex One-Dimensional Warranties

231

Table 6.3 Average Replacement Cost to the Buyer Under Partially Renewing Combination FRW/PRW, Exponential Distribution

2.0 2.5

£[/?(Y)]

E(Y)

A(. 25, 1)

.8339cb .8639cb

2.25 2.75

.3706cb .3141cb

From Table 6.3, we see that for an item with an average lifetime of 2.0 years, the average cost to the buyer per year is about 37% of the purchase price of the item, when the item is covered under free-replacement warranty for the first three months and under linear pro-rata warranty for the remaining nine months of a one-year warranty period. For an item with an average time to failure of 2.5 years, this cost is about 31.4% of the original purchase price of the item. Note that these average annual costs for unwarrantied items would be 50% and 40% of the purchase price for the 2.0-year and 2.5-year items, respectively. Example 6.4 [Weibull Distribution; Policy 7a] For the Weibull distribution, closed-form expressions do not exist for either the renewal function or the distribution of the excess random variable. Since an exact analytical solution is therefore not possible, we take a numerical approach to determining expected buyer’s cost. We again consider |x = 2.0 and 2.5, Wx = 0.25, W = 1.0, and values of the Weibull shape parameter of p = 2 and 4. E(Y) is calculated using the table of the renewal function given in Appendix C. £[7?( Y)] is estimated by means of a simple simulation, the details of which are given in Chapter 11. The results of the simulation are given in Table 6.4. In this table, E[-]

Table 6.4 Simulation Results: Buyer’s Replacement Costs, Weibull Distribution p

M -

£[*(*)]

S.E.

2

2.0 2.5 2.0 2.5

.9330cb .9567cb .9902cb .9960cb

.000952 .000775 .000330 .000214

4

Chapter 6

232

denotes the estimated expected cost based on the simulation results and S.E. denotes the observed standard error of E[ ]/cb. The magnitude of the standard errors would indicate that the estimates may be in error by at most about .002 for p = 2, and about .0006 for p = 4. We can now estimate buyer’s average annual cost, A{.25, 1), for this warranty with Weibull distributed times to failure. The results are given in Table 6.5. In the table, M(.25) is obtained from Appendix C and the remaining items are calculated from Equations (6.11) and (6.12). Again, these numbers should be compared with 0.5cb for |x = 2.0 and to 0.4cb for |x = 2.5. We see that this warranty is worth only a few percent of the purchase price to the buyer when p = 2 and one percent or less when p = 4. The Buyer's Life Cycle Cost

We now look at the life cycle cost (LCC-I) to the buyer over the period [0, W], assuming that L > W. Let Z(L; W1, W) be the (random) total replacement cost to the buyer over the period [0, L), and let £(L; Wu W) denote the expected value of Z(-). Then the total life cycle cost to the buyer, including the cost of the initial purchase, say Cb(L; Wx, W), is given by cb + Z(L; Wu W), so that the buyer’s expected life cycle cost for this warranty over the period [0, L ] is given by

(6.13) As in the analysis of the renewing PRW, a conditional argument can be used to evaluate £(•). Let Y1 be the time from initial purchase until the first purchased replacement. The variable Yx in this analysis is called the first purchase interval. Here the generic purchase interval Y is the same random variable as that considered in analysis of the free-replacement warranty in Chapter 4, except that the relevant time increment is Wx instead

Table 6.5 Average Replacement Cost to the Buyer Under Partially Renewing Combination FRW/PRW, Weibull Distribution 3 2 4

2.0 2.5 2.0 2.5

Af(.25)

E(Y)

A(. 25, 1)

.01222 .007833 .0001648 .00006749

2.0244 2.5196 2.00033 2.50017

.4609cb .3797cb .4950cb .3984cb

Complex One-Dimensional Warranties

233

of W. Note that this is the first instant at which the warranty renews under Policy 7a. It follows that if we condition on Y1; we have

(6.14) where /?(•) is the replacement cost function defined in Equation (6.8). The analysis proceeds as in similar previous cases: Unconditioning (and noting that FY(y) = 0 for 0 < y < W1), we obtain (6.15) where (6.16) This is a standard renewal equation (See Chapter 3), the solution to which is expressed in terms of the ordinary renewal function My(*) corresponding to Fy(-) as

(6.17) the last expression being obtained by integration by parts. Substitution of h(L) from Equation (6.16) and R(y) from Equation (6.8) into this expression now yields, from Equation (6.13),

(6.18)

234

Chapter 6

This may also be expressed, by use of Equation (6.9), in terms of the distribution of the excess random variable. The result is E[Cb( L ; Wl9 W)]

(6.19) Unfortunately, neither of these expressions for the buyer’s expected cost of the partially renewing combination FRW/PRW warranty can be evaluated analytically for nearly any life distribution. There are, however, various approximations available. (See the notes at the end of this chapter.) One approach (Nguyen and Murthy [1]) is based on bounds for £(•)> namely,

(6 .20) From this we obtain (6.21)

This is not a great improvement as far as an exact solution is concerned, since neither £[R(T)] nor My(*) can be evaluated analytically for most life distributions. However, closed-form expressions for both are available for the exponential distribution, and both can be approximated or evaluated numerically fairly easily for other distributions. Approximations for M Y(L) are given in Equations (4.20)-(4.22). Using the last of these in Equation (6.19) leads to the approximation

(6 .22)

Complex One-Dimensional Warranties

235

We note that this result is exact for the exponential distribution. It is not of much help in the case of many other important life distributions, however, since it requires the distribution of the excess random variable and the first two moments of Y. In cases where these cannot be determined analytically, the approximation of Equation (6.21) may be used with My(L) approximated by Equation (4.21) if F(*) is NBU, and by Equation (4.20) otherwise. Alternatively, Equation (4.22) may be used with the required V(Y) given by

(6.23) where |x and a 2 are the mean and variance of the random variable X (the lifetime of an individual item), and M(-) is the ordinary renewal function associated with the distribution function of X. This form is particularly useful since /g M{t) dt has been tabulated for many important life distributions (See Baxter et al. [2]). Example 6.5 [Exponential Distribution] For the exponential distribution, exact results can be determined. An expression for E[F(Y)] is given in Example 6.3, along with numerical values for the cases |x = 2.0 and 2.5 (X = .5 and .4, respectively), Wx = 0.25, and W = 1.0. In addition, we have £(Y) = (1 + XWy/X, V(Y) = 1/X2, and

the last result being exact rather than an approximation in the case of the exponential distribution. Values for these functions and the buyer’s expected life cycle costs for L = 7.5 and 15.0 are given in Table 6.6. Example 6.6 [Weibull Distribution] We again consider the Weibull distribution with P = 2 and 4, and |x = 2.0 and 2.5. As in the previous examples, we take W = 1 and W1 = 0.25. We use the approximation of Equation (6.21) and take L to be 7.5 and 15. Values of F[F(Y)], obtained by simulation, are given in Examples 6.3 and 6.4; approximate values for My(L) for the Weibull distribution may be obtained by use of Equation (6.23), with values of M(W1) (also needed for evaluation of E(Y) = |x[l + M(WX)]) obtained from tables or computer evaluation and values of the integral of this function obtained from

Chapter 6

236

Table 6.6 Life Cycle Cost to the Buyer for the Partially Renewing FRW/PRW, Exponential Distribution

2.0 2.5

L

E{Y)

V(Y)

My (L)

E(R)

£[Cb(L, Vfu M0]

7.5 15.0 7.5 15.0

2.25 2.25 2.75 2.75

4.00 4.00 6.25 6.25

3.228 6.562 2.641 5.368

.834 .834 .864 .864

3.605cb 6.389cb 3.213cb 5.569cb

Baxter et al. [2], Finally, the variance of X is obtained for the Weibull distribution as

with X = r ( l + l/p)/|x, as before. Values of the means and variances required in the computations are given in Table 6.7; approximate expected life cycle costs to the buyer, obtained by use of Equation (6.21), are given for the stated combinations of parameter values and warranty terms in Table 6.8. (Note: Values of Jo M{t) dt obtained from the tables of Baxter et al. [2] are accurate for W1 = 0.25 to only one significant digit. This should not greatly affect the final result, however, since this term is only a very small contribution to V(Y).) The Se lle r's Expected Profit

We begin with the long-run average profit per item. As before, we use cs to denote the average cost to the seller of supplying an item. It follows that the (asymptotic) value of the seller’s average cost per unit of time is simply cjyjl . The long-run expected profit to the seller, say W), is

Table 6-7 Means, Variances, and Estimated Values of R(Y) for the Weibull Distribution p

M *

(T2

E(Y)

V(Y)

E(R)

2

2.0 2.5 2.0 2.5

1.093 1.708 .315 .492

2.222 2.722 2.227 2.727

1.049 1.677 .311 .487

.934 .957 .990 .996

4

Complex One-Dimensional Warranties

237

Table 6.8 Approximate Life Cycle Cost to the Buyer for the Partially Renewing FRW/PRW for the Weibull Distribution L = 15.0

L = 7.5

p 2

M * 2.0 2.5 2.0 2.5

4

My(L)

E[Cb(L, Wu W)}

My{L)

E[Cb(L, Wu W)]

2.982 2.369 2.900 2.283

3.752cb 3.245cb 3.866cb 3.272cb

6.357 5.124 6.268 5.034

6.904cb 5.881cb 7.202cb 6.011cb

the difference between income (i.e., buyer’s cost) and average cost per unit. From Equation (6.11), it follows that (6.24) where 2s[/?(Y)] is given by Equation (6.10). Total profit to the seller over the life cycle of the item, say I1(L; Wu W), can be calculated similarly. The expected total number of items supplied in the interval [0, L] is 1 + M(L), each at cost cs. Total expected profit in this period is the difference between buyer’s expected total cost and seller’s total expected cost. Thus, (6.25) 6.2.5

C o st A nalysis o f the Fully Renew ing C o m b ina tio n FRW/PRW

The fully renewing combination warranty was given in Chapter 2 as Policy 7. Under full renewing, the warranty clock is set to zero upon failure of an item at any time prior to W from the time of sale. The policy statement is as follows: Policy 7

COMBINED FREE-REPLACEMENT AND PRO-RATA POLICY : Under this policy the manufacturer agrees to provide a replacement free of charge up to time Wl from the time of initial purchase; any failure in the interval Wl to W {> Wx) is replaced at a prorated cost. The proration may be either linear or nonlinear. In either case, the replacement item is covered under a new warranty identical to the original one.

In our analysis, we again consider only linear proration. Some of the results of this section can be extended to nonlinear proration as well by analyses along the lines of those of Chapter 5.

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We note at the outset that the buyer’s cost may be expected to be less for this warranty than for the partially renewing combination, since with the full renewing version more free replacements may occur. Correspondingly, the seller’s profit may be expected to be less, all other things being equal. We analyze this warranty as in the previous section, looking first at the cost per unit of time for the buyer, the buyer’s life cycle cost, and then the seller’s profit. Here, however, one can easily calculate the average per-item cost and profit, and these will be included in the analysis as well. As has often been the case, many of the results, while providing a formal solution to the problems, will be found to be mathematically intractable for most life distributions. Bounds and approximations will be found to be useful in most applications and will be provided for some of the cases considered. Additional results along these lines may be found in the references cited in the notes at the end of this chapter. The Buyer's Average Cost of Replacement

We consider both average cost per item provided and the average cost to the buyer per unit of time. We begin with the buyer’s average cost per item. Since the warranty renews every time a replacement item is provided (whether free or at some cost to the buyer), the cost to the buyer can be expressed as a function of X , the time to failure of the item being replaced. Let S(A') denote the selling price of a replacement when the item in service fails at age X. Then (6.26) The average selling price, i.e., the average cost per item to the buyer, is therefore

(6.27) where, as before,

is the partial expectation of X.

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This result is, of course, the same as the expression of Equation (6.7), which gives the average per-unit cost to the buyer for items sold under comparable rebate-type warranty terms. Note that this expectation may also be written as (6.28) Equations (6.27) and (6.28) provide alternate expressions for asymptotic (as L —> oo) average per-item costs. The average cost to the buyer per unit of time is now easily obtained as (in the notation of the previous section) (6.29) Example 6.7 [Exponential and Weibull Distributions] For the cases considered previously, with again W1 = .25 and W = 1, values of E[5(2^)] can be found in the last column of Table 6.2. Division by p gives A( W l9 W). For purposes of comparison with previous results, the values are given in Table 6.9. Comparable results for the partially renewing version of this warranty are given in Table 6.3 for the exponential distribution and in Table 6.5 for the Weibull. By comparison with Table 6.9, we see that the fully renewing combination FRW/PRW costs the buyer slightly less than the corresponding partially renewing combination warranty, as expected. For relatively shorterlived items, the savings would be more.

Table 6.9 Average Replacement Cost

to the Buyer Under Fully Renewing Combination FRW/PRW, Exponential and Weibull Distributions

A(.25, 1)

p 1 2 4

2.0 2.5 2.0 2.5 2.0 2.5

.3679cb .3127cb .4595cb .3788cb .4945cb .3982cb

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Chapter 6

The Buyer's Life Cycle Cost

The analysis of the buyer’s LCC also parallels that of the partially renewing combination warranty. We again define Z(-) to be the replacement cost to the buyer and £(•) to be its expectation, but with both interpreted here as costs under the fully renewing combination warranty. Similarly, Cb(L, Wl9 W) is the buyer’s (random) life cycle cost under this warranty. Since the warranty process renews on failure of an item regardless of when it fails, in this case we condition on X x, the time to first failure, rather than on the purchase interval Y. The resulting expression analogous to Equation (6.14) is

(6.30)

where S(-) is the cost of the first failure, as given in Equation (6.26). In this case, unconditioning leads to

(6.31)

where

(6.32)

This is again a renewal integral equation. The solution is

(6.33)

Substituting H{L) from (6.32) and .S'(x) from (6.26), and using the fact that the buyer’s expected cost is £[Cb(L, W1, IT)] = cb + £(L, VTX, IT),

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we find

(6.34)

Alternatively, this may be written as

(6.35) Again, closed-form solutions do not exist for most life distributions of interest. Nguyen and Murthy [1] provide bounds and approximations for this case as well. These may be expressed in terms of either the ordinary renewal function M () or the renewal function My(*) associated with the random variable indicating the purchase interval. Bounds based on the former, which are usually found to be easier to evaluate, are given by

(6.36) This leads to the approximation (6.37)

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Alternative bounds are

(6.38) and (6.39) the latter being applicable only for life distributions that are NBU. The midpoints of the intervals defined by the endpoints of these bounds may be used as alternative approximations to (6.37). Continuing the analysis of the previous examples, we analyze the exponential distribution and the Weibull distribution with P = 2 and 4 in Examples 6.8 and 6.9. As before, in both cases we consider jx = 2.0 and 2.5, with Wx = .25, W = 1. Example 6.8 [Exponential Distribution] For the exponential distribution, exact expressions for the buyer’s life cycle costs can again be determined. It is instructive to compare these with the approximations. For this purpose, we use the approximation of Equation (6.37). For the exponential, it is easily seen that

and

Exact and approximate values for buyer’s expected cost are given in Table 6.10 for L = 7.5 and 15.0. Note that these approximations, based on Equation (6.37), significantly overestimate the actual life cycle cost to the buyer. (In fact, in most cases the values exceed the buyer’s cost for the corresponding partially renewing combination, given in Table 6.5. This cannot be the case, since more free replacements will be provided under the fully renewing warranty than under an otherwise equivalent warranty

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Table 6-10 Approximate and Exact Values of Life Cycle Cost to the Buyer for the Fully Renewing FRW/PRW, Exponential Distribution

E[Cb(L, Wu W)] L 7.5 15.0

2.0 2.5 2.0 2.5

E(S)

M{L)

Exact value

Approximation

.7357cb .7817cb .7357cb .7817cb

3.75 3.00 7.50 6.00

3.538cb 3.156cb 6.298cb 5.501cb

3.627cb 3.236cb 6.386cb 5.581cb

that is only partially renewing.) In practice, caution in using the approximations is suggested. Example 6.9 [Weibull Distribution] For the Weibull distribution, the approximation of Equation (6.37) gives anomalous results and the lower bound of Equation (6.39), which is appropriate since the distribution is NBU for p = 2 and 4, is used instead. (The results of Table 6.6, which gives approximate values for the partially renewing version of this warranty may be used as approximate upper bounds, since costs to the buyer for the fully renewing version must always be less.) Lower bounds for expected life cycle cost to the buyer, based on Equation (6.39), are given in Table 6.11 for the parameter values and warranty terms considered in our previous examples. The values of A(.25, 1) are obtained from Table 6.9. For purposes of comparison, the bounds are given for the exponential distribution (p = 1) as well. In all cases, the upper bound may be obtained by adding c5 to the values given in Table 6.11; the corresponding approximation may be obtained by

Table 6.11 Lower Bounds for Life Cycle

Cost to the Buyer for the Fully Renewing FRW/PRW, Weibull Distribution Lower bound for

E[Cb(L, p

2 4

2.0 2.5 2.0 2.5

W)]

L = 7.5

L = 15.0

3.45cb 2.84cb 3-71cb 2.99cb

6.89cb 5.68cb 7.42cb 5.97cb

Chapter 6

244

adding .5cb to the tabulated value. For P = 2 and 4, both of these calculations will yield values greater than the approximate values for the corresponding partially renewing FRW/PRW given in the previous tables. In fact, this is also true of the lower bound for p = 4, |x = 2.0 (given as 7.42 in Table 6.11). As noted previously, this cannot occur, since the fully renewing version is necessarily less costly to the buyer. This further suggests that caution be used in applying these approximations, unless great precision is not required. In many cases, simulation methods for estimation of warranty costs may be preferred. This approach will be discussed in some detail in Chapter 11. The Seller's Expected Profit

We look first at the seller’s long-run average profit. Because of the fully renewing feature of the warranty, the initial period is analogous to the ordinary free-replacement warranty. As in the case of the FRW, the number of free replacements per purchase is a geometric random variable, with expectation F{W^)I[ 1 - F(Wj)]. It follows that the expected cost to the seller is cs/[l - F(Wj)] per purchase. By an analysis analogous to that of the partially renewing combination warranty, we find W), the longrun expected profit as L —> oc to the seller per sale, to be (6.40) where E(R) is the expected cost of a purchase, here given by

(6.41) As in the case of the partially renewing warranty, the expected total profit to the seller over the life cycle of the item is given by (6.42) with E[Cb(L , Wu W)] in this case given by Equation (6.35). 6.3 6.3.1

C UM ULA TIV E W ARRANTIES Structure o f C um ula tive W a rra nties

In Chapters 4 and 5 and in the previous section of this chapter, we have been concerned with basic consumer warranties: free replacement, pro

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rata, and a combination of both types of warranty terms. These are the types of warranties typically offered on a wide range of products, and they usually involve purchases of one or, at most, a few items (e.g., one automobile or a set of two or four tires as replacements on wearing out of the originals). In these cases, warranties are invariably dealt with on an individual basis. Commercial and government transactions, on the other hand, often involve purchases of whole batches of items (e.g., fleets of automobiles, hundreds or thousands of tires purchased by a government agency or an auto rental firm). In such situations, dealing with warranties on an individual basis may be a tedious process and may involve a significant administrative cost, both to buyer and to seller. Cumulative warranties, under which the warranty covers the total service time (or usage, etc.) of an entire batch of items, have been proposed as an alternative. In this section, we look at a number of cumulative warranties based on the standard freereplacement, pro-rata, and combination structures previously analyzed. The notion of a cumulative warranty was first introduced, at least in concept, in the 1970s in the context of military acquisition of goods and equipment (Trimble [3]; whether or not such policies were ever actually implemented is not known). More recently, another version of this type of warranty has been considered in commercial applications, where the warranty has been called a “fleet warranty” (Berke and Zaino [4]). The general structure of a cumulative warranty is that the product is sold in batches of n items and, rather than providing a warranty of length W on each individual item, the batch as a whole is warranted to provide a total service time of nW units. Warranty terms may be FRW, PRW, or some form of combination warranty analogous to those discussed in the previous section. Under a cumulative free-replacement warranty, the consumer is supplied with n items at cost nch (where ch may be different from the price that would be charged for purchase of a single item). Upon failure of the last of this batch of items, the total service time of the n items is calculated. If this is at least nW, the seller has no further warranty obligation, and the buyer must purchase any replacements at full price. If the total service time is less than nW, free replacements are provided by the seller, one at a time. As these fail, their lifetimes are added to the previous total, and the process continues until a total service time of at least nW is attained. Under a cumulative pro-rata warranty, the process is the same as that just described except that any necessary replacements are provided at prorated cost rather than at no cost to the buyer. Under a cumulative combination warranty, replacements would be provided free until a total service time of nW1 (where Wx is the length of the free-replacement portion

246

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of the warranty on a single item, as in the previous section), and at prorated cost if the total service time exceeded nW1 but was less than nW. In practice, there are many ways in which policies having these general features could be implemented, and as a result many complications may be encountered in attempting such implementation. Several cumulative warranties were described in Chapter 2. We shall analyze a few of these and will restate the specific warranty terms for each in order to provide explicit descriptions prior to introduction of the cost models. The various approaches embodied in these statements will serve to illustrate some of the practical difficulties. In general terms, the difficulties are as follows: 1. If all of the items in a batch are used simultaneously, any warranty claims will not be settled until after failure of the last item. This could be a long time after the initial purchase. Furthermore, replacements are usually needed after each failure. 2. If k < n of the items are used simultaneously, with the remainder of the items in the batch retained for spares, similar settlement and replacement problems ensue. 3. A possible solution to these two problems is to agree on a fixed maximum time at which any warranty claim will be settled, regardless of whether or not all items have failed. The difficulties with this approach are in determining both the time and amount of settlement that would be fair to both buyer and seller. (Presumably some credit would have to be given for expected future service time of items that had not yet failed at the time of settlement.) 4. Ordinarily it would be the responsibility of the buyer to track and record times to failure or usage at failure of individual items. In most cases, the seller would have no way of checking this and would have to assume that it was done honestly. 5. Many other bookkeeping problems may be encountered. For example, batches purchased at different times may complicate the process. 6. The many ways in which such policies may be implemented make it difficult for both buyer and seller to evaluate and compare alternatives. In addition, very little analysis of these types of warranties has been done. Furthermore, as we shall see, the cost models for the few warranties of this type that have been analyzed involve even more complex renewal functions, further complicating the comparison of policies. In spite of these shortcomings, the notion of a cumulative warranty is an appealing one in certain situations involving batch purchases. An advantage to the buyer is that items sold under cumulative warranty will usually be sold at a lower price than that of individually warranted items. The biggest advantage to the seller, and the reason that it would be possible to sell the items at a lower price, is that fewer warranty claims would result,

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because a few unusual early failures could be offset by the excess life of some longer-lived items. A potential advantage to both buyer and seller is that warranty claims need not be made on each failure, but instead are made only upon failure of the last item in the batch or at some other agreed upon time, thus potentially reducing the administrative costs of both. Another consideration that is, in a sense, an advantage to both buyer and seller is that items under cumulative warranty may be offered with a longer warranty period because fewer claims would be expected. This is inherently advantageous to the buyer. It is advantageous to the seller in that it may be a valuable advertising strategy and may provide a competitive edge in negotiating contracts for the purchase of large batches of items. Furthermore, the length of the warranty period could be made a function of the number of items purchased under cumulative warranty, since for most life distributions the average number of claims per unit should decrease as n increases. In the remainder of this section, we will look at two approaches to formulating and analyzing cumulative warranties. The first is due to Guin [5] and parallels the analyses of Chapters 4 and 5. The second, due to Berke and Zaino [4], is based on a guaranteed minimum mean time to failure (MTTF), and it is somewhat related to certain aspects of the reliability improvement warranty (RIW), to be discussed in the next chapter. In fact, as noted previously, the notion of a cumulative warranty may have first been suggested in the context of RIW as used in military procurements. It was mentioned in a Headquarters, U.S. Air Force, Directorate of Procurement Policy Document (Trimble [3]). 6.3.2

C o st M odels fo r C um ula tive W a rra nties

In this section, we analyze the cumulative FRW of Policy 14 and the cumulative PRW of Policy 16, using the basic techniques, including the renewal function approach, used in Chapters 4 and 5 in the analysis of the basic (noncumulative) versions of the FRW and PRW. Cumulative combination warranties have not been analyzed in detail, but many of the results of the previous section should be extendible in a similar fashion. Analysis of the Cumulative FRW

In this section, we analyze the simplest version of the cumulative freereplacement warranty. We assume that the warranty is nonrenewing, that the item in question is nonrepairable and, in looking at life cycle costs, that replacements are made instantaneously. Models for assessing per-unit as well as life cycle costs will be derived. The resulting models will be used to compare the cumulative FRW with the FRW analyzed in Chapter 4.

248

Chapter 6

(In this context, the FRW of Chapter 4 will be called the standard FRW, as opposed to the cumulative FRW under consideration here.) Under a cumulative warranty, whether or not the buyer is entitled to a warranty benefit depends on the total service time of the lot of n items initially purchased, i.e., on Sn = 2 "=1 X h where X t is the lifetime of the ith item. The version of the cumulative FRW to be analyzed is the following: Policy 14 CUMULATIVE FRW: A lot of n items is warrantied for a total (aggregate) period of nW. The n items in the lot are used one at a time. If Sn < nW, free replacement items are supplied, 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 nW. Per-Unit Cost to the Seller

As was the case for the standard FRW, the important random variable in analysis of the cumulative FRW is N (also denoted N(W, n)), the number of replacements supplied under warranty. The distribution of N is given by

(6.43) where F^n\- ) is the «-fold convolution of F( ) with itself, and, for j = 1, 2

,...,

(6.44) The expected number of replacements needed, which we denote MCF(nW), is given by

(6.45)

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Note that when n = 1, MCF(*) reduces to A/(-), the ordinary renewal function associated with F(*). The expected per-batch cost to the seller is now easily calculated as the total cost of supplying the original batch of n items plus the expected cost of replacements, namely cs[n + MCF(nW)], where, as usual, cs is the average cost to the seller of supplying a single item. The expected cost per unit, say E[CS(W, ri)\, is therefore (6.46) Note that this compares with the per-unit cost under the standard FRW of (6.47) where M( ) is the ordinary renewal function associated with F(*). For purposes of computation, it is also useful to note that, since the ordinary renewal function (see Chapter 3) is (6.48) we can write (6.49) Example 6.10 [Exponential Distribution] The CDF for the exponential distribution is given, for x > 0, by

For purposes of comparison with previous results, we consider X = .5 and .4 (corresponding to (x = 2.0 and 2.5, respectively) and W = 2. Since very few claims would be expected under cumulative warranty with these choices of X and W (and since a shorter warranty period relative to the mean would seem appropriate), we also consider X = 1.25, 1.00, and 0.80 (|x = 0.8, 1.0, and 1.25). In all cases, lots of size n = 2, 3, 4, and 5 will be analyzed along with the standard FRW (n = 1).

Chapter 6

250

In the analysis, we require the &-fold convolutions of F(-) with itself. It is well known (see, for example, Kapur and Lamberson [6], p. 25) that for the exponential distribution, the distribution of Sk is a gamma distribution with parameters X and k , with CDF given by

for* > 0. This function is extensively tabulated and is also easily computed directly by use of most standard computer statistical packages. Equation (6.49) can then be used to calculate MCF(nW). The ratio of expected costs to cs, the seller’s per-unit cost of supplying an item to the buyer, is tabulated in Table 6.12. Note that, as expected, the seller’s cost per unit is uniformly decreasing as lot size (n) is increased, and it is always less than the case n = 1, which corresponds to the standard FRW. The Weibull distribution employed in previous examples is not easily analyzed here, there being no closed-form expression for convolutions, and is omitted. Note: Another realization of the cumulative FRW, exactly paralleling the standard FRW, would be to replace the entire batch of items if < nW. In this case, the renewal function in the cost model would be the ordinary renewal function associated with F(,l)(-). This is not a policy likely to have much appeal to the seller, however, since it would entail a large potential loss (though with very small probability, if the proper choice of W is made).

Table 6.12 Factors, E[CS(W, n)]/cs, for Calculating the Seller’s Expected Cost of Cumulative FRW, Exponential Distribution, W = 1 Lot size (n) 0.80 1.00 1.25 2.00 2.50

1

2

3

4

5

2.250 2.000 1.800 1.500 1.400

1.791 1.568 1.401 1.184 1.125

1.628 1.416 1.267 1.093 1.055

1.543 1.337 1.198 1.055 1.028

1.490 1.287 1.156 1.034 1.015

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Per-Unit Cost to the Buyer

The initial cost to the buyer is simply ncb, where cb here is the price per unit when sold in lots of size n under cumulative free-replacement warranty (which would be expected to be less than the corresponding selling price under standard FRW). The unit cost to the buyer can be calculated as ncb divided by the number of items supplied, namely n + MCF(nW). For the exponential distributions and choices of n given in Example 6.10, the buyer’s cost per unit can be calculated as the ratio of cb to the values tabulated in Table 6.12. The Buyer's Life Cycle Cost

Life cycle costs involve much more complicated renewal functions. This is due to the fact that purchase instances occur when the last item in the batch fails, which may not be the item in service at time nW from purchase. This happens if the first k items, for some k < n, provide nW units of service time. It follows that the relevant excess random variable in this case is the residual life of the item in service at time nW after purchase only if Sn < nW (i.e., if N > 1). If Sn > nW (so no replacements are required under cumulative FRW), however, then the excess random variable of interest is the residual life of the item in service at time nW after purchase plus the lifetimes of any remaining items in the batch. Denote this excess random variable y*(nW). Then the purchase interval is Z = nW + y*(nW). The distribution of Z is obtained as follows: y*(fzW) is the time to the first purchase after nW. Let y(nW) be the time from nW until the first failure after nW. Then y is the usual excess random variable, i.e., the residual life of the single item in service at time nW from the initial batch purchase. i%(-), the distribution function of y, is given in Chapter 3. We next consider y*. Let J be the number of items from the original batch not yet used at time nW (0 < / < n). Note that if / = 0, then y* = y. If / = j > 0, however, then y* = y + S;, where Sj = Ui=1 X h with X l9 X 2, . . . , Xj being the lifetimes of the j items used from the original lot to replace items that fail after time nW from the initial purchase. Thus, we can write (6.50) and the distribution of Z can be determined by finding its conditional distribution given J = j and unconditioning. This requires the distribution of J. The possible values of J are 0 , The value J = n -

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Chapter 6

1 is equivalent to the event that the first item put into service has a lifetime in excess of nW. Thus, (6.51) Similarly, / = j (j = 1, . . . , n - 2) implies that the number of items used in [0, W) is exactly n - j. Thus, for these values, (6.52) where

is the A>fold convolution of F(-) with itself. The final term is (6.53)

By definition of y*, the conditional distributions of y* given J are (6.54) where F * G () is the convolution of F(-) with G(). We now obtain the unconditional distribution of y* as (6.55) Finally, since Z = nW + y*, Fz(*) is a simple translation of /%*(•). The result is (6.56) We now look at the buyer’s life cycle cost (LCC), say Cb(L; W , n), assuming that the buyer continues to purchase batches of n items over some extended period L. This process generates a sequence Z1? Z2, . . . of purchase intervals. It follows that the number of such purchases in [0, L) is MZ(L), where Mz(-) is the renewal function associated with Fz(*), the CDF of Z. This provides a formal solution to the problem of determining the buyer’s LCC, namely (6.57)

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Explicit expressions for A/z(-) have not been obtained for standard life distributions, and simulation is suggested as a means of evaluating the expected buyer’s cost in any specific application. The Se lle r's Long-Run Expected Profit

As usual, we deal with expected values and calculate profit as the difference between income (the buyer’s expected cost) and the seller’s expected total cost in [0, L). Income to the seller is realized at the outset and at each purchase interval thereafter, i.e., at times 0, Z1? Z x + Z2, . . . . The income in each instance is nc5, with the total income given by (6.57). Since items are used one at a time, the fact that they are purchased in batches is irrelevant, and the expected cost to the seller in [0, L) is simply c8[l + M(L)], i.e., the cost per item times the total expected number of items to be supplied over the life cycle. (Note that in practice, unless L is large, considerable variability may occur in actual costs for both buyer and seller because of end effects. Since each purchase is assumed to be a batch of size ft, a large cost increment would be involved if a purchase were necessary just prior to L.) Indifference Prices

In Chapter 4, an approach that was suggested for determining the value of a free-replacement warranty to the buyer was the calculation of an indifference price, i.e., a differential pricing structure such that a buyer would be indifferent between purchasing an item without warranty at cost cu or with warranty at cost c£ > cu. Similarly, an indifference price for the seller is a selling price c£* such that long-run profit would be the same whether items were sold with FRW at c£* or without warranty at cu. In the case of a cumulative FRW, three such comparisons may be made for both buyer and seller: no warranty versus cumulative FRW, standard FRW versus cumulative FRW, and no warranty versus standard FRW. Denote the buyer’s indifference prices for these three situations c£ us (= ci of Chapter 4), c£uc, and c£sc, respectively. (The fact that all of these are functions of W, L, and the parameters of the life distribution, and that the first two are also functions of ft, is suppressed in this notation.) The first of these indifference prices was given in Chapter 4 as (6.58) The price c£ uc at which in the long run a buyer is indifferent between an unwarrantied item at cost cu and a cumulative warranty at per-unit cost

254

Chapter 6

cumulative warranty > standard warranty, with the reverse inequalities holding for the seller’s costs. This imposes a similar relationship on the indifference prices. In principle, discounting could be introduced into the analysis of LCCs and indifference prices. Again, formal results along these lines could be obtained, but explicit closed-form expressions could not. Finally, we note that there are many other schemes possible for implementing a cumulative FRW. For example, n items may be purchased and used simultaneously. When the first failure in this group occurs, a replacement would presumably be required. At this point, however, it would not be known, under cumulative warranty, whether this item will be provided free or paid for by the buyer— in fact, this will not be known until the last item fails. In practice, it would be necessary for the buyer and seller to have negotiated an agreement regarding the supply of and payment for needed replacements at the time of the initial purchase. Policy 15 of Chapter 2 is an attempt to alleviate this difficulty by purchase of a lot of n items, of which a number k are put into immediate service and the remaining n - k items are held in reserve as spares for use in replacing failed items. What happens when these are exhausted (assuming

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that the total service time does not exceed nW, i.e., that the warranty is still in force) would still have to be worked out between the buyer and seller. Cost models for cumulative warranties of this type have not been investigated. Analysis of the Cumulative PRW

Here we consider purchases of batches of n items with the entire batch warranted for a total cumulative period of n W, but under pro-rata warranty terms. We again distinguish between a cumulative PRW and a PRW on a single item by calling the latter a standard PRW. The standard PRW was analyzed in detail in Chapter 5. We consider the rebate form of the cumulative PRW, given in Chapter 2 (with slight modification) as follows: Policy 16 CUMULATIVE PRW: A lot of N items are purchased at cost ncb and warranted for a total period nW. The n items may be used either individually or in batches. The total service time Sn is calculated after failure of the last item in the lot. If Sn < nW, the buyer is given a refund in the amount of cb(n — SJW ), where cb is the unit purchase price of the item. As in analysis of the standard PRW, an important element of the analysis here is the rebate function, q{-\ W, n). This is given in the preceding policy statement as (6.61)

Other forms of the rebate function could be considered as well— for example, cumulative versions of the standard PRW structures analyzed in Chapter 5. The analyses would parallel those of Chapter 5 with modifications similar to those to be given in what follows. In the remainder of this section, we analyze the buyer’s per-unit and life cycle costs and the seller’s per-unit and life cycle profit for the rebate form of the cumulative PRW. The Buyer's Per-Unit Cost

The expected cost per batch to the buyer is calculated as nch minus the expected rebate. Division by n will provide the per-unit expected cost. Calculation of the expected rebate requires the CDF of Sn, which we denote

256

Chapter 6

F(n\-), and related functions. The result is

(6.62) where |x ^ is the partial expectation with respect to F ^ * ) , given by (6.63) The actual buyer’s cost per unit for a given batch of n items is (6.64) From (6.62), it follows that the expected per-unit cost to the buyer is (6.65) Example 6.11 [Exponential Distribution] We again consider the exponential distribution with W = 1 and the same choices of X and n as in Example 6.10. The distribution of Sn is a gamma distribution with parameters n and X, i.e., the probability density function of Sn is

x > 0. The partial moments involve incomplete gamma functions, which are easily evaluated in this case since n is an integer. The result is

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257

Thus, for the exponential distribution,

Values of E[Ch(W , n)\/cb are given in Table 6.13. Note that, as expected, costs to the buyer are lowest for the standard PRW (n = 1) and increase as n increases. The Se lle r's Per-Unit Profit

Let cs be the seller’s per-unit cost of supplying an item in lots of size n. Since cost to the buyer is income to the seller, it follows from Equation (6.65) that the seller’s per-unit expected profit, ir(W, n), is simply ( 6 .66)

The quantity in brackets in (6.66) was calculated for the exponential distribution in Example 6.11 and tabulated in Table 6.13. The cumulative PRW may be evaluated and compared with the standard PRW (n = 1) in terms of expected profit by substituting these values along with the relevant costs into Equation (6.66). It is important to note that any conclusions drawn from such numerical results are valid only for the exponential distribution. As we have seen in previous chapters, in other cases the results can be quite different.

Table 6.13 Factors, E[Ch(W, n)]/cb, for Calculating the Buyer’s Expected Cost of Cumulative PRW, Exponential Distribution, W = 1 Lot size (/z) 0.80 1.00 1.25 2.00 2.50

1

2

3

4

5

.571 .632 .688 .787 .824

.652 .729 .796 .896 .927

.690 .766 .846 .940 .964

.713 .805 .877 .963 .980

.728 .825 .897 .975 .989

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Chapter 6

Life Cycle Costs

In analyzing life cycle costs, we assume that a replacement batch of size n is purchased upon failure of the last item in a batch, and that this process is repeated over a life cycle of length L. (Equivalently, we assume a renewing cumulative PRW.) As in Chapter 5, LCCs are analyzed by considering first the expected replacement cost to the buyer, R (•), conditional on the total service time, say SnA, of the first lot purchased. Under this warranty, the buyer receives a credit of ncb( 1 - SnA/nW) if 5„tl < nW, so his net purchase price on the next batch is ncb minus this amount. If S„,i> nW , any additional batches needed are purchased at full price. It follows that the required conditional expectation, denoted = jc)], is given by

(6.67) Unconditioning in this expression, we obtain

(6 .68) Applying renewal theoretic arguments as in Chapter 5, we find

(6.69) where M„(*) is the ordinary renewal function associated with F(n)(*). The total expected LCC to the buyer, including the initial purchase, is ncb 4- E[R{L)]. This expression can be used to obtain the seller’s life cycle profit as well. These functions cannot be evaluated for standard life distributions and simulation is again suggested. Formal solutions could also be obtained for discounted life cycle costs and for indifference prices. These are also analytically intractable. Finally, we note that if items are not used sequentially, replacements will be needed prior to failure of the last item in the batch. This poses the same administrative problems as discussed at the end of the last section in

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259

the context of the cumulative FRW. For the cumulative PRW, an attempt to deal with these problems is given in Policy 17. Many other approaches are possible. Policies such as these have not been analyzed. 6.3.3

G ua rante ed M ean Tim e to Fa ilure

In formulating cumulative warranties, an alternative to basing the warranty terms on the total accumulated service life of a batch of items is to consider instead the mean time to failure (MTTF) |x of the items. Warranties that guarantee the MTTF of a batch of items have been called fleet warranties (Berke and Zaino [4]). This notion is particularly applicable if the seller is a vendor of components used in the manufacture of a larger system and the buyer is the manufacturer of the final product. Here we consider the “fleet” version of the FRW. This would be appropriate, for example, for sales of a nonrepairable component to a manufacturer of a repairable system (i.e., a system that is at least repairable to the extent that this component is replaceable on failure). Operationally, terms of the fleet FRW require no action on the part of the seller if the true mean exceeds the guaranteed MTTF, which we denote jjl g . If the true mean is less than |xG, then compensation in the form of a number K of free replacements is provided by the seller, where K is a function of the difference between \lg and the actual mean |x. In practice, of course, the true mean will not be known and must be estimated from data ultimately obtained by the buyer as the items in the batch are put into use. This introduces a problem previously encountered in implementing the cumulative warranty concept, namely, the potentially extensive amount of time that may be required until the last item in the lot fails. An approach to dealing with this problem, more appropriate for guaranteed MTTF than for the cumulative FRW of the previous section, is to censor the data, that is, specify at the outset an observation period of length T, say, and estimate the average time to failure on the basis of data collected during the observation period. There are a number of ways of dealing with censored data; Berke and Zaino [4] suggest that the estimate of |x be based only on the lifetimes of those items actually failing during the observation period. A potentially more significant problem in either case (complete or censored data) is that the result is an estimate of |x and, as such, is subject to sampling variability (the n items being a sample). This must be taken into account as well when calculating expected costs. The approach proposed by Berke and Zaino [4,7] is as follows: We assume that a batch of n items is purchased at cost to the buyer of cb per item. All n items are put into service and replaced upon failure. We

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assume that the process has been in continuous operation for a time t, with t sufficiently large so that the process may be considered to be in equilibrium. An observation window of length T is selected and the lifetimes of all items that fail in the interval [t, t + T] are recorded. (Note that this requires that the start times of items in service at time t be determined.) The compensation to the buyer for this period, if any, is determined as a function of |1, the estimated MTTF, calculated as the average lifetime of all items that fail in the interval [t, t + T]. In principle, this compensation may be realized in any number of ways, for example, a cash rebate, a discount on future purchases (a kind of PRW), or by provision of a number of free replacements (a form of FRW). Berke and Zaino choose the last form and calculate the number of replacements, K = K(T), as

(6.70) where N(T) is the number of failures in [t, t + T]. As already noted, the estimator p used in calculating K(T) is (6.71) where N t = N^T) is the number of times the ¿th item is replaced in [t, t + T] and X i} is the lifetime of the yth item in service in the sequence beginning with item i. (Thus, if we number the items in service at time t from the start of the process from 1 to n, then X n is the lifetime of the first such item, given that it fails before time t + T, X 12 is the lifetime of its replacement, if any, X 35 is the lifetime of the fourth replacement of the third item in the original ordered set, and so forth.) Note that N( T) = 2 ?«i N{ T) .

Although the estimator of Equation (6.71) may not be optimal, it has some reasonable properties. Berke and Zaino [7] cite some empirical evidence that it performs well for data from Weibull distributions. They also show that the estimator is consistent and nearly unbiased. (See Chapter 12.) We assume in this analysis that the X tj are independent and identically distributed with finite third moment P3 = E{X3i}), mean p and variance a 2. Using the estimator of Equation (6.71), we may now rewrite (6.70) as (6.72)

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where (6.73) Note that Q is a random sum of random variables. It can be shown, using a renewal theoretic argument (Berke and Zaino [4]), that Q is asymptotically (as t and T —» °o) normal and has mean and variance given approximately (for large T) by (6.74) and

(6.75) It follows that the distribution of K(T), given that K(T) > 0 (i.e., that Q > 0), may be approximated by a truncated normal distribution. From this and the formula for the mean of a truncated normal (see Johnson and Kotz [8]), we now obtain (6.76) which may be approximated by use of the asymptotic results of (6.74) and (6.75). The approximate expected total cost to the buyer, E[Ch(T\ n , |xG)], and expected profit to the seller, t t (T; n , |xG), are now easily determined as functions of E[K(T)]. The results are (6.77) and (6.78)

Chapter 6

262

where, as before, cs is the unit cost to the seller of supplying items in batches of size n. Example 6.12 [Exponential Distribution] We again consider the exponential distribution with |x = 0.8, 1.0, 1.25, 2.0, and 2.5. Suppose that |xG = 1.0 and T = 15. For the exponential distribution, cr2 = U \2 = |x2 and 1x3 = 6 (jl 3, and the asymptotic results of (6.74) and (6.75) reduce to quite simple expressions. Values of E[K(T)], calculated using the approximation of (6.69), are given in Table 6.14 for n = 1, . . . , 5, 10, 20, 50, and 100. Substitution of these values along with values of the seller’s unit cost cs and selling price cb into (6.77) and (6.78) will yield approximate average buyer’s cost and seller’s profit per batch for items sold under cumulative (fleet) warranty with free-replacement terms. The tabulated results show the patterns one would expect. For values of |x considerably in excess of the guaranteed MTTF, very few free replacements will be required. For values of |x at or below |xG, the number of free replacements required may be significant and increases with the number of items in the lot. Note that n = 1 in this illustration is not truly a fleet warranty, but it is still different from the standard FRW in that the number of replacements here is based on the mean life of the items rather than on the number of failures. Thus, in comparing these results with those of the standard or

Table 6.14 Expected Number of Free Replacements, E[K(T)\, Under Fleet Warranty with |xG = 1, T = 15, and Selected Values of True MTTF and Batch Sizes n Mean time to failure

n

0.80

1.00

1.25

2.00

2.50

1 2 3 4 5 10 20 50 100

4.202 7.818 11.462 15.140 18.812 37.511 75.004 187.500 375.000

1.545 2.185 2.626 3.090 3.454 4.885 6.909 10.924 15.449

.395 .286 .196 .134 .091 .013 0 0 0

.016 .001 0 0 0 0 0 0 0

.009 0 0 0 0 0 0 0 0

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cumulative FRW (where the expected number of free replacements required would be calculated by use of renewal functions), quite different results might be expected. Much remains to be done in application and analysis of this as well as the other cumulative warranties discussed in this chapter and the references cited. NOTES

Section 6.1 1. The first analysis of a combination warranty is due to Heschel [9]. A first-failure model was used to provide a first approximation to determining the cost of a combination FRW/PRW assuming exponential lifetimes. Section 6.2 1. The two renewing versions of the combination FRW/PRW are due to Nguyen and Murthy [1]. The policies are discussed in some detail in the article and cost analyses from the buyer’s and seller’s points of view are given. The validity of the approximation given in Equation (6.22) is proven in their Lemma 3. 2. The alternate bounds and approximations, including that of Equation (6.23) , were suggested by Frees and Nam [10]. Their analysis includes a comparison of several results of this type as well as additional numerical results for the exponential and gamma distributions. 3. Some additional results and further discussion of combination warranties are given in Biedenweg [11], Thomas [12], and Ritchken [13]. Biedenweg’s analysis of total life cycle costs is somewhat more complex than the models of this section. Thomas deals with optimization problems in the case of nonrepayable items. Ritchken corrects an error in Thomas’s formula and derives the average per-item cost for the fully renewing combination warranty. 4. Patankar and Mitra [14] considered the effect of unclaimed warranty benefits on the seller’s cost for combination warranties. Section 6.3 1. The first attempt at structuring and analysis of cumulative FRW and PRW policies is due to Guin [5]. A number of policies, including Policies 14-18 of Chapter 2 and several others, were developed, and a few were analyzed by means of renewal functions and a limited number of simulation runs. Policies having a fixed cutoff point (either in time or number of failures), after which a warranty settlement is made, are also discussed. A great deal remains to be done in defining

264

2.

Chapter 6

warranty terms and especially in modeling and analysis of cumulative warranties if such policies are to be used in practice. There are many unresolved problems in the guaranteed MTTF approach. These include conceptual, analytical, and statistical areas. For example, the notion is applicable in most of the situations dealt with by Guin (see the previous note), and similar questions with regard to implementation arise. Furthermore, alternative characteristics to the mean, for example, the median or some other fractile, may be appropriate in some applications. Many extensions of the modeling and analysis efforts begun by Berke and Zaino [4] are needed. These will involve some unique applications of renewal theory. Some difficult statistical problems, in data analysis and estimation theory, also require careful analysis.

EXERCISES

6.1.

Calculate the buyer’s and seller’s expected costs for the rebate combination lump-sum warranty of Policy 3c with W, and a, as in Example 6.1 for the following distributions with p = 2.0 and 2.5 in each case: Exponential distribution:

Gamma distribution:

Lognormal distribution:

For the gamma and lognormal distributions, determine the parameters so that a 2 = 1.0929 and 0.5891 when p = 2.0 and a 2 = 1.7077 and 0.9205 when p = 2.5. (These are the variances for the Weibull distributions of Example 6.1 when the Weibull p is 2 and 4, respectively.)

Complex One-Dimensional Warranties

6.2.

6.3.

6.4.

6.5.

6.6. 6.7. 6.8.

265

Calculate expected costs as in Example 6.1 for the Weibull distribution with |x = 2.0 and 2.5 for both p = .50 and .75. Repeat Exercise 6.1 for the gamma and lognormal distributions with a 2 = 20.0 and 7.321 when |x = 2.0 and a 2 = 31.25 and 11.441 when |x = 2.5 (equivalent to Weibull variances when p = .50 and .75, respectively). Extend the results to include the mixed exponential distribution, given by

with 0 < p < 1. Use the values p = .05 and .20 in the analysis. Compare the results of Exercise 6.1 and Example 6.1 with those of Exercise 6.2. Note that those in the first group are DFR distributions, except for the exponential, which has a constant failure rate, while those in the second group are IFR distributions. What do you conclude about the effect of this on costs? Derive expressions for the seller’s expected cost per unit when items are sold under Policy 3d when lifetimes have exponential, mixed exponential, and lognormal distributions. Calculate values for this cost using the parameter values of Exercises 6.1 and 6.2, as appropriate. (Note that the mixed exponential is always DFR, and hence the mean-variance combinations of 6.1 are not possible in this case.) Suppose that a warranty consists of a 100% rebate in the period up to time Wj from purchase of a product, a 50% rebate from time Wx to W2, with 0 < w 1 < W2 ^ W/2, and a prorated rebate from W2to W. Assume that the policy is nonrenewing. Give an expression for the rebate function q(-) for this warranty and derive the seller’s and buyer’s cost models. Determine explicit expressions for the buyer’s and seller’s costs of the warranty in Exercise 6.5 for the exponential, mixed exponential, and lognormal distributions. Calculate expected costs for buyer and seller for the policy of Exercise 6.5 using the distributions and parameter values of Exercises 6.1 and 6.2. Derive a model for the seller’s expected cost of the combination FRW/PRW assuming that the warranty renews only in the FRW portion of the warranty and is nonrenewing if a failure occurs in the PRW portion.

266

6.9.

Chapter 6

Determine the distribution of the excess random variable for the shifted exponential distribution with distribution function

Extend the results of Example 6.3 to this distribution. 6. 10. Determine an exact expression for F[S(20], given in Equation (6.27), for the lognormal distribution, and use the result to obtain an approximation of the expected cost to the buyer based on Equation (6.37). 6 . 11. Repeat Exercise 6.10 for the mixed exponential distribution. 6 . 12. Determine explicit expressions for the seller’s expected profit under Policy 7 for the distributions of the previous two exercises. 6.13. Select appropriate parameter values (e.g., those used in earlier exercises) and use the results of the previous three exercises to conduct a numerical study of the buyer’s cost and the seller’s profit under this warranty policy. 6.14. Define at least three additional cumulative warranty policies as generalizations of rebate lump-sum, FRW, and PRW policies. 6.15. Discuss the various possibilities with regard to renewing and partial renewing for the policies defined in Exercise 6.14. What would be involved in deriving cost models for these policies? 6.16. Discuss possible difficulties in implementation of the policies of the previous two exercises. How might the policies be revised to overcome these difficulties? 6.17. Show that if F(t) is the gamma distribution with parameters X and P, then F (n)(i) is a gamma distribution with parameters X and ftp. (Hint: Use moment generating functions.) 6.18. Use the result of Exercise 6.17 to calculate expected per-unit cost to the seller for batches of size n = 2, 3, 4, and 5 of items sold under cumulative warranty Policy 14, assuming gamma distributed lifetimes, with parameter values as in Exercises 6.1 and 6.2. 6.19. Calculate per-unit costs to the buyer under the assumptions of Exercise 6.18. 6 .20. Calculate the expected cost per unit to the buyer for the cumulative PRW of Policy 16, assuming gamma distributed lifetimes with |x = 2.0 and 2.5 and p-values as in Exercises 6.1 and 6.2. 6 . 21. Calculate the seller’s per-unit expected profit for the policy and assumptions of Exercise 6.20. 6 . 22. Determine 1x3 = E (X 3) for the gamma, Weibull, and lognormal distributions, and use the results to calculate costs of the guaranteed MTTF warranty as in Example 6.12, using the parameter values of

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Exercises 6.1 and 6.2. Compare the results for the different distributions and parameters used in this exercise and Example 6.12. REFERENCES 1. 2.

3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14.

Nguyen, D. G., and Murthy, D. N. P. (1984). Cost analysis of warranty policies, Naval Research Logistics Q., 31, 525-543. Baxter, L. A., Scheuer, E. M., Blischke, W. R., and McConalogue, D. J. (1981). Renewal Tables: Tables o f Functions Arising in Renewal Theory, Tech. Rept., Decision Systems Dept., Univ. of Southern California, Los Angeles, CA. Trimble, R. F. (1974). Interim Reliability Improvement Warranty (RIW) Guidelines, HQ USAF Directorate of Procurement Policy Document, Dept, of the Air Force, Washington, D.C. Berke, T. M., and Zaino, N. A. (1991). Warranties: What are they? What do they really cost?, 1991 Proc. Ann. Reliab. and Maintain. Symp., 326-330. Guin, L. (1984). Cumulative Warranties: Conceptualization and Analysis, Doctoral Dissertation, Grad. School of Bus. Admin., Univ. of Southern Calif., Los Angeles, CA. Kapur, K. C ., and Lamberson, L. R. (1977). Reliability in Engineering Design, John Wiley and Sons, Inc., New York. Berke, T. M., and Zaino, N. A. (1993). Some renewal theory results with application to fleet warranties, submitted to Naval Res. Log. Q. Johnson, N. L., and Kotz, S. (1970). Continuous Univariate Distributions—7, John Wiley and Sons, Inc., New York. Heschel, M. S. (1971). How much is a guarantee worth?, Indust. Eng., 3, 14-15. Frees, E. W., and Nam, S. H. (1988). Approximating expected warranty costs, Management Science, 34, 1441-1449. Biedenweg, F. M. (1981). Warranty Policies: Consumer Value vs. Manufacturer Cost, Doctoral Dissertation, Dept, of Operations Research and Statistics, Stanford Univ., Stanford, CA. Thomas, M. U. (1983). Optimum warranty policies for nonrepayable items, IEEE Trans. Reliab., 32, 282-288. Ritchken, P. H. (1985). Warranty policies for non-repairable items under risk aversion, IEEE Trans. Reliab., 34, 147-150. Patankar, J. G ., and Mitra, A. (1989). “Effects of warranty execution under various rebate plans,” presented at TIMS XXIXth International Meeting, Osaka, Japan.

7

Reliability Improvement Warranties

7.1 CONCEPTS

The previous three chapters have dealt with many different versions of the basic free-replacement and pro-rata warranties and various combinations and extensions thereof. Although the emphasis in these chapters was, for the most part, on applications involving consumer products, all of these types of warranties are used as well in commercial and governmental purchases of such goods and also in the purchase of goods and equipment that are sold only to commercial and government organizations. The last type of warranty considered in Chapter 6, cumulative versions of the basic FRW and PRW, was considered to be appropriate only in commercial and government sales. The reliability improvement warranty, commonly known simply as RIW, is another class of warranty policies that are applicable only in these types of sales, where they are employed mainly in the context of expensive, repairable items. Examples of applications are commercial and military aircraft, radar units, ship gyroscopes, various types of ship and aircraft navigation systems, and many other complex military and nonmilitary equipment. The intent of an RIW is to provide an incentive to the seller to improve the reliability of the product, thereby reducing long-run repair and maintenance costs. Including product reliability in the warranty coverage considerably complicates the process, since it is first necessary to define this term and then, in practice, to estimate it, usually from data obtained under operational conditions. The measure usually used to assess product reliability and improvements therein is the mean time between failures (MTBF), and most RIWs include an MTBF guarantee. (In fact, some authors define RIW strictly in terms of MTBF. We use the term to include improvement in any agreed-upon measure of reliability.) 269

270

Chapter 7

The warranties on the complex systems covered under RIW are often quite complex as well. As opposed to a consumer-type warranty, where very large numbers of items with identical warranties are sold, the RIW will typically be a unique warranty contract, covering far fewer items and carefully negotiated by the seller and buyer. Furthermore, it follows from the comments of the previous paragraph that the characteristics on which the warranty terms are based are usually quite different under RIW— for example, MTBF or mean time to repair, rather than (or in addition to) calendar time or usage. As a result, the models and analysis of the previous chapters, which are based primarily on multiple realizations of the same process and long-run average outcomes, are not appropriate. In fact, cost models for the RIW have developed along quite different lines. In this chapter, we present a brief history of the reliability improvement warranty, a more detailed look at some of the specific features of “typical” RIWs, and a discussion of a few of the cost models that have been proposed for analysis of some types of RIW. We conclude with some notes providing a few additional observations on RIW and the problems of determining warranty terms as well as costs, and we give references to many additional sources of further information on RIW. Throughout the chapter, the emphasis will be on applications in military procurement in the United States, since this is the area in which the concept has found the most widespread use, and to which most of the published literature is devoted. Some additional material on RIW applications and many additional references are given in Chapter 13. 7.2 THE HISTORY OF RIW In this section we provide a brief outline of the development of the RIW concept and its use, particularly in military acquisition. Since its inception, many concerns about its use have been expressed. The key issues among these that are related to the subject of this book are the following: 1. The need for assessing life cycle costs 2. The need for adequate and realistic cost models 3. The problem of obtaining valid and reliable data, in sufficient quantities, for estimating these models The objective of our short history is to provide some insight into the nature of these problems. Additional details regarding the early history of RIW can be found in Proceedings of the Failure Free Warranty Seminar [1], Schmidt [2], Shmoldas [3], and Gándara and Rich [4]. Some details on more recent experience with RIW use may be found in Guglielmoni

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[5], Yuspeh [6], Dhillon [7], and many of the recent references listed at the end of this chapter. 7.2.1

Evolution of the RIW Concept

Although RIW is now widely used in military applications, the first use of this type of warranty appears to have been in purchases of aircraft by commercial airlines. The successful use of an RIW by Pan American World Airways in the purchase of Boeing 747s in the late 1960s is discussed by Hiller [8] and Shmoldas [3]. Commercial use of RIW to date (as far as one can tell from published literature) has apparently been limited to applications of this type. In military procurement, in the United States, the RIW was initially called a “failure free” or “standard” warranty (Trimble [9]). Some versions of this warranty had been introduced at about the same time as the inception of the airline RIW (Gregory [10], Klaus [11]). The failure-free warranty was apparently initially a basic FRW, without the explicit notion of reliability improvement. It was first offered by Lear Siegler (though not accepted by the government) in 1964 on gyroscopes that were bid with a 5-year, 5000-operating-hour warranty. The seller warranted the gyros to be failure free, with failed items being replaced or repaired, at the seller’s option (Schmidt [2]). The first actual use of an RIW, in 1967, also in a Lear Seigler contract, was in procurement of a gyroscope for the F -lll aircraft (Schmidt [2], Markowitz [12], Gándara and Rich [4]). This warranty did include a guaranteed MTBF provision, and it was apparently quite successful. Schmidt [2] notes that a target MTBF of 400 hours was exceeded— the estimated actual MTBF was 531 hours— and that a 40% reduction in maintenance costs per operational hour was achieved. The RIW came into wider use during the next several years (see Gándara and Rich [4]), with a guaranteed MTBF as the predominant feature. At about this time, the notion of a cumulative warranty was also introduced, e.g., on Air Force F-16 components (Gándara and Rich [4]). (Cumulative warranties are not a principal feature of the RIW generally and will not be included in our cost models or discussion here. They are discussed in Chapter 6.) Following its introduction during the period from 1969 to the mid-1970s, the RIW received increasingly wider use. This was primarily the result of ever increasing pressure from Congress, culminating, in 1983 through 1985, with the passage of some key legislation, particularly Section 794 of the FY 1984 Department of Defense Appropriation Act, which required that all weapons systems and components be covered by warranty in future procurements. The concern was not only with improving reliability and

272

Chapter 7

maintainability, but in assuring that the government did not continue to carry the entire risk for errors in design and manufacture of weapons and other expensive systems. Section 794 specifically required that contractors for weapons systems (1) guarantee that the design and manufacture of each component and the system itself conform to the contract performance requirements; and (2) guarantee that the systems and each component be free of all defects in materials and workmanship that would cause the system to fail to meet these requirements. In the event of any system failure, the contractor was to either repair or replace the system or any parts necessary to achieve the required performance at no cost to the government, or, if the contractor failed in this, to pay whatever cost the government incurred in obtaining the necessary repairs (Yuspeh [6]). Section 794 was quickly implemented by the Department of Defense; guidelines were given in an Action Memorandum by the Deputy Undersecretary of Defense for Acquisition Management (Gilleece [13]). The RIW is now firmly established as a factor in military acquisition. As it continues in use, its evolution will undoubtedly also continue. Continued efforts to address the cost modeling and data needs will be required as well.

7.2.2

Product Perform ance A greem ents

A broader term that has recently been used to describe any warranty or guarantee that includes an incentive to achieve or improve product performance or reliability is product performance agreement— PPA (Fleig [14]). This class of warranties includes many types of contracts, RIW being just one warranty in one of four classes: 1. 2. 3. 4.

Federal acquisition regulation agreements—covering inspection, supplies, design, performance specification, and technical data Contractor repair agreements—covering rewarranty of repaired or overhauled equipment, reliability guarantees, repair/exchange agreements, and RIW Field measurements agreements—covering MTBF verification tests, availability guarantees, RIW with MTBF guarantee, logistics and other cost guarantees, mean time to repair, and similar guarantees Special features agreements—covering guarantees on characteristics such as ultimate life, commercial service life, software, test and repair improvement, and other unusual or unique features

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It is asserted (Fleig [14]) that, overall, “PPA’s deliver higher reliability, reliability growth, lower net life-cycle costs, and improved availability.” 7.3

TYPICAL RIW FEATURES

The original intent was to apply RIW in situations where operational (field) reliability, support costs, and potential reliability growth were all reasonably predictable. In such applications, many versions of RIW, with a wide variety of terms and features, were developed. Characteristics common to almost all applications include the following: 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

Complex equipment A specified performance requirement Ability to evaluate field or operational reliability Potential for reliability growth Supplier provision of field maintenance and repair Relatively long warranty terms (typically three to five years) Requirement for engineering analysis of failures Requirement for design changes to correct defects A fixed price contract Contractor fees based on demonstrated reliability improvements

Each bidding situation involving RIW will ordinarily include many additional unique characteristics as well. Two important features that are used as further incentives to the contractor to improve reliability are the following: 1. The MTBF guarantee, with agreed upon criteria for determining whether or not the minimum required MTBF has been achieved 2. Guaranteed turnaround time (TAT), either on average or on an individual basis, depending on the application For both guaranteed MTBF and guaranteed TAT, RIW contracts commonly require either that spares be consigned (at contractor’s expense) for use as replacements if the guarantee is not met or that monetary damages be paid by the contractor (Newman and Nesbitt [15]). Many additional RIW provisions and the challenges they present to contractors are discussed by Bonner [16]. The types of equipment for which RIW (as opposed to the many other types of warranties in use in the military) is appropriate include items that 1. 2.

Are self-contained Are easily transportable or in installations that can readily be serviced by the contractor

274

3. 4. 5. 6. 7. 8. 9.

Chapter 7

Are not prone to failures induced by other components or sources Have well-defined failure modes Can be ordered in sufficient quantity to make RIW cost effective Are sealed or otherwise protected from unauthorized access Can be expected to have high utilization Can be monitored by means of an attached usage indicator Are such that failure and operational data can be furnished to the contractor on a regular basis.

These conditions, of course, are appropriate in the context of nonmilitary applications as well, and not all of them need apply in every case. The constraints are not overly restrictive, and many items routinely purchased in commercial and government transactions, as well as many custommade items, would qualify. 7.4 C O ST M O DELS FOR RIW

Cost models for RIW reflect to some extent the wide diversity in terms of RIW contracts and the many cost elements that may be identified in the complex situations in which such warranties are employed. The models that have been developed, in fact, are often oriented to specific applications. In the remainder of this chapter, we look at a selection of these models. 7.4.1

The Life C yc le C o st C o nce p t in RIW

Attempts to model the costs associated with reliability improvement warranties have focused on two important aspects, identification and assessment of important cost elements, and determination of life cycle costs. In this context, we note that the life cycle in question is almost always the useful life of a particular piece of equipment, rather than the total cost of an original item and its replacements over an extended period of time. Thus, we deal with the life cycle cost concept denoted LCC1 in previous chapters. The importance of dealing with life cycle costs becomes apparent when the cost of designing for reliability and producing reliable equipment is considered. Many reasons for unreliability can be identified. When dealing with complex equipment, the context in which RIW was introduced, these include the following: 1. 2. 3.

State-of-the-art capability requirements Testing in environments that do not represent operational conditions Lack of other incentives for reliability

The last comes about because reliability is often expensive to achieve; to maximize profits, contractors aim for the lowest acceptable reliability. (In addition, the future spare parts market may be very profitable.)

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These incentives for unreliability are mitigated when one looks at LCC, because, in the long run, the cost of achieving reliability is offset by the reduced cost of maintaining the system. The result of this trade-off is shown graphically in Figure 7.1, which shows total cost to the seller, cs, as a function of reliability. This total cost is expressed as the sum of initial cost, which includes the cost of reliability, and support costs for the system (and, implicitly, the cost of reliability improvement). As a result of these cost relationships, the contractor is given an incentive to find the combination of initial and support costs that will minimize his total cost and hence maximize his profit. Ideally, contract terms should be such that this optimal value (of MTBF or any other appropriate reliability measure) is specified as the reliability goal. The preceding analysis assumes that the selling price cb, including profit, remains constant, as shown in Figure 7.1, i.e., in government parlance, that the contract is a firm fixed price contract. This is, in fact, one of the common (though not universal) conditions for RIW procurements.

Figure 7.1 Contractor’s cost versus reliability.

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Because of considerations such as these, estimated or predicted life cycle costs are commonly used in negotiating RIW contracts and in evaluating a contractor’s performance. The usual approach is to compare LCC with and without warranty. 7.4.2

Life C yc le C ost M odels fo r RIW

As noted, the key considerations in determining the cost of items purchased under RIW are identification and estimation or prediction of the various cost elements. We look at several of the many such models that have appeared in the literature. For additional discussion of models of this type, see Dhillon [7]. Typically, the models are derived by looking at total cost to the buyer, including all cost elements that the contractor (i.e., seller) would expect to incur and would therefore include in pricing the item. Thus, the buyer's cost model will usually include specific seller’s cost elements plus (usually exlicitly stated) profit factors. As a result, the seller’s total cost and profit can usually be deduced from the model expressed from the buyer’s point of view, and we will not explicitly state these separately. Model 1 [Basic RIW Cost Model] The first detailed analysis comparing life cycle costs of warrantied and unwarrantied items (Balaban and Reterer [17-19]) involved the purchase of n items of electronic equipment, of which n0 < n were to be used simultaneously, with the remaining items being retained as spares. The warranty period is W, and it is assumed that the items may be in service for a period L > W; L is taken to be the life cycle of the item. We use Cb(W) to denote the (random) total cost to the buyer in the interval [0, W] of an item purchased under RIW at time 0; CU(W) is the cost of an unwarrantied item over the same time period. The basic LCC models express the expected total cost to the buyer as the total of original purchase price plus the expected cost of failures plus the cost of maintenance support. The latter two cost factors may be broken down into many individual cost elements. The following models are based on the Balaban and Reterer approach. In these models, certain costs are amortized (linearly, when specified) over the entire lifetime of the items. The amortization factor is A = WIL. The cost of an unwarrantied item is modeled as (7.1) where cu is the purchase price per unit of an unwarrantied item, cx is the initial support cost for the batch of n items (which includes the cost of test

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equipment, handbooks and other materials, training costs, etc.), cR(W) is the total cost to the buyer of reliability modifications in the period [0, W], cs is the average recurring support cost per unit of time (including administrative costs, cost of retraining, etc.), and cM(W) is the total user maintenance cost in the period [0, W]. Note that this model may be modified in an obvious way if something other than straight-line amortization is desired. Discounting to present value may easily be introduced into the model as well. In calculating the cost of warrantied items, it is assumed that the seller will provide repairs and reliability modifications and that when the contractor’s bid is prepared, these costs will be included and profit will be calculated as a function of these as well as the cost of supplying the items. It is also assumed that the cost cb of a warrantied item may be higher than that of an unwarrantied item, even though the additional costs just mentioned are included separately. This is due to the fact that the seller may spend more on development in order to provide a more reliable item initially and thereby reduce the possibility of future more costly repairs and modifications. This provides the incentive for reliability improvement. (Note that inclusion of these elements in the original bid is appropriate for fixed-price contracts, which is one of the conditions for RIW application listed in the previous section.) We now calculate the expected cost to the buyer under RIW as

(7.2) where n1 is the number of items purchased under warranty, cRx(W) is the contractor cost for reliability modifications over the period [0, W] (discounted and amortized), cd(W) is the total contractor’s expected direct cost of repair, r is the contractor’s risk factor per year, a(W) is the warranty period expressed in years, P is the contractor’s profit factor (expressed as a proportion), and Cj1, cs\ and (^ (W ) are the values of cx, cs, and cM(W) under warranty. Note that n1, the number of items purchased under warranty, may be different from the number without warranty because of warranty services supplied by the contractor. The same is true of support and maintenance costs. The cost elements in these models may be further broken down into many additional cost factors. In fact, this is necessary in determining the total LCC of any complex system. To pursue the analysis further, it is necessary to specify a failure distribution F(-) for the items. Balaban and Reterer [17,19] take this to be an exponential distribution with parameter

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X. Since the failure rate X is generally not known, a prior distribution on X may be specified as well; Balaban and Reterer use a discrete distribution on a set of possible values X2, X2, . . . , for this purpose. The total expected cost of item failures may then be calculated as n0khWcf, where n0 is the number of items in operation at any given time, X is the average failure rate, h is the average number of operating hours per unit of time, and cf is the user’s direct cost of failure. They also develop a heuristic model for expressing the cost of reliability modification (i.e., the cost of reducing the failure rate from X to AfX, with M < 1). A shifted exponential is used to model the time required to implement a modification. This ultimately provides a functional form for cR( ). In the end, many cost elements still must be estimated, and management decisions for both buyer and seller require that this be done very carefully. Methodologies used for cost estimation of this type are beyond the scope of this book. Given appropriate estimates of the costs in models (7.1) and (7.2), one can estimate the value of the warranty by calculating an indifference price as was done for some basic consumer warranties in Chapters 4 through 6. This is done by equating the right-hand sides of Equations (7.1) and (7.2) and solving for ch as a function of cu. This yields the purchase price per unit for warrantied items at which the life cycle cost would be equivalent to that of unwarranted items purchased at a unit cost of cu. Example 7.1 [Balaban and Reterer] The following illustration is based loosely on the example of a purchase of magnetic drums discussed by Balaban and Reterer [17,19]: Electronic equipment is offered without warranty at $18,000 per unit. The buyer wishes to purchase 1000 units and assumes a life cycle of L = 5 years. The alternative is to purchase the units at cost cb each with a warranty of W = 5 years. (Thus, the amortization factor is W/L = 1.) We assume the number of operational hours per month to be h = 75, giving a total of 4500 operational hours per unit over the five-year life cycle. The expected number of failures per unit is therefore 4500/530 = 8.5. Assume that the total cost of reliability modification is estimated to be $3,500,000 in cost plus fees if the warranty coverage is not selected by the buyer and $2,500,000, exclusive of fees, if the items are purchased under RIW. We take the buyer’s initial and recurring support costs to be cx = $50,000 and cs = $500 per month without warranty and Cj1 = $10,000 and csx = $800 under RIW. Suppose that the buyer’s cost per failure for unwarranted items is estimated to be $870, comprising $650 in labor, $20 shipping, and $200 in materials, and that these costs for a warrantied item are $49 in user labor cost and $10 in user shipping cost. Finally, suppose that the supplier’s

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labor, shipping, and material costs are $480, $20, and $125, respectively, under RIW, and that his risk factor is r = .03 and profit is bid at P = .10. This gives expected total maintenance costs for the five-year life cycle/warranty period to be 8.5(870) = $7400 per unit for unwarranted items, for a total of cm (5) = $7,400,000 for 1000 units; cM2(5) = 8.5(59)(1000) = $501,500 for 1000 unwarranted units; and estimated direct cost to the seller of repairs under warranty of cd(5) = 1000(480 + 20 + 125)(8.5) = $5,312,500. The total expected cost for 1000 items purchased without warranty is found to be

Under warranty, the expected cost is

Equating the warranted and unwarranted costs, we find the indifference price to be c£ = $18,685. The buyer is better off in the long run purchasing the warranty if the price per unit of the warranted item is $18,685 or less; otherwise, the unwarrantied item will be less expensive in the long run. Now suppose that the buyer can reduce the average cost per failure from $870 to $750. We then have cM(5) = $6,395,000 and find the expected cost without warranty to be $27,955,000, giving cl = $17,660. Thus, the price that the buyer should be willing to pay for a warranted item is less than that of an unwarrantied item. The reason for this seemingly strange result, of course, is that all of the cost of the warranty is being passed along to the buyer along with a fairly hefty fee. Note that the key difference in this example is not the reliability improvement (in fact, we assumed the same average failure rate in both cases), but the fact that the seller assumes the responsibility for maintenance when the item is purchased with warranty. A more detailed and complex analysis is required to include both the improvement in reliability through time and

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the possibility that this improvement may depend on whether or not the RIW is in effect. Model 2 [RIW with Spares Shipment Guarantee] Gates et al. [20] discuss related cost models with additional features. We consider first the basic RIW with spares guarantee. The objective of this warranty is to reduce the user risk of low equipment availability. The contractor agrees to ship a spare within a specified time after each failure. A storage depot is set up for this purpose, at which nd items are initially placed, with nd determined as a function of the predicted MTBF and average turnaround for replacing failed items. The contractor is assessed damages for stockouts at the depot (i.e., inability to replace failed items on a timely basis). The expected cost to the buyer for this RIW is given by (7.3) where P' is a fixed fee (profit to the contractor) in dollars, cw is the expected cost to the contractor associated with the warranty, U is the average utilization rate of the items per unit of time, |x is the anticipated MTBF, Z)S(W) denotes the damages assessed the contractor for stockouts during the warranty period, and the remaining symbols are as previously defined. In addition to the term for damages, model (7.3) differs from (7.2) primarily in that the contractor’s fee is fixed rather than being expressed as a proportion of certain costs and cw includes several items expressed explicitly in (7.2). It also differs in that W = L is assumed and usage rates (necessary for determining costs in applications— see Example 7.1) are explicitly included here. Model 3 [RIW with Guaranteed Turnaround Time] In Model 2, damages for nonavailability of equipment were based on stockouts, the occurrence of which may dramatically affect the turnaround time (TAT) for replacement of failed items. Since TAT itself is the critical factor affecting availability, an alternative is to model this explicitly. Gates et al. [21], suggest a simple revision of (7.3), namely (7.4) where Dt(W) denotes expected damages to be assessed in the event of failure to meet the guaranteed TAT, ig, and the remaining symbols are as for Model 2. It is suggested in [21] that Dt(W) be determined as a linear

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function of the difference between tg and the experienced average TAT (given that this exceeds tg). Model 4 [Warranty Price Model] We now discuss a model, due to Balaban [22] and Balaban and Meth [23], that provides an analysis of the profit factor P in Model 1. The basic idea is to express the price of the warranty itself as the sum of a fixed cost cf plus the cost of warranty service, all multiplied by the profit factor, which is expressed as a function of the assumed failure rate XB at which the contract is bid and a number of other cost elements. The cost of warranty service is approximated by the product of expected number of repairs and the average cost per repair. The resulting basic “Warranty Price Model,” expresses the expected warranty cost cw as (7.5) where UB is the total utilization (assumed for bidding purposes) over the warranty period, cr is the average cost of a repair, and P(XB) is the contractor’s profit if the contract is bid at failure rate XB. In analyzing the contractor’s profit, it is assumed that the average failure rate is a random variable A with specified prior distribution GA(*). The seller’s expected profit is modeled as a function of cs(X), the total warranty cost for failure rate X, and PA(X), the probability of award if the contract is bid at X. The result is

(7.6) where P is a profit factor applied to all bids, cL is a cost factor associated with losing an award (e.g., employee termination costs, cost of unused inventory and facilities, and so forth), and cB is the cost of preparing the warranty bid. This result is a useful conceptual model for analysis of profit. The difficulty in practice is determining the functions involved in the analysis (e.g., cs(-), PA(-), Ga(-)). Another problem is the use in the model of an average failure rate. More realistically, one should model the change in X through the warranty period. Some related results along these lines will be given in the section on MTBF guarantees to follow. Model 5 [Cost of Reliability Modification] Reliability modification requires both an engineering design change and implementation of the required change. We assume that a design modi-

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fication to fix some identified failure mode, and thereby increase system reliability, has been determined. We are concerned with economic incentives for the contractor to develop and implement this reliability modification. An approach developed by Chelson [24] is based on the fact that a reduction in the number of required repairs can result in increased profits. Thus, we wish to compare the total ultimate cost to the contractor with and without the reliability modification. These are functions of the average repair cost per unit without the reliability modification, denoted cri, and this average after the modification is made, denoted cr2. In developing models for cri and cr2, it is assumed that units fail only when operating and then at a constant rate X (i.e., that the time to failure is exponentially distributed), that all random variables are independent, that units are repaired at a constant cost cx before modification, at constant cost c2 after modification, and modified at constant cost cm, and that repaired units are installed in a sufficiently short time after repair so that installation may be considered to be instantaneous. As usual, we take W to be the length of the warranty period. U is taken to be the average utilization rate during the warranty period. Further, we take k l9 X2 to be the failure rates of the identified failure mode before and after modification and \ 0 to be the system failure rate before modification. The expected repair cost without the reliability modification is given by (7.7) where cH is a handling cost per failure and tm is the time at which the reliability modification is to be implemented (if it were to be implemented). With reliability modification, the expected cost of repair is

(7.8) where (7.9)

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and cK is the cost of modification kits to correct units that had not been repaired and modified during the remainder of the warranty period. The rationale for cT2 is as follows: The first term is the cost of repair plus the cost of reliability modification for a returned system, times the probability that it is returned at some time between tm and the end of the warranty period. The second term accounts for repair costs on subsequent returns. The third term is the expected handling cost, and the final term is the cost of supplying modification kits for units that are never returned during the remainder of the warranty times the probability of that occurring. To complete the analysis, we assume that the procurement is for n units and that all have been delivered prior to time tm from the initial order time. (If this is not the case, a delivery rate function can be included to account for this. See Chelson [24].) The total expected cost of future repairs is then simply ncn without the reliability modification, and ncX2 + cf with the modification, where c{ is a fixed nonrecurring cost associated with the reliability modification. Finally, let Cs be the cost savings for implementation of the reliability modification. We have (7.10) Example 7.2 (Chelson [24]) Suppose that we have delivered 250 units of a system having an MTBF of 1000 hours, with 50% of the failures being due to an identified failure mode and that we wish to determine the expected cost savings given that the necessary modification to “fix” this mode is implemented. Assume that W = 60 months, tm = 12 months, cf = $25,000, cm = $150, cH = $200, cx = c2 = $300, cK = 0 (or, equivalently, that all units will have been returned for repair prior to the end of the warranty period, so that no modification kits will be required), and U = .1. Finally, suppose that \ 0 = .001, = .0005, and X2 = .00001 (these being rates per hour and hence multiplied by 720U = 12 to obtain rates per month). [Note: The presentation here has been somewhat modified from that given by Chelson. In particular, the assumption that all units are delivered at the outset has been added.] We now obtain

giving costs without and with reliability modification of

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and

The expected cost saving for the reliability modification is therefore

(See Chelson [24] for a more detailed treatment of this example, including various choices of input values and an analysis of the effect of these on the estimated cost saving.) 7.4.3

C ost M od els fo r RIW w ith G ua rante ed MTBF

The true effect of reliability improvement is found in warranties that specify a guaranteed MTBF. The next two models deal with certain aspects of RIW that include this feature. The basic notion of the MTBF guarantee is stated in Policy 19 of Chapter 2. The key features of policies of this type are as follows: (1) guaranteeing that the MTBF will not be below a specified value |xG; (2) determining required engineering changes so that the MTBF equals or exceeds the guaranteed value if it initially fails to do so; (3) retrofitting existing units to conform to the required engineering changes; and (4) consigning spares for the buyer’s use until such time as the MTBF guarantee is met. The intent is to negotiate an agreed-upon MTBF, a methodology for estimating the actual MTBF based on field data, and a course of action to be taken by the contractor should the MTBF guarantee not be met. The models to be given in what follows relate to a generalization of a version of the RIW stated in Chapter 2 as follows: Policy 20

RELIABILITY IMPROVEMENT WARRANTY: Under this policy, the manufacturer agrees to repair or provide replacements for any failed parts or units until time W after purchase. In addition, the manufacturer guarantees the MTBF of the purchased equipment to be as follows: no MTBF is guaranteed until time Wx after date of first production delivery; during the period from Wxto W2after first delivery, the MTBF is guaranteed to be at least Mx\ from W2 to W3 the MTBF is guaranteed to be at least M2\ and from W3to W the MTBF is guaranteed to be at least M3 (with 0 < Wx < W2 < W3 < W and 0 < Mx < M2 < M3). If

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during any period the MTBF guarantee is not met, the manufacturer will provide, at no cost to the buyer, engineering changes and product modifications as necessary to achieve the MTBF requirements.

Balaban [25] discusses RIWs of this type (e.g., with Wl = 12 months, W2 = 24 months, W3 = 36 months, and W = 48 months) in a military procurement. Modeling and analysis of RIWs of this complexity have not been done, as far as we know. Specification of both the warranty terms and the probabilistic structure necessary for such an analysis in sufficient generality have not been accomplished. A few results toward this end, however, are available. We state two of them. The first is a model for estimating the number of consignment spares required for satisfying an MTBF guarantee. The second is a warranty pricing model with MTBF guarantee. Model 6 [Estimation of Required Number of Spares] This model is not specifically a cost model, but a model for estimating a significant cost determinant, the number of spares needed as replacements to service the warranty. In fact, at least three versions of the model have been proposed. Balaban [25] suggests that spares (“loaners”) be provided at the seller’s expense and consigned for use by the buyer in the event that evidence becomes available that the true MTBF is less than |xG, the guaranteed MTBF. The number of additional consignment spares, Nc, required at any given time is calculated as (7.11) where £ is the estimated MTBF (estimated by an agreed upon method) at the time of the detemination and As is the number of spares required prior to that time. This approach to determining spares requirements is attributed to the airlines. Gates et al. [20] use instead (7.12) where Nt is a target number of spares (determined on the basis of the anticipated MTBF) purchased initially by the buyer. Spares remaining at the end of the contract period become the property of the buyer. These warranties entail considerable risk on the part of both buyer and seller. For example (Gates et al. [20]), if Nt = .2n, which is typically roughly the case, and (1 = .67|xG, then the cost to the seller is .1ncs, where

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cs is the seller’s production cost per unit, or 10% of the entire production cost, which, under a fixed-price contract, would be paid for by the seller. On the other hand, the buyer has a considerable risk as well, e.g., that the reliability goal may not be achieved (in fact, the achieved |x in the example is one-third low) and that the estimated mean may not reflect the true MTBF. The intent of RIW is to negotiate a means of sharing these risks equitably. Another problem in applying (7.11) or (7.12) is determining an initial target level. Balaban [25] suggests a probabilistic approach, selecting Nt so that the probability that more than this amount will be required is a specified value p. The result is (7.13) where nYis the number of installed units, h is the number of operational hours in the time period required to obtain and install a replacement, |xG is the guaranteed MTBF, and zp is the p-fractile of the standard normal distribution. This formula is based on the normal approximation to the Poisson distribution, which is the distribution of number of failures in a time period if the time to failure is exponentially distributed. If at some point the estimated MTBF, |i, is calculated and it is found that (1 < (jl g , then the number of spares is increased to Nt(|i), so that the number of additional spares required is (7.14) (If the opposite inequality holds, so that Nc calculated from (7.14) is negative, spares may be returned to the contractor.) Example 7.3 (Balaban [25]) Suppose that n = 100, h = 60, and |xG = 500 hours. Take p = .95; then zp = 1.645. (This value may be found in any table of the standard normal distribution.) We obtain 17.7 so 18 spares, or a little under 20% of the original number purchased, are required. If the MTBF is estimated at some time during the warranty period and found to be |1 = 400, then the number of spares required becomes

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At(400) = 21.4, so 4 additional spares are required. Under typical RIW terms, these are furnished at the seller’s expense. If at a later date, (1 increases to 450, we would have At(450) = 19.3, so 2 fewer spares would be required. Model 7 [Cost of MTBF Guarantee] Gates et al. [20] consider a warranty pricing model similar to Models 2 and 4, but incorporating the notion of consignment spares as a cost factor as well. The model is (7.15) where cs is the cost per consignment spare, nc is the number of consignment spares required, and the remaining terms are as previously defined. The value of nc may be calculated as indicated in the various versions of Model 6. Costs of repair versus reliability modification are discussed in some detail by Gates et al. [21]. Reliability growth may be incorporated into the model in various ways. In the context of RIW, it is appropriate to assume that reliability improvements occur in increments, corresponding to “fixing” failure modes that are identified at distinct points in time. Gates et al. [21] incorporate reliability improvements into the model as follows: We assume an initial failure rate of X0, at time 0, and that improvements (reducing the failure rate) take place at times tu t2, , tk, with tx < t2 < * * * < tk < W. Each improvement is assumed to result in an incremental decrease in the previous failure rate, the decrease in the ith interval being AX,-, and to be introduced into the population of items in use at the rate at which items fail. (Thus, AXZis the failure rate associated with the ith failure mode identified.) As a result, the failure rate function X(i) is of the form

(7.16)

The impact of this on RIW costs and trade-offs between cost of improvement and cost of future repairs are discussed in Reference [21]. Clearly, the decision regarding the trade-off is quite time dependent— it would not make sense to implement an improvement, even at modest cost, if there

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were only a short time remaining in the warranty period. This is also discussed along with a number of other factors that influence the implementation decision. 7.4.4

A na lysis o f Policy 20a

From the seller’s point of view, a factor in pricing items sold with RIW is that the warranty periods are ordinarily of sufficiently long duration so that failures may be expected to occur even when reliability goals (e.g., MTBF guarantees, reliability growth, etc.) are met. It is essential to select a pricing level so that these anticipated costs will be recovered. This is the thrust of many of the models discussed previously in this chapter. Another approach (Kruvand [26]) is based on the concept that under these conditions an “allowable” number of failures should be paid for by the buyer, with the seller paying for failures only if this allowable number is exceeded. Policy 20a of Chapter 2 states this explicitly as follows: lot of N items is purchased with individual warranty periods W. Items that fail prior to W are repaired or replaced at the buyer’s expense until k such failures occur, after which the manufacturer will repair or replace failed items until each of the N items in the lot and its replacements achieves a total service time of W. P olicy 2 0 a: A

Here it is implicitly assumed that the warranty period W is considerably larger than the guaranteed MTBF . Thus, it makes sense for the buyer to pay for a certain number of replacements, since W7(jl g failures would be expected even if the MTBF guarantee were met. The seller then is responsible for the cost of any failures in excess of this (or any other agreed upon) amount. Model 8 [Per-Item Cost with Allowable Failures] In analyzing this warranty policy, we look at the costs per item (that is, the initial cost plus the cost of replacements) to the buyer and seller. Let N be the number of failures in the period [0, W], with N = Nh -b Ns, where Nb = Ab(W) and Ns = NS(W) are the numbers of replacements paid for by the buyer and seller during the warranty period, respectively. Suppose that cs is the average cost to the seller of supplying an item, c5 is the selling price of the item, and cr is the replacement cost (to either party). (cs, cb, and cr may include many individual cost elements such as those in the models discussed previously.) Expected costs to seller and buyer are then given by (7.17)

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and (7.18) To complete the analysis, we need the expected values of Ns and Nb, which are, in fact, partial expectations of N. Let X l9 X 2, . . .be the lifetimes of the initial item and its replacements. Suppose that the X t are independent and identically distributed with distribution F(-). Then E(N) = M(W) is the ordinary renewal function associated with F(-), and the usual renewal theoretic analysis gives

(7.19) where F (y)(-) is the /-fold convolution of F(-) with itself. In addition, since P(NS = 0) = P(Nb < k) = P(N < k), and P(NS = /) = P(N = j) for / < k, we have (7.20) To allow for the possibility of reliability improvement, we may assume, for example, that F,(i) < F,_ :(i) for all i, or that fx, > ix,-!, where |x, is the MTBF of the ith item in the sequence. The preceding results must then be modified accordingly, for example, by replacing F(;)(-) by the convolution of F^*) to Fy(-). Example 7.4 [Exponential Distribution] Suppose that an item is purchased with a 1000 (operating) hour warranty and a guaranteed MTBF of |xG = 200 hours. In addition, suppose that item lifetimes are exponentially distributed (so \ = .005, if the true mean is |xG). Then the expected number of failures during the warranty period would be 5. If we use the analysis of Example 7.3 to determine the number of items to be supplied on a probabilistic basis with p = .95, we obtain a total of 5 + 1.645(5)1/2 = 7.24. Thus, reasonable choices for k appear to be 5, 6, and 7. For these choices of k , we now calculate expected costs to the buyer and seller assuming that the MTBF guarantee is met exactly, that the true mean is 25% below the guaranteed value (|x = 150, or X = .00667), and that the guarantee is exceeded by 25% (|x = 250, X = .004)

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If the X t are exponential with common parameter X, then the /-fold convolution of F(-) with itself is a gamma distribution, with CDF given by

In addition, the corresponding renewal function is M(x) = kx. Thus, M(W) = M(1000) = 6.67, 5.00, and 4.00 for the selected values of X. The desired expected costs are calculated by use of Equations (7.17)-(7.20). The results are given in Table 7.1. The increase in expected cost to the seller if the MTBF guarantee is not met is apparent in these results. 7.4.5

C o n fid e n c e Bounds on C osts and Profits

The cost models discussed in the previous sections are expected-value models, that is, they express expected costs in terms of various quantities that are assumed known or are themselves expected values. In reality, costs are functions of many random variables, which may be only partially predictable. If the (joint) distributions of these cost determinants can be reasonably guessed, it may be possible to determine the probability distribution of the buyer’s and seller’s costs and thereby calculate probability limits for these quantities. Another approach is to express the variance of cost in terms of the variances of the individual cost elements and use asymptotic statistical theory to determine approximate probability limits. Methods for accom-

Table 7.1 Buyer’s and Seller’s Expected Cost per Unit, Policy 20a, Exponential Distribution Buyer’s expected cost

k

X = .0067

X = .0050

X = .0040

5 6 7

cb -1- 1.37cr cb + 2.30cr cb + 3.34cr

cb -1- 2.20cr cb -1- 3.08cr cb + 3.81cr

cb + 2.52cr cb -1- 3.14cr cb + 3.56cr

Seller’s expected cost 5 6 7

cs -1- 5.30cr cs + 4.37cr cs 4- 3.33cr

cs + 2.80cr cs + 1.92cr cs + 1.09cr

cs + 1.48cr cs -1- 0.86cr cs + 0.44cr

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plishing this are discussed in Chapter 12. To illustrate how such techniques may be applied in costing RIW, we look at a simplified analysis of the warranty pricing model given in Equation (7.5). The analysis is based on an approach discussed by Balaban and Meth [23]. Model 9 The warranty pricing model of Equation (7.5) expresses warranty cost to the buyer as the expected seller’s cost E(CS) times a profit factor P. The seller’s expected cost is (7.21) where cf is a fixed cost element, Xis the failure rate, U is the total utilization during the warranty period, and cr is the average cost of repair. The Balaban-Meth analysis is as follows: Suppose that U is controlled to the extent that it can be assumed constant and that c{ is also a known constant, but that the failure rate and repair costs may vary during the course of the warranty period. The model deals with expected values of these random variables; we now introduce variances of these quantities as well. Denote the random variables in question A and Cr (with £(A) = X and E(Ct) = cr). Suppose that the variances of these random variables are and (7?, respectively, and that they are independent. Then it can be shown (see Chapter 12) that the variance of the seller’s cost, say o^, is given approximately by

(7.22) Furthermore, if profit, say II (also a random variable), is calculated as a proportion P of cost, then

(7.23) Since under quite general conditions these quantities will be asymptotically normally distributed, the formulas (7.22) and (7.23) can be used to determine approximate probability limits on cost and profit. For example, 95% probability limits on cost are given by (7.24)

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and on profit by (7.25) Similar results can be obtained if other cost elements are considered to be random variables and for other cost models as well. Additional details and a general approach to determining asymptotic results along these lines will be given in Chapter 12. Example 7.5 (Balaban and Meth [23]) The following example loosely follows the analysis of Balaban and Meth. We suppose a sale of 500 units, each with a utilization rate of 50 hours per month and a warranty of 48 months. This gives a total utilization of U = 500(50)(48) = 1,200,000 hours during the warranty period. Suppose that £(A) = A = .005 per hour, cta = .0008, cr = $400, a r = $80, and cf = $250,000. Then

so that the standard deviation of seller’s cost is a s = $619,500. The expected cost is

and the 95% probability interval is

Thus, the seller may be 95% certain that the cost of warranty will be between $1,436,000 and $3,764,000. If the contract is bid with a profit factor of 10%, the seller is 95% certain that the actual profit will be between $143,600 and $376,400. NOTES

Section 7.2 1. The DoD Appropriations Act for FY 1984, which effectively forced the military to require warranties on weapons systems, is discussed in some detail by Yuspeh [6] as well as in several of the more recent

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references on RIW. Yuspeh includes comments on concerns of both the DoD and industry and on some subsequent legislation. 2. Another categorization of warranties used by the government is given by Rotz [27]. Types of warranties include (1) warranty of design, (2) warranty of manufacture, (3) performance warranty, (4) warranty of workmanship and material, and (5) warranty of line-replaceable units, modules, and parts. Still another approach to classification is given by Guglielmoni [5], who identifies 16 types of warranties and other incentives for improving reliability. 3. Markowitz [28] discusses the history of RIW from 1966. Additional discussion and further information on the pioneering Lear Siegler contract may be found in Markowitz [29]. Section 7.3 1. The initial experience of the government and Lear Siegler in using the RIW in military procurement is discussed in some detail by Harty [30]. The RIW agreement itself is also spelled out in considerable detail. It is interesting to note that the concepts of MTBF and TAT guarantees and the notion of a cumulative warranty were embodied in the RIW from the outset (though not necessarily in the cost models). 2. Early experience of the U.S. Air Force with RIW on the F-16 and a widely used tactical navigation system (ARN-118 TACAN) are discussed by Balaban et al. [31]. 3. Additional discussion of the features and conditions for use of RIW may be found in Reterer [32]. 4. Factors to consider in determining the duration of the warranty period are discussed at length by Bonner [16] and Springer [33]. 5. Some important management considerations from the seller’s point of view are discussed by Schmidt [34]; Kowalski and White [35] consider management aspects from the buyer (government) point of view. 6. PPAs and a computerized decision support system for warranty analysis are discussed by Dizek [36]. Section 7.4 1. The reliability/life cycle cost trade-off was key to analysis of RIW since its inception. See, for example, Balaban and Reterer [18,19]. Many additional references up to 1980 on determining the life cycle cost of RIW and related issues may be found in Dhillon [37]. 2. Figures similar to Figure 7.1 have appeared in a number of articles on RIW, e.g., Shorey [38], Gates et al. [21], Toohey and Calvo [39], and Marshall [40]. The cost trade-offs noted in Section 7.4.1 are discussed in some detail in these references. Of most interest to cost modeling is the work of Marshall [40]. Key to the analysis is a model for the cost trade-off that includes discounting to present value.

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3.

Folse [41] has also modeled life cycle costs with and without RIW. A number of modeling approaches are taken, and discounting is included here as well. In addition, a statistical discriminant analysis for selection of items for procurement under RIW is proposed. 4. Many additional analyses of the role of guaranteed MTBF in the selection and evaluation of warranty policies have appeared in the literature. Some additional sources are Bazovsky [42], Metzler [43], Reterer [32], and Newman and Nesbitt [15]. Methods of assessing reliability improvement are discussed in most standard reliability texts. See also Barton [44] for application in RIW. 5. The analysis of the warranty structure proposed by Kruvand [26], which is discussed in Section 7.4.4, is original but does not address exactly the warranty he presented. His approach was actually a cumulative warranty. Cost models for that warranty have not been investigated. Miscellaneous Notes 1. Some other types of warranties used in military procurement are discussed by Lunsford et al. [45]. 2. In the discussions of the models in this chapter (and the previous chapters), it is almost always implicitly assumed that the parameter values (e.g., MTBF, X in the exponential distribution, etc.) required for determining costs, etc., are known. In fact, this is probably rarely the case. In practice, one needs to estimate many quantities from available information. Some approaches to dealing with estimation problems will be discussed in Chapter 12. Some of the data problems that one may expect to encounter are discussed by Day and McIntyre [46], Glaser [47], Flieg [14], and Rotz [27]. 3. Related instrumentation problems associated with the collection of valid and reliable data are discussed by Story [48]. 4. The use of RIW in foreign sales is discussed by Shelton and Paxman [49]. EXERCISES

7.1. 7.2. 7.3.

Recalculate the buyer’s expected cost in Example 7.1 with contractor’s risk factors of r = .01 and .05 and with profit factors of P = .50 and .15. Comment. Repeat the calculations of the previous exercise with each remaining variable increased by 10%. Do the same with each variable decreased by 10%. Which variables most affect expected costs? Analyze the profit function P(XB) of Model 4 with P = .10, CL = $200,000, and CB = $100,000. Suppose that X is uniformly distributed over the interval (.0015, .0025) (so that |i ranges from 400 to

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667 hours), i.e.,

for .0015 < X < .0025. Suppose that cs(X) is an increasing function of X in the interval (.0015, .0025), e.g., cs(X) = a(X - .0015), and that PA(X) is a decreasing function, e.g., PA(X) = b/X, for some a > 0 and b < .0015. Derive an expression for the contractor’s profit as a function of X, choose values of a and b that will give reasonable results, and calculate P(XB) for a range of bids at failure rates XB. 7.4. Consider alternative priors GA(*) in the analysis of Exercise 7.3. Possible choices are the exponential distribution with mean |x = 500, a gamma distribution, and a triangular distribution with peak at X = .002. 7.5. Suggest alternatives for the functions cs(X) and PA(X) in Exercise 7.3, and complete the analysis for the suggested functions. 7.6. Combine the alternatives suggested in Exercises 7.4 and 7.5 and analyze the various resulting models. 7.7. Calculate the seller’s expected profit for each of the models of the previous four exercises for a range of parameter values. Prepare tables of reasonable alternatives, and discuss the conditions under which the models appear to provide realistic results. 7.8. Perform a numerical investigation of the warranty cost cw given in Equation (7.5) for the “realistic results” of Exercise 7.7 and selected values of the remaining parameters in (7.5). 7.9. Recalculate expected repair costs in Example 7.2 assuming that the distribution of time to failure F0( ) is lognormal, Weibull, and gamma (see Exercise 6.1), rather than exponential. Use the X-values given. For each distribution, take |jl = 1/X0 = 1000 and ct = .15p. Select parameter values corresponding to this mean and standard deviation, and complete the calculations. 7.10. Repeat the calculations of Exercise 7.9 with p, = 900 and 1100. Calculate expected costs for each distribution and each new set of parameter values, and compare the results with those of Example 7.2 and Exercise 7.9. 7.11. Suppose in Example 7.3 that we wished to be 99% confident that the number of spares required does not exceed Nt (i.e., p = .99). Calculate Nt(500) and 7Vt(400) for this case. Repeat the calculations for p = .90.

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7.12.

Suppose that F(t) in Model 8 is a gamma distribution with parameters X and 0, given by

t > 0. Then F(n\t) is a gamma distribution with parameters X and Use this result, with X = .0025, 0 = .50 and X = .01, p = 2.00, to calculate the expected buyer’s and seller’s costs under Policy 20a analogous to those in Table 7.1. Show that for the gamma distribution p = p/X. Repeat Exercise 7.12 with p = 150 and 250 for each of the two P-values given. az0.

7.13. 7.14.

REFERENCES

1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

Proc. Failure Free Warranty Seminar (1973). U.S. Navy Aviation Supply Office, Philadelphia. Schmidt, A. E. (1976). A View o f the Evolution of the Reliability Improvement Warranty, Rept. 76-1, Defense Systems Mgmt. College, Ft. Belvoir, VA. Shmoldas, A. E. (1977). Improvement of Weapon System Reliability Through Reliability Improvement Warranties, Rept. 77-1, Defense Systems Mgmt. College, Ft. Belvoir, VA. Gándara, A., and Rich, M. D. (1977). Reliability Improvement Warranties for Military Procurement, Rept. No. R-2264-AF, The RAND Corp., Santa Monica, CA. Guglielmoni, P. B. (1986). An approach to selecting warranties/incentives, 1986 Proc. Ann. Reliab. and Maintain. Symp., 166-170. Yuspeh, A. R. (1986). Legislation on weapon system warranties, 1986 Proc. Ann. Reliab. and Maintain. Symp., 438-442. Dhillon, B. S. (1989). Life Cycle Costing, Gordon and Breach Science Pub., New York. Hiller, G. E. (1973). Warranty and product support. The plan and use thereof in a commercial operation, in Proc. Failure Free Warranty Seminar, U.S. Navy Aviation Supply Office, Philadelphia. Trimble, R. F. (1974). Interim Reliability Improvement Warranty (RIW) Guidelines, HQ USAF Directorate of Procurement Policy Document, Dept, of the Air Force, Washington, D.C. Gregory, W. M. (1964). Air force studies product life warranty, Aviation Week and Space Tech., 2 Nov. 1964.

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11. 12. 13.

14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26.

297

Klause, P. J. (1970). Failure-free warranty idea lauded, wider use desired, Aviation Week and Space Tech., 9 Feb. 1970. Markowitz, O. (1976). Aviation supply office FFW/RIW case history #2, Abex pump, 1976 Proc. Ann. Reliab. and Maintain. Symp., 357362. Gilleece, M. A. (1984). Proposed Guarantee Guidance to Implement Section 794 o f the Department o f Defense Appropriation Act for FY 1984—ACTION MEMORANDUM, Office of the Under Secretary of Defense, 19 Jan. 1984. Fleig, N. G. (1986). The United States Air Force Product Performance Agreement Center, 1986 Proc. Ann. Reliab. and Maintain. Symp., 449-453. Newman, D. G., and Nesbitt, L. D. (1978). USAF experience with RIW, 1978 Proc. Ann. Reliab. and Maintain. Symp., 55-61. Bonner, W. J. (1976). A contractor view of warranty contracting, 1976 Proc. Ann. Reliab. and Maintain. Symp., 351-356. Balaban, H., and Reterer, B. (1973). The Use o f Warranties for Defense Avionics Procurement, ARINC Research Pub. No. 0637-02-11243, Annapolis, MD, June, 1973. Balaban, H., and Reterer, B. (1973). Life-Cycle Cost Implications in the Use of Warranties for Avionics, ARINC Tech. Perspective No. 8, Annapolis, MD, July, 1973. Balaban, H., and Reterer, B. (1974). The use of warranties for defense avionics procurement, Proc. 1974 Ann. Reliab. and Maintain Symp., 363-368. Gates, R. K., Bortz, J. E., and Bicknell, R. S. (1976). A quantitative analysis of alternative RIW implementations, Proc. Nat. Aerospace and Electron Conf., 1-8. Gates, R. K., Bicknell, R. S., and Bortz, J. E. (1977). Quantitative models used in the RIW decision process, Proc. 1977 Ann. Reliab. and Maintain. Symp., 229-236. Balaban, H. S. (1976). Controlling risks in reliability improvement warranties, presented at Military Oper. Res. Soc. Balaban, H. S., and Meth, M. A. (1978). Contractor risk associated with reliability improvement warranty, Proc. 1978 Ann. Reliab. and Maintain. Symp., 123-129. Chelson, P. O. (1978). Can we expect ECP’s under RIW?, Proc. 1978 Ann. Reliab. and Maintain. Symp., 204-209. Balaban, H. S. (1975). Guaranteed MTBF for military procurement, Proc. 10th International Logistics Symposium, August, 1975. Kruvand, D. H. (1987). Army aviation warranty concepts, 1987 Proc. Ann. Reliab. and Maintain. Symp., 392-394.

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27.

Rotz, A. O. (1986). Warranty pricing with a life cycle cost model, 1986 Proc. Ann. Reliab. and Maintain. Symp., 454-456. Markowitz, O. (1975). Failure free warranty/reliability improvement warranty— buyer viewpoint, Trans. 20th Ann. ASQC Tech. Conf., 89-97. Markowitz, O. (1977). Continuous reliability improvement assurance within operations and logistics, Proc. 1977 Reliab. Conf. for the Elec. Power Industry, 100-105. Harty, J. C. (1971). Practical life cycle cost/cost of ownership type procurement via long term/multi year “failure free warranty” (FFW), showing trial procurement results, Annals o f Reliab. and Maintainability, 10, 241-251. Balaban, H., Cuppett, D., and Harrison, G. (1979). The F-16 RIW program, Proc. 1979 Ann. Reliab. and Maintain. Symp., 79-82. Reterer, B. L. (1976). Considerations for effective warranty application, Proc. 1976 Ann. Reliab. and Maintain. Symp., 346-350. Springer, R. M., Jr. (1977). Risks and benefits in reliability warranties, J. Purchasing and Materials Mgmt., 13, 8-13. Schmidt, B. A. (1979). Preparation for LCC proposals and contracts, Proc. 1979 Ann. Reliab. and Maintain. Symp., 62-66. Kowalski, R., and White, R. (1977). Reliability improvement warranty (RIW) and the army lightweight doppler navigation system (LDNS), Proc. 1977 Ann. Reliab. and Maintain. Symp., 237-241. Dizek, S. G. (1986). Automated decision support for warranty selection, 1986 Proc. Ann. Reliab. and Maintain. Symp., 460-465. Dhillon, B. S. (1981). Life cycle cost: A survey, Microelectron. Reliab., 21, 495-511. Shorey, R. R. (1976). Factors in balancing government and contractors risk with warranties, 1976 Proc. Ann. Reliab. and Maintain. Symp., 366-368. Toohey, E. F., and Calvo, A. B. (1980). Cost analysis for avionics acquisition, 1980 Proc. Ann. Reliab. and Maintain. Symp., 85-90. Marshall, C. W. (1981). Design trade-offs in availability warranties, 1981 Proc. Ann. Reliab. and Maintain. Symp., 95-100. Folse, R. O. (1977). Quantification o f Selection Criteria for Reliability Improvement Warranty Contracts, Doctoral Dissertation, Dept, of Quantitative Methods, Louisiana State Univ., Baton Rouge, LA. Bazovksy, I. (1968). Appraisal of guaranteed MTBF warranty programs, Ann. Assurance Sciences, 1, 256-265. Metzler, E. G. (1974). Forcing functions integrate R & M into design— DoD TACAN procurement policy on reliability and maintainability, Proc. 1974 Ann. Reliab. and Maintain. Symp., 52-55.

28. 29. 30.

31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43.

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44.

299

Barton, H. R., Jr. (1985). Predicting guaranty support using learning curves, 1985 Proc. Ann. Reliab. and Maintain. Symp., 354-356. 45. Lunsford, J. D., Berk, J. H., and Nixon, D. E. (1986). Warranting conventional munitions, 1986 Proc. Ann. Reliab. and Maintain. Symp., 443-447. 46. Day, R. C., and McIntyre, L. E. (1978). RIW data collection and reporting method, Proc. 1978 Ann. Reliab. and Maintain. Symp., 56-72. 47. Glaser, A. J. (1981). A data information system for RIW contracts, Proc. 1981 Ann. Reliab. and Maintain. Symp., 139-143. 48. Story, J. K. (1991). A new methodology for performance instrumentation vis-a-vis warranty requirements, 1991 Proc. Ann. Reliab. and Maintain. Symp., 352-356. 49. Shelton, D. K., and Paxman, R. G. (1982). A reliability warranty concept for the FMS environment, 1982 Proc. Ann. Reliab. and Maintain. Symp., 34-39.

8

Two-Dimensional Warranty Policies

8.1 IN TRO D UC TIO N

In Section 2.2.3, we discussed two-dimensional warranties and defined four different free-replacement policies (Policies 8-11) and a family of pro-rata policies (Policy 12). A two-dimensional warranty policy differs from the one-dimensional policies discussed in Chapters 4-7 in two respects: 1. The warranty is characterized by a two-dimensional region in a plane as opposed to an interval in one dimension. 2. The item failures are events occurring randomly on this two-dimensional plane. Many products are sold with Policy 8 (e.g., automobiles, components of aircraft), but we are unaware of any manufacturer offering the other types of policies. Policy 8 tends to favor the manufacturer, while Policy 9 tends to favor the consumer. In contrast, Policies 10 and 11 offer a sensible compromise and are suggested for consideration by practitioners. An approach to modeling the first failure by a two-dimensional joint distribution function was discussed in Section 2.7. Modeling of subsequent failures depends on the nature of the rectification action. In the case of nonrepairable items, the only option available is replacement of failed items by new ones. In this case, the subsequent failures occur according to a two-dimensional renewal process if the time to replace is sufficiently small, so that it may be reasonably assumed to be zero. The preceding way of modeling item failures involves a two-dimensional point process formulation. An alternate approach is to model item failures using essentially a one-dimensional point process formulation. In Section 301

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8.2 we discuss this approach and contrast it with the two-dimensional approach. Sections 8.3 and 8.4 deal with a variety of two-dimensional warranty policies based on the one-dimensional approach. In Section 8.3, we examine free-replacement policies and derive expressions for expected warranty cost per unit sale. The study of life cycle cost for these models is done in Section 8.4. The remaining sections deal with analysis of twodimensional warranties based on two-dimensional models for item failures. Section 8.5 deals with expected cost per unit sale for the free-replacement (FRW) policy, and in Section 8.6 we study the expected life cycle costs for the same set of policies. The analysis of pro-rata (PRW) warranty policies is done in Section 8.7. We conclude with a brief discussion of further topics in two-dimensional warranty policies in Section 8.8. 8.2

MODELING ITEM FAILURES

In this section we discuss the one-dimensional approach to modeling item failures and contrast it with the two-dimensional approach. 8.2.1 The One-Dimensional Approach

Let X c(t) and Yc(t) denote the age and usage obtained for the item currently in use at time t. Let Y(t) denote the total usage that a buyer has had from the current plus earlier items over the interval [0, i), with the first sale taking place at time t = 0. If no item failures have occurred in [0, i), then X c(t) = t and Yc(t) = Y(t). This is also true for the case where all failed items are repaired and the repair time is assumed to be zero. In contrast, if the item is not repairable and there have been one or more failures in [0, t), then X c(t) < t and Yc(t) < Y(t). In the one-dimensional approach, one models Yc(i) as a function of X c{t). This relationship characterizes item usage as a function of the age of the item. We assume that the relationship is linear with a nonnegative coefficient R. That is,

(8. 1) Here R represents the usage per unit time, or usage rate, and may vary from user to user. We model R as a random variable with density [distribution] function g(/*)[G(r)], i.e.,

Different forms of G(r) reflect different usage rates across the population of buyers. Some suitable forms for g(r)[G(r)] are the following:

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1. R uniformly distributed over [yb y j , with 0 < yj < y^ < oo, i.e.,

2.

3.

R = yi + A[y^ - y j, with 0 < y, < y^ < a>, and A a random variable with a Beta density given by

where B (p , q) is the Beta function. Note that R assumes values in the interval [7,, 7J . R distributed according to the Gamma distribution function, i.e.,

and T(p) is the gamma function. Note: In (1) and (2), R is bounded by y^, whereas in (3) it can assume any finite value. Conditional on R = r, let \(t\r) St denote the probability that the current working unit at time t will fail in the small interval [t, t + Si) i.e., we assume failures to occur according to a Poisson process with intensity function \(i|r), t > 0. We model \(t\r) by a relationship

(8 .2) where the function v|;(jc, y) is an increasing function of both x and y. This implies that the probability of item failure increases with its age and usage. A special case is the linear form given by (8.3) with the parameters 0, > 0 for 0 < / < 3. Conditional on R, if 02 and 03 are zero, the intensity function depends only on the usage rate and not the age or the total usage. If 0j and 03 are zero, then there is no effect of usage on item failure, and in this case the model reduces to the usual onedimensional formulation with time or age being the only independent variable in the model formulation. If 02 is zero, then again the model reduces

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to a one-dimensional case, with usage being the one-dimensional independent variable. Let N(t) denote the number of failures over [0, t). This is a random variable. Conditional on R = r, it is a Poisson process with intensity X(t\r) given by (8.2) or by (8.3) for the special case. In other words, N(t) is a conditional Poisson process. Let 2V(f|r) denote the number of failures over the interval [0, t) conditional on R = r. Once we obtain the distribution function for N(t\r), then by combining it with G(r) we can at least conceptually obtain the distribution function for N(t). The form of the distribution function for N(f|r) depends on the rectification action used for failed items. In the remainder of the chapter, we assume that the time to repair or replace is sufficiently small that it can be considered as being zero. We consider two cases: REPAIRABLE ITEMS [Minimal Repair] Here, whenever an item fails, it is rectified through minimal repair. As a result, conditional on R = r, we have and In this case, it follows that N(i|r) is a nonstationary Poisson process with intensity function (8.4) As a result (see Chapter 3), the conditional probability distribution of N(t\r) is given by (8.5) and the probability distribution of N(t) is given by

( 8 .6) NONREPAIRABLE ITEMS [Replacement with New] Since failed items cannot be repaired, all failed items need to be replaced by new ones. Let F(t\r) denote the distribution function of X ^r, the time to first failure conditional on R = r. It is easily seen that this is given by (8.7)

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Since all subsequent failure times have the same distribution function, we see that A(i|r) is a renewal process associated with the distribution function F{t\r). As a result, we have

(8.8) where F^n\t\r) is the «-fold convolution of F(t\r) with itself. From this we can obtain the probability distribution for N(t) from (8.6), using (8.8). The actual number of failures over the warranty would depend on the warranty region. We discuss this in Section 8.3. 8.2.2

The Tw o-Dim ensional A pproach

Let {Xu Yx) denote the time to first failure and the item usage at first failure. Similarly, let (Xh Y,), i > 2, denote the time interval between ith and (i - l)st failure and the item usage between the two failures. We model (Xh Y,) through a bivariate distribution function. For each i > 1, let Ff(x, y ) denote the distribution function for (Xh Y,), i.e., (8.9) The form of F,-(jt, y) depends on the nature of the rectification actions. We consider two cases. NONREPAIRABLE ITEMS [Replacement with New] In this case, whenever an item fails it must be replaced by a new one. We assume that item failures are all independent. As a result, (Xh Y,), i > 1, is a sequence of independent and identically distributed random variables with a common two-dimensional joint distribution function F(x, y ), and item failures over the plane are modeled by a two-dimensional renewal process of the form discussed in Section 3.9. As we shall see, the number of failures over the warranty region depends on the shape of the region and in most cases can be obtained from the counts for the two-dimensional renewal process associated with F(x, y) and some related one-dimensional point processes. REPAIRABLE ITEMS [All Repaired Items Identical] Here (X ly YJ represents the time and usage at first failure, with joint distribution function given by F(x, y). Since all repaired items are identical, they have the same joint distribution function Fr(x, y), which may be different from F(x, y). In other words, (Xh Y,), i > 2, are random variables from a distribution Fr(x, y). As a result, failures over the two-dimensional plane occur according to a modified renewal process discussed in Section

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3.9. As we shall see, the number of failures over the warranty region depends on the shape of the region. In most cases this can be obtained from the counts for the just mentioned two-dimensional modified renewal process. 8.3

FREE-REPLACEM ENT POLICIES [O NE-DIM ENSIO NAL APPROACH]

In this section we study Policies 8-11 of Chapter 2. In all of these policies, the manufacturer has to either repair or provide new replacements for all items whose failures occur within the warranty region, at no cost to the buyer. We derive expressions for the expected cost of servicing the warranty per unit sale as a function of the warranty parameters. We first consider the case where failed items are repairable and are minimally repaired after each failure. Later we discuss the nonrepairable case, where, whenever a failure occurs under warranty, the failed item is replaced by a new one. Let cr and cs denote the cost of each minimal repair and each replacement, respectively. 8.3.1 A na lysis o f Policy 8

Policy 8 was defined in Chapter 2 as follows: Free-Replacement Policy Under this policy the manufacturer agrees to repair or provide a replacement for failed items free of charge up to a time W or up to a usage U, whichever occurs first, from the time of the initial purchase. The warranty region is a rectangle [0, W) x [0, U) as shown in Figure 8.1. VF and U are the parameters of the policy.

Conditional on the usage rate R = r, the warranty ceases (see Figure 8.1) at time X r, given by

(8 .10) when r > yl, and at time W when r < y 1, where y l is given by

( 8 . 11) Also shown in Figure 8.1 is a random variable Zr, which represents the time at which the first failure outside the warranty period occurs. Note that Z r can be either smaller or larger than W.

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Figure 8-1 Warranty region for Policy 8. Let N(W, U) denote the number of failures under warranty and let N(W, U\r) denote the number of failures under warranty conditional on R = r. Then, we have ( 8 . 12)

Thus, once we obtain an expression for the distribution function for N(W , U\r), we can obtain an expression for the distribution function for N(W, U). REPAIRABLE ITEMS [Minimal Repair] Since minimal repair makes the failed item operational without affecting the failure rate, the relationship between N(W, U\r) and iV(i|r) is given by (8.13)

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Since iV(f|r) is a nonstationary Poisson process with intensity function X(t|r) (given by (8.4)), the expected value of N(W, U\r) is given by

(8.14)

Using the conditional expectation argument (see Appendix A), we have the expected number of failures over the warranty region given by

or

(8.15) Using k(t\r) given by (8.4), we have

(8.16) with X r and yl given by (8.10) and (8.11), respectively. The expected warranty service cost per unit sale, E[CS(W, U)], is given by (8.17) Important Note for Examples: For the numerical examples in this chapter, the unit for usage U is 104 miles and for W is years. As a result, W = 1 and U = 2 corresponds to a time limit of one year and a usage limit of 20,000 miles. The unit for r is 104 miles/year.

309

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Example 8.1 [G(y); Uniform Distribution] Let 7, = 0.0 and = 5.0 (x 104 miles/year). This implies that the mean usage per year (i.e., average number of miles per year) is 25,000 miles. We assume the following parameter values: and We consider a range of values for U and W in the interval 0.5 to 2.0. Table 8.1a shows E[N(W, U)] for different combinations of W and U. These were obtained by numerical computation using (8.16). Consider the case W = U = 0.5—that is, a warranty period of six months and a usage limit of 5000 miles. The expected number of repairs per 10,000 items is 14. With W held at 0.5 and U increasing, the expected number of repairs per 10,000 items increases and equals 35 when U = 2 years. Now consider the case where U is held at 0.5 and W is increasing. The expected number of repairs per 10,000 items increases and equals 24 when W = 2 ( x 104) miles. The reason for a larger value for U = 2.0 and W = 0.5 as opposed to U = 0.5 and W = 2.0 is as follows. Since the mean usage is 2.5 and 7I = 1, we have, for W = U = 0.5, that the probability of the warranty ceasing due to the usage limit being exceeded is higher than that due to the age limit being exceeded. Hence, increasing U with W fixed at 0.5 implies greater warranty coverage (on the average) to the user than increasing W and fixing U at 0.5. As a result, the expected number of failures in the former case is larger than in the latter case. Consider the following special cases: Case (1): and This implies that item failure rate depends only on the usage rate and not on age or total usage. As a result, \(t\r) is a constant— in other words,

Table 8.1a

[Example 8.1] w

u 0.50 1.00 1.50 2.00

Expected Number of Failures Under Warranty for Policy 8

0.50

1.00

1.50

2.00

.0014 .0024 .0031 .0035

.0018 .0034 .0048 .0061

.0021 .0040 .0059 .0077

.0024 .0046 .0069 .0091

310

Chapter 8

item failures, conditioned on R = r, occur according to a stationary Poisson process. Table 8.1b gives E[N(W, U)\ for this case, and as expected the values are smaller than the corresponding entries of Table 8.1a. Case (2): and This implies that item failure rate depends on the usage rate r and the total usage Yc(i) but not on age. Table 8.1c gives E[N(W, U)] and, as is to be expected, the values are smaller than the corresponding entries of Table 8.1a. Example 8.2 [G(r); Gamma Distribution] Suppose that G(r) is given by a Gamma distribution with parameter p. In this case, the mean usage rate is p. We choose p = 2.5 so that the results are comparable with those of Example 8.1. We consider the special case 02 = 03 = 0. This is Case (1) of the previous example and we have X(t\r) = 0O + 0ir. From (8.15) we have, after some

Table 8.1b Expected Number of Failures Under Warranty for Policy 8 [Example 8.1: Case (1)] w

u 0.50 1.00 1.50 2.00

0.50

1.00

1.50

2.00

.0011 .0017 .0021 .0023

.0013 .0022 .0029 .0034

.0014 .0025 .0033 .0040

.0015 .0026 .0036 .0044

Table 8.1c Expected Number of Failures Under Warranty for Policy 8 [Example 8.1]: Case (2)]

W

U 0.50 1.00 1.50 2.00

0.50

1.00

1.50

2.00

.0009 .0014 .0017 .0018

.0013 .0021 .0027 .0033

.0015 .0026 .0036 .0043

.0018 .0031 .0043 .0053

Two-Dimensional Warranty Policies

311

simplification,

where

is the Gamma distribution function. Standard mathematical tables (Abramowitz and Stegun [1]) give values of this for a range of x and p. Using (8.17) one can obtain E[CS(W, U)]. Table 8.2 gives E[N(W, U)} for the following parameter values: and On comparison with the corresponding values in Table 8.1b, we see that for a given W and £/, the expected number of failures for R distributed uniformly with mean 2.5 is approximately the same as that for R having a Gamma distribution with the same mean. NONREPAIRABLE ITEMS [Replacement with New] In this case, whenever an item fails under warranty it is replaced by a new one. Conditional on R = r, N(W, U\r) is related to N(t\r) according to (8.13) with A(t|r) a renewal process as discussed in Section 8.2.1. The distribution function of N(t\r) is given by (8.8). From this it follows that

Table 8.2 Expected Number of Failures Under Warranty for Policy 8 [Example 8.2]

w

U 0.50 1.00 1.50 2.00

0.50

1.00

1.50

2.00

.0011 .0018 .0021 .0023

.0013 .0023 .0030 .0036

.0013 .0024 .0034 .0042

.0013 .0025 .0036 .0045

312

Chapter 8

the expected value of N(W, U\r) is given by (8.18) where M(t\r) is the conditional renewal function associated with the distribution function F(t\r), i.e., (8.19) Using the formula for conditional expectation (see Appendix A), we have

(8 .20) with X r and 7I given by (8.10) and (8.11), respectively. The expected warranty service cost per unit sale, E[CS(W, U)], is given by

(8 .21) with E[N(W, U)] given by (8.20). Example 8.3 [G(y); Gamma Distribution] Suppose that G(r) is given by a Gamma distribution with parameter p. We consider the special case 62 = 63 = 0, so that \(t\r) = 60 + 6xr. This implies that, conditioned on 7, the failure rate is constant. As a result, the expected number of replacements by new items under warranty is the same as the number of minimal repairs in Example 8.2. Hence, the expected number of repairs is given by

where

Two-Dimensional Warranty Policies

313

is the Gamma distribution function. Table 8.2 gives E[N(W, U)] for 0O = .003, 0j = .007, and 02 = 03 = 0.0. Using this, E[CS(W, l/)], the expected warranty service cost, can be obtained from (8.21). 8.3.2

A na lysis o f Policy 9

Policy 9 was defined in Chapter 2 as follows: Free-Replacement Policy Under this policy the manufacturer agrees to repair or provide a replacement for failed items free of charge up to a minimum time W from the time of the initial purchase and up to a minimum total usage U. The warranty region is given by two strips as shown in Figure 8.2. W and U are the parameters of the policy.

Conditional on the usage rate R = r, the warranty ceases (see Figure 8.2) at time W when r > yl = U/W and at time X r = Ulr when r < yl. This is different from Policy 8 studied earlier. Since the expected number of

Figure 8.2 Warranty region for Policy 9.

314

Chapter 8

failures under warranty for both the repairable and nonrepairable cases can be obtained using the approach used in Policy 8, we omit the details of the derivation and simply state the final results. REPAIRABLE ITEMS [Minimal Repair] The expected number of failures under warranty is given by

(8 .22) Using X(t\r) given by (8.4), we have

(8.23) with X r and yl given by (8.10) and (8.11), respectively. The expected warranty service cost per unit sale, E[CS(W, [/)], is given by (8.17) with E[N(W, U)] is given by (8.23). Example 8.4 [G(r); Uniform Distribution] Let G(r) be uniform with y, = 0.0 and yu = 5.0. Consider the parameter values of Example 8.1, i.e., and Table 8.3 shows £[Af(\y, i/)] for different combinations of W and U. These were obtained by numerical computation using (8.23). On comparing these results with those of Table 8.1, we see that, for a given (W, U) combination,

This is to be expected, as the warranty region for Policy 8 is a proper subset of the warranty region for Policy 9. As a result, N(W, JJ) for Policy 9 is stochastically greater than that for Policy 8, and hence the result. Consider the case W = U = 1. Then E[N(W, U)] = .0024 for Policy 8 and .0712 for Policy 9. This corresponds to a 30-fold increase. Note that under Policy 8 the buyer had a maximum coverage for 1 year or 20,000

Two-Dimensional Warranty Policies

315

Table 8.3 Expected Number of Failures Under Warranty for Policy 9 [Example 8.4]

w

U 0.50 1.00 1.50 2.00

0.50

1.00

1.50

2.00

.0197 .0658 .1421 .2484

.0255 .0712 .1467 .2522

.0341 .0794 .1545 .2594

.0454 .0904 .1652 .2697

miles, whereas under Policy 9 the buyer has a minimum coverage of 1 year and 20,000 miles. As a result, for a high intensity user the warranty would cease at time W under Policy 9 and well before W under Policy 8. For a low intensity user, the warranty would cease at W under Policy 8 and well beyond W under Policy 9. Hence, the dramatic increase in the mean number of repairs under Policy 9. N O N R E P A Y A B L E ITEMS [Replacement with New] The expected number of replacements under warranty is given by (8.24) with X r and 7I given by (8.10) and (8.11), respectively, and M(t\r) given by (8.19). The expected warranty service cost per unit sale, E[CS(W, U)], is given by (8.21) with £[A(W, U)] given by (8.24). Example 8.5 [G(r); Gamma Distribution] We consider the special case of Example 8.3 and the parameter values used in that example. Table 8.4 gives E[N(W, U)] obtained using (8.24). Comparing these results with those of Table 8.3, we see that the expected number of replacements under Policy 9, for a given (W, U) combination, is larger than that for Policy 8 for reasons discussed in Example 8.4. 8.3.3

A na lysis o f Policy 10

Policy 10 was defined in Chapter 2 as Free-Replacement Policy Under this policy the manufacturer agrees to repair or provide a replacement for failed items free of charge up to a time Wx from the time of the initial purchase provided the total usage at failure is below U2, and up to a time W2

Chapter 8

316

Table 8-4 Expected Number of Failures Under Warranty for Policy 9 [Example 8.5]

w 0.50 1.00 1.50 2.00

0.50

1.00

1.50

2.00

.0026 .0033 .0043 .0055

.0048 .0052 .0052 .0066

.0072 .0074 .0078 .0083

.0095 .0097 .0100 .0104

Figure 8.3 Warranty region for Policy 10.

Two-Dimensional Warranty Policies

317

provided the total usage at failure does not exceed Ux. The warranty region is shown in Figure 8.3. Wu W2, Ul9 and U2 are the parameters of the policy.

Conditional on the usage rate R = r, the warranty ceases (see Figure 8.3) at time W2 when r < 7I; at time X rl, given by (8.25) when 7I < r < y2; at time Wl when y2 < r < y3; and at time X r2, given by (8.26) when 73 < r < 00, with 7I, y2, and 73 given by and

(8.27)

Following the approach used in the analysis of Policy 8, we have the following results: REPAIRABLE ITEMS [Minimal Repair] The expected number of failures under warranty is given by

(8.28) where K(t\r) is given by (8.4). The expected warranty service cost per unit sale, E[CS(WX, W2, Ul9 U2)], is given by (8.29)

Chapter 8

318

Example 8.6 [G(r); Uniform Distribution] We consider the special case where Wx = .5W2 and U1 = .5f/2. The parameters 0f. (1 < / < 4) are the same as in Example 8.1. Table 8.5 gives E[N(W, U)] for different combinations of W2 and f/2. Note that the warranty region for Policy 10 is a subset of the warranty region for Policy 8. Hence, E[N(W, U)] for Policy 8 is larger than E[N(.5W, .5U, W, U)] for Policy 9. This is confirmed by a comparison of Tables 8.1 and 8.5. NONREPAIRABLE ITEMS [Replacement with New] The expected number of replacements under warranty is given by

(8.30) with M{t\r) given by (8.19), X rl and X r2 given by (8.25) and (8.26), and y l, y2, and y3 given by (8.27). The expected warranty service cost per unit sale, E[CS(WX, W2, Ul9 U2)\, is given by (8.31)

Table 8.5 Expected Number of Failures Under Warranty for Policy 10 [Example 8.6]

/ /^ 0.50 1.00 1.50 2.00

0.50

1.00

1.50

2.00

.0012 .0018 .0021 .0024

.0016 .0028 .0038 .0045

.0018 .0034 .0049 .0061

.0021 .0039 .0057 .0073

319

Two-Dimensional Warranty Policies

Example 8.7 [G(r); Gamma Distribution] We consider the parameter values of Example 8.2. Table 8.6 gives the expected number of replacements under warranty for the case = .5 W2 and t/j = .5U. On comparison with Policy 8, we see that E[N(W, U)] for Policy 8 is larger than £[Af(.5W, .5t/, W, U)] for Policy 10, as expected. 8.3.4

A na lysis o f Policy 11

Policy 11 was defined in Chapter 2 as follows: Free-Replacement Policy

Under this policy the manufacturer agrees to repair or provide a replacement for failed items free of charge up to a maximum time W from the time of the initial purchase and for a maximum total usage of U. Let X = time since purchase and Y = total usage at failure. The item is covered under warranty if Y + (Ul W)X < U. If Y + ( U/W)X > U, then the item is not covered by the warranty. The warranty region is given by the triangle shown in Figure 8.4, and the parameters of the warranty are W and U.

Conditional on R = r, the warranty ceases (see Figure 8.4) at X r given by (8.32) Derivations similar to those given previously yield the following results: REPAIRABLE ITEMS [Minimal Repair] The expected number of failures under warranty is given by (8.33) with k(t\r) given by (8.4) and X r given by (8.32). Ta b le 8.6

[Example 8.7]

Expected Number of Failures Under Warranty for Policy 10

w2

U2 0.50 1.00 1.50 2.00

0.50

1.00

1.50

2.00

.0009 .0016 .0021 .0024

.0011 .0019 .0026 .0032

.0012 .0020 .0028 .0035

.0012 .0022 .0030 .0037

320

Chapter 8

Figure 8.4 Warranty region for Policy 11. The expected warranty service cost per unit sale, E[CS(W, U)], is given by (8.17) with £[N(W, U)] given by (8.33). Example 8.8 [G(r); Uniform Distribution] The parameters 0,(1 < / < 4) are the same as in Example 8.1. Table 8.7 gives E[N(W, U)] for different combinations of W and U.

Table 8.7 Expected Number of Failures Under Warranty for Policy 11 [Example 8.8]

w

U 0.50 1.00 1.50 2.00

0.50

1.00

1.50

2.00

.0009 .0014 .0017 .0019

.0012 .0021 .0027 .0033

.0015 .0026 .0035 .0043

.0017 .0030 .0041 .0052

Two-Dimensional Warranty Policies

321

Table 8.8 Expected Number of Failures Under Warranty for Policy 11 [Example 8.9]

w u

0.50 1.00 1.50 2.00

0.50

1.00

1.50

2.00

.0007 .0011 .0013 .0015

.0009 .0015 .0019 .0022

.0010 .0017 .0022 .0027

.0011 .0019 .0025 .0030

Note that the warranty region for Policy 11 is a subset of the warranty region for Policy 8 and of that for Policy 10 when Wl = .5W2, Ut = .5U2, W2 = W, and U2 = U. Hence, E[N(W, [/)] for Policy 11 cannot exceed E[N(W, U)] for Policy 8 and E[N(.5W, .5U, W, U)] for Policy 9. The results of Tables 8.1, 8.5, and 8.7 confirm this for this example. NONREPAIRA BEE ITEMS [Replacement with New] The expected number of replacements under warranty is given by (8.34) with X r given by (8.10) and M(t\r) by (8.19). The expected warranty service cost per unit sale, E[CS(W, [/)], is given by (8.21) with E[N(W, U)] given by (8.34). Example 8.9 [G(r); Gamma Distribution] We use the parameter values of Example 8.2. Table 8.8 gives E[N(W, U)], the expected number of replacements under warranty, for different combinations of (W, U). Note that E[N(W, U)] for Policy 11 is smaller than E[N(W, U)] for Policy 8 and E[N(.5W, .5(7, W, U)] for Policy 9 for reasons discussed in the previous example. The results of Tables 8.3, 8.6, and 8.8 confirm this. 8.4

LIFE CYC LE C O STS [O NE-DIM ENSIO NAL APPROACH]

Let L denote the product life cycle. We shall confine our discussion to nonrepairable items. Whenever an item fails outside the warranty region we assume that the buyer purchases a replacement item at full price. This is called a repeat purchase. Let K(L) denote the random number of repeat purchases over the product life cycle. The total cost to the manufacturer from each new purchase and subsequent repeat purchases over the product

Chapter 8

322

life cycle is also a random variable. Let LCC(L; parameters) denote the expected value of this life cycle cost. In this section we derived expressions for the manufacturer’s expected cost for Policies 8-11. 8.4.1

A n a ly sis o f P o lic y 8

Let K(L\r) denote the number of repeat purchases over the life cycle conditioned on R = r. We first obtain an expression for E[K(L\r)]. Using conditional expectation, we obtain E[K(L)\ by the relation (8.35) In Section 8.3.1 we introduced the random variable Z r. It represents the time to first failure outside warranty (see Figure 8.1). As a result, it also represents the time between purchases conditional on R = r. We need to consider the two cases: (1) r < yl and (2) r > y l, where yl is given by (8.11). Note that Z r can be written as (8.35) where (8.37) and X r is given by (8.10). \r represents the time at which the warranty ceases, and £ is the time to first failure afer \ r given r (see Figure 8.1). Hence, the distribution for Z r is the same as that for £ except for a shift Of XrSince all failures over [0, \r) are replaced by new items, the failures over this interval occur according to a renewal process with the distribution function for time between failures, F(t\r), given by (8.7). As a result, £ is the excess age or residual life of the item in use at \ r for this process. From the results of Section 3.5.2, we have the distribution F^(t\r) of £ given by (8.38) where M(t\r) is the renewal function associated with F(t\r) and is given by Equation (8.19).

Two-Dimensional Warranty Policies

323

Let Fz (t\r) denote the distribution of Z r. Then from (8.36) and (8.37) we have (8.39) Since Zr is the time between purchases, K(L\r) is the number of renewals in [0, L), with time between renewals distributed according to Fz {t\r). As a result, the expected value of K(L\r) is given by (8.40) where Mz (t\r) is the renewal function associated with Fz {t\r). From (8.35) we have (8.41) As a result, the expected life cycle cost LCC(L; W, U) is given by (8.42) with E[CS{W, £/)] given by (8.17). Note that, in (8.42), E[K(L)\ is also a function of the warranty parameters W and U. 8 .4 .2

A n a ly sis o f P o lic y 9

The approach here is identical to that for Policy 8 except that x, is given by (8.43) instead of (8.37) (see Figure 8.2). As a result, LCC(L; W, U) is given by (8.42) with E[K{L)\ obtained from (8.41) using (8.43) instead of (8.37) and E[CS(W, t/)j given by (8.21) with E(N(W, U)] given by (8.24). 8 .4 .3

A n a ly s is o f P o lic y 10

The approach is similar to that in Section 8.4.1 for Policy 8 except that we need to consider the following four cases separately:

Chapter 8

324

1. 0 < r < yl 2. yl < r < y2 3. y2 ^ r < y3 4. 73 < r < 00 where 7I, 72, and 73 are given by (8.27). (See Figure 8.3.) The time at which the warranty ceases, is given by

(8.44)

with X rl and X r2 given by (8.25) and (8.26), respectively. As a result, the expected life cycle cost LCC(L; Wu W2, U1, U2) is given by

(8.45) with E[K{L)] obtained from (8.41) using (8.44) instead of (8.37) and £'[CS(W'1, W2, Uu U2)} given by (8.31). 8 .4 .4

A n a ly sis o f P o lic y 11

Here again, the approach is similar to that for Policy 8. X Y(see Figure 8.4) is given by (8.32). As a result, the expected number of repeat purchases over the product life cycle, E[K{L)\, is given by (8.41) but with xr = X r instead of by (8.37). The expected life cycle cost LCC(L; W, U) is given by (8.46) with £[CS(W, U)] obtained using (8.33) in (8.17). Comment: It is impossible to obtain, analytically, expressions for £[^(L )]. In practice, one would need to use some computational procedure to evaluate these expressions. A numerical scheme, in the case of Policy 8, would be to obtain E[K(L\r)\ (from (8.39) and (8.40)) for a range of values of r and then obtain £[iC(L)] from (8.41) using numerical integration. This would involve considerable computational effort. An alternate approach is via simulation; we discuss this in Chapter 11.

Two-Dimensional Warranty Policies 8 .5

325

FR EE-R EPLA C EM EN T P O LIC IES [TW O -D IM E N S IO N A L A PPRO A C H]

In this section we derive expressions for the expected cost of servicing warranty per unit sale with item failures modeled by a two-dimensional point process. We consider nonrepairable items, so that, whenever a failure occurs under warranty, the failed item is replaced at no cost to the buyer. We also assume that the time to replace is negligible. As such, twodimensional renewal theory plays an important role in the analysis. We commence with some preliminary mathematical results, which we shall use later in the analysis of Policies 8-11.

8.5.1 P re lim in a ry A n a ly sis

The age and usage of the ith item at failure is given by {Xh 7,) with a joint distribution function (8.47) Associated Univariate Processes

Define Stf* and S*2) as follows: (8.48) Here {Xt\ i > 1} is a sequence of independent and identically distributed random variables with distribution function Fx (x) = F(x, °°), the marginal distribution function of X. As such, it defines a univariate renewal process. Similarly, the sequence {7,; i > 1} defines another renewal process with distribution function FY(y) = F(oo, y), the marginal distribution function of 7. As indicated in Section 3.9, the analysis of the two-dimensional renewal process can be carried out in terms of the two univariate renewal processes generated by the sequences {^} and {7,}. The associated counting variables are (8.49)

Chapter 8

326 Expected Number of Renewals in [0, x) x [0, y)

Let N(x, y ) denote the number of renewals over [0, x) x [0, y). In Chapter 3 we showed that (8.50) and also derived the distribution of /Vfr, v}. Let M(x, y) denote the expected number of renewals over the rectangle [0, x ) x [0, y), i.e., M(x, y) = E[N(x, y)]. From Chapter 3, we have (8.51)

8 .5 .2

A n a ly sis o f P o lic y 8

Let 7V( = 7V(W, U)) be the number of failures under warranty. Since the warranty region is the rectangle [0, W) x [0, U), N is the same as N(x, y) given by (8.50) with x = W and y = U. Hence, the expected number of failures under warranty is given by (8.51) with x = W and y = U. As a result, the expected cost of servicing warranty per unit sale, E[CS(W, t/)], is given by (8.52) where M(W, U) is obtained from (8.51) and cs is the cost of each replacement. Example 8.10 [Beta Stacy Distribution] Suppose that the failure distribution is given by the Beta Stacy distribution (see Example 3.6), i.e., the density function/( jc , y) for (Xh Y,-), / > 1, is given by

where x > 0; 0 < y < and a, b , a, , 0l5 02 > 0. From Chapter 3 (Example 3.6), we see that the two parameters that affect the usage rate (defined by E[Yi]/E[Xi]) in a significant way are a and . We shall consider three different sets of values for a and corresponding to different usage intensities. Let the remaining parameter values

Two-Dimensional Warranty Policies

327

be as follows: and For a and ()>, the three different sets of values are Set (a): Set (b): Set (c):

a = 2.65865 a - 1.99395 a = 1.32930

and = 1.18180 and 4> = 2.36371 and = 4.72742

Note that the age of the item [X)\ and the usage [Y,] at failure are random variables. Their mean values for the above sets of parameter values are as follows: Parameter Set (a) Set (b) Set (c)

Mean age E[Xi] 4.0years 3.0years 2.0years

Mean usage £[Yt] 2.0( x 104 miles) 3.0(x 104 miles) 4.0( x 104 miles)

The mean usage at failure varies from 20,000 miles for set (a) to 40,000 miles for set (c), and the average age at failure varies from 4.0 years for set (a) to 2.0 years for set (c). As mentioned earlier, ^ [ Y j / f i ^ ] is an indicator of the usage intensity. This is smallest for parameter set (a) and largest for parameter set (c). As a result, we can interpret set (a) as corresponding to light usage, set (b) to moderate usage, and set (c) to heavy usage. Table 8.9 shows the expected number of failures under warranty for the three parameter sets and for different combinations of (W , U). These were obtained using a computational procedure, the details of which can be found in Iskandar [2]. A two-dimensional plot of these results is shown in Figure 8.5. For set (a), we see that the expected number of failures under warranty is very small (.0001) for W = 0.5 and U = 0.5. When U is increased with W unaltered, the value does not change. This is because of the low usage intensity. Very few users will have a usage that exceeds 0.5 units over a time period of 0.5 units, and hence there is no advantage to such users to have U increased beyond 0.5 with W = 0.5. On the other hand, increasing W from 0.5 and leaving U unaltered at 0.5 results in a larger value (.0165) for the expected number of failures under warranty, implying a better

328

Chapter 8

Two-Dimensional Warranty Policies

329

Figure 8.5 Plot of E[A] as a function of W and U [Policy 8; Beta Stacy distribution—parameter sets (a)-(c)].

warranty coverage for the consumer. This is to be expected, since most users are covered for failures over the entire period up to W = 2 and very few would have their warranty cease earlier due to exceeding the usage limit of 0.5. A better appreciation of this is obtained from Figure 8.6, which shows the contour plots for E[N(W, U)] for set (a). In this case, the contours go from .001 to .041 in steps of .005. For W < 0.85, the expected number of failures under warranty is < .0001, irrespective of U. This has significant implications from a marketing viewpoint. For example, the manufacturer, instead of offering a warranty with W = U = 0.5, can offer a warranty with W = 0.5 (i.e., failures covered for six months) and U —> 00 (i.e., no limit on the mileage). This can be used as a marketing tool provided the usage intensity of the consuming population is low, so that parameter set (a) is appropriate. Note that as the usage intensity increases, the contour corresponding to .0001 moves closer to the vertical axis. For the manufacturer to offer unlimited usage but have the expected number of failures under warranty less than .001, the manufacturer has to decrease the war-

330

Chapter 8

Table 8-9 Expected Number of Failures Under Warranty for Policy 8 [Example 8.10] w

Vv

u 0.50 1.00 1.50 2.00

0.50

1.00

1.50

2.00

.0001 .0002 .0007 .0001 .0003 .0013 .0001 .0003 .0018 .0001 .0003 .0020

.0013 .0027 .0080 .0020 .0051 .0165 .0020 .0069 .0245 .0020 .0078 .0316

.0061 .0109 .0258 .0110 .0219 .0537 .0135 .0316 .0812 .0137 .0394 .1075

.0165 .0258 .0461 .0316 .0529 .0969 .0425 .0782 .1478 .0479 .1008 .1975

Parameter Set Set Set Set Set Set Set Set Set Set Set Set

(a) (b) (c) (a) (b) (c) (a) (b) (c) (a) (b) (c)

Figure 8.6 Contour plots for E[iV] as a function of W and U [Policy 8; Beta Stacy distribution—parameter sets (a)-(c)].

Two-Dimensional Warranty Policies

331

332

Chapter 8

ranty period (W) from .85 to .66 if the usage intensity is characterized by set (b) and to .43 if the usage intensity is characterized by set (c). In general, the consuming population is comprised of people with different usage rates. If one divides the population into three groups (low, medium, and high) based on the intensity of usage, then one can model the warranty claims for the three groups by the three sets that have been mentioned. Let p a, p b, and p c (/?a + p b + Pc = 1) denote the fraction of low, medium, and high intensity users across the population. Then, with the warranty parameters W = U = 1, the expected warranty cost per unit to the manufacturer, say 8, is given by

where c is the cost of each replacement. If the manufacturer knows the fraction of the different types of users, the extra amount that the manufacturer should charge for items sold with warranty is 8. In this case, one sees that the manufacturer is breaking even in terms of the warranty cost, but consumers with low and medium intensities are subsidizing high intensity usage consumers. In general, the manufacturer does not have information about the fractions of different types of users. If the extra amount charged to all consumers buying the product with warranty is the warranty cost for high usage intensity— i.e., the manufacturer charges .0165c instead of 8— then the selling price of warrantied items goes up. The profits of the manufacturer also go up due to the fact that warranty costs for consumers with low and medium intensity usages are smaller than that for high intensity usage consumers. In addition, higher values for 8 can result in fewer sales. Example 8.11 [Multivariate Pareto Distribution of the Second Kind] Suppose that the failure distribution is given by the multivariate Pareto distribution of the second kind (see Example 3.8), i.e., the density function f i x , y) for (X,, Y,), i > 1, is given by

where x > y > 02>«i> «2 > 2, p2 < 1, and /0(-) denotes the modified Bessel function of order zero (see Abramowitz and Stegun [1]). A two-dimensional plot of E[N] as a function of W and U is shown in

Two-Dimensional Warranty Policies

333

Figure 8.7 Plot of E[N] as a function of W and U [Policy 8; multivariate Pareto

distribution].

Figure 8.7 and a contour plot is shown in Figure 8.8 for the following set of parameter values: and This set of values corresponds to the mean age and the mean usage (number

334

Chapter 8

Figure 8.8 Contour plots for £[7V] as a function of W and U [Policy 8; multivariate Pareto distribution].

of miles traveled) at failure to be .4 years and 3818 miles, respectively. Note that since f ( x , y ) is zero for x < 0Xor y < 02, EjA/] is zero for W < and U < 02. Table 8.10 shows for different combinations of (W, U). Example 8.12 [Bivariate Exponential Distribution] We consider the bivariate exponential distribution function (see Hunter [3]), i.e., the density function /(*, y) given by

Two-Dimensional Warranty Policies

335

Table 8-10 Expected Number of Failures Under Warranty for Policy 8 [Example 8.11] u

w 0.50 1.00 1.50 2.00

0.50

1.00

1.50

2.00

0.8329 1.2904 1.2300 1.2316

0.9249 2.0606 2.7844 2.8927

0.9395 2.1831 3.5403 4.3971

0.9441 2.2064 3.7153 5.0700

where In {•} is the modified Bessel function of the first kind of order n (see Abramowitz and Stegun [1]). Let /(/?, q) denote the two-dimensional Laplace transform of f ( x , y), i.e.,

For the bivariate exponential, this is given by

From this, M(/?, q), the Laplace transform of M(x, y), is found to be

On taking the inverse Laplace transformation, we have

where i

Chapter 8

336 with

E[7V] is obtained from M(jt, y) with x = W and y = U. 8.5.3

Analysis of Policy 9

Let N (= N(W, U)) denote the number of failures under warranty. Then N is related to the two univariate renewal processes and N^2) by the relation (8.53) which can also be rewritten as (8.54) with x = W and y = U. As a result, the expected number of failures under warranty is given by (8.55) where MX(W) and M y{U) are obtained from (8.56) and (8.57) and M (W , U) is obtained from (8.51). As a result, the expected warranty cost per unit sale is given by (8.58) Example 8.13 [Beta Stacy Distribution] We consider the three parameter sets of Example 8.10. Table 8.11 gives E[N(W, U)] for different combinations of W and U for the three parameter

Two-Dimensional Warranty Policies

337

Table 8-11 Expected Number of Failures Under Warranty for Policy 9 [Example 8.13] w

u 0.00 0.50 1.00 1.50 2.00

0.00

0.50

1.00

1.50

2.00

Parameter

.0000 .0000 .0000 .1540 .0976 .0708 .3416 .2141 .1544 .5502 .3419 .2453 .7750 .4791 .3421

.0001 .0003 .0020 .1540 .0977 .0728 .3416 .2141 .1564 .5502 .3419 .2473 .7750 .4791 .3442

.0020 .0080 .0487 .1547 .1029 .1188 .3416 .2170 .2018 .5502 .3430 .2922 .7750 .4792 .3888

.0137 .0487 .2311 .1616 .1354 .2938 .3443 .2410 .3690 .5505 .3591 .4519 .7750 .4884 .5416

.0467 .1527 .5314 .1862 .2244 .5763 .3587 .3139 .6320 .5565 .4164 .6954 .7758 .5310 .7660

Set (a) Set (b) Set (c) Set (a) Set (b) Set (c) Set (a) Set (b) Set (c) Set (a) Set (b) Set (c) Set (a) Set (b) Set (c)

sets. The values are larger than the corresponding values for Policy 8 because the warranty region for Policy 8 is a subset of the warranty region for Policy 9. The two-dimensional plots of E[N(W, £/)], as a function of W and [/, are shown in Figure 8.9, and the corresponding contour plots are shown in Figure 8.10. The figures are self-explanatory. Note that in this case either W or U can be zero, in which case the policy reduces to a one-dimensional policy and can be analyzed more easily by dealing with the appropriate marginal distribution function as opposed to the bivariate distribution function. The case where W = 0 corresponds to the policy being based solely on a usage limit. We shall compare the results of this policy with those of Policy 8 for parameter set (b) and W = U. The ratio of expected numbers of failures is a good indicator for the comparison. This ratio decreases roughly from 100 for W = U = 0.5 to 13 for W = U = 1.0, to 6 for W = U = 1.5, and to 4 for W = U = 2.0. This is to be expected, for the following reason. For small W and U, under Policy 8, the manufacturer’s obligations cease very early, with very few claims. Under Policy 9, the manufacturer has to service the warranty for a long time for some of the consumers due to very low usage rates and similarly service a certain number of consumers for

338

Chapter 8

Two-Dimensional Warranty Policies

339

Figure 8-9 Plot of £]7V] as a function of W and U [Policy 9; Beta Stacy distribution—parameter sets (a)-(c)]. very high total usage due to the time limit not being exceeded. As W and U increase, the number of such consumers decreases, and hence the ratio also decreases. If one divides the population into three groups (low, medium, and high) based on the intensity of usage as with the previous policy, then the expected warranty cost per unit to the manufacturer, 8, with W = U = 1, is given by

If the manufacturer has information about these fractions, then the premium for items sold with warranty must equal 8. Note that under this policy, consumers with high and medium intensities are subsidizing low intensity usage consumers. This is just the opposite of the previous policy. If the manufacturer does not have information about the fraction of different types of users and charges a premium equal to the warranty cost for consumers with high usage intensity (i.e., .2018c) instead of 8, then the selling price goes down. In this case, the manufacturer’s profits also

340

Chapter 8

Two-Dimensional Warranty Policies

341

Figure 8.10 Contour plots for E[N] as a function of W and U [Policy 9; Beta Stacy distribution—parameter sets (a)-(c)]. go down, because warranty costs for consumers with low and medium intensity usages will be larger than those for consumers with high usage intensity. 8.5.4

Analysis of Policy 10

Let N (= N(Wl9 W2, Ui, U2)) denote the number of failures under warranty. This quantity can be expressed in terms of failures over three rectangles (see Figure 8.3) and is given by the relation (8.59) The expected number of failures under warranty is given by (8.60) using M{x, y) from (8.51).

Chapter 8

342

As a result, the expected warranty cost per unit sale is given by

(8.61) Example 8.14 [Beta Stacy Distribution] We consider again the same set of parameter values. Table 8.12 shows £[N(W!, W2, U l9 U2)] for W1 = .5W2 and Ux = .5U2, and for a range of (W2, U2) combinations. We see that the expected number of failures under warranty is smaller than the corresponding values for Policy 8. This is to be expected, as the warranty region for Policy 10 is a subset of the region for Policy 8. The expected number of failures under warranty for Policy 10 is roughly half that under Policy 8. This has a significant impact on the selling price and on serving the interests of both low and high intensity usage consumers.

8.5.5

Analysis of Policy 11

Define + ( UIW)Yiy i > 1. {Tt \ i > 1} is a sequence of independent and identically distributed random variables with distribution function FT(t)

Table 8.12 Expected Number of Failures Under Warranty for Policy 10 (Wx = .5W2 and Ux = .5U2) U2 0.50 1.00 1.50 2.00

0.50

1.00

1.50

2.00

Parameter

W. Formally, the results are as follows: Case (1): 0 < t < W _ Analogous to a notation used previously, let H(t; W y U) = P(Z > t). Conditioned on the first failure occurring with X 1 = x and Y x = y, we have, from a fairly straightforward probabilistic argument,

On removing the conditioning, we have

(8.71) Case (2): W < t < oo Again conditioning on the first failure with X 1 = x and Y x = y, we have

Chapter 8

346

On removing the conditioning, we have

(8.72) The expected life cycle cost is given by (8.73) where MZ{L) is obtained from (8.70) using H(t) from (8.71) and (8.72), and £[C S(W, i/)] is given by (8.52). 8.6.2

A na lysis o f Policy 9

Note that in this case H(t) = 0 for t < W. For t ^ W , H(t) is obtained as follows. As before, by conditioning on the first failure with X 1 = x and Yi = y, we have

On removing the conditioning, we have

(8.74)

Two-Dimensional Warranty Policies

347

The expected life cycle cost is given by (8.75) where MZ(L) is obtained from (8.70) using H{t) from (8.74), and £[CS(W, t/)] is given by (8.58).

8.6.3

A na lysis o f Policy 10

We follow the same procedure. H{t\ W,, W2, Ux, U2) is obtained by considering the following three cases separately: Case (1): 0 < t < Wt As before, by conditioning on the first failure with = x and Yt = y, we have

Unconditioning, we have

(8.76)

Chapter 8

348

Case (2): W, < t < W2 Again, conditioning on the first failure with X x = x and

= v, we have

Unconditioning, we have

(8.77) Case (3): W2 ^ t < °° Again, conditioning on the first failure with

= x and Yx = y, we have

Two-Dimensional Warranty Policies

349

Unconditioning, we have

(8.78) The expected life cycle cost is given by (8.79) where MZ{L) is obtained from (8.70) using H{t) from (8.76)-(8.78), and E[CS(WU W2, Uu U2)] is given by (8.61).

8.6.4

A na lysis o f Policy 11

We follow the same procedure. We first consider the case W ^ t < °° and then the case 0 s K W. Define y = U/V. Case_(l): I f s K °o_ Let H2(t; W, U)) = H(t; W, U) denote the probability that Z > t. By conditioning on the first failure with X x = x and Y t = y, we have

Chapter 8

350

On removing the conditioning, we have

(8.80) Case (2): 0 < t < W Again conditioning on the first failure with X 1 = u and Y{ = y, we have

On removing the conditioning, we have

(8.81) where

W, U) is given by

(8.82) with 772(t; W, U) given by (8.80). The expected life cycle cost is given by LCC(L; W,U) = [MZ(L) + 1]E[CS(W, U)]

(8.83)

Two-Dimensional Warranty Policies

351

where MZ(L) is obtained from (8.70) using H(t) from (8.80) and (8.81), and E[CS{W, U)] is given by (8.67). Comment: It is impossible to obtain the expected life cycle costs analytically even for the exponential case. Direct computation would require the evaluation of multiple integrals, and this would involve considerable effort. An alternate approach is to obtain estimates using the simulation approach. This is discussed in Chapter 11. 8.7

PRO-RATA POLICIES [TWO-DIMENSIONAL APPROACH]

In Chapter 2, we defined a two-dimensional pro-rata warranty policy as follows: Policy 12 PRO-RATA REPLACEMENT POLICY: Under this policy the manufacturer agrees to refund the buyer a fraction of the original purchase price should the item fail before time W from the time of the initial purchase and the total usage at failure is below U. The fraction refunded depends on W — X l and U — Y1. Let the warranty region be denoted by il and let R(x, y) denote the refund conditional on X 1 = x and Y1 = y. Then R(x, y) is nonincreasing in x and y for (x, y) E il and 0 for (jc, y) £ ft. Different forms of il and R(x, y) define different policies. We consider three policies (called Policies 12c, 12d, and 12e), which are as follows: Policy 12c The warranty region il is given by

and the rebate function is given by (8.84)

where cb is the buyer’s purchase price per unit. Policy 12d The warranty region is given by

352

Chapter 8

and the rebate function /?(*, y ) is given by (8.85)

where

with flj and fl2 given by

and

Under this policy, the refund is full should the item failure correspond to a point in fix and partial if in il2. Policy 12e The warranty region il is given by

and the rebate function given by

(8 .86) We consider both nonrenewing and renewing rebate policies. In a nonrenewing rebate policy, the buyer is refunded an amount R(x, y) depending on the age and usage of the item at failure, and he has no obligation to buy a replacement unit. In a renewing rebate policy, the buyer is provided with a replacement item at a cost cb - R(x, y) should the item fail under warranty, and a new warranty is issued. Hence, in this case the warranty

353

Two-Dimensional Warranty Policies

ceases only when the original or a replacement does not fail in the warranty region. 8.7.1

Unified Approach

We consider a unified approach that allows us to treat the renewing and the nonrenewing rebate warranty policies as two special cases. If item /, j > 1, fails under warranty with X t = x and Yz =y, the buyer has the choice of either (1) buying a replacement item at cost cb - R( j c , y) or (2) obtaining a refund in the amount of R(x, y). Note that i = 1 corresponds to the first sale at full price, and values of i > 2 correspond to replacement items supplied at rebate.) If the buyer chooses alternative (1), the replacement item is covered by a new warranty identical to that of the original item. If the buyer chooses alternative (2), he receives a refund and the warranty is terminated. The warranty also terminates when an item fails outside the warranty period for the first time. We assume that the buyer’s decision to choose (1) or (2) occurs randomly. Let Ui denote the buyer’s decision at the failure of the /th item. Ui is a random variable defined as

The probability of J7f- = 1, conditional on X t =

jc ,

Yz — y, is given by (8.87)

Given {Xt = x, Y, = y}, f/z is independent of previous failure history. We assume that p(x, y) is nonincreasing in x and y and that 0 < p(x, y) < 1 if ( j c , y) E il and p(x, y) = 0 if ( j c , y) £ il. Two special cases of interest are 1. p(x, y) = 1 for ( j c , y) E il. This corresponds to the renewing rebate policy. 2. p(x, y) = 0 for ( j c , y) E il. This corresponds to the nonrenewing rebate policy. 8.7.2

Analysis [General]

In this section, we carry out the analysis to obtain the expected warranty cost per item sold at full price for the general case. Note that the warranty

354

Chapter 8

ceases if an item fails under warranty and the buyer does not buy a replacement, or if the item fails outside warranty for the first time. In other words, the warranty terminates whenever t/f- = 0 for i > 1. Define N ( = N(W, U)) as follows: ( 8 . 88)

N is the number of items needed to service the warranty (i.e., the original item plus the number of replacement items under warranty). Since (Xi9 Yi9 £/,), i > 1, is a sequence of independent trivariate random variables and since the event {N = n} depends only on the information {{Xi9 Yh Ui); i = 1 , 2 , . . . , n}, it follows (Ross [4], p. 58) that A is a stopping time for (Xh Yh U)), i = 1 , 2 , . . . . Let Q ( = Q(W, U; p )) denote the total refund for an item sold at full price. Q is equal to the total refund resulting from the original item and all replacement items. As such, it is given by (8.89) The expected value of Q is given by (8.90) and represents the expected warranty service cost per item sold at full price. Since A is a stopping time, using Wald’s equation (Ross [4], p. 58) we have (8.91) We now derive the expressions for £]A'] and E[R(X{, Yj)]. Since item failures occur independently with an identical warranty for each replacement item, we have (8.92) where (8.93)

Two-Dimensional Warranty Policies

355

This is a geometric distribution, and, as a result, we have (8.94) E[R(XU Yj)] is given by (8.95) (Note: p(x, y) and later.) 8.7.3

y ) are zero outside ft for the cases considered

Analysis [Policies 12c-12e]

In this section we carry out the analysis to obtain the expected warranty cost per unit sold. Policy 12c We assume p(x, y) to be of the form (8.96) This implies that the buyer’s decision to purchase a replacement for an item failing under warranty is not affected by the age and/or usage of the failed item. As mentioned earlier, p = 0 and 1 correspond to nonrenewing and renewing rebate policies, respectively. Using (8.96) in (8.93), we have, from (8.94), (8.97) Using (8.84) in (8.95), we have

(8.98)

356

Chapter 8

As a result, from (8.91), the expected warranty cost per item sold at full price is given by

(8.99) Special Cases p = 0 [Nonrenewing PRW]:

p = 1 [Renewing PRW]:

Example 8.16 [Beta Stacy Distribution] Suppose that the failure distribution is given by the Beta Stacy distribution. We consider the three sets of parameter values of Example 8.10. Tables 8.14 and 8.15 give the ratio E[Q]/ch for a range of W and U values for nonrenewing PRW (p = 0) and the renewing PRW (p = 1). The cost difference between the two is negligible for small W and U. As W and/or U becomes large, the difference becomes significant and the cost for the renewing case is greater than for the nonrenewing case for obvious reasons. Note that for W = 0.5, increasing U from 0.5 to 1.0 or more does not serve the consumer’s interest for reasons discussed in Example 8.10. On comparing Tables 8.9 and 8.14, we see that the expected costs under Policy 8 are always larger than those under nonrenewing Policy 12c (i.e., p = 0) for a given set of parameters, a given (W, U) pair, and c5 = cs. This is to be expected since the expected cost under Policy 8 > M(W, U) cs > F(W, U)cb > expected cost under Policy 12c. Policy 12d We assume p{x, y ) to be of the form

(8 .100)

Two-Dimensional Warranty Policies

357

Table 8.14 E[Q\/cb for Nonrenewing PRW Policy 12c

w

u 0.50 1.00 1.50 2.00

0.50

1.00

1.50

2.00

.00001 .00002 .00007 .00001 .00003 .00013 .00001 .00004 .00019 .00001 .00004 .00023

.00013 .00027 .00087 .00022 .00054 .00180 .00027 .00075 .00270 .00029 .00090 .00353

.00064 .00118 .00317 .00119 .00240 .00665 .00157 .00352 .01012 .00178 .00448 .01347

.00179 .00303 .00670 .00352 .00627 .01413 .00489 .00936 .02162 .00585 .01220 .02899

Parameter Set Set Set Set Set Set Set Set Set Set Set Set

(a) (b) (c) (a) (b) (c) (a) (b) (c) (a) (b) (c)

with p x < p 2. This implies that the buyer is more likely to buy a replacement with refund if failure occurs in il2 than in IV The special cases resulting are as follows: 1. Pi = p 2 = 1. This corresponds to renewing rebate policy with 100% refund in the first region and partial refund in the second region. Table 8.15 E[Q]!cb for Renewing PRW Policy 12c W

U 0.50 1.00 1.50 2.00

0.50

1.00

1.50

2.00

.00001 .00002 .00007 .00001 .00003 .00013 .00001 .00004 .00019 .00001 .00006 .00023

.00013 .00027 .00088 .00023 .00054 .00183 .00027 .00075 .00277 .00029 .00091 .00364

.00064 .00119 .00326 .00121 .00246 .00703 .00159 .00364 .01102 .00180 .00467 .01508

.00182 .00311 .00703 .00363 .00662 .01564 .00511 .01016 .02535 .00615 .01356 .03606

Parameter Set Set Set Set Set Set Set Set Set Set Set Set

(a) (b) (c) (a) (b) (c) (a) (b) (c) (a) (b) (c)

358

Chapter 8

2. Pi = Pi = 0- This corresponds to nonrenewing rebate policy with 100% refund in the first region and partial refund in the second region. 3. Pi = 0 and p 2 = 1. This corresponds to a combined policy— nonrenewing rebate warranty with 100% refund in the first region and renewing rebate warranty with partial refund in the second region. Using (8.100) in (8.93), we have, from (8.94),

(8 .101) From (8.95), we have E[R(XU Yx)] given by

(8 . 102) where 4>(jc, y) is the same as in (8.85). As a result, from (8.91), the expected warranty cost per item sold at full price is given by

(8.103) Special Cases 1- Pi = Pi = 1:

2. Pi = p 2 = 0:

359

Two-Dimensional Warranty Policies

3. Pi = 0 and p 2 = 1:

Example 8.17 [Beta Stacy Distribution] Suppose that the failure distribution is given by the Beta Stacy distribution. We consider the three sets of parameter values of Example 8.10. Let Wx = .5W2 and Ux = ,5U2. Tables 8.16 and 8.17 give the ratio E[Q\/ cb for a range of W and U values for the nonrenewing PRW (px = p 2 = 0) and the renewing PRW (px = p2 = 1). On comparing Table 8.16 with Table 8.14, we see that the expected cost for Policy 12c is always smaller than the corresponding cost for Policy 12d. This is because in Policy 12d the refund is full when the item fails early in its life, whereas it is always less than full under Policy 12c. Policy 12e We assume p(x, y) to be of the form given by (8.96). Using this in (8.93), we have, from (8.94), (8.104)

T a b le 8.16 Ux = .5 U2) U2 0.50 1.00 1.50

2.00

E[Q\/cb for Nonrenewing PRW Policy 12d (Wx = .5W2 and

0.50

1.00

1.50

2.00

.00002 .00004 .00017 .00002 .00007 .00033 .00002 .00008 .00045 .00002 .00008 .00052

.00032 .00067 .00214 .00051 .00130 .00441 .00055 .00178 .00657 .00057 .00206 .00853

.00156 .00290 .00775 .00286 .00587 .01620 .00358 .00853 .02456 .00379 .01073 .03259

.00440 .00742 .01616 .00852 .01527 .03399 .01160 .02270 .05190 .01334 .02940 .06947

Parameter Set Set Set Set Set Set Set Set Set Set Set Set

(a) (b) (c) (a) (b) (c) (a) (b) (c) (a) (b) (c)

Chapter 8

360

Table 8-17 E[Q\lch for Renewing PRW Policy 12d (W1 = .5W2 and Ux = .5 U2) U2 0.50 1.00 1.50 2.00

0.50

1.00

1.50

2.00

.00002 .00004 .00017 .00002 .00007 .00033 .00002 .00008 .00045 .00002 .00008 .00052

.00033 .00067 .00216 .00052 .00131 .00449 .00055 .00179 .00674 .00057 .00207 .00881

.00157 .00293 .00795 .00289 .00600 .01712 .00363 .00881 .02672 .00385 .01117 .03651

.00447 .00762 .01694 .00880 .01612 .03762 .01212 .02462 .06085 .01401 .03269 .08641

Parameter Set Set Set Set Set Set Set Set Set Set Set Set

(a) (b) (c) (a) (b) (c) (a) (b) (c) (a) (b) (c)

where F^x) and F2(y) are the marginal distribution functions of F(x, y ). E[Q\ is obtained using (8.95) and (8.104) and is given by

(8.105) As in Policy 12c, p = 0 and p = 1 correspond to nonrenewing PRW and renewing PRW, respectively. Example 8.18 Suppose that the failure distribution is given by the Beta Stacy distribution. We consider the three sets of parameter values of Example 8.10. Tables 8.18 and 8.19 give the ratio E[Q\/ch for a range of W and U values for the nonrenewing PRW (p = 0) and the renewing PRW (p = 1). On comparing Tables 8.18 and 8.11, we see that the expected costs under Policy 9 are greater than the corresponding costs under Policy 12e, for reasons discussed in Example 8.17, when cb = cs. 8.8

ADDITIONAL TOPICS

In this section we list some extensions and additional topics for further study.

361

Two-Dimensional Warranty Policies

Table 8.18 E[Q\/cb for Nonrenewing PRW Policy 12e

w

U 0.50 1.00 1.50 2.00

0.50

1.00

1.50

2.00

.06919 .04472 .03296 .14471 .09448 .06963 .21925 .14492 .10723 .29065 .19488 .14502

.06932 .04563 .04024 .14472 .09498 .07547 .21925 .14510 .11169 .29065 .19490 .14821

.07055 .05170 .07561 .14519 .09946 .10668 .21930 .14811 .13874 .29065 .19662 .17118

.07517 .07004 .15133 .14791 .11451 .17619 .22043 .15995 .20193 .29081 .20544 .22807

Parameter Set Set Set Set Set Set Set Set Set Set Set Set

(a) (b) (c) (a) (b) (c) (a) (b) (c) (a) (b) (c)

Table 8.19 E[Q]/ch for Renewing PRW Policy 12e W

U 0.50 1.00 1.50 2.00

1. 2.

0.50

1.00

1.50

2.00

.08083 .04933 .03544 .20547 .11740 .08144 .39082 .20582 .13765 .67281 .32093 .20602

.08105 .05063 .04518 .20549 .11844 .09165 .39082 .20640 .14794 .67281 .32105 .21579

.08316 .05952 .10405 .20695 .12786 .15715 .39106 .21561 .22025 .67281 .32885 .29474

.09125 .08981 .33344 .21525 .16240 .41114 .39733 .25399 .50164 .67445 .36993 .60609

Parameter Set Set Set Set Set Set Set Set Set Set Set Set

(a) (b) (c) (a) (b) (c) (a) (b) (c) (a) (b) (c)

Our pro-rata warranty models are based on the two-dimensional approach to modeling item failures. One can also build models based on a one-dimensional approach. The study of more complex warranty policies— e.g., two-dimensional combined policies— is an open topic for research.

Chapter 8

362 3. 4.

In the case of one-dimensional policies, there is considerable work done on optimal warranty servicing strategies (see Chapter 9). In contrast, no such work has been carried out for the two-dimensional case. The analysis of warranty policies with different types of repair is also an open topic for research.

NOTES

Section 8.2 1. The one-dimensional approach, in the context of warranties, was first proposed by Moskowitz and Chun [5]. Their formulation is a special case of a more general formulation given by Murthy and Wilson [6]. Sections 8.3 and 8.4 1. The results of this section are based on Murthy and Wilson [6] and Murthy et al. [7]. Sections 8.5 and 8.6 1. The results of this section are based on Murthy et al. [8, 9]. The use of simulation for estimating life cycle costs is discussed in Chapter 11. 2. We have confined our analysis to warranties with warranty regions as shown in Figures 8.1-8.4. One can study warranties with more complex regions as well; the complexity of the analysis would depend on the shape of the region. Wilson et al. [10], deal with one such warranty, the analysis of which involves a two-dimensional delayed renewal process. 3. For more on two-dimensional distributions suited for modeling, see Johnson and Kotz [11]. 4. Singpurwalla [12] models failures using a two-dimensional Poisson process. Section 8.7 1. The results of this section are based on Iskandar et al. [13]. EXERCISES

8.1.

8.2.

[Policy 8a] Consider a two-dimensional FRW policy with warranty region as shown in Figure 8.11. Using the one-dimensional approach to modeling failures, obtain the seller’s expected cost per unit for the case where failures are repaired minimally. Compare the results with those for Policy 8. [Policy 9a] Consider a two-dimensional FRW policy with warranty region as shown in Figure 8.12. Using the one-dimensional approach to modeling failures, obtain the seller’s expected cost per unit for the case where failed items are repaired minimally. Compare the results with those for Policy 9.

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363

Figure 8.11 Warranty region for Policy 8a. 8.3.

[Policy 8a] Consider a two-dimensional FRW policy with warranty region as shown in Figure 8.11. Using the two-dimensional approach to modeling failures, obtain the seller’s expected cost per unit for the case where failed items are always replaced by new items. Compare the results with those for Policy 8.

Figure 8.12 Warranty region for Policy 9a.

364

8.4.

8.5. 8.6. 8.7.

8.8.

Chapter 8

[Policy 9a] Consider a two-dimensional FRW policy with warranty region as shown in Figure 8.12. Using the one-dimensional approach to modeling failures, obtain the seller’s expected cost per unit for the case where failed items are replaced by new items. Compare the results with those for Policy 9. Consider the case where items are nonrepairable and sold with FRW Policy 8a. Obtain the seller’s expected life cycle cost based on item failures being modeled by the two-dimensional approach. Consider the case where items are nonrepairable and sold with FRW Policy 9a. Obtain the seller’s expected life cycle cost based on item failures being modeled by the two-dimensional approach. In the case of automobiles, sometimes a failure is fixed by the user or by a private mechanic, rather than by the dealer. The seller incurs no cost when this happens. Consider FRW Policy 8, with item failures modeled by the one-dimensional approach. Conditional on the usage rate being R = r, let p r denote the probability that a failure is fixed by the buyer at no cost to the seller. Calculate the seller’s expected cost per unit. A failure close to the end of the warranty period often results in no claim because the buyer feels that it is not worth the effort. One can model this, for FRW Policy 8, as in Figure 8.13. Suppose that a failure occurring in Region II indicated in Figure 8.13 results in a claim with probability p and no claim with probability 1 - p, and

Figure 8.13 Claim and no-claim regions for Policy 8.

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365

that all failures in Region I result in a warranty claim. Using the one-dimensional approach, calculate the seller’s expected cost per unit. 8.9. Repeat the analysis of Exercise 8.8 for PRW Policy 12. 8.10. Many items (e.g., automobiles, appliances) are not used continuously over the warranty period. Consider the case where the item is used periodically, i.e., it is in use for a period 7\ and idle for a period 72, with this pattern repeating itself. When the item is idle, there is no usage and we assume that the item does not deteriorate when idle. Suppose that items are repairable and are sold with FRW Policy 8. Assume that Tx and T2 are much smaller than W. Using the one-dimensional approach and assuming that failed items are always minimally repaired, obtain an expression for the seller’s expected cost per unit. REFERENCES

1. 2. 3. 4. 5. 6. 7. 8. 9.

Abramowitz, M., and Stegun, I. A. (1965). Handbook o f Mathematical Functions, National Bureau of Standards Applied Math. Series, No. 55, U.S. Govt. Printing Office, Washington, D.C. Iskandar, B. P. (1991). A Two Dimensional Renewal Function Solver, Research Report No. 4/91, Department of Mechanical Engineering, The University of Queensland, St. Lucia, Australia. Hunter, J. J. (1974). Renewal theory in two dimensions: Basic results, Adv. Appl. Prob., 6, 376-391. Ross, S. M. (1972). Applied Probability Models with Optimization Applications, Holden-Day, San Francisco. Moskowitz, H., and Chun, Y. H. (1988). A Bayesian Approach to the Two-Attribute Warranty Policy, Paper No. 950, Krannert Graduate School of Management, Purdue University, West Lafayette, IN. Murthy, D. N. P., and Wilson, R. J. (1991). Modelling two dimensional warranties, in Proc. 5th. Int. Conf. on Appl. Stoch. Models and Data Analysis, Granada, Spain, pp. 481-492. Murthy, D. N. P., Wilson, R. J., and Iskandar, B. P. (1993). Twodimensional failure free warranties: one-dimensional point process models; in preparation. Murthy, D. N. P., Wilson, R. J., and Iskandar, B. P. (1993). “Two dimensional warranty policies: A mathematical study,” in Proc. Tenth ASOR Conf. , Perth, Australia. Murthy, D. N. P., Iskandar, B. P., and Wilson, R. J. (1990). Twodimensional failure free warranties: Two-dimensional point process models, Operations Research, under review.

366

Chapter 8

10.

Wilson, R. J., Murthy, D. N. P., and Iskandar, B. P. (1993). An extension to 2-D failure free warranty, in preparation. Johnson, N. L., and Kotz, S. (1972). Distributions in Statistics: Continuous Multivariate Distributions, John Wiley and Sons, Inc., New York. Singpurwalla, N. D. (1987). A Strategy for Setting Optimal Warranties, Report GWU/IRRA/Serial TR-87/4, The George Washington University, Washington D.C. Iskandar, B. P., Murthy, D. N. P., and Wilson, R. J. (1991). “Two dimensional rebate policies,” in Proc. Eleventh ASOR Conf., Gold Coast, Australia.

11. 12. 13.

9

Warranty Servicing

9.1

INTRODUCTION

When a product is sold with warranty, the manufacturer has to service it as per the terms stated in the warranty policy. This consists of actions such as providing a refund or replacing a failed item by a new one in the case of nonrepairable items and either repairing or replacing a failed item in the case of repairable items, given that the failure occurred while the item was covered under warranty. These actions are called warranty servicing. Earlier chapters dealt with warranty servicing for a single item, using simple service strategies— either always repair or always replace. In this chapter we focus our attention on warranty servicing of multiple items sold either as a single lot or continuously over the product life cycle, more complex servicing strategies involving a choice between repair and replacement, and some related problems. The outline of the chapter is as follows. Sections 9.2 and 9.3 deal with warranty servicing for nonrepairable items. In Section 9.2, we study the nonrenewing pro-rata warranty, where the manufacturer offers an unconditional rebate when items fail under warranty. In order to meet this payment, the manufacturer must set aside a fraction of the revenue generated by sales. This is called warranty reserving. We examine warranty reserving for both a single lot sales and for continuous sales over the product life cycle. In Section 9.3, we study the free-replacement warranty and obtain expressions for the expected number of replacements needed over time to service the warranty. This section is closely related to Chapters 4 and 5 but differs in that here we deal with multiple sales, either as a single lot or distributed over time, whereas in Chapters 4 and 5 we confined our attention to single-item sales. For repairable products sold with free-replacement warranty, the manufacturer has the option of either repairing a failed item or replacing it 367

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with a new one. In Section 9.4, we study the case where the failed item is always repaired. We derive expressions for the expected number of repairs over time. In Section 9.5, we study a variety of repair-replacement strategies with relatively simple structure. The objective is to formulate a strategy that will minimize the expected cost of servicing the warranty. In most applications, the repair cost is uncertain. In these cases, repair/ replace decisions must be based on estimates of these costs. In what is called a repair limit policy, a failed item is repaired only if the estimated cost of repair is below a specified limit. In Section 9.6, we study an optimal repair limit policy to minimize the expected cost of servicing the warranty. Finally, in Section 9.7, we discuss a variety of topics of relevance to servicing warranty. These include the following: 1. When a failed item is returned to the manufacturer for repair under warranty, the buyer is deprived of its usage for a period of time (called repair service time). This period consists of the waiting time for repair plus the time to repair and test before the repaired item is returned to the buyer. If the repair service time is large, it can result in customer dissatisfaction. Also, for certain products (e.g., mainframe computers used in banking or airline operations, major components of aircraft) the warranty terms may include a penalty if the repair service time exceeds the specified value stated in the warranty policy. The repair service time can be reduced by (1) increasing the number of repairmen; (2) use of loaners; and (3) proper management of inventory of new and repaired items. These issues are closely related to the topics of queues and logistics. 2. In many cases, not all claims are exercised, and this affects the warranty servicing costs. 3. Occasionally, a manufacturer is forced to recall all items belonging to one or more batches due to manufacturing defects or to recall all items produced because of a design fault. Potential warranty costs may be a factor in such decisions. 4. An interesting situation arises when the warranty servicing is done by a third party, e.g., retailers of the product or a servicing agency. This raises many additional issues, such as fraud, incentives, etc. We implicitly assume that all warranty claims are exercised (except for a subsection of Section 9.7) and that the time to repair or replace is negligible unless stated otherwise. We use the same notation as in previous chapters, introducing new notation as and when needed. 9.2

WARRANTY RESERVES [PRW POLICIES]

In this section we consider nonrepayable items sold with a nonrenewable pro-rata warranty policy, under which the manufacturer refunds a fraction

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of the purchase price should the item fail within the warranty period. We first consider a single lot sale and later the case where sales occur continuously over the product life cycle. In a single lot sale, the manufacturer sells a lot of N items, which are put into use immediately. Examples of this range from fleet purchases of automobiles, buses, and so forth, to the sale of bulbs to replace all the lights in a factory at regular intervals. In the case of continuous sales, we assume that the items are put into use immediately after purchase. Note that in both cases each item is covered by an individual warranty. As a result, the analysis in this section makes extensive use of the results of Chapters 4 and 5. 9.2.1

Single Lot Sales

Basic Formulation

Let Y, denote the refund for the /th item in the lot. We assume that the refund is linear, i.e., (9.1) where is the age of item i at failure. The mean |x(W) and the variance d 2(W) of Yi are given by (9.2) and (9.3) with

given by

Since items fail independently, the total warranty refund, TR(W), is a random variable given by (9.4)

370

Chapter 9

To meet these payouts, the manufacturer must set aside a sum R(W) as warranty reserves from the revenue generated from the sale of the lot. If R(W) is selected to equal the total expected refund, then (9.5) This implies that the manufacturer must set aside a fraction y of the sale price of each item to create the warranty reserve; y is given by (9.6) Note that the reserve will not cover the total payout if TR(W) exceeds the reserve amount R(W). In order to evaluate the probability of this happening, we require the distribution function for TR(W). This can be obtained in terms of the convolutions of the distribution functions of Yf, which in turn are related to the distribution functions of the X t. Although in principle this is straightforward, it is analytically intractable for even the simplest form of F(x). However, if N is large and we assume that items fail independently, then by the Central Limit Theorem, TR(W) is approximately normally distributed with mean and variance given by A|x(W) and Mx2(W), respectively. In this case, selecting reserve R(W) = A jjl (W) implies that the probability of the total payout exceeding the reserve is approximately .5. By increasing R(W), this probability can be reduced. The argument is as follows. The normal approximation can be used to obtain a 100(1 - a)% prediction interval for TR(W). This is given by (9.7) where za/2 is the a/2-fractile of the standard normal distribution. It follows that if the warranty reserve R(W) is chosen according to the relation (9.8) then by proper choice of e (> 0), the probability of the total payouts exceeding the reserves can be made as small as desired. However, this implies setting aside a larger fraction of the revenue generated by sales to service the warranty.

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Warranty Servicing

Example 9.1 [Exponential Distribution] Suppose that the failure distribution is exponential with parameter X, i.e.,

We take the unit for time to be years, so that X has dimension (year)-1. From (9.2) and (9.3) we have

and

Note that in both |x(W) and a(W ), X and W appear as a product. Hence, the warranty reserve is a function of this product. Let N = 100 and ch = $10. Values of R(W) (in dollars) obtained from (9.8) with s = 0 for two sets of X and W (in years) are shown in Table 9.1. Note that the mean time to failure is 2 years for X = .5 and 10 years for X = .1. The ratio of the expected warranty service costs between a high reliability product sold with shorter warranty (X = .1 and W = 1) and a low reliability product sold with longer warranty (X = .5 and W = 2) is roughly 8. Table 9.2 shows R(W) and P(e), the probability of the payout TR(W) exceeding the warranty reserve R(W ), as a function of e for the case X = .5 and W = 1 year. Note that as e increases, P(e), the probability that the Table 9.1 Warranty Reserve R(W) (in $) [Lot Size = 100; Exponential Failure Distribution; Example 9.1]

W (years)

X (year-1)

1

2

0.10 0.50 1.00 2.00

48 213 367 567

213 367 567 754

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Chapter 9

Table 9.2 Warranty Reserve R(W) and Probability of Payouts Exceeding Warranty Reserves P(s) vs. e [Example 9.1] 8 R(W) P(e)

0.0

213 .500

0.1

221 .460

0.5 252 .301

1.0

292 .159

1.5 331 .067

2.0 370 .023

2.5 409 .005

3.0 449 .001

warranty reserve is not adequate to meet the payout, decreases, as expected. Investment of Warranty Reserves

In the previous subsection, the warranty reserves /?(W) are set aside at the time of the sale although payouts occur over a period W subsequent to the sale. The reserves can be invested (e.g., in the short-term money market or as interest bearing deposits) to generate income. This implies that the total amount of funds to be set aside as reserves is smaller than it would be if the reserves were not invested. In order to compute the reserves needed, a model for returns on investments is required. We consider two model formulations for this purpose. Let I(t) denote the value at time t of an investment of value I0 at time t = 0. For Model I, I(t) is given by (9.9) and for Model II, it is given by (9.10) where (> 0) is the rate of return on investment. Let Ri(W; ), the reserves set aside (and invested), be selected to equal the expected total payout over the warranty period. For Model I, the payout 1 - x/W for an item failing at time x (0 < x < W) can be met by an amount (1 - xl W) exp(-x) of the initial reserves invested for a period r. A sa result, it is easily seen that (9.11) for Model I and (9.12)

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for Model II. Note that i?i(W; ) is a function of and decreases as 0 for 0 < a < T, then the following hold: 1.

2.

If crF(W) < csG(W), then a. If B(T; W) > cs/cr, then there exists a unique a* < T such that the optimal replace/repair strategy is to replace for failures in (0 , T - a*] and to repair for failures after T — a*. Furthermore, a* is the solution of B(a; W) = cjcr. b. If B(T, W) < cs/cr, then the optimal replace/repair strategy is always to repair. If crT(W) > csG(W), then the optimal replace/repair strategy is always to replace. ■

In general, it is difficult to test the condition B'{a; W) > 0. The following proposition gives a simple condition to ensure that B '(a; W) > 0. Proposition 9.2 (Nguyen and Murthy [3]) If g(t) > f{t) for 0 < t < T + W, then B'(a\ W) > 0 for 0 < a < T. ■ Example 9.12 [Weibull Distribution] Let F(t) and G(t) be Weibull distributions with the same shape parameter, i.e.,

Chapter 9

398

and

Suppose that X = 1.0, \0 = 1.4, and p = 1.6. This corresponds to a mean time to failure of 0.8966 for new items and 0.6404 for repaired items. Let T = 1.0 year. Figure 9.4 shows B(a; W) as a function of a for various choices of W. It is seen that B '(a ; W ) > 0 f o r 0 < a < T and W < 0.5. Hence, using Proposition 9.1, the optimal repair/replace strategies can be obtained. A numerical approach is necessary to obtain a*. Table 9.10 shows the optimal repair/replace strategy for a range of cjcs and W values. For W = 0 (Policy 1), the optimal strategy is to always repair for cr/cs < .75. For cr/cs > .80, the optimal strategy is to replace all failures occurring in the interval [0, T - a*) and to repair all failures occurring in the interval [T - a*, T). Note that a* decreases as cr/cs increases and becomes zero (that is, always replace) when cr/cs = 1. This is intuitively obvious when the failure rate is increasing, as is the case for the Weibull distribution with p > 1. When cr/cs > 1, the optimal strategy is always to replace. This is as expected, for if the cost of repair is more than the price of a new unit and repaired items are inferior to new ones, then it makes

F ig u re 9-4

Plot of B(a; W) vs. a [Example 9.12].

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Warranty Servicing

Table 9.10 Optimal Repair/Replace Decision [Repaired Items Identical but Different from New Items; Example 9.12] .70 .75 .80 .85 .90 .95

Always repair Always repair

Always repair a* = .72 a* = .41 a* = .22

Always repair a* = .19

II

o

W = 0.50

p*

W = 0.25

II

a* = .50 a* = .35

q

W = 0.00

* 3

cr/cs

o

II

f} is given by (9.74)

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403

from which we have the conditional expectation (9.75) The expected value of N2(W) can be obtained by using a renewal theoretic argument, conditioning on U1. Note that

(9.76) Using (9.73) and (9.75) in (9.76) and removing the conditioning, we have

which can be simplified to yield (9.77) This completes the proof. 9.6.2

Expected Warranty Service Cost per Unit Sale

Note that since repair is carried out only when the estimated cost of repair is less than ft, the expected cost of each repair carried out, £, is given by

(9.78) The failed item always being repaired, irrespective of the repair cost, corresponds to ft being ». In this case, the expected cost of repair is given by (9.79) and it is easily seen that

CT > 1 .

404

Chapter 9

Let oo(ft; W) denote the expected warranty service cost per unit sale. Since each replacement costs cs and the average cost of each repair is £, we have (9.80) Using (9.71), (9.77), and (9.78) in (9.80), we have, after some simplification,

(9.81) Optimal ft

The optimal ft, if it exists, is the value of ft that minimizes a>(ft; W). This can be obtained from the usual first-order condition given by (9.82) In general, it is difficult to obtain ft* analytically. One needs to use a computational scheme to obtain ft*, using (9.82). The following proposition gives bounds for ft*: Proposition 9.4 (Murthy and Nguyen [5]) 1. 2.

If F(t) if IFR, then 0 < ft* < cs. If F(t) if DFR, then ft* > cs.

The proposition states that if the failure rate is decreasing, then it is worth spending more than the replacement cost for repair, as the repaired item has a smaller failure rate and hence is more reliable than a new item. However, this situation happens seldom in the real world. For increasing failure rate, repaired items are less reliable than new items, and hence the optimal repair limit must be less than the price of a new unit. Example 9.13 [Weibull Distribution] We consider the Weibull distribution with parameters X and p, i.e.,

Suppose we take p = 2 and X = 0.886. This results in F(t) having an increasing failure rate with mean time to failure of 1.0. Let cs = 1.0.

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405

Suppose that the repair cost distribution H (z) is also a Weibull distribution with parameters 0 and X. We consider the following three sets of parameter values. a. b. c.

0 = 0.5 and X = 2.0 This implies a decreasing repair cost rate. 0 = 1.0 and X = 1.0 This implies a constant repair cost rate. p = 3.0 and X = 0.883. This implies an increasing repair cost rate.

The preceding values of X were chosen so that the expected repair cost cr is equal to 1. The corresponding values of £ will depend on ft, but they are always less than cr. Figure 9.6 shows the optimal ft (ft*) as a function of W for these three sets of parameters values. These were obtained using a computational scheme to find the solution to (9.50) by an iterative gradient method. Note that ft* is always less than cs (= 1), as is to be expected. Also, note that ft* decreases with increasing W. The results imply that the longer the

Figure 9.6 Plot of ft* vs. W [Example 9.13].

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406

warranty period, the smaller the repair limit, which in turn implies that the failed unit is more often replaced by a new. This is to be expected. We compare the optimal repair limit replacement policy with the following two service strategies: 1. 2.

Always repair: This corresponds to £ = o°, and the expected service cost is given by co(oo; W). Always replace: This corresponds to £ = 0, and the expected service cost is given by co(0; W).

If F is IFR, then it is easily seen that o>(0; W) < o)(°°; W) for all W. Hence, we compare the expected cost W) with co(0; W). Define the percentage savings, t i(W), as

Figure 9.7 shows t i(VF) for different p as a function of W. For a given p, decreases with W , as is to be expected. For a given W, t j(W) decreases as p increases, as is to be expected for the following reason. When p is less than 1, the repair cost rate is decreasing. The smaller the value of p, the greater is the probability of the repair being carried out at a small cost. When p is greater than 1, the repair cost rate is increasing. This implies that the probability of the repair cost being small decreases with increasing P, and hence the advantage of repair over replacement is reduced. Thus, ri(W) decreases as p increases. t ](W)

9.7 ADDITIONAL TOPICS

In this section we discuss a variety of topics of relevance to warranty servicing. 9.7.1

Repair Facility Planning

Repair facility planning involves issues such as the number of repairmen needed to service the warranty, inventory of components for repair replacements, and strategies to reduce the mean repair service time. We give an overview, indicate some of the salient features of different problems, and relate them to relevant literature. Readers should consult the notes at the end of the chapter for relevant references.

407

Warranty Servicing

Figure 9.7

Plot of t i(W )

v s.

W [Example 9.13].

The Repairman Problem

The repair facility can be viewed as a service station with failed items arriving randomly over time. The repair time can be either deterministic or random. In either case, the service facility can be viewed as a queuing system with single input and the number of servers equal to the number of repairmen. The statistical characterization of the arrival rate of items depends on the number of items in use and the individual ages of these items. Since these change randomly, the arrival rate changes stochastically and is not easily characterized. This makes it impossible to use queuing models, since they deal with simpler characterizations of the arrival rates. However, the problem is similar to a machine-interference problem where a group of machines break down repeatedly and are made operational by repairing failed machines. Service repair time depends on the arrival rate and the number of repairmen. If the number is small, then the service repair time increases.

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Increasing the number of repairmen decreases this at the expense of additional labor cost. Determining a sensible trade-off between the service repair time and the cost of labor can only be done through simulation studies. Another interesting feature of warranty service is that the demand on the service facility changes over the product life cycle, as illustrated in Examples 9.3 and 9.4. This implies that to optimize the repair facility operation, one should have either a variable size labor force or use an overtime option. Also, for many heavy and nonportable items, the repairman has to travel to different locations to provide warranty service. This adds an extra dimension to operations management. The literature on the traveling repairmen problem is highly relevant to the analysis of this situation. Inventory of Spares

As seen from Example 9.3, the expected demand for spares varies over the product life cycle. The inventory level of spares depends on the frequency of replenishment and the safety-stock level selected. This frequency, in turn, can vary over the life cycle. A simple inventory policy is the following: The inventory is updated every A units of time. The quantity added to the inventory is selected so that the level of inventory after receipt of the order equals the demand at the next ordering point plus the safety stock at some selected level. The safety stock must be sufficient to ensure that the probability of the inventory being reduced to zero between ordering points is small. The extensive literature on inventory control is relevant to this problem. If the item is not repairable, the manufacturer has to carry new items for use as replenishments. The need for replacements occurs randomly, and the expected demand changes over the product life cycle. Determining the level of replacement items to be held in stock is similar to the stocking of spares discussed earlier. Often components are repairable. In this case, when a failed component is replaced by a new one, the manufacturer can repair the failed item at a later time and add it to the inventory of repaired spares. As a result, there is an inventory of new spares and an inventory of repaired spares. The optimal levels for each are critical for minimizing the expected cost of operating the repair service facility. There are many papers under the heading of “Logistics” that deal with the problem of spares (new and used) for the effective maintenance of systems and are relevant to analysis of the operation of the repair service facility. Our models give estimates for the mean number of repairs over time. If the item has j components, then one can build a model for the distribution

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of different components that need to be replaced whenever the item fails. From this model, one can evaluate the mean number of the different types of components needed to service the warranty. Finally, some items (or components) tend to deteriorate when in storage. In this case, we have an extra dimension added to the decision problem. The literature on inventories or perishable items is of some relevance. Use of Loaners

For certain expensive items, the downtime cannot exceed some specified value. Often the warranty contract specifies a penalty should this happen. One way for the manufacturer to reduce the probability of this happening is to have a stock of loaners, which are issued to the owners of failed items when they are undergoing repair. This implies additional servicing costs, and the manufacturer must determine the optimal number of loaners to be held in stock. Again, because of the complexity of the model needed to study this problem, the optimal number can only be determined by use of simulation studies. 9.7.2

Warranty Claims Not Exercised

Up to this point, we have considered the servicing of items that fail under warranty and implicit in our analysis is the assumption that all claims under warranty are exercised. If a fraction of warranty claims are not exercised, then the analysis needs to be modified. The modification needed to include this requires a model for the fraction of claims not exercised. Let e(x) denote the probability that the warranty claim is not exercised if the age of the item at failure is x (0 < x < W). One can choose many different forms for the function e(x). One simple form is

This implies that all items that fail before age / are returned for rectification under warranty; for items that are of age greater than / at failure, only a fraction of the failed items are returned. This is a reasonable model, because it accounts for the fact that for items failing very close to the end of the warranty period there is very little [great] incentive for the owner to exercise the warranty claim under pro-rata [free-replacement] warranty. This probability can easily be incorporated in the models of earlier sections for expected cost analysis (e.g., see Patankar [25]). If some of the claims are not valid, one can use a similar approach.

Chapter 9

410 9.7.3

Product Recall

Thus far our discussion has focused on warranty servicing where a failed item is returned by the owner for rectification under warranty. Occasionally, the manufacturer has to recall either a fraction or all of the items sold for some rectification action. The recall of only a fraction of the items arises when items are produced in batches and some of the batches are defective due to inferior component(s) being used and this is not detected under quality control. The manufacturer is held responsible for damages resulting from such defective components, either under express or implied warranty. A hypothetical example is the following. Due to poor quality of insulation, certain batches of a domestic appliance (e.g., an electric frying pan) are prone to result in an electrical shock under normal use. In this case, under the terms of implied warranty, the manufacturer can be held liable for damage caused by such defective items, and it is more economical for the manufacturer to recall items from such batches. A total recall situation usually arises because of poor design specifications that can lead to malfunction and serious damage under certain conditions and are discovered only after the items have been produced and sold. A hypothetical example of this is where the brakes of an automobile malfunction under certain conditions of driving. In such cases, the manufacturer can be held responsible for damages caused under the terms of warranty for fitness. Under the conditions just discussed, the manufacturer has the option of recalling the items, either for replacement of defective components or for replacement of one or more old components by newly designed ones. The optimal decision depends on many factors. Failure to act can result in huge payouts. On the other hand, any recall is not only costly, but it can do serious damage to the reputation of the manufacturer and may affect sales for a long time. The literature on this topic is very limited. Dardis and Zent [26] carry out a simple cost-benefit analysis of the Pinto recall. In their model they incorporate various costs, such as tracing of items, rectification, etc. Some analytical research has been carried out on the logistics of product recall and optimal strategies for recall, such as items being returned to the manufacturer or to some retail-level outlets. There is considerable scope and need for further research on this topic. 9.7.4

Servicing by a Third Party

For many products, it is physically impossible for the manufacturer to service item failures under warranty in either a single central facility or a

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chain of facility centers operated by the manufacturer. In such cases, the manufacturer might delegate the warranty servicing to either the retailer or some independent servicing operator to service all item failures under warranty. This raises many interesting problems, and we indicate two of them: 1. If payment to the service agent is based on the number of claims serviced, there is the possibility of the agent acting in a fraudulent manner, e.g., overservicing, repairing items failing outside warranty and reporting these as failures under warranty, to name a few. 2. On the other hand, if the agent is paid a fixed amount, then the agent can do an unsatisfactory job and minimize his expenses. This action can affect the reputation of the product and, in the long run, the profits of the manufacturer. The literature on this topic is again very limited. NOTES

Section 9.1 1. Effective service support for products has assumed an important role in the strategic marketing of products. For more on this, see Lele and Karamarkar [6]. 2. Planning to provide effective product service is a topic of increasing attention. Bleuel and Bender [7] discuss various issues in product service planning. Section 9.2 1. The warranty reserves problem has received considerable attention. See, for example, Menke [8], Amato and Anderson [9], and Thomas [1]. These all deal with single lot sales and model warranty reserves to equal the expected payouts. The results for continuous sale and prediction intervals are new. Section 9.3 1. This section is based on a model from Nguyen [10]. Section 9.4 1. The minimal repair case is based on Nguyen and Murthy [11]. Biedenweg [12] discusses the case where repaired items are identical but different from new. 2. Gerner and Bryant [13] develop a regression model for the demand for repair. They model failure in terms of the probability of the item failing at each attempt at use. Section 9.5 1. Section 5.1 is based on Nguyen [10] and Section 5.2 on Nguyen and Murthy [14]. For more on repair vs. replace strategies, see Nguyen

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412

[10] and Biedenweg [12]. In particular, Biedenweg discusses issues such as existence of optimal strategies. 2. Nguyen and Murthy [3] deal with a model that includes the use of repaired and new items in the servicing strategy. Section 9.6 1. This section is based on Murthy and Nguyen [5]. Section 9.7 1. Most books on logistics discuss the planning of spares and repairs; see, for example, Hutchinson [15]. For the traveling repairman problem, see Agnihotri [16]. Inventory management has also received a good deal of attention; see, for example, Brown [17] and Hadley and Whitin [18]. For more on loanders, see Karmarkar and Kubat [19]. 2. For more on warranty claims not always exercised, see Patankar [20]. 3. Chandran and Lancioni [21] and Fisk and Chandran [22] discuss the topic of product recall. Min [23] deals with strategies for recall based on a mathematical programming formulation. The model proposed in Tapiero and Posner [24] is appropriate for describing the building up of reserves where large claims can occur, as is the case in product recall situations. EXERCISES

9.1. 9.2. 9.3. 9.4.

Derive Equations (9.66)-(9.69). Prove Proposition 9.1. Prove Proposition 9.2. Consider Example 9.12. Let p = 1, so that F{t) and G{t) have exponential distributions with parameters Xand \ 0, respectively, i.e., and t > 0. Show that if a* (the optimal a, which minimizes a>(a; T, W) given by Equation (9.65)) exists, it is given by

9.5.

Compare the results of Exercise 9.4 with those of Example 9.12 with X and X0 being the reciprocals of the means of F(t) and G(i), respectively.

Warranty Servicing

9.6. 9.7.

413

Prove Proposition 9.4. Consider Example 9.13. Let 0 = 1, so that F(t) is the exponential distribution. Show that ft* (the optimal ft, which minimizes co(d; W) given by Equation (9.81)) is equal to cs.

REFERENCES

1. 2. 3. 4. 5. 6.

7. 8.

9. 10. 11. 12. 13. 14.

Thomas, M. U. (1989). A prediction model for manufacturer warranty reserves, Man. Sci., 35, 1515-1519. Baxter, L. A., Scheuer, E. M., Blischke, W. R., and McConalogue, D. J. (1982). On the tabulation of the renewal function, Technometrics, 24, 151-156. Nguyen, D. G., an Murthy, D. N. P. (1986). An optimal policy for servicing warranty, J. Oper. Res. Soc., 37, 1081-1088. Mahon, B. H., and Bailey, R. J. M. (1975). A proposed improved replacement policy for army vehicles, Oper. Res. Quar., 26, 477-494. Murthy, D. N. P., and Nguyen, D. G. (1988). An optimal repair cost limit policy for servicing warranty, Math. Comp. Modelling, 11, 595599. Lele, M. M., and Karmarkar, U. S. (1983). Good product support is smart marketing, Harvard Bus. Rev., 61, No. 6 , 124-132. Bleuel, W. H., and Bender, H. E. (1980). Service-MarketingEngineering Interactions, AMACOM, American Management Association. Menke, W. W. (1969). Determination of warranty reserves, Man. Sci., 15, B-542-B-549. Amato, H. N., and Anderson, E. E. (1976). Determination of warranty reserves: An extension, Man. Sci., 22, 1391-1394. Nguyen, D. G. (1984). Studies in Warranty Policies and Product Reliability, Doctoral Dissertation, The University of Queensland, Australia. Nguyen, D. G., and Murthy, D. N. P. (1984). A general model for estimating warranty costs for repairable products, HE Trans., 16, 379-386. Biedenweg, F. M. (1981). Warranty Analysis: Consumer Value vs. Manufacturers Cost, Doctoral Dissertation, Stanford University, Stanford, CA. Gemer, J. L., and Bryant, W. K. (1980). The demand for repair service during warranty, J. Business, 53, 397-414. Nguyen, D. G., and Murthy, D. N. P. (1989). Optimal replace-repair strategy for servicing products sold under warranty, Euro. J. Oper. Res., 39, 206-212.

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15.

Hutchinson, N. E. (1988). An Integrated Approach to Logistic Management, Prentice-Hall, Englewood Cliffs, NJ. Agnihotri, S. R. (1988). A mean value analysis of the traveling repairman problem, HE Trans., 20, 223-229. Brown, R. G. (1967). Decision Rules for Inventory Management, Dryden Press, Hinsdale, IL. Hadley, G., and Whitin, T. M. (1963). Analysis o f Inventory Systems, Prentice-Hall, Englewood Cliffs, NJ. Karmarkar, U. S., and Kubat, P. (1983). Value of loaners in product support, HE Trans., 15, 5-11. Patankar, J. G. (1978). Estimation o f Reserves and Cash Flows A ssociated with Different Warranty Policies, Doctoral Dissertation, Clemson University, Clemson, SC. Chandran, R., and Lancioni, R. A. (1981). Product recall: A challenge for the 1980’s, Int. J. Phys. Distribution Mat. Man., 11, 46-55. Fisk, G., and Chandran, R. (1975). How to trace and recall products, Harvard Bus. Rev., 53, No. 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). Warranty reserving, Naval Res. Log. Q., 35, 473-479. Patankar, J. G., and Mitra, A. (1989). Effects of warranty execution under various rebate plans, presented at TIMS XXIX Int. Meeting, Osaka. Dardis, R., and Zent, C. (1982). The economics of the Pinto recall, J. Cons. Affairs, 16, 261-277.

16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26.

10

Warranty and Engineering

10.1

INTRODUCTION

When a product is sold with warranty, additional costs in the form of warranty servicing costs are incurred. From the analysis of Chapters 4-9, we see that the expected value of this warranty cost depends on the failure distribution of the product. This in turn depends on the engineering of the product. As a result, the expected warranty cost can be influenced by engineering decisions made with regard to product design, development, and manufacture. Higher reliability, or a lower failure rate, can be achieved through improvements in design or manufacture. Such improvements reduce the expected warranty cost, but at the expense of increased development and manufacturing costs. Thus, optimal engineering decisions must be based on a sensible trade-off between development and manufacturing costs on the one hand and expected warranty cost on the other. In this chapter we study this topic and discuss various engineering decision problems that take into account the warranty costs for the product. The outline of the chapter is as follows. In Section 10.2, we discuss the different stages of product engineering that a product must go through before it is released on the market. The actions at each stage affect the failure of items over the warranty period and hence the warranty servicing costs. In Section 10.3, we examine various options at the design level and discuss some optimal design decision problems. Section 10.4 deals with some problems at the manufacturing level and related decision issues. In Section 10.5, we study the problem of testing items before sale when the item has a bathtub failure rate history. As a result of the testing, items failing prematurely either do not reach the consumer or are fixed before they are sold. This involves additional costs, but reduces failures under 415

416

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warranty and hence the expected warranty costs. Finally, in the last section, we discuss some additional related topics. 10.2

ENGINEERING OF PRODUCTS

Before a product (consumer durable or industrial) is released on the market, many stages of decision making are involved. Some of these deal with economic and marketing aspects—e.g., the sale price, amount spent on advertising, and so on— and others with technical or engineering aspects. Decisions with regard to engineering aspects are made at three levels: 1. 2. 3.

Design Manufacture Presale testing

and are sequentially linked, as shown schematically in Figure 10.1. The design stage is the first step in the engineering of a product. Traditionally, product design has dealt mainly with the functional aspect of the product, without much consideration given to manufacturing and postsale maintenance or servicing issues. Design decisions are mainly technical in nature and are done at two levels, the component level and the system level. At the component level, the designer deals with issues such as the maximum acceptable stress to which a component of a mechanical product can be subjected. At the system level, issues such as choice of components and subsystems to ensure some specified system performance are dealt with. In using this traditional approach, problems often arose at the manufacturing level that required design modifications. This type of design iteration can be very costly. As a result, the design philosophy has been changing to incorporate manufacturing aspects into the design phase. With the advent of computer integrated manufacturing, this integration has become more common and, as a result, fewer design changes are needed. However, the incorporation of postsales economic issues, resulting from the consequences of the design, into the decision making process at a design level is a relatively new concept, the importance of which is more and more becoming recognized. This approach requires a strong interaction between design, manufacturing, marketing, and sales. Thus, many groups or departments of a manufacturing organization must be involved cooperatively at very early stages of product development. In Section 10.3, we discuss some models for optimal decision making at the design stage, which take into account the postsale warranty costs. The manufacturing phase deals with processes and input material needed to translate a conceptual design into a physical end product for sale. Be-

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Figure 10.1 Design (I)-Manufacture (II)-Sales-Service (III) sequence. cause of inherent variations in input material and in the process itself, not all end product items conform to design specifications. Items that do not perform satisfactorily, because they do not conform to the required design specifications, are defective items. Defective items have higher failure rates than nondefective items and have a significant impact on warranty cost. By proper design of the process and effective control, the fraction of defective items produced can be reduced. However, it is not possible to completely eliminate the production of defective items. One way of reducing defective items reaching the market is to inspect and test items on

418

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completion of the final stage of manufacturing. These actions reduce expected warranty cost but increase the manufacturing cost. In Section 10.4 we discuss two models of the decision process at the manufacturing level that are based on a trade-off between these two costs. Many manufactured products have a high failure rate in the early stages of product life for a variety of reasons. These include substandard components, errors in assembly, improper machine setup or calibration, damage during manufacturing, and so forth. Such failures, also called “infant mortality,” are due to “teething problems” and are rectified through appropriate actions such as replacing defective components or reassembly of the product. With usage, the failure rate decreases initially as the teething problems are discovered and rectified. With the passage of time, the failure rate remains nearly constant for a certain time period, after which it begins to increase due to wear and the aging process. As a result, a plot of the failure rate through time has a bathtub shape, as discussed briefly in Chapter 3. For products with a bathtub failure rate curve, the probability of an item failing at a young age is high. This may be a major contributing factor to high warranty costs. Once the item survives the initial period, corresponding to infant mortality, the failure rate becomes relatively small. To reduce warranty cost, the manufacturer can use burn-in. This involves putting each item into use (e.g., on a test bed) for a certain period of time before it is sold. Should the item fail during burn-in, it is repaired if it is repairable and junked otherwise. Thus, the items sold are ones that have either survived the burn-in or have had early failures fixed before sale. As a result, the expected number of returns under warranty is reduced, as is the expected warranty cost. The burn-in is worthwhile only if the reduction is greater than the cost of the burn-in program. We discuss this problem in Section 10.5. 10.3

WARRANTY AND OPTIMAL DESIGN

In this section, we discuss alternate options regarding actions available to a manufacturer at the design stage that could influence the product failure distribution. We assume that the form of the distribution is fixed and that the manufacturer’s actions affect the values of one or more of the parameters of the distribution function. We first consider the case where no product development is required. We assume that the manufacturer can design the product so that its reliability can assume any value within a specified interval. The manufacturing cost depends on the reliability chosen. For example, a higher item reliability might require components with tighter functional specifications or more

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419

reliable components. In either case, the resulting cost is higher. Later we consider the case where the desired product reliability is higher than the actual reliability of the product. In this case, reliability improvements must be attained through a research and development effort. We conclude with some additional topics of relevance that link warranty and design. 10.3.1

Optimal Reliability Choice and Allocation

Let F(jc ; 0) be the product time-to-failure distribution, and suppose that the manufacturer can choose the value for the parameter 0 (within limits) at the design stage. We assume the following: 1.

0 is a scalar, with smaller values of 0 corresponding to more reliable products. (The extension to a vector 0 is straightforward.) 0 can take on any value in the interval 0 _ < 0 < 0 + . 0 _ and 0 + reflect the limits for the allowable and achievable reliability. 2 . Cm( 0), the manufacturing cost per unit, is a continuous function of 0 with dCm(0)/ 0 and dj > 1. This implies that as x decreases (so that the product becomes more reliable), the unit cost increases. Under the stated assumptions, it follows that the unit cost per system is given by

where 0 is the set

Note that the failure rate of the system is a constant r, given by

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421

As part of design specification, we want to ensure that the failure rate r is not greater than a specified value y. This will ensure a certain minimum reliability for the product. Suppose that the item is sold with a free-replacement warranty (FRW) with warranty period W. We examine two cases. Case 1 [Repairable System] Suppose that the manufacturer can repair a failed system by replacing only a single failed component, say component 7, and that the cost of this is Cmy(0y) + ch, where Cmy(0y) is the cost of the new component and ch is the average extra cost incurred in handling the warranty claim. Since components fail independently and component failure distributions are exponential, the expected number of component 7 failures, 1 < 7 < /, over the warranty period W , is given by W0y. As a result, the expected cost of servicing the warranty per unit sale is given by

From (10.1), we have

Case 2 [Nonrepairable Items] Suppose that a failed system cannot be repaired by replacing failed components (or in any other way) or that such repairs would cost more than a new system. In this case, whenever a system fails, the entire system must be replaced by a new one. This is done at a cost Cm(0). Component failures are assumed to occur as in Case 1. As a result, we have

For both cases, the optimal 0* is obtained by minimizing £[CS(0 ; W)] subject to the constraints. We illustrate these results by means of a numerical example with J = 4. The cost parameters and the limits on 0y(per year), 1 < 7 < 4, are shown in Table 10.1. The parameters were selected so that (1) when component reliability is minimum (i.e., 0y = 0/", 1 < 7 < 4), the cost per unit is the same for all four components, and (2) the cost per unit for component 7 with reliability 0y+ - 80, for a given 80, is decreasing in 7. This implies

Chapter 10

422

Table 10.1 Model Parameter Values for Example 10.1

i

1

2

3

4

ai dj b,

0.20

0.30

3.00 8.40

0.24 2.50 8.64

8.80

0.40 1.50 8.87

Ö;-

0.00

0.00

0.00

0.00

0/

0.50

0.50

0.05

0.50

Table 10.2

2.00

Optimal Reliability Allocation [Example 10.1] Case 1: Repairable Items

7

e?

e2*

0*

0:

2 e;

Cm(0*)

io(0*; W)

£[CS(0*; W)]

2 1

0.260 0.162

0.334 0.224

0.398 0.281

0.454 0.333

1.446

44.9 60.0

30.6 22.4

75.5 82.4

1.000

Case 2: Nonrepairable Items 7

e?

ei

0?

04*

2 0;

Cm(0*)

(©*; W)

E[Cs(0*; W)}

2 1

0.219 0.168

0.282 0.226

0.338 0.279

0.387 0.327

1.226

1.000

49.4 60.0

72.9 65.9

122.3 125.9

e

=

[6 j, e2, e3, e j .

that for the same reliability, cost per unit for component 4 > cost per unit for component 3 > cost per unit for component 2 > cost per unit for component 1. The warranty period is W = 1 year, and ch is taken to be $10. We consider two values of y, viz., y = 1/year and 2/year. The former corresponds to a product with greater reliability. Table 10.2 shows the optimal reliability allocation (obtained using a standard computational procedure). The optimal allocation of reliability to different components is shown in Figures 10.2 and 10.3 for the repairable and the nonrepairable cases. We make the following observations regarding these results: 1.

The optimal cost per unit for y = 1 is greater than that for y = 2. This is to be expected, since the design requirements specify a more reliable product when y = 1.

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423

Figure 10-2 Optimal failure rate allocation [repairable products].

2.

3.

For both cases the constraint is tight when 7 = 1 and W is small. Furthermore, whenever the constraint is not tight, 0* decreases with W. This is to be expected, since W increasing corresponds to a longer warranty period and, to reduce the expected warranty cost, the reliability must improve or, equivalently, 0* must decrease. For case (1), individual 0* may increase or decrease with W when the constraint is tight. This is again to be expected, since the warranty cost depends on each 0y. In contrast, for case (2), the individual 0* do not change with W when the constraint is tight, because the warranty cost depends on 2 0y as opposed to the individual 0y.

10.3.2

Optimal Product Development

One way of reducing the expected warranty cost is to improve product reliability. This involves research and development, subjecting the product to a sequence of test-fix-test-fix iterations. During this process, the product is tested until a failure mode appears. Design or engineering modifications

424

Figure 10.3

Chapter 10

Optimal failure rate allocation [nonrepairable products].

are then made as attempts to eliminate the failure mode, and the product is tested again. As this continues, the product reliability improves. The development program costs money, and the process of improvement is uncertain. Thus, to decide on optimal development programs, one needs to build models for product improvement that take into account the underlying uncertainty. Using these, one can decide on optimal development plans, which minimize the total expected cost of development plus the servicing cost for warranty. This determines when the improvement process is terminated. We propose two different models for the reliability improvement process, denoted Models I and II. In the remainder of this subsection we confine ourselves to the special case where the product failure distribution is exponential. For both models, this implies that, with development, the failure rate decreases.

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425

We define the following development costs: Cd = cost of running the development program per unit of time CD = expected cost of each design modification to fix a failure mode If t is the duration of the development program and N( t ) is the number of design modifications made, then the cost of the development program is given by (10.3) In Model I, t is deterministic and N( t ) is a random variable; in Model II, the reverse is true. Let 0 be the failure rate at the end of the development program. In both Models I and II, the failure rate at the end of development is a deterministic function of the program development period. (A third model that might be considered is one in which the failure rate at the end of the development period is a random variable.) Finally, we assume that (1) the product is sold with a one-dimensional free-replacement (FRW) policy with warranty period W\ and (2) the item is repairable, and the manufacturer always chooses to minimally repair items that fail under warranty. Let cr be the expected cost of each repair and 5 the total number of items sold. Then the expected warranty cost a)(0; W) is given by (10.4) We now discuss the two different models for product development. Model I In this model, failures (and hence modifications to eliminate them through design) are assumed to occur according to a nonhomogeneous Poisson process with intensity v(t). v(t) is called the modification rate. v(0) is the failure rate before the start of the development program, and the aim of the program is to reduce this quantity. v(t) is a nonincreasing function of t. The number of modifications made over (0, i), N(t), is a random variable with (10.5)

426

Chapter 10

If the development program is stopped at time t , then the failure rate for the product is given by ( 10. 6)

Since v(i) is a nonincreasing function of t, larger values of t imply smaller or equal failure rates. In other words, the longer the duration of product development, the smaller the failure rate for the product at the end of the development program. The expected cost of the development program is given by (10.7) The duration of the development program, t , is to be optimally selected to minimize the total expected cost of product development and expected warranty costs. Let £[C s(t ; W)] be this total expected cost. Then from (10.4) and (10.7) we have

( 10 .8 ) The optimal t , t *, is the value of t that minimizes £[CS(t ; W)\. If t * exists, it can be obtained by the usual first-order condition. Various forms for v(t) have been proposed. We indicate one proposed by Crow [1], and given by (10.9) where X > 0 and 0 < p < i. In the literature on reliability growth, this model is called the nonhomogeneous, Poisson process, reliability growth model. It is the stochastic version of a fairly well-known deterministic model for reliability growth proposed by Duane [2]. Example 10.2 Suppose that the item in question is an expensive industrial product sold with a long warranty period. Let the modification rate v(t) be given by (10.9) with X = 1/week so that time during development is measured in weeks. Let CD = $1000, Cd = $100/day, cr = $100, and 5 = 1000. Table 10.3 gives t *, the optimal development period (in weeks), and the expected cost to the manufacturer, £[C s(t *; W)], for different combinations of W (in years) and p. Note that for a given p, as W increases the optimal development period also increases. This is to be expected,

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427

Table 10.3 Optimal Product Development-—t * vs. W and ß [Example 10.2] ß 0.4 0.5

W (years)

T* (weeks)

e(T*) (per year)

E[Cs(t *; W)] (x 100)

3 5 7 3 5 7

50 71 89 59 87

0.038 0.031 0.027 0.065 0.054 0.047

213 281 339 331 448 549

112

since one needs a more reliable product to reduce the expected warranty cost. For a given W, as 0 increases, again the development period increases. Model II Here we assume that the development program is stopped after n modifications and the problem is to decide on rc*, the optimal value of n. We assume that the mean life of the product at the end of the development period, |x(n), is equal to the mean time between the nth and (n + l)st modifications, if testing were to continue. As a result, if Sn denotes the time to the nth modification, then

( 10 . 10 ) and the failure rate at the end of the development is given by

( 1 0 . 1 1) As in Model I, we assume that failures (and modifications) occur according a nonhomogeneous Poisson process with intensity function v(i). As a result, the probability of n modifications in (0, t) is given by

( 10. 12) where A(t) is given by (10.5).

428

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Since Sn < t if and only if N(t) > n, we have

If the modification rate v(t) is given by (10.9), then, after some simplification (see Murthy and Nguyen [3]), we obtain (10.13) and, as a consequence, (10.14) Since 0 < p < i , |x(/t) increases with n , as expected. The manufacturer’s total expected cost, E[Cs(n; W)], is given by (10.15) and, using (10.11), (10.13), and (10.14), we have (10.16) The optimal n, n*, is the value of n that minimizes E[Cs(n; W)]. Example 10.3 Suppose that the modification rate v(t) is given by (10.9) with \ = 1/ week, so that time during development is measured in weeks. Let the parameters of the model be the same as in Example 10.2, namely, CD = $1000, Cd = $100/day, cr = $100, and 5 = 1000. Table 10.4 gives /i*, the optimal number of developments, which minimizes E[Cs(n\ W)], for different combinations of W and (3. Also shown are the values of E[Cs(n*; W)].

429

Warranty and Engineering Tab le 10-4 ß 0.4 0.5

Optimal Product Development— n* vs. W and (3 [Example 10.3] W (years) 3 5 7 3 5 7

n* 4 4 5 7 8

10

0(n*) (per year)

E[Cs(n*, W)] (x 100)

0.033 0.033 0.025 0.063 0.056 0.045

188 255 307 314 430 528

Note that for a given (3, as W increases, n* also increases, as expected, since one needs a more reliable product to reduce the expected warranty cost. For a given W, as (3 increases, again n* increases. 10.3.3

Redundancy

Thus far in this section we have focused our attention on improving item reliability by either choosing more reliable components or by initiating a development program. An alternate way to improve item reliability is through redundancy. Typically, redundancy involves duplicating one or more of the components. This is possible only for certain components, namely those for which incorporation of such duplication is permissible by the functional design of the item. Two types of redundancy have been used for improving item reliability. These are (1) active redundancy and (2) passive redundancy. In the case of active redundancy, the component as well as its duplicate are in use and the system functions as long as one of them is working. Thus, if the individual failure times of the two are X 1 and X 2, respectively, then the failure time, as a pair, is given by

In contrast, in the case of passive redundancy, at the start only one component is in use. When it fails, the duplicate (or spare) is automatically switched on. As a result, the failure time, as a pair, is given by

The design of passive redundancy is more complex, as it requires a switching mechanism. If the switch is itself imperfect, then the failure time, as a pair,

430

Chapter 10

is given by

where p is the probability that the switch is in failed state when needed. In this section, we study the effect of passive redundancy with perfect switching on the manufacturer’s total expected cost. We consider a component of a complex system that is a critical component of the system in the sense that the system becomes inoperative whenever the component fails. Suppose that the component is nonrepayable and cheap. Let Cm denote the unit manufacturing cost for the component, and let the failure distribution be given by F{t). Assume that handling and labor cost for replacing a failed component, ch, is either comparable to or large relative to Cm. The item is sold with a FRW policy with warranty duration W. If there is no redundancy built into the system, then the manufacturer’s expected cost per unit sales is given by (10.17) where M{t) is the renewal function associated with F(t). With passive redundancy and a perfect switch, the distribution function of time between failures is given by F(2)(i) = F(t) * F(t), where * is the convolution operation. As a result, the expected number of claims under warranty (resulting when both components fail) is given by M2(W), where M2(0 is the renewal function associated with F(2)(i). The cost of each module used (consisting of two components, each costing Cm, and the switching mechanism, costing 8Cm) is (2 + 8)Cm. As a result, the manufacturer’s expected cost per unit sale is given by (10.18) Building in redundancy is the optimal strategy if (10.19) From (10.17) and (10.18), we have the following result: Proposition 10.1: Building in redundancy is the optimal strategy if

431

Warranty and Engineering

The implication of this is that redundancy is worthwhile if ch/Cm is high and 8 is small. Example 10.4 Let F(t) be the exponential distribution with mean time to failure |x = 4 years, i.e., F{t) = 1 - e~tl4. As a result, F(2)(i) is distributed according to a Gamma distribution with mean time to failure of 8 years. Table 10.5 shows E[CS(W\ Redundancy)]/cm for two different ratios of ch/Cm and selected values of W and 8. We first consider the case ch/Cm = 2, i.e., handling cost is twice the cost per unit. For W = 1 year, the optimal strategy for all three values of 8 is that there be no redundancy built in. However, for W = 2 and 3 years, building in redundancy results in lower cost if 8 is small (i.e., 0.0 or 0.1). When 8 = 0.5, the advantage from fewer warranty claims on the average is negated by the increase in the module price due to the high cost of the switch when W = 4 years, redundancy is the optimal strategy for all three values of 8. For the case ch/Cm = 5 (i.e., handling cost five times the unit cost of a component), the results are different. For W = 1 year, redundancy yields lower expected cost for 8 = 0.0 or 0.1. For W = 2, 3, and 4 years, the optimal strategy for all 8 is to build in redundancy. Table 10.5 £[Cs(W)]/cm With and Without Redundancy [Example 10.4] (a) cJCm = 2.0 Redundancy

1 2

1.75* 2.50 3.25 4.00

2.0799 2.3679* 2.7231* 3.1353*

3 4

8 = 0.50

2.1826 2.4771* 2.8412 3.2638*

2.6199 2.9139 3.3135 3.7773*

t

8 = 0.00

o-H o II

No redundancy

6©

w

(b) cJCm = 5.0 Redundancy

8 = 0.00

1 2

2.50 4.00 5.50 7.00

2.1864* 2.6438* 3.2655* 3.9868*

3 4

indicates lower cost

ol-H Ö II

No redundancy

60

w

2.2890* 2.7530* 3.3836* 4.1152*

8 = 0.50 2.6997 3.1898* 3.8559* 4.6288*

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Note: We have considered the case where the redundancy involves only the duplication of the component, i.e., the system includes the component plus one spare. One can consider redundancy involving two or more spares installed in either passive or active mode as well. 10.3.4

Other Design Issues

One way of reducing warranty cost is to build diagnostic features into the item so that, for most common types of failures, the user is guided to rectifying the problem. Failures that cannot be fixed by the user need to be returned to the manufacturer for rectification. A typical example of such diagnostic features being built into an item is a modern photocopier. Building in such features increases the cost of the item and is worthwhile only if the reduction in the expected warranty cost is more than the additional costs incurred. In some items, the design includes either a warning signal that requires action on the part of the consumer to avoid a failure (e.g., engine oil being low, and unless action is taken immediately, there is the possibility of engine failure) or a mechanism that automatically initiates action to minimize item failure (e.g., a fuse acting as a protective device against electrical damage). Finally, in the context of servicing warranty, effective modular design helps in two ways. Locating a fault in a failed item becomes easier, and the downtime for repair is reduced, because a failed module can be replaced by a working loaner. 10.4

MANUFACTURING AND WARRANTY

Because of variability in the manufacturing process, the quality of items produced varies. In the simplest characterization of this quality variation, items can be classified as being either nondefective or defective. An item is called nondefective if it meets the specified design and performance standards, and defective if it does not. The production of defective items is random. The characterization of this random process depends on whether the items are produced continuously or in batches. It is useful to identify two types of defective items, which we call Type I and Type II. A Type I defective item is not operational when put into use (e.g., a defective assembly or defective component that makes the item inoperable). A Type II defective item is operational when put into use but has performance characteristics (e.g., mean time to failure) that are considerably inferior to those of a nondefective item. Type I defectives can be detected either by inspection or testing for a very short (or nearly zero)

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time duration. In contrast, a Type II defective item can be detected only through item testing for some significant time period. Here we confine our attention to defective items of Type II, and we refer to them simply as defective items. The expected warranty servicing cost that may be incurred when a defective item reaches the consumer is considerably more than the corresponding cost for a nondefective item. Hence, warranty servicing costs may be reduced by reducing ( 1) the number of defective items produced, or (2) the number of defective items reaching consumers. The former is achieved by controlling the process; we discuss a model when this is achieved through lot sizing. The latter is achieved by inspection and life testing to weed out defective items. Both these forms of action involve additional costs and are justified only when the reduction in the expected warranty costs exceeds the additional cost involved in controlling quality. In this section, we first consider a model (Model I) where inspection and testing are used to reduce the number of defective items reaching the consumer. Later we consider another model (Model II) involving lot sizing. We confine our attention to batch production; i.e., items are produced in batches as opposed to being produced in a continuous mode. We consider both FRW and PRW. 10.4.1

Model I [Quality Control Through Inspection and Testing]

We assume that items are produced in batches and that each batch contains N items. K is the total number of batches produced. We assume K to be large. The items are sold individually, with warranty. Product Quality [Production of Defective Items]

We assume that the production process is in a steady state condition and that each item produced is nondefective [defective] with probability p x [p2\ with p x + p 2 = 1. If p2 = 0, then no defective items are produced. In this case, there is no need for any quality control scheme, as all items meet the desired specification. In practice, because of variability in the production process, we have 0 < p2 < 1, with smaller p 2 implying better product quality. Let F^t) [F2(t)] denote the failure distribution function for nondefective [defective] items. Let/)(i), F,(i), and r^t) denote, respectively, the density function, survivor function, and the failure rate associated with F,(i), i = 1, 2. These are related as follows: /¿(t) = dFi(t)ldt, F,(i) = 1 - F,(i), and ri{t) = fi{t)!Fi{t). Note that defective items are inferior to nondefective items. We characterize this as follows: r^t) « r2(t), 0 < t < ; i.e.,

434

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defective items have a high failure rate in relation to that for nondefective items. This implies that

i.e., the mean time to failure for defective items is much smaller than that for nondefective items. Although the items are produced in batches, items are picked randomly for use, whether for sale or for replacement under warranty. As a result, the distribution of time to failure, F(r), of an item picked randomly for use is given by

( 10 .20 ) Note that in our model formulation, p x and p 2 are constant. Quality Control Scheme

We consider a simple scheme for quality control characterized by two parameters, n (0 < n < N) and T (>0) as follows: A sample of n items is taken randomly from each batch and tested on a test bed for T units of time. In reliability testing, T is also known as the burn-in time. If no item fails (i.e., no item in the sample has life less than T ), then the items in the batch are released for use with no further testing. However, if one or more should fail, then all the items in the batch are subjected to a burnin of period T, and those which survive are released for use. In other words, 100% testing is used should an item from the sample fail under test, and all items that fail under test are scrapped. The rationale for this scheme is that a batch that contains few defective items will, on the average, have no failures when a sample is tested, and hence there is no need for the testing of the whole batch. In contrast, a batch with a high number of defectives will result, on average, in one or more failures in the sample tested, and this in turn will result in the whole batch being tested to weed out some of the defective items. Since defective items are inferior, on average a greater faction will fail during testing than will fail for nondefective items. As a result, the quality of items released for use under the quality control scheme is better than with no testing. We assume that batches released with 100% testing are not differentiated from batches released with only a sample being tested. (This avoids the separate bookkeeping that would be required if the two types of batches were differentiated.) However, the failure distributions of items from the two types of batches are different, as are the probabilities

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435

of randomly selected items being nondefective. In the next section, we obtain the probabilistic characterization of defective and nondefective items released for use and the cost of testing. The optimal quality control scheme is characterized by n* and T*9 the values that minimize an appropriate cost criterion. For the rebate form of the FRW with minimal repair, we use the asymptotic total cost per unit (released for sale) as the cost criterion rather than the total expected cost per unit. For the PRW policy with replacement, we again use the asymptotic total cost per item, with items chosen randomly from a “large” stockpile of items (released for sale). The reason for these choices is that they model realistic situations and result in analytically tractable expressions. We assume the following: 1. All failures under warranty result in warranty claims. 2. All claims are valid, and failures are rectified as per warranty terms. 3. A claim is made as soon as an item fails. 4. The time to rectify is relatively short compared with the mean time between failures, so that it can be treated as zero. Analysis of the Quality Control Scheme

As indicated earlier, the control scheme alters the quality of items released for use. In this section we characterize the quality of items released and the cost aspects of quality control. Output Quality Note that the items released for use are a mixture of four types, which we shall denote Types A, B, C, and D. They are as follows: Type A:

Nondefective items not subjected to testing. The failure distribution is given by F^t). Type B: Defective items not subjected to testing. The failure distribution is given by F2(t). Type C: Nondefective items that survive the testing. The distribution function, F^t), t > 0 , is given by

( 10 .21 )

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436

The failure density f x{t) and the failure rate r^t) are given respectively, by

( 10 .22 ) and (10.23) Type D:

Defective items that survive the testing. The distribution function, F2(r), t > 0 , is given by ( 10.21) with F ^ ) replaced by F2(*). The failure density f 2{t) and the failure rate r2{t) are given by (10.22) and (10.23) with f x(-) and Fa(*) replaced by / 2( ) and F2(-), respectively.

Let q denote the probability that a batch is released for use without 100% testing— in other words, no item in the sample tested fails. Conditional on the number of nondefective items in the sample being J = j {1 < j < n}, the probability of the batch not being tested 100% is given by (10.24) The probability that the sample contains j nondefective items is given by (10.25) As a result, we have

which can be simplified to yield (10.26) and (10.27)

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437

Note that, for a given batch, the number of items released and the proportion of the different types (Types A -D ) are random quantities. Let Nf, A-5, TVf, and Nf denote the number of Type A, Type B, Type C, and Type D items in the ith batch released. It is easily seen that (10.28) (10.29) (10.30) and (10.31) The number of items released in batch i is given by (10.32) and the number of items scrapped in batch i is given by (10.33) Note that A, = 0 if batch i is not subjected to 100% testing. Using (10.28)(10.31), we have, from (10.32) and (10.33), (10.34) and (10.35) Manufacturing Cost per Item Released The manufacturing cost is the sum of the production cost and the cost of testing. Let Cmbe the production cost per item. If no quality control scheme is employed, then the manufacturing cost per item is simply the production cost per item. With quality control, we have, in addition, the cost of testing, and this is a function of the parameters n and T of the quality control scheme. Let yK(n, T ) denote the manufacturing cost per item released. In this section, we derive an expression for \imK^ x[yK(n, 71)]. The number of batches subjected to 100% testing, K , is a binomial (K , q) random variable, i.e.,

438

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k = 0, 1, . . . , K. The number of batches not subjected to 100% testing is K — K. As indicated earlier, Nh the number of items discarded in batch / (1 < / < A), is a random variable. The total cost of manufacturing K batches consists of the following: 1. 2. 3.

Production cost: KNCm Testing cost: A7VC, + {K - K}nCh where Ci { = Ci(T) = i + ct>2r , ()>! > 0 and (J>2 ^ 0} is the cost of testing each item Scrapping cost: NtCd, where Cd is the cost of scrapping a failed item

The total number of items released for use is given by KN - 2 -Li N(. As a result, the cost per item released is

(10.36)

Note: This ratio is infinite with probability (1 - a)NK, which tends to zero as K —> oo. As K —» oo, we have from the weak law of large numbers (see, for example, Theorem 4.2.1 of Heathcote [4]) that

and

w here---- > indicates convergence in probability. As a result, we have

(10.37)

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439

Note that Q and a are functions of T and that no quality control corresponds to n = T = 0. In this case y(0, 0) is simply Cm. Free-Replacement Policy [Minimal Repair]

Under this policy, the manufacturer agrees to repair an item if it fails within the warranty period [0, W], and the item is subjected to minimal repair. Let cr be the cost of repair and handling for each warranty claim. The expected warranty servicing cost per unit depends on the type of item. The expected warranty servicing cost for a Type A item is given by the expected number of repairs within warranty period times cr, i.e., by cr j y rx(t) dt = cT{ - In FX(W)}. Similarly, for Type B, C, and D items, the expected warranty servicing costs for items are giveji by cr /o' r2(t) dt = cr{ - In F2(W)]1 S™ f x{t) dt = cr{ - In [F,{T + W) IFX{T)}, and cr ^ f2(t) dt = cr{ — In [F2( T + W) /F2(T)}, respectively. The warranty cost per item released is given by c t

(10.38) where Nf{ = number of type * (* = a, b, c, d) item failures for batch i. Again, as K —> we have from the weak law of large numbers that

for * = a, b, c, d. It is easily shown that

and

440

Chapter 10

As a result, we have (10.39)

where (10.40) (10.41) (10.42) and (10.43) The total cost to the manufacturer per item released is the sum of the manufacturing cost per item released and the warranty servicing cost per item released and converges in probability to (10.44) with y(n, T) given by (10.37) and T) given by (10.39). Without quality control, we have n = T = 0. In this case (10.45) Optimal Quality Control Scheme The manufacturer’s asymptotic cost per item released, CS(W; n , T), is a nonlinear function of the quality control scheme parameters n and T. The optimal quality control scheme is given by n* and T*, which minimize Cs(n, T). Note that this is a mixed mathematical programming problem, since n is an integer and T is a continuous variable. A necessary condition

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for the existence of a local minimum is that n* and T* satisfy the following: (10.46) and (10.47) Even for the simplest failure distributions, it is not possible to obtain n* and T* analytically. It is therefore necessary to use a computational procedure to obtain them. In this context, the following result is of particular significance. Proposition 10.2 (Murthy et al. [5]) The optimal n, n*, is either 0 or N. In other words, the optimal scheme involves either no inspection or full ( 100%) inspection. Special Case: Exponential Failure Distribution Let Fj(i) and F2(t) be exponential distributions with parameters and \ 2, respectively, with \2 > K > 0- 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 this case, the failure rates of Type A and C items are \ l9 and those of B and D are \ 2. 7 (w, T) is given by (10.37) with (10.48) From (10.39), we have (10.49) with qa, . . . , qd given by (10.40)-(10.43), in which a is given by (10.27) and q by (10.26). We have the following two propositions: Proposition 10.3 Murthy et al. [5]) We have n* = N (i.e., 100% inspection is optimal) if

for some T > 0.

442

Chapter 10

Comments: 1. 2.

The inequality is more likely to be satisfied if (i) Cm, Cd, and Ct(Y) are small relative to Wcr; (ii) \2 ~ is large; and (iii) p x is close to .5 (i.e., 50% defectives!). If the inequality is satisfied for p x = p, then it is also satisfied for p < P i 2 5 1 - p-

Proposition 10.4 (Murthy et al. [5]) We have n* = 0 (i.e., inspecting no items is optimal) if

Comments: 1.

The inequality is more likely to be satisfied if (a) Cm, Cd, and Q(0) are large relative to Wcr; (b) \2 _ Is small; and (c) p x is close to 0. This is just the reverse of Proposition 10.3. 2. The inequality is always satisfied when p is very close to zero. In this case, since the majority of the items are defective, there is no advantage in doing 100% testing to weed the defectives out. The more sensible option is to scrap the whole batch. 3. If the inequality is satisfied for p l = p, then it is also satisfied for Pi ^ P •

Pro-Rata Warranty

Under this policy, the manufacturer agrees to refund a fraction of the original sales price c5 to the consumer if the item sold fails within the warranty period [0, W]. We assume linear proration. The item for use can be of Type A, B, C, or D. The warranty servicing cost per item sold of Type A is given by the sum of the amount refunded and the handling cost for each warranty claim. If ch is the cost of handling for each warranty claim, then for Type A items the expected warranty servicing cost is given by (10.50) Similarly, for Type B, C, and D items, this cost is given by (10.50) with M 0 replaced b y / 2(0 , / i ( 0 >a n d / 2(i), respectively. In a similar manner to that used in the last section, we have that the expected warranty servicing cost per unit released converges in probability

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to

(10.51) as K —> oo, with qa, . . . , qd given by (10.40)-(10.43). The total cost to the manufacturer per item released, CS(W; n , T), is the sum of the manufacturing cost per item released (given by (10.37)) and the asymptotic warranty servicing cost per item released (given by (10.51)). With no quality control, n = T = 0. In this case,

(10.52) Optimal Quality Control Scheme As with the free-replacement policy, CS(W; n , T), the manufacturer’s asymptotic cost per item released, is a nonlinear function of the quality control scheme parameters n and T. The optimal quality control scheme is given by n* and T*, which minimize CS(W; n , T). This is again a mixed mathematical programming problem, since n is an integer and T is a continuous variable. A necessary condition for the existence of a local minimum is that n* and T* satisfy (10.46) and (10.47). Proposition 10.5 (Murthy et al. [5]) The optimal n, n*, is either 0 or N. In other words, the optimal scheme involves either no inspection or full ( 100%) inspection. Special Case: Exponential Failure Distribution Let Fx(r) and F2(t) be exponential distributions with parameters Kx and respectively, with X2 > K > 0* T) is given by (10.37) with a given by (10.27). From (10.51) we have, (10.53)

Chapter 10

444

with (10.54) (10.55) qa, . . . , #d given by (10.40)-(10.43), a given by (10.27), and q given by (10.26). We have the following two propositions: Proposition 10.6 (Murthy et al. [5]) We have n* = N (i.e., 100% inspection is optimal) if

for some T > 0. Comments: 1.

2.

The inequality is more likely to be satisfied if (i) Cm, Cd, and C, are small relative to cb and ch and (ii) p x is close to .5. If the inequality is satisfied for p l = p, then it is also satisfied forp < Pi
1.25. The reason for this is as follows. Since all items released for sale are rectified through minimal repair, the warranty servicing cost increases rapidly with W for any defective item released. As a result, as the warranty period increases, longer testing is needed to reduce the number of defective items being released. This is seen more clearly in Table 10.7, where as W increases, T* increases and the expected number of defective items released decreases. Table 10.6 n *, T* 9 and C£W; n*, t *) vs. W

w

n*

T*

CS(W; n*, T*)

1 2

0 100 100 100

0.0000

415.00 514.30 567.13 606.47

3 4

0.2259 0.4231 0.5343

Chapter 10

446

Figure 10.4

T* vs. W [FRW policy with minimal repair].

The optimal testing period varies from 0.2259 year for W = 2 years to 0.5343 year for W = 4 years. Obviously, it is not possible to test items for such long periods. However, life testing is usually caried out in an accelerated manner. Under accelerated life testing, the item is subjected to a harsher environment, which hastens the aging process. As a result, testing for one unit of time in the accelerated mode corresponds to P(>1) units of testing under normal conditions. This implies that testing for T/(5 units

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Table 10.7 Expected Number of Nondefective and Defective Items per Batch

Expected number of items per batch of 100 items Weeded out during testing

w 1 2 3 4

Nondef.

Def.

Released for use

Total

Nondef.

Def.

Total

75.000 71.687 68.915 67.399

25.000 10.128 4.603 2.950

100.000

0.000

0.000

0.000

3.317 6.085 7.601

14.872 21.397 22.050

18.185 26.282 29.651

81.815 73.718 70.349

of time in the accelerated mode corresponds to testing for T units under normal conditions. If testing in the accelerated mode is done with p = 50, testing for 0.5343 year in the normal mode would require accelerated testing for only approximately four days. This practical solution corresponding to the optimal period may be obtainable. The effect of p 1 on n* and T* with the remaining parameters held at their nominal values is shown in Figures 10.5 and 10.6. Note that for small p x (close to 0) and largep x (close to 1), n* = 0, as expected from Proposition 10.7. For W = 1, n* and T* are zero for all values of p x. For W = 2, n* = 100 for .48 < P i< .92. Over this range, the optimal T* first increases and then decreases, as shown in Figure 10.6. The reason for this is as follows. When p x = 1, no item is defective, and hence there is no need for testing. As p x decreases (for .92 < p x < 1), the fraction of defectives is sufficiently small so that testing to weed out defective items is not justified, and as a result, T* = 0. As p 1 decreases still further (for .48 < p x < .92), the fraction of defectives increases, and testing is worthwhile. Note that testing for a longer time reduces the warranty cost per item but increases the manufacturing cost per item, since a greater fraction of items fail during testing and are discarded. The reason that T* increases as p x decreases in the interval .69 < p l < .92 is that the increase in the manufacturing cost per item is less than the decrease in the warranty cost as p 1 decreases. In the interval .48 < p x < .69, T* decreases with p x decreasing, because the increase in the manufacturing cost per item is more than the reduction in the warranty cost per item. Finally, for 0 < p 1 < .48, the fraction of defectives is very high. The cost of any testing is not worth the reduction in the warranty cost, and hence T* = 0. As W increases, the interval over which n* = 100 also increases, and T* has a shape similar to that for W = 2 except that the

448

Chapter 10

Figure 10.5 n* vs. p x [FRW policy with minimal repair].

values for any given p l are increasing with W. This is to be expected for reasons discussed earlier. Finally, Figure 10.7 shows the effect of p x on CS(W; n*, T *) for the four different values of W. With no testing, the expected cost CS(W; 0, 0) is a linear function of p x. Since n* = 0 for p x = 0 and 1, CS(W; 0 , 0) as a function of p x is given by the straight line joining CS(W; 0, 0) for p x = 0 and p x = 1. Note that when n* = 100, CS(W; n*,

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Figure 10-6 T* v s . p A[FRW policy with minimal repair]. T*) is below this line as seen clearly for W = 4. Also, note that CS(W; n* decreases with p x increasing, as expected. PRW Policy [Linear Rebate] Table 10.8 gives n *, T*, and CS(W; n*, T *) for the four different warranty periods. For W = 1 to 4, /i* = 100, implying 100% testing. Figure 10.8 shows T* as a function of W. For W < 0.5, T* = 0. This is to be expected, since small W implies smaller warranty service cost, even with all defective

450

Figure 10-7 CS(W; n*, T*)

Chapter 10

v s.

p l [FRW policy with minimal repair].

items released, and the cost of testing is not justified. For 0.5 < W < 6.0, n* = 100, so that 100% testing is the optimal strategy. In contrast to the minimal repair case, T* increases with W for 0.5 < W < 1.7 and decreases for 1.7 < W < 6.0. For 0.5 < W < 1.7, the testing period increases with W because the warranty cost for released defective items increasingly dominates the cost per item released. Thus, by reducing the number of released defective items through increased testing, the war-

Warranty and Engineering

Figure 10.8

T* vs. w [p r

w

451

policy].

Table 10.8 n*, T , and CS(W; n*, T*) vs. W w

n*

r

CS(W; n*, r )

1 2

100 100 100 100

0.1539 0.1871 0.1565 0.1090

560.03 650.45 720.22 778.35

3 4

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ranty cost can be decreaseed by more than the increase in the manufacturing cost per item released. For 1.7 < W < 6.0, the influence of the warranty cost for released defective items decreases while the influence of the other costs increases, resulting in the testing period decreasing as W increases. This continues until, for W > 6.0, testing is again not justified, so that T* = 0. Table 10.9 shows the expected number of defective items released and weeded out in each batch. We now study the effect of p x on n* and T* with the remaining parameters held at their nominal values. Figures 10.9 and 10.10 show the influence of p x on n* and T*. These are similar to Figures 10.5 and 10.6. The reason for the variations in T* (as a function of p x) is the same as that for the FRW case. 10.4.2

Model II [Quality Control Through Lot Sizing]

We assume that items are produced in lots of size L. The product quality depends on L, as will be indicated later in the section. K is the total number of batches produced. We assume K to be large. The items are sold individually with warranty, and the warranty period is W. Product Quality

We assume that at any given time the manufacturing process can be in one of two states, “in control” or “out of control.” The process starts in control, and after the production of an item, it can switch from in control to out of control with probability 1 - q or stay in control with probability q. We assume that the process produces only nondefective items when in control and only defective items when out of control. Once the process enters the Table 10-9 Expected Number of Nondefective and Defective Items per Batch

Expected number of items per batch of 100 items Weeded out during testing

Released for use

w

Nondef.

Def.

Total

Nondef.

Def.

Total

1 2

2.273

11.490 13.172 11.633 8.834

13.763 15.927 13.944 10.451

72.727 72.245 72.689 73.383

13.510 11.828 13.367 16.166

86.237 84.073 86.056 89.549

3 4

2.755 2.311 1.617

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Figure 10-9 n* vs. p l [PRW policy].

out-of-control state, it stays there until the process is stopped and reset so that it is back in control. We assume, as in the previous model, that nondefective items have failure distribution Fx{x\ 0X) and defective items have failure distribution F2(x; 02). However, we do not carry out any inspection or testing to weed out defective items, because the mechanism to prevent defective items from reaching the consumer is through lot sizing.

454

Figure 10.10

Chapter 10

r* vs. p x [p r

w

policy].

Before a lot production starts, the process is checked to make sure that it is in control. This involves time and effort and costs an amount Cf if the process is in control and Cf + j) (j) > 0) if it is out of control. This cost is called the setup cost. The process is not interrupted until the production of a lot of size L is completed. As a result, should the process change from in control to out of control, then a certain number of items will be defective. Let Nt be the number of nondefective items in lot i. For Ni to take the value j, 0 < j < L, the first j items must be nondefective and the process must switch from in control to out of control after the

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production of the yth item. As a result, we have (10.56)

From this, we have that the expected number of nondefective items in a lot is given by

(10.57) Let p denote the fraction of nondefective items in a lot. We find

(10.58) Manufacturing Cost per Item

The manufacturing cost per lot consists of (1) the setup cost and (2) the cost of material and labor to produce a unit. The setup cost is Cf with probability qL (corresponding to the process being in control at the end of producing a lot) and is Cf + t ] with probability 1 - qL. Let a denote the cost of material and labor per unit. As a result, the expected manufacturing cost per item is given by

(10.59) FRW Policy [Minimal Repair]

Here, whenever an item fails under warranty, it is minimally repaired and returned to the owner. Let cr be the expected cost of each repair. If the item is defective, then the expected warranty service cost per item is given

456

Chapter 10

by

where r2{x) is the failure rate associated with the failure distibution of defective items, given by F2(x). Similarly, if the item is nondefective, the expected warranty service cost per item is given by

where rx{x) is the failure rate associated with the failure distribution of nondefective items, given by F ^ jc). The expected warranty service cost per lot is given by (10.60) where F[A^] is given by (10.57). From this we have the expected warranty cost per item given by

(10.61) where p is given by (10.58). The manufacturer’s total expected cost per item, E[CS{W\ L)], is the sum of the expected manufacturing cost per item plus the expected warranty cost per item and is given by

(10.62) Optimal Lot Size The optimal lot size, L*, is the value of L that minimizes E[CS(W; L)]. If q is very close to 1, then qL can be approximated by (10.63)

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Using this approximation and treating L as a continuous variable, the approximate lot size L** can be obtained by solving dE[Cs(W ; L)\ldL = 0. This yields

(10.64) PRW Policy [Linear Rebate]

Here, whenever an item fails under warranty, a fraction of the sale price is refunded. We assume that the rebate is linear. Let cb be the sale price per item. For a nondefective item, the expected refund is given by

Similarly, for a defective item, this quantity is given by

The expected warranty service cost per lot is given by

(10.65) where is given by (10.57). From this we find the expected warranty cost per unit to be

( 10.66)

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Chapter 10

As a result, the manufacturer’s expected cost per item, E[CS(W ; L)], is given by

(10.67) where Cm is given by (10.59). Optimal Lot Size The optimal lot size, L*, is the value of L that minimizes E[CS(W ; L)] given by (10.67). If q is very close to 1, then, using the approximation given in (10.63), the approximate lot size L** is

( 10.68) where (10.69) and (10.70) Example 10.6 Suppose the failure distributions of both nondefective and defective items are exponential distributions, with parameters = 0.1 and \ 2 = 1.0, respectively. This implies that the mean time to failure is 10 years for nondefective items and 1 year for defective items. We assume that the lot size cannot exceed a specified upper limit Lm. Let Lm = 100, so that the optimal lot size is constrained by the relation L* < Lm = 100. The nominal values for the remaining parameters are taken to be q = .99, Cm = $5.00, Cs = $50.00, = $10.00, cr = $5.00, 0X = .95, 02 = 0.04, cb = $30.00, a = 1.0, and k = 1.0. We consider four different values for the warranty period W, ranging from 1 to 4 years.

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We also include the case W = 0, which corresponds to the product being sold with no warranty. FRW Policy [Minimal Repair] Table 10.10 gives values of L* and E[CS(W ; L*)], obtained by evaluating E[CS(W; L)\ for L = 1, . . . , Lm; L** and E[CS(W; L**)], obtained by using the approximation given by (10.64); and £[CS(W; Lm)], the cost per unit if the lots are of size Lm( = 100). The percentage reduction in cost, RC, given by

is also shown in Table 10.10. For W = 0, we have L* = Lm. As W increases, L* decreases, since a longer warranty period implies increased warranty costs for defective items released. Hence, smaller lot sizes are required to ensure that the expected fraction of defective items released is smaller. The percentage reduction, RC, with L = L*, increases with W, indicating that lot sizing becomes more critical as the warranty period increases. Note that the error between L* and L** (obtained using the continuous approximation) decreases as W increases and that L** is always less than L*. We now study the effect of q on L*, holding the remaining variables at their nominal values. From Figure 10.11, we see that there is a critical value ^(dependent on W) such that, for q < q, L* = Lm, and for q < q < 1, we have L* < Lm. For q = 1, we have L* = Lm. Note first that if q = 1, then the process is always in control. Reducing the lot size increases the manufacturing cost per item with no effect on the warranty cost per item, and so L* = Lm. Now, as q decreases with L kept at Lm, Table 10.10 Exact and Approximate Optimal Lot Sizes and Related Costs [FRW Policy]

0

w L* E[CS(W ; L*)] L** E[CS(W; L**)] E[CS(W; L J ] RC (%)

100

5.5634

100

5.5634 5.5634

0.00

1

2

3

4

58 7.8439 47 7.8739 8.0383 2.42

38 9.5526 33 9.5758 10.5131 9.14

30 11.0515 27 11.0701 12.9879 14.91

26 12.4429 23 12.4653 15.4628 19.53

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Figure 10.11

l

* v s . q for Example 10.6 [FRW: minimal repair].

the expected fraction of defectives in a lot increases, resulting in a higher warranty cost per item. By decreasing L as well, the warranty cost can be reduced, but the manufacturing cost per item will then increase. Decreasing L is justified if the reduction in warranty cost compensates for the increase in manufacturing cost. This occurs, so that L* decreases as q decreases until some value (depending on W) is reached. At this point, the increase in manufacturing cost is greater than the decrease in the warranty cost as q decreases, resulting in higher values for L*. Consequently, L* increases to the maximum Lm as q decreases from this point and then, for q < q,

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461 3 3

L* is constrained to be the maximum Lm. Note that q decreases as increases. This is to be expected, since the warranty costs increase as increases. The expected total cost per item with optimal lot size, E[CS(W ; L*)], is plotted versus q Figure 10.12. For a given W, E[CS(W ; L*)] decreases as q increases. This should obviously be the case, since as q increases there is a lower probability of the process changing from in control to out of control. Similarly, for a given q, E[CS(W ; L*)] increases as W increases.

Figure 10.12 E[CS{W\ L*)]

v s.

q for Example 10.6 [FRW: minimal repair].

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PRW Policy [Linear Rebate] Table 10.11 shows L* and E[CS(W; L*)] obtained by evaluating E[CS(W ; L)] for L = 1, . . . , Lm; L** and E[CS(W :; L**)] obtained by using the approximation given by (10.68); £[CS(W; Lm)], the cost per unit if the lots are of size Lm ( = 100); and RC, the percentage reduction in cost, defined earlier. The results are similar to those for the FRW policy. Note that for the nominal values used, for W = 1,2, and 3, the optimal lot sizes are smaller than those for the FRW policy and larger for W = 4. Also, the percentage reduction, RC, is larger than that for the FRW policy for W = 1, 2, and 3 and smaller for W = 4. Again, the approximate optimal lot sizes are close to the optimal lot sizes. The influence of q on L* is shown in Figure 10.13 and is similar to that for the FRW policy shown in Figure 10.11. 10.5

PRESALE TESTING

When the failure rate of a product has a bathtub shape, failure in the early period of usage (the infant mortality region) can be high. As a result, the fraction of items returned under warranty in a short period subsequent to the sale can be high. This not only implies high warranty costs, it also results in a low product reputation. Burn-in is a testing program (similar to that used in Model I of the previous section) that aims to eliminate the frequently occurring early failures before items are sold. The rationale for the burn-in is as follows: For nonrepairable products, items with very short life should not be sold. For repairable products, failures occurring during the initial phase, corresponding to the infant mortality region, are rectified cheaply by testing and fixing before sale as opposed to fixing under war-

Table 10.11 Exact and Approximate Optimal Lot Sizes and Related Costs [PRW Policy]

0

w L* E[CS(W ; L*)\ L** E[CS(W; L**)] E[CS(W; L J ] RC (%)

100

5.5634

100

5.5634 5.5634

0.00

1

2

3

4

38 9.9418 33 9.9658 10.8939 8.74

31 12.2504 27 12.2722

28 13.9478 25 13.9714

27 15.3331 24 15.3620

14.1262

16.2893 14.39

17.8809 14.25

13.28

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463

Figure 10.13 L* vs. q for Example 10.6 [PRW: linear rebate].

ranty, which entails not only additional costs but affects the good will of the consumer. Although the testing scheme is similar to that discussed in Model I of the previous section, the motivation is different. In Model I, the testing was done to weed out defective items. If the proportion of defective items is very low, then there is no need for testing. Here, in contrast, all items are similar, and the testing is to eliminate early failures before items are sold.

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With burn-in, the expected warranty cost is reduced. However, this is at the expense of the costs involved in the burn-in. In this section we develop a model to obtain the optimal burn-in period, which achieves a trade-off between these two costs. 10.5.1

Burn-In

Let F(t) denote the failure distribution for the item. It is assumed that the failure rate r(t) has a bathtub shape with r(t) strictly decreasing over 0 < t < Tm with (Tm > 0). If an item is subjected to a burn-in for a period t , the failure distribution of the item after burn-in, FT(i), is given by (10.71) Let / T(r) and rT(t) denote the failure density function and the failure rate associated with the distribution function FT(r). Let i|/( t ) be the expected cost per item when the burn-in is of duration t . This depends on whether the product is repairable or not. For a repairable product, we assume that all failures during burn-in are fixed minimally. As a result, we have (10.72) where, as before, Cm is the manufacturing cost per item and cr is the expected cost of each repair; cx is the fixed setup cost of burn-in per unit, and c2 is the cost per unit time of burn-in per unit. Note that i|/( t ) is an increasing function of t . For a nonrepairable product, the burn-in cost per unit is 1.

2.

cx + c2t, 0 < t < t , if the item fails at age t during burn-in cx + c2t , if the item survives the burn-in

As a result, the expected burn-in cost per unit is given by

Since the probability of a unit surviving the burn-in is F ( t ) , and since costs must be recovered from items sold, we have the expected cost per unit

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given by

(10.73) Note that i|/( t ) is an increasing function of t in this case as well. 10.5.2

Total Expected Cost per Unit

The manufacturer’s expected cost per unit, E[CS(W; t )], is given by (10.74) where i|/(t ) is given by (10.72) or (10.73) depending on the type of product, and co(W; t ) is the expected warranty cost per unit. a)(W; t ) depends on the type of warranty policy, the nature of the product (repairable or nonrepayable), and the type of rectification action. In the sections that follow we consider four different cases. 10.5.3

FRW [Repairable Product]

We assume that all failed items returned under warranty are fixed minimally. As a result, the expected warranty cost per unit is given by

(10.75) where ch is the handling cost for each claim. Using (10.72) and (10.75) in (10.74), we have

(10.76) 10.5.4

FRW [Nonrepayable Product]

In this case, whenever an item fails under warranty, it is replaced by a new one at a cost of i|/(t ) + ch per replacement. The expected number of

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replacements under warranty occur according to a renewal process. As a result, the expected number of replacements under warranty is given by Mt (W), the renewal function associated with the distribution function FT(f). The expected warranty cost is given by (10.77) Using (10.73) and (10.77) in (10.74), we have

(10.78)

10.5.5

PRW [Linear and Lump-Sum Rebates]

Under this policy, whenever an item fails under warranty, the manufacturer refunds an amount R(t) given by (10.79) where 0 < / c < l , 0 < a < l , and ch is the sale price per unit. Two special cases are Case 1: Linear rebate: a = 1 and k = 1. Case 2: Lump-sum rebate: a = 0 The warranty cost per unit is the expected refund per unit, given by

By use of (10.71) and (10.79), this can be rewritten as

(10.80)

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Using (10.72) and (10.80) in (10.74), we have

(10.81) 10.5.6

Optimal Burn-In

The optimal t * is the value of t that minimizes (10.76), (10.78), or (10.81), depending on the policy, t * can be obtained by the usual first-order condition. Note that, for obvious reasons, t * must always be less than Tm . Example 10.7 Let F(t) be given by

with X1? X2 > 0 < Pi < 1, 02 > 1, and 0 < p < 1. When p = 0, F(t) has an increasing failure rate, and when p = 1, it has a decreasing failure rate. When 0 < p < 1, it has a bathtub failure rate. Consider p = 0.1, \ x = 4, X2 = 0.08, = 0.5 and 02 = 3, with cost parameters Cm = $5, cr = $2, cx = $0.20, c2 = $5, ch = $10, and S = $20. Figure 10.14 shows t * as a function of W for the four cases considered, namely, 1. 2. 3. 4.

FRW FRW PRW PRW

policy policy policy policy

[item repairable] [item nonrepairable] [lump-sum rebate: a = 0 and k = 1] [linear rebate: a = 1 and k = 1]

To study the variations in the magnitude of savings in expected total cost, define t i(W) as

where o)„(W) is the expected warranty cost if no burn-in is employed. Thus, t ](W) is the relative savings using burn-in as opposed to not using it. Figure

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468

Figure 10.14

t

* vs. W for four different warranty policies.

10.15 shows a plot of as a function of W for the four different cases. is negative for certain values of W because of the fixed burn-in cost Ci, if c1 is zero, then t i(W) is always positive and approaches 0 as W approaches 0 or oo.

t ](W)

10.6

ADDITIONAL TOPICS

Some issues that we have not covered are the following: 1. Certain products are designed differently for different usage. For example, washing machines for domestic use are different from those for commercial use (e.g., in laundromats, hospitals, or hotels). Designing for different usages and taking this into account in selecting warranty items is a topic that we have not discussed. 2. Usually a warranty becomes null and void when an item is subjected to misuse or abuse, e.g., a domestic washing machine being used in a commercial application. Design and use of features to detect this would reduce warranty payouts as long as the user cannot tamper with the detector. One example is the odometer for monitoring miles that a car is driven (although dishonest used-car dealers can and do reset the odometer).

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Figure 10.15 r\* vs. W for four different warranty policies.

NOTES

Section 10.2 1. There are many books which deal with the design of reliable systems. See, for example, Kapur and Lamberson [6] and Dhillon and Reiche [7]. Design in the context of warranty is discussed in Marshall [8]. Section 10.3 1. Our presentation of reliability choice and allocation is based on Nguyen and Murthy [9]. 2. Our presentation of optimal product development is based on Murthy and Nguyen [3]. For more on reliability growth, see Dhillon [10]. 3. The literature on design based on redundancy is very extensive. See, e.g., Goldberg [11] and Tillman et al. [12]. 4. Diagnostic design is a relatively new concept, and very little work has been done. See Hegde and Kubat [13] and Malcolm and Foreman [14]. Section 10.4 1. Model I is based on Murthy et al. [5]. Model II is based on Djamaludin et al. [15].

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2.

Marucheck [16] discusses acceptance sampling schemes incorporating resulting liability. Goel and Coppola [17] deal with acceptance sampling of components when the failure distribution is exponential. See Lie and Chun [18] for a different model formulation. 3. There are many papers dealing with screening of parts; see Morrison [19]. Ritchken et al. [20] and Tapiero and Lee [21] deal with quality control, product design, and servicing. 4. Mann [22] and Mann and Saunders [23] deal with problems in which the warranty period is based on the outcome of testing a sample from the batch of items. Section 10.5 1. The model presented is based on Nguyen and Murthy [24]. 2. The literature on burn-in is very extensive—see Leemis and Beneke [25]. EXERCISES

10.1. 10.2.

10.3. 10.4. 10.5. 10.6. 10.7. 10.8.

Derive conditions similar to those of Proposition 10.1 when redundancy is passive and consists of two spare units and the switch is perfect. Derive conditions similar to those of Proposition 10.1 when redundancy is passive and consists of one spare unit and the switch is imperfect in the sense that it operates [fails to operate] when required with probability (1 - q) [q\. Prove Proposition 10.2. Prove Proposition 10.3. Prove Proposition 10.4. Prove Proposition 10.5. Prove Proposition 10.6. Prove Proposition 10.7.

REFERENCES

1. 2. 3.

Crow, L. H. (1974). Reliability analysis for complex repairable systems, in Reliability and Biometry, F. Proschan and R. J. Serfling (eds.), SIAM, Philadelphia, pp. 397-410. Duane, J. T. (1964). Learning curve approach to reliability monitoring, IEEE Trans. Aerospace, 2, 563-566. Murthy, D. N. P., and Nguyen, D. G. (1987). Optimal development testing policies for products sold with warranty, Rel. Eng., 19, 113— 123.

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4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21.

471

Heathcote, C. R. (1971). Probability: Elements o f the Mathematical Theory, George Aleen and Unwin, London. Murthy, D. N. P., Djamaludin, I., and Wilson, R. J. (1993). Product warranty and quality control, Reliability and Qual. Eng., in press. Kapur, K. C ., and Lamberson, L. R. (1977). Reliability in Engineering Design, John Wiley and Sons, New York. Dhillon, B. S., and Rieche, H. (1985). Reliability and Maintainability Management, Van Nostrand Reinhoid, New York. Marshall, C. W. (1981). Design trade-offs in availability warranties, in Proc. Annual Rel. and Maint. Symp., 95-100. Nguyen, D. G., and Murthy, D. N. P. (1988). Optimal reliability allocation for products sold under warranty, Eng. Opt., 13, 35-45. Dhillon, B. S. (1980). Reliability growth: A survey, Microelectronics and Rel., 20, 743-751. Goldberg, H. (1981). Extending the Limits o f Reliability Theory, John Wiley and Sons, New York. Tillman, F. A., Huang, C. I., and Kuo, W. (1977). Optimization techniques for system reliability with redundancy: A review, IEEE Trans. Rel., 26, 148-155. Hegde, G. G., and Kubat, P. (1989). Diagnostic design: A product support strategy, Euro. J. Oper. Res., 38, 35-43. Malcolm, J. G., and Foreman, G. L. (1984). The need: Improved diagnostic— rather than improved R, in Proc. Annual Rel. and Maint. Symp., 315-322. Djamaludin, I., Murthy, D. N. P., and Wilson, R. J. (1993). Quality control through lot sizing for items sold with warranty, Int. J. Prod. Econ., in review. Marucheck, A. S. (1987). On product liability and quality control, HE Trans., 19, 355-360. Goel, A. L ., and Coppola, A. (1979). Design of reliability acceptance sampling plans based upon prior distribution, in Proc. Annual Rel. and Maint. Symp., 34-38. Lie, C. H ., and Chun, Y. H. (1987). Optimal single-sample inspection plans for products sold under free and rebate warranty, IEEE Trans. Rel., R-36, 634-637. Morrison, A. J. (1983). Application of a parts screening program, in Proc. Annual Rel. and Main. Symp., 231-235. Ritchken, P. H., Chandramohan, J., and Tapiero, C. S. (1989). Servicing, quality design and control, I1E Trans., 21, 213-220. Tapiero, C. S., and Lee, H. L. (1989). Quality control and product servicing: A decision framework, Euro. J. Oper. Res., 39, 61-73.

472

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22.

Mann, N. R. (1970). Warranty periods based on three ordered sample observations from a Weibull propulations, IEEE Trans. Rel., R-19, 167-171. Mann, N. R., and Saunders, S. C. (1969). On evaluation of warranty assurance when life has a Weibull distribution, Biometrika, 56, 615625. Nguyen, D. G., and Murthy, D. N. P. (1982). Optimal burn-in time to minimize cost for products sold under warranty, HE Trans., 14, 167-174. Leemis, L. M., and Beneke, M. (1990). Burn-in models and methods: A review, I1E Trans., 22, 172-180.

23. 24. 25.

11

The Simulation Approach to Warranty Analysis

11.1

INTRODUCTION

As indicated in Sections 2.3 and 2.4, the analytical approach involves (1) building a mathematical model, (2) analysis of the model, and (3) interpretation of the results. The analysis can be either mathematical (usually called simply “analytical”) or computational. The analytical approach yields solutions (or final results) in closed-form mathematical expressions. The advantage of this is that it allows for parametric study since the solutions are explicit functions of model parameters. An illustrative case is the following: In many warranty studies, one is interested in the expected number of failures over an interval [0, W) as a function of model parameters. Consider the case where the first item is put in use at t = 0; all item failures over [0, W) are rectified through replacement of each failed item by a new one, and F(t; 0) characterizes the failure distribution, with 0 being the model parameter. From Chapter 3, we see that failures can be modeled by a renewal process with time between renewals given by F(t; 0). The expected number of failures over [0, W) is given by M (W ; 0), where M(t; 0) is the renewal function associated with F(t; 0). M(-; 0) is obtained as the solution of the renewal equation. For special forms of F(t; 0), M(t; 0) can be obtained analytically. For example, when F(t; 0) is the exponential distribution, M (t; 0) = 0i, so that one may carry out a study of M (t; 0) as a function of the parameter 0. Unfortunately, it is impossible to carry out an analysis using analytical methods for most model formulations. For example, in the preceding illustrative case, for most forms of F{t\ 0), one needs to use a computational approach because no closed-form expression exists for M (t; 0). In Section 473

474

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3.6, we discussed numerical and simulation approaches to obtaining the renewal function. Similarly, for many models in Chapters 4-10, we had to resort to the numerical approach to carry out the analysis and optimization. The numerical approach to analysis was seen to be extremely involved for some of the models. We give four examples. 1. In Section 5.4.1, the life cycle cost to the buyer of items sold under renewing linear pro-rata warranty was considered. The result, given in Equation (5.47), is an integral that is not easily evaluated, either analytically or numerically, for most important life distributions. There are a number of other instances of such functions in the analysis of the basic FRW and PRW given in Chapters 4 and 5, many of which are both analytically intractable and not easily handled numerically over the ranges of integration that are of interest in applications. 2. Combination warranties, discussed in Chapter 6, are even more difficult to analyze. For example, the average cost to the buyer for items purchased under renewing combination FRW/linear PRW is given in Equation (6.10). This expression is evaluated analytically for the exponential distribution in Example 6.3. Simulation results for the Weibull distribution (the details of which will be given later in this chapter) are discussed in Example 6.4. 3. Analysis of cumulative warranties (Section 6.3) leads to many renewal functions, partial expectations, and renewal-type equations for very complex random variables. Examples are the buyer’s expected life cycle cost for the cumulative FRW, given in Equation (6.57), and the buyer’s expected life cycle replacement cost under cumulative PRW, given in Equation (6.69). 4. In Section 8.6, we were interested in obtaining the expected number of repeat purchases over the product life cycle. This required the solving of two-dimensional integral equations involving functions of five variables and, as remarked there, these are impossible to obtain analytically and are difficult and time consuming to obtain using the numerical approach. In all of the preceding examples, the simulation approach offers an alternate method for obtaining the solution to the problem without the need of solving complicated mathematical formulations. In this chapter we focus our attention on the simulation approach to analysis of warranty costs. The outline of the chapter is as follows. In Section 11.2, we give a brief introduction to simulation methodology and its use in warranty studies. It is worth noting that, although the simulation approach can be used in cases where other approaches (e.g., analytical or numerical) do not work, the approach has limitations. This issue is also discussed in Section 11.2. Fol-

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475

lowing this, we discuss simulation modeling and some of the different simulation languages in Section 11.3. Since our models involve one- and two-dimensional distribution functions for characterizing item failures, the success of the simulation approach depends critically on the efficiency with which random numbers from a given distribution function can be generated on a computer. This topic is discussed in Section 11.4. In Section 11.5, we briefly discuss simulation on a microcomputer and give an illustration using Minitab. In Sections 11.6 and 11.7, we discuss obtaining the one- and two-dimensional renewal functions by simulation. In Section 11.8, we give a fairly general simulation package for one-dimensional warranty cost analysis, which allows the user to specify arbitrary distribution functions for time to failure. In Section 11.9, we give some examples to illustrate the use of the simulation approach in model analysis. Finally, we conclude with some general comments on building expert systems for warranty studies.

11.2 SIMULATION METHODOLOGY

We illustrate the simulation methodology by considering the illustrative case discussed in Section 11.1. We are interested in obtaining the expected number of failures over some interval, say [0, T), using the simulation approach. We assume that failures occur independently according to a distribution function F(t; 0) and that failed items are replaced instantaneously by new ones.

11.2.1

Basis of Simulation

The basis of simulation is to obtain an estimate of the quantities in question based on statistics gathered from a large number of statistically independent time histories of item failures over [0, T) generated on a computer. Note that this is distinct from statistical estimation based on lifetimes or other actual data, which will be discussed in Chapter 12. Simulation proceeds as follows: For each time history t = 0 corresponds to starting a new warranty period (or cycle) with a new item. Each time history can be viewed as a realization of a renewal process with t = 0 being a renewal point. (The process of generating a time history is discussed in the next subsection.) Let K denote the number of independent time histories generated, and let N ^T) be the number of renewals in [0, T) in the ith simulation run (i.e., the ith time history generated). N ^T) is a random

Chapter 11

476

variable with mean M(T; 0) and variance a 2( / ’; 0) given by

( 11. 1) and

( 11.2) where F^\t; 0) is the j-fold convolution of F(t; 0) with itself. The following is an estimator of the expected number of failures over [0, T): (11.3) This estimator is a function of the statistics obtained from the time histories of failures simulated on the computer. Since the expected number of failures is M (T; 0), M(T\ 0) can be viewed as an estimator for M(t; 0) for t = T. Alternate estimators for M(t\ 0) will be discussed in Section 11.6. 11.2.2

Generating Time Histories to Obtain Af,(r)

Note that the time between renewals is a sequence of independent and identically distributed random variables with distribution function F(t; 0). The logic flow chart to obtain N^T) is shown in Figure 11.1. The variable TIME corresponds to time and is set to zero at the start of each simulation run. It is advanced in each iteration by a random amount X , which represents the time between renewals. X is generated using a random number generator. (The process of generating random numbers according to a specified distribution F(t; 0) will be discussed in Section 11.4.) Each simulation run (or time history) ends when TIME exceeds T and yields a value Ni(T) for the ith simulation run. At the end of K independent simulation runs, we have the statistic N ^T), 1 < / < /C, and the estimate M (T; 0) is obtained using (11.3). 11.2.3

Statistical Analysis of the Estimate M(T\ 0)

Since the simulation runs are statistically independent, the A^T) are a sequence of independent and identically distributed random variables. It

The Simulation Approach to Warranty Analysis

F ig u re 11.1

477

Logic flow diagram to generate time histories for obtaining N¿(T).

is easily shown that (11.4) and (11.5) In other words, the estimator is an unbiased estimator (see Chapter 12) that converges to the true value in the mean-square-error sense as n —» o°. In addition, as will be shown in Chapter 12, one can obtain confidence limits for M(T; 0) as follows: An approximate 100(1 - a) confidence

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478

interval for the expected number of failures is given by

( 11.6) where (11.7) and zi _ a /2 is the (1 - a/2)-fractile of the standard normal distribution. (See Chapter 12, particularly Sections 12.3.2 and 12.4.2.) 11.2.4

Computational Aspects

From a computational point of view, the effort involved depends on (1) K , the number of independent runs; (2) T, the length of time for each simulation; and (3) the complexity of generating the random variable X and the expected number of times X is generated in each simulation run. The precision of the estimate increases as K increases. However, this results in an increase in the total computation involved. Thus, one needs to strike a suitable trade-off between precision and computational effort. 11.3

SIMULATION MODELING

In Chapters 2 and 3, we discussed the use of mathematical models for warranty studies and the process of building such models. In Chapters 4-10, we dealt with models for various aspects of warranty. These models are suitable for either analytical or numerical approaches to analysis. As such, the model formulations were constrained by the limitations of such approaches. For example, we implicitly assumed that all users are similar, so that item failure does not vary from user to user. If we relax this assumption, then the model formulation becomes more complex and also more difficult to solve, either analytically or numerically. When a simulation approach is used, no such constraints need to be imposed. Models for simulation are built in terms of simple equations and logical expressions, and they are organized in such a way that the operations on the computer correspond to operations in the real world. As such, they are built in a series of sections. Each section is described in a straightfor-

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479

ward manner (using simple equations and logical expressions), without undue concern for the complexity introduced by having many such sections. The terms simulation model and simulation modeling mean the following: A mathematical model developed for study by simulation is called a simulation model, and the process of building such a model is called simulation modeling. In practice, a special terminology has evolved for use in simulation modeling. We describe this in the next subsection.

11.3.1

Simulation Terminology

In simulation modeling, the real world relevant to the problem under study is viewed as a system consisting of many sections, each interacting with the other. Each section is described in terms of entities, and the term attribute denotes a property of an entity. Any process (internal or external) that causes changes in the system over time is called an activity. The initiation, alteration, or conclusion of an activity is called an event. The term state o f the system is used to mean a description of all the entities, attributes, and activities as they exist at a given point in time. The evolution of the system is studied by following the changes in the state of the system. Entities and attributes make up the static structure of a simulation model. They describe the state of the system but not how the system operates. This is the task of activities and events. The key to simulation is the model builder’s ability to organize system events so that the order in which they are executed on the computer corresponds to the order in which they occur in the real world. This preserves the time relationships between simulated and real events. Events that cause changes to the state of the system can either occur continuously over time or at discrete points in time. When changes occur at discrete points in time, the simulation is called discrete simulation. In discrete simulation, time is broken into intervals. The beginning and end of each interval correspond to the occurrence of an event. These intervals are not necessarily equal and can be random variables. The flow of time is achieved by updating the clock at the time instants at which events occur. In the remainder of the chapter, we will confine ourselves to discrete simulation, since simulations for warranty studies are characterized by events occurring discretely over time. In the case of the illustrative example of Section 11.1, the manufacturer and consumers (who have bought the product) would be called entities; items sold or used as replacements would be called activities; and the sale and the return of failed items (for replacement by new ones) would be called events.

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480 11.3.2

Building a Simulation Model

The building of a simulation model involves the following two steps: Step 1 [Model Statement] This involves breaking the system into sections and identifying entities, activities, and events. Step 2 [Computer Implementation] Implementation of the model involves writing a computer program to generate a time history of changes based on the model statement of Step 1. This consists of 1. 2. 3.

Writing a set of event programs to describe the system operating rules Defining lists and matrices that store information Writing an executable routine that directs the flow of information and control

11.3.3

Event Versus Activity Oriented Simulation

In discrete simulation, events and activities play a major role. The key element of simulation is the simulation algorithm, the structure of which is shown in Figure 11.2. Basically, the procedure involves finding the next potential event and executing it, if possible. The execution of an event involves advancing the clock and changing the state of the system. In a compuer, the state is represented by numerical values and is called the system image. Changes in the system image over time correspond to changes in the real world. The key factor in simulation is the control of sequences that cause changes to the system image. There are two ways of executing this control, which lead to two approaches to simulation: event oriented simulation and process oriented simulation. In event oriented simulation, the emphasis is on event finding, and the simulation algorithm automatically obtains the next event when all the state changes have been completed. The user supplies event “routines” that give a detailed description of state changes taking place at the time of each event. In process oriented simulation, the emphasis is on a process, that is, a time-ordered sequence of events, separated by the passage of time, which describes the entire experience of an entity. The user supplies a process “routine” for each different process in the simulation model. The subtle difference is that, in event oriented simulation, the times at which system changes occur are treated as characteristics of the activities, while in process oriented simulation these times are treated as attributes of entities. The difference is important, as some languages (to be discussed in Section 11.3.5) are based on the event oriented approach, while others

The Simulation Approach to Warranty Analysis

481

Figure 11-2 Simulation algorithm for discrete simulation. are based on the process oriented approach. The difference is in the control of program execution, but the final simulation remains the same. 11.3.4

Verification and Validation

Verification is a testing procedure that establishes the faithfulness of the computer output (obtained from Step 2 of Section 11.3.2) to the response of the model (stated in step 1). Validation refers to testing in order to establish the credibility of the simulation model. Here the testing involves comparing the computer output with real data. 11.3.5

Simulation Languages

Though almost any type of programming language can be used to implement a simulation model, various special languages have been developed

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for simulation applications. These languages provide the user with a variety of services; they are designed to ease the job of translating a model statement (Step 1 of Section 11.3.2) into a computer program (Step 2 of Section 11.3.2). They contain (a) status descriptions, which are definitions of the essential elements of the model statement; (b) procedures for modifying the state of the model; and (c) procedures for controlling and observing model behavior. In addition to these, extensive error checking exists to prevent misuse of the language, as well as routines for debugging and locating logical errors. For discrete event simulation, many different simulation languages have been developed over the years. The advantages of a simulation language over a general purpose language are the following: 1. Simulation languages automatically provide most of the features needed for modeling. Hence, less programming time is required, and, in general, the code is smaller. 2. Simulation languages provide basic blocks especially suited for execution of Step 1 (model statement) and Step 2 (translating into a program) of Section 11.3.2. 3. Simulation models written in simulation language are easier to change. 4. Simulation languages provide better error detection and hence a lesser chance of an error being undetected. On the other hand, simulation languages do have some disadvantages in relation to general purpose languages. These are as follows: 1. 2.

Simulation languages lack the flexibility of general purpose languages. In general, they are less efficient compared with an efficient code in a general purpose language.

It is not possible to discuss the different discrete simulation languages in detail. We give a brief outline of some of the more commonly used ones. GPSS/V and GPSS/H [General Purpose System Simulator]

GPSS/V and GPSS/H are process oriented simulation languages. The latter is a compiled language as opposed to the former, which is interpretive. As a result, simulation models written in GPSS/H run faster than those written in GPSS/V. The GPSS/H language consists of more than 60 standard statements, many of which have a pictorial representation (called a block), which is suggestive of an operation performed by the statement. Building a GPSS model can be thought of as combining a set of standard blocks into a block diagram that represents the path taken by a typical entity during simulation. Once a block diagram is constructed, the translation into a program is fairly straightforward using the standard statements.

The Simulation Approach to Warranty Analysis

483

Simscript 11.5

Simscript II.5 is a process oriented or event oriented simulation language; however, because of the generality of the process approach, the use of event approach is not essential. The English-like and free-form syntax makes programs written in Simscript II.5 easy to read and almost selfdocumenting. The main features, viz., a general process approach, sophisticated data structures, and powerful control statements, make this language especially well suited for large, complex simulation models that are not queuing oriented. SLAM II [Simulation Language for Alternative Modeling]

SLAM II allows one to build a process oriented model, an event oriented model, or a combination of the two. The building of a process model begins with a graphical network diagram (involving nodes and branches) of the system being simulated. For example, a node might represent the creation of an entity, and a branch may correspond to the passage of time. The network description is translated into a program using an equivalent set of SLAM II statements. 11.3.6

Simulation for Warranty Studies

Simulation for warranty studies depends on the goal or aim of the study. When the aim is to obtain expected warranty cost per unit sale or over a product life cycle, one needs to simulate the time history for item purchases, failures, and replacements for one consumer only. When the aim is to evaluate servicing strategies (for example, use of loaners) one needs to simulate the time histories for item purchases, failures, and replacements for all consumers. This implies keeping track of item failures for each consumer. Although most special simulation languages can be used for simulating time histories for warranty studies, the special structure of the models (e.g., models in the study of two-dimensional warranties) makes general purpose languages attractive because of their flexibility. Hence, in most of the remainder of the chapter we confine ourselves to the use of FORTRAN as the language for building and simulating models. 11.3.7

Disadvantages of the Simulation Approach

The disadvantages of simulation are the following: 1. The results of the simulation approach only yield statistical estimates. The precision of the estimate can be improved by increasing the number of simulation runs, but this requires increased computational effort, and hence increased cost.

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2. The simulation (as well as the numerical) approaches yield answers for a specific set of model parameter values. The simulation (or computation) must be repeated if the parameter values are altered. 11.4

RANDOM NUMBER GENERATION

The success of the simulation approach depends critically on the ability to generate random variables with specified characteristics in an efficient manner. In this section we briefly discuss the generation of such random numbers. 11.4.1

One-Dimensional Random Number Generator [Using

FMÌ

We commence with the case where F(x) is a uniform distribution over [0, 1], which we denote X ~ U{0, 1), given by ( 11. 8)

We shall denote random variables generated according to this distribution by Uh i > 1. Following this, we shall indicate methods for obtaining random variables X t from F(x) for a variety of distribution functions of interest in warranty studies. Uniform Distribution

The great majority of random number generators in use are linear congruential generators (LCGs). Here, a sequence of integers Zl5 Z2, . . . is obtained by the recursive formula (11.9) where m, a, c, and Z0 are nonnegative integers with 0 < m, a < m, c < m, and Z0 < m. m is called the modulus; a is called the multiplier; c is the increment; and Z0 is called the seed or the starting value. Note that Z, is an integer assuming values in [0, m - 1], i/f-is obtained by the relationship ( 11. 10)

Starting with different values of Z0 generates different sequences. For a given Z0, the sequence repeats itself when i becomes sufficiently large.

The Simulation Approach to Warranty Analysis

485

This defines a cycle, and the cycle continues to repeat itself. The length of the cycle is called the period of the generator, and for LCGs the period is at most m. By proper choices of m, a, and c, the period of an LCG can be made equal to m. The proper choice of these values is machine dependent, and we shall not go into the details of computer implementation. This information may be found in Law and Kelton [1]. Other types of uniform random number generators are mixed generators, multiplicative generators, and composite generators, to name a few. Further details of these can also be found in Law and Kelton [1]. Nonuniform Distributions

We shall confine our attention to the generation of a continuous random variable with distribution function F(*). The most common approach is the inverse transform approach. The method is efficient only if F _1, the inverse function of F, can be easily evaluated. The algorithm for obtaining Xj is as follows: Generate ~ U(0, 1). Compute X t from the relation X t = F~1(Ui).

Step 1: Step 2:

Many other methods for obtaining X with distribution F(jc) are available, but they will not be discussed here. We give the algorithms for obtaining X /s for a variety of distribution functions F(jc) of interest in warranty studies. Uniform

[a < X < b] Generate Ut ~ f/(0, 1). Compute X t from the relation X t = a + (b - a)Ut.

Step 1: Step 2: Exponential

[with mean 0]

Step 1: Generate Ut ~ U{0, 1). Step 2: Compute X t from the relation X t = - 0 lr\{U^. k-Erlang Step 1: Step 2: Weibull Step 1: Step 2:

[with mean 0] Generate t/iy ~ ¿7(0, 1), 1 < j < k. Compute X t from the relation X ( = -(Q/k) In n * = 1(f/ y )[shape parameter p and scale parameter \] Generate i/f- ~ ¿7(0, 1). Compute X t from the relation X t = X[ —ln(t/f-)]1/p.

Chapter 11

486

Normal [iV(0, 1)] We describe the polar method, which generates random variates in pairs (Xn , X i2). The algorithm is as follows: Generate f/, and U{0, 1). Let Wg = U2 + V 2. Step 2: If Wi > 1, go back to Step 1. Otherwise, let Z, = V ( - 2 In W^/W^ X n = (2Ut - 1)Zf- and X a = (2V§ - l) Z ,

Step 1:

Normal

[iV(|x, a 2)]

Step 1: Generate Y( ~ A(0, 1). Step 2: Compute X { from the relation X t = |x + cry,. Lognormal Step 1: Step 2: 11.4.2

[shape parameter a and scale parameter

jjl ]

Generate y, ~ N(\l , a 2). Compute X t from the relation X t = exp(y/).

One-Dimensional Random Number Generator [Using h(x)]

As discussed in Chapter 3, when failed items are repaired minimally, then item failures occur according to a nonstationary Poisson process with intensity function h(x), x > 0. A variety of methods have been developed for generating event times when events occur according to a nonstationary Poisson process. (Note that for a stationary Poisson process, the interevent times are exponentially distributed and hence can be obtained from the algorithm indicated in the previous subsection.) We shall give an algorithm for a method known as the thinning method. It is based on the “acceptance-rejection approach” and requires h(x) to be bounded from above. Let h* = max{/i(jc)}. Let X {_x be the time instant of the (i - l)st event, and assume that this has been validly generated. The algorithm gives X h the time instant for event i (the next event), as follows: Step Step Step Step

1: 2: 3: 4:

Set x = X i_l. Generate Ux and U2 ~ U(0, 1). Replace x by x - (1/h*) In U1. If U2 ^ h(x)lh*, return X t = x. Otherwise, go back to step 2.

The Simulation Approach to Warranty Analysis

487

We assume X 0 = 0, since this corresponds to the item being new at t = 0 in our simulation studies. 11.4.3 Two-Dimensional Random Number Generator [Using F[x, y)]

The study of two-dimensional warranties via the simulation approach requires generating pairs of random variables (Xh Y,) from a two-dimensional distribution function F(x, y). Note that F(x, y ) can be written as

Using the conditional distribution, we generate (Xh Y,) separately as two univariate random variables. The algorithm is as follows: Step 1: Step 2: 11.5

Generate X t ~ Fx {x), obtaining the value x{. Generate Y, ~ FY\x {y\X = 2Q.

SIMULATION ON A MICROCOMPUTER

For many “small” problems, simulation on a PC or other microcomputer may be appropriate. This approach would typically involve the use of a “canned” package along with a minimal amount of relatively simple programming. There are many statistical and other program packages that lend themselves well to this type of analysis. What is needed for this approach is a program that will generate random variables from a variety of life distributions and will also enable the user to program the various additional computations that are required in warranty analysis. Programs of this type, which are ordinarily written either on a word processor or while running the package itself and then stored, are called Macros. With the capability of writing and saving Macros, it is often relatively easy to simulate renewal functions and many of the related processes encountered in the warranty cost models, as well as to make repeated runs, as necessary, quite easily. In this section, we illustrate this approach using Minitab, which is a widely available statistical program package with considerable statistical analysis capability, and in which Macros are relatively easily written and stored. 11.5.1

Building a Microcomputer Simulation Model

Note that the approach to building a simulation model outlined in Section 11.3, involving the two steps of building a model and computer imple-

488

Chapter 11

mentation, is applicable to simulation on a microcomputer as well. The difference is that the second step will typically involve a relatively simple set of instructions (i.e., the Macro) rather than an extensive and complex computer code. Thus, the simulation languages previously discussed would ordinarily not be used in employing statistical packages for simulation; instead, instructions employing program commands or using some simplified versions of, e.g., FORTRAN or BASIC, would be used. In addition, random number generation would proceed as programmed by the supplier of the statistical software package, rather than by programming the algorithms discussed in Section 11.4. Typically, such packages include options for selecting among many life distributions. It is important to note that caution is required in using microcomputers for simulation, as there are some special problems encountered when using this type of equipment because of short word length, and some microcomputer software and hardware have apparently provided less than satisfactory solutions. This problem is discussed by Law and Kelton [1], Section 7.5, and in the several references on the subject that they cite. 11.5.2

Illustration: A Minitab Macro

Minitab is a general purpose statistical package for data manipulation, editing, and analysis. This software is widely used in academic institutions and in industry as well, and it is available on a wide variety of systems. It is command-driven, relatively easily learned, and includes a wide variety of statistical analyses. (See [2]. Note that later versions of Minitab are menu-driven but can be used as command-driven as well.) Data may be entered into a matrix (called a Minitab Worksheet) in several ways, including randomly generated numbers. Distributions that may be used to generate data include the exponential, Weibull, gamma, lognormal, normal, uniform, beta, and many others. Sequences of commands and loops may be created (either in Minitab or by use of a word processor) and stored. (See [2] for details.) This sequence of commands may then be executed a specified number of times, with the results entered into a Minitab worksheet for further analysis or stored on a diskette or hard disk for later analysis. Because of these features, Minitab lends itself quite well to small-scale simulation studies. We illustrate the use of Minitab by means of a simulation study of Policy 7a, discussed in Example 6.4. The function to be simulated is £[R(y)], the expected replacement cost to the buyer per unit for items sold under combination FRW/linear PRW, assuming a Weibull distribution of time to failure. The formal analytical solution for this problem is given in Equation (6.10). This function is used in the analysis of Policy 7a, in which free

The Simulation Approach to Warranty Analysis

489

replacements are provided up to time W1? and replacements are provided at cost (X - Wx)ch!{W - Wx) for failed items having failure times X in the interval (WU W], where ch is the selling price of the item. Replacements of items whose lifetimes exceed W are purchased at full price. In this analysis, Y is the interval between actual purchases, i.e., free replacements are excluded in averaging costs. The buyer’s replacement cost, R(Y), is given in Equation (6.8). To simulate this process, it is necessary to generate a set of random variables X , check each to determine if AT < W1, and generate additional random variables for those cases for which this is so (i.e., for those cases in which free replacements are provided), continuing the process until the total service time in each instance exceeds Wx. Finally, replacement costs must be calculated and averaged to form the estimate of £[/?(Y)], which we denote £[/?(Y)]. A Minitab Macro that will accomplish this simulation assuming Weibull lifetimes is given in Figure 11.3. (The Weibull distribution function is given in Example 6.1.) The Macro, in fact, consists of three parts, each of which is itself a Minitab Macro. The first is entitled ER-WEIB. MTB, and is run by entering the command EXEC ‘ER-WEIB’ K where K is the number of times the macro is to be executed. ER-WEIB, in turn, calls the additional Macros PR1.MTB and PR2.MTB. Inputs to the program are K1 = number of observations to be generated (with standard PC Minitab configurations, K1 of about 4000 or 5000 is the practical maximum); K2 = p (the Weibull shape parameter); K3 = X (the Weibull scale parameter); K4 = Wx\ and K5 = W. Macro ER-WEIB first runs Macro PR1. PR1 generates K\ Weibull random variables with parameters p = K2 and X = K3, selects all results that exceed Wx = K4, and sets all values that exceed W to W. Next, PR2 is called several times. PR2 generates additional Weibull variables and adds the results to those in the initial set whose values were found to be less than Wx. This is done a maximum of three times, which assures (in any realistic warranty application) that it is highly unlikely that any results will be less than Wx. Finally, ER-WEIB calculates descriptive statistics (including mean, median, max, min, standard deviation, etc.) for the set of K\ calculated R-values. Note that this Macro may easily be modified to simulate other distributions as well. This is done by changing the subcommand “WEIB K2 K3” in PR1 and PR2 to specify alternative distributions in the Minitab library (and changing the parameter specifications, K1 and X2, and, if necessary, a few other statements, accordingly).

Chapter 11

490 EXEC *PR1* EXEC ’PR2’ K7 EXEC ’PR2’ K7 EXEC *PR2* K7 LET C2=(C2-K4)/(K5-K4) LET K3=l/K3 PRIN K1-K7 DESC C2 END

Ca) ER-WEIB.HTB

LET K3=l/K3 RAND K1 Cl; WEIB K2 K3. COPY Cl C2; OMIT Cl (0:K4). CODE (K5:99999)K5 C2 C2 LET K6=0 LET K7=(C0UN(C2XK1) END

(b) PR1.MTB

COPY Cl Cl; USE Cl (0:K4). N Cl K6 RAND K6 C3; WEIB K2 K3. LET C1=C1+C3 COPY Cl C3; OMIT Cl (0:K4). CODE (K5:99999)K5 C3 C3 STAC C2 C3 C2 LET K7=(C0UN(C2XK1) END

(c) PR2.MTB

Figure 11.3 Minitab Macros for evaluation of /?(T), Weibull distribution. Example 11.1 In Example 6.4, the values of the Weibull shape parameter used were P = 2 and 4. Values of A were chosen so that the mean times to failure were |x = 2.0 and 2.5. The combination warranty analyzed was for Wx = 0.25 and W = 1.0. A sample run of ER-WEIB using these parameter values and K \ = 4000 is given in Figure 11.4. Note that both the Minitab commands and the output results are listed. C2 is the column that contains the final R(T)-

491

The Simulation Approach to Warranty Analysis MTB > M TB > MTB > M TB > M TB > M TB > MTB > MT B > MT B > SUBO MT B > SUBO MTB > MTB > MTB > MTB > MT B > MTB > SUBO MTB >

N MTB > SUBO MTB > MTB > SUBO MTB > MTB > MTB > MTB > MTB > MTB > SUBO MTB > N MTB > SUBO MTB > MTB > SUBO MTB > MTB > MTB > MTB > MTB > MTB > MTB > MTB > K1 K2 K3 K4 K5 K6 K7 MTB >

LET Kl=4000 L ET K2=2 L ET K 3 = . 44312 LET K 4 = . 25 LET K5=l EXE C ’E R - W E I B ’ EXE C ’P R 1 ’ LET K 3 = l/K3 R A N D K1 Cl; W E I B K2 K3. COP Y Cl C2; O M I T Cl (0:K 4 ). CODE (K5:9 9 9 9 9 )K5 C2 C2 LET K6=0 LET K 7 = ( C 0 U N ( C 2 X K 1 ) END EXE C ’P R 2 ’ K7 COPY Cl Cl; U SE Cl (0:K 4 ) . N Cl K6

= 5 8 RAND K6 C3; WEIB K2 K3. LET C1=C1+C3 COPY Cl C3; OMIT Cl (0:K4). CODE (K5:99999)K5 C3 C3 STAC C2 C3 C2 LET K7=(C0UN(C2) oc?

( 12.20) and

( 12.21) Example 12.8 [Partial Moment, Exponential Distribution] In analysis of the expected cost of the PRW, an important quantity was the partial expectation |xw. For the exponential distribution, this was given

534

Chapter 12

(see Example 5.1) by

where W is the length of the warranty period. Suppose we estimate \l w based on X = 1IX, which is both the moment and ML estimator of X. The estimator is

We consider this as a function of Y = X, i.e., (Lw = g(JQ. Thus, E(X) = p, and V(X) = v 2/n = 1/nX2. We find

so, since |x = 1/X,

Similarly,

Thus,

Note that |iw is asymptotically unbiased since the last term in this expression tends to zero as n —» 0. Determine |x and a 2, and derive moment estimators of X and p. Derive the maximum likelihood estimators of X and p for the gamma distribution. (See Johnson and Kotz [8 ], Section 7.2, regarding solution of the likelihood equations.) Suppose time to failure is lognormally distributed, with density given by

t > 0. Determine the first two moments of the distribution, and derive moment estimators of r\ and 0 . Find the maximum likelihood estimators of t ] and 0 in the lognormal distribution. Discuss the relationship between these and the estimators of |x and a 2 in the normal distribution. Suppose that F(-) is the mixed exponential distribution, given by

t > 0. Show that the rth moment about zero is

Statistical Estimation of Warranty Costs

12.9. 12.10. 12.11. 12.12.

Derive moment estimators of Xl5 \ 2>and P based on the first three moments. (Note that the solution is not unique.) Derive the likelihood equations for the mixed exponential distribution, and discuss their solution. Write a program for the solution of the likelihood equations for the Weibull distribution. Write programs for the solution of the likelihood equations for the gamma and mixed exponential distributions. The following are data on time to failure (years) of a shipboard radar system: 0.610 1.377 1.109 1.361 1.836 2.034 1 .6 8 6 0.992 1.631 2.236 3.574 1.498 0.334

12.13.

571

3.074 1.846 2.740 3.134

1.861 0.303 0.401 2.223

2.170 1.995 0.307 2 . 2 2 0 1.902 1.218 3.447 2.536 1.127 2.298 0.322 0.380 1.665 1 . 2 2 2 0.535 2 . 2 1 1

Estimate the parameters of the failure distribution, assuming that it is Weibull. Calculate both the moment and ML estimates, and compare the results. The following are failure data in months of a high-intensity flood lamp:

OO

2.61 2.25 3.30 3.06 1.81 1.30 4.67 10.08 2.74 1.35 4.41 1 0 . 8 8 2.42 5.24 3.64 1.37 9.71 4.82 2.94 2.92 0.71 3.16 1.93 3.80 3.77 2.31 2

12.14. 12.15. 12.16.

Estimate the parameters of the failure distribution, assuming that it is lognormal. Suppose that the distributional assumptions were reversed in the previous two exercises. Estimate the parameters of the “wrong” distribution, and compare the results. Use the results of Exercises 12.12 and 12.13 to estimate the renewal function M(t) for t = .5(.5)2.0. Do the same using the results of Exercise 12.14, and compare the results. Forty-five engine components were subjected to extreme conditions of temperature and pressure and then run in a testing device

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until they failed. Failure times (days) were as follows: 0.90 0.30 2.31 0 . 1 0 1.29 0.23 0 . 1 2 0.48 0.15 0 .0 0 0.13 1.17 0.73 1.93 0.72 0.28 0.08

12.17. 12.18.

12.19. 12.20.

12.21.

12.22. 12.23.

2.36 1 . 0 2 0.82 1 . 0 1 0 .0 0 0 . 2 0 0.54 0.81 0.13 2.65 0 . 1 2 0.18 0.85 0.30 0.81 0.07 0.48 0.25 0.81 0.18 0.79 2 . 0 2 0.23 0.91

0.53 1.55 0.28 0.37

Estimate the parameters of the failure distribution, assuming that it is a gamma distribution. Estimate the corresponding renewal function for t = .5(.5)2. Extend the analyses of Exercises 12.14 and 12.15 to include the gamma distribution and the data of Exercise 12.16. For which of the preceding three data sets would it be possible to estimate the parameters of the mixed exponential distribution? Extend the analyses of the previous exercises, in so far as is possible, to this case as well. Using the maximum likelihood estimates of Exercise 12.12, determine asymptotic 95% confidence intervals for X and p. Suppose that the failure times of the engine components of Exercise 12.16 were exponential rather than gamma. Calculate the ML estimate of X. Use the result to determine a 90% confidence interval for component reliability R , defined to be R = P (T > 0.50 days). (Note that these are test data obtained under accelerated conditions.) Suppose F(t) is a Weibull distribution and that we wish to estimate xp, the /?-fractile of the distribution, defined by the relationship p = F{xp). Derive an expression for xp. The ML estimator x*p of xp is obtained by substituting the ML estimators X*, p* of X, p in this expression. Determine the asymptotic variance of jc * in terms of the asymptotic variances and covariance of X* and p*. Use the result of Exercise 12.21 to obtain a 90% lower confidence bound for the .99-fractile based on the data of Exercise 12.12. Suppose that lifetimes are IID and are exponentially distributed with parameter X, but that the data are grouped into k classes. Suppose that the classes are of equal width 8 , with the lower limit of the first class being x0, so that the ith class is the interval [jc0 + (i - 1)8, jc0 + i8 ), i = 1 , . . . , k. We observe nk, where Hi is the number of lifetimes in the ith class. Write the likelihood function in terms of data of this type, and determine the likelihood equation for estimating X.

Statistical Estimation of Warranty Costs

12.24. 12.25. 12.26.

12.27. 12.28. 12.29.

12.30. 12.31.

12.32. 12.33. 12.34. 12.35.

573

Devise a computerized scheme for solving the likelihood equation of the previous exercise. Select three sets of grouped data from Davis [44] and calculate the ML estimate of X using the routine of the previous exercise. Suppose that a renewal function M(t) is approximated by

Let |i and x and Y > y

Symbols a n d N otation

657

Joint probability density functionfor (X , Y) (= d2F(x, y)/dx dy) Failure rate function (= / ( x, y)/F(x, y)) M o d e lin a C o sts

Seller’s total warranty cost per item (random variable) Buyer’s total cost per item (random variable) Buyer’s purchase price Manufacturing cost per item Expected life cycle cost A n a ly sis

Sum of failure times and usages for items 1 to n (S* = 2?-i X, and 5 2 = 2"=1 Y,) Distribution function of (.S',1,, 5^) [F(n)(x, y) = F(x, y ) * F(x, y) * ■ ■ ■ * F(x, y)] Bivariate Laplace transform of F(x, y) Marginal distribution functions Renewal function associated with F(x, y) max{n: < x) max{n: s p 2 y} m inj^^, ^ 2)} [number of failures in [0, x) x [0, y)] m ax{^1}, A^2)} SYMBOLS SPECIFIC TO CHAPTER 4

Total sales Warranty execution function Parameter of warranty execution function Proportion of valid claims (Section 4.7.4) Proportion of invalid claims Proportion of valid claims rejected (Section 4.7.4) Proportion of invalid claims rejected Expected seller’s cost per item Discount function Life distribution of items leading to invalid claims

658

Symbols and Notation

SYM BO LS SPEC IFIC TO CHAPTER 5

a: X, E[Cb(Ii+1, W)] CbO Cb(T, W) C s(0

UCB K(W) H{L) K(L, W) h(t)

= cr/cb [repair cost relative to replacement] (Section 5.2.4) Age at failure for Item /, Expected cost to buyer of item h +1 (= cb - q{X,)) Total cost to the buyer under renewing PRW Expected short term average cost per unit to buyer Total cost to the supplier under conditional PRW = E[Cb(T, W)] Number of replacemnets = /o cb min{x/VY, 1} dF = E[Cb(L, W)]/cb = cbix,/W

SYM BO LS SPECIFIC TO CHAPTER 6 C o m b ina tion W a rra nties

a: W: Cb(a, W) Cs(a, W) Cb(W) CS(W) Z(L; W) i(L; W) S(X) A(W) Y R(Y) rr(W) n(L ; W) E[Cb(L, W)]

[al5 a 2, . . . , a*], parameters of combination lump-sum rebate policy [Wl9 W2, . . . , Wk\ , parameters of combination lump-sum rebate policy Buyer’s cost per item (random variable) Manufacturer’s cost per item (random variable) Buyer’s cost per item (random variable) Manufacturer’s cost per item (random variable) Buyer’s total replacement cost over (0, L] (random variable) Expected value of Z(L, W) Selling price of a replacement if X is the age of failed item Average cost to the buyer per unit of time Interval between purchases Buyer’s replacement cost Long-run profit to the seller Total expected profit over [0, L] Buyer’s expected life cycle cost

C um ula tive W a rra nties

n : Number of items supplied k : Number of items used simultaneously MCF(nW): Expected number of replacements under cumulative warranty

Symbols and Notation

E[CS(W, n)]: ^b,uc*

c b ,sc:

q(Sn; W, n) E[Ch(W , *)] t t (W, w) Z £[Cb(L; W, /i)] 7?(L) Pc K(T) Vq ° 2q C\y{T\ n, p,G) t t (T; n, pc)

Expected cost per unit to the supplier Indifference price [comparing cumulative warranty with no warranty] Indifference price [comparing cumulative warranty with standard warranty] Rebate function under cumulative warranty Buyer’s expected cost per unit Manufacturer’s profit per unit Purchase interval Expected life cycle cost to the buyer Replacement cost to buyer under cumulative PRW Guaranteed MTTF Estimated MTTF Number of replacements (= max{0, Q}) Mean of Q Variance of Q Total cost to the buyer under fleet warranty Manufacturer’s expected profit under fleet warranty

SYM BO LS SPECIFIC TO CHAPTER 7 M odel 1

n: CU(W): A: |1: cu: Cj: CR(W): cs: cM(W): cd(W): r:

Number of items in batch Cost to the buyer of an unwarranted item Amortization factor [ = W/L] Estimate of |x Purchase price per item Initial support cost for a batch of n items Total cost to the buyer of reliability modification Average recurring support cost per item Total user maintenance cost over [0, W] Contractor’s total expected cost of repair Contractor’s risk factor

M odels 2 and 3

U: DS(W): P': Cw-

659

Average utilization rate of items per unit time Damages assessed for stockouts during warranty period Fixed fee Expected cost to contractor associated with the warranty

Symbols and Notation

660

tg: Turnaround time [TAT] Dt(W): Expected damages to be assessed in the event of failure to meet the guaranteed TAT M odel 4

Cf- Fixed cost Cw« Expected warranty cost UB: Total utilization Contract bid for failure rate Contractor’s profit with failure rate \ B p m . A: Average failure rate (random variable) Ga(-): Distribution function for A cB: Cost of preparing warranty bid cl : Cost factor associated with losing an award Probability of contract award M odel 5

cn : cr2: cH: Cs: tm: cK:

Average repair cost without reliability modification Average repair cost after modification Handling cost Cost savings from implementation of reliability modifications Time at which reliability modification is implemented Cost of modification kits

M odel 6

|xG: |i: Nc: Ns:

Guaranteed MTBF Estimate of MTBF Number of additional consignment spares Number of spares required

M odel 7

See Models 2 and 4 M odel 8

NS(W):

Number of replacements paid for by the manufacturer (random variable) Nb(W): Number of replacements paid for by the buyer (random variable) nt: Target number of spares

Symbols and Notation

661

SYM BO LS SPECIFIC TO CHAPTER 8

Xc(t): Yc(t): R: G(r): \(t\r): N(t\r): N(t): Z r.

Age of item in use at time t Usage obtained from item in use at time t Usage rate per unit time (random variable) Distribution function for R Failure rate conditional on R = r Number of failures in [0, t) conditional on R = r Number of failures in [0, t) Time at which first failure occurs outside warranty conditional on R = r Z : Time between repeat purchases H{f, W, U): Distribution function for Z K(L\r): Number of repeat purchases over product life cycle conditional on R = r K(L): Number of repeat purchases over product life cycle N(W, U): Number of failures under warranty P(x, y): Probability of a repeat purchase if age and usage at failure is (X, y) Ur Binary random variable indicating repeat purchase or not Q(W, U): Total refund for an item sold at full price

SYM BO LS SPEC IFIC TO CHAPTER 9

N TR(W) R(W0 R(W-, 8)

Ki(W; ) 7(4») v(0 N(W ) NS(W) p(0 N'(W) NR(W0 Pr(0

ECR(W) ECR(W; 8)

Number of items sold (single lot sales) [ = S0] Total warranty refund Warranty reserves (= £[TR(W)]) Discounted warranty reserves Rate of return on investment Warranty reserves with investment of reserves Ratio of warranty reserve to selling price (per item) Refund rate Number of replacements under warranty Number of spaces needed under warranty for a lot of N items (random variable) Expected demand rate for spares Number of repairs [replacements] under warranty Number of repairs under warranty for a lot A items (random variable) Expected demand rate for repairs Total expected cost of repairs Total discounted expected cost of repairs

662

Symbols and Notation

L NR(W; L) ECR(W; L) ECR(W, L; Ô) a a* oo(a; W) «((a*; r , W)/a)(0; T, W )] Distribution for cost of each repair Repair limit Optimal repair limit Expected cost of each repair [= z dH(z)/H(ft)\ Number of times failed items are replaced under warranty Number of items failed items are repaired under warranty Time interval between two replacements of failed items Distribution function for U Expected warranty servicing cost per item 100[1 W)/(o(0; W)] Probability that warranty claim is not exercised if age at failure is x

SYM BO LS SPECIFIC TO CHAPTER 10 W a rra nty and Design

«;•

c m(0) Cd Cd co(e, w ) v(t) m Cs(t ; W) n Cs(n; W) P

reliability parameter for component j (1 < j < /) Manufacturing cost for component y (1 < y < /) Manufacturing cost per item Development cost per unit time Expected cost of each design modification Expected warranty cost per item Modification rate for product improvement Duration of development program Number of modifications in (0, t) (random variable) Cost to manufacturer per item (random variable) Number of modifications Cost to manufacturer per item (random variable) Probability that the switch is in working state

Symbols and Notation

663

W a rra nty and M a nufa cturing

Pi P2 m

Fi{t) N n T K Ni Ni k Q cd Cw,K(n ’ T) Cw(n, T) CS(W; n, T) 9 Ct T) L L* Ni C\V,L CS(W ; L)

Probability that an item is nondefective Probability that an item is defective Failure distribution function for a nondefective item Failure distribution function for a defective item Lot size Sample size Duration of life testing Number of batches produced Number of items released in batch i Number of items scrapped in batch i Number of batches subjected to 100% testing Cost of testing per item tested Cost of scrapping per item scrapped Warranty cost per item released Asymptotic warranty cost per item released Asymptotic total manufacturer’s cost per item Probability that the manufacturing process stays in control Setup cost if process state is in control Additional setup cost if process state is out of control Lot size Optimal L Number of nondefectives in lot i Expected warranty cost per item Total expected manufacturer’s cost per item

W a rra nty and Presale Testing t : Burn-in time [presale testing] cu c2: Parameters for burn-in cost a)(W; t ) : Expected warranty cost per item Cs(W;t ): Total expected manufacturer’s cost per item

SYM BO LS SPEC IFIC TO CHAPTER 11

M(t; 0): M{t): Milt): M(x, y): Mt(x, y ):

Renewal Estimate Estimate Renewal Estimate

function associated with F{t\ 0) of M{t\ 0) of M{t; 0) based on method /, 1 < i < 7 function associated with F(x, y) of M (x, y) based on method /, 1 < / < 2

Symbols and Notation

664

Z: MZ(L ): MZ(L):

Time between repeat purchases Expected number of repeat purchases over product life cycle Estimate of MZ(L)

SYM BO LS SPEC IFIC TO CHAPTER 12

0: X» • • • , 0(*i, . . . , X n): t (0): T(XU X n): MSE: BAN: UMVUE: L (-):

IV

iC

X: s2: Mr: ML: , X n): Rt Afi(i) M2{i) M3(t) V,(t) F(W) %C,(W)] fl(VT)

Failure distribution parameter Failure times of n items Estimator of 0 Function of parameter 0 Estimator of t (0) Mean squared error Best asymptotically normal Uniformly minimum variance unbiased estimator Likelihood function rth moment about the mean rth moment about zero Sample mean Sample variance rth sample moment Maximum likelihood ML estimator Reliability of item [Probability that item will last for period T\ Estimator of R T Linear estimator of M(t) Consistent parametric estimator of M(t) Nonparametric estimator of M(t) Variances of estimator 1< i< 3 Estimate of F(W) Estimate of £[CS(W)] Estimate of expected profit per item, tt

SYM BO LS SPECIFIC TO CHAPTER 14

S(W, cb): /: f/( ): Zw: Zf:

Total sales Initial wealth Utility function Monetary return if item doe£ not fail under warranty Monetary return if item fails under warranty

Symbols and Notation

0: : n (W, Cb; 6): DRM: ^drnrr 4>: 0: q^ qv

665

Consumer’s perceived probability of item failure under warranty True probability of item failure under warranty Expected profit per item Dispute resolution mechanism Cost of DRM Precision of DRM Bias in DRM Probability of ruling in favor of buyer when quality is high Probability of ruling in favor of buyer when quality is low

APPENDIX A

Basic Results from Probability Theory

A.1

RANDOM VARIABLES

A random experiment is an experiment whose outcome cannot be determined in advance. The set of all possible outcomes is called the sample space and usually is denoted by if. A random variable X is a function that assigns a real value to each outcome in if. For example, consider an item that can be in one of two states, working or failed. The random variable X can be assigned the value 1 when the item is in the working state and the value 0 when it is in the failed state. A.2

DISTRIBUTION FUNCTION

The distribution function F(jc) of the random variable X is defined as

A random variable X is said to be discrete if takes only a countable set of values. Let this set be denoted by {xx, x2, . . . , xn}, where n can be infinite. Then

A random variable is said to be continuous if there exists a function/(x), called the density function, such that

667

Appendix A

668 or equivalently,

Some examples of discrete and continuous distribution/density functions are: Poisson: Geometric: Binomial: Exponential: Gamma: Weibull: Normal: A.3

M O M ENTS OF RANDOM VARIABLES

The yth moment of the random variable X , My, is given by

provided that the integral or sum exists. Of particular interest are the mean (|x) and the variance (a2), given by

The moments may be obtained from the moment generating function, given by

Basic Results from Probability Theory

669

and the yth moment is given by

A.4

C HARA C TERISTIC FUNCTION AND LAPLACE TRANSFO RM

The characteristic function cf)(i) is defined as

where i = V - 1. The Laplace transform of a density function f(x ),f(s )y of a nonnegative random variable X is given by

when the integral exists. f(x) can be obtained from f(s ) by the inverse Laplace transform, given by

The Laplace transform is very useful in obtaining the density function of sums of nonnegative independent random variables and in solving certain types of integral equations. A.5

TWO OR MORE RANDOM VARIABLES

We shall confine our discussion to two random variables, denoted X and Y. The extension to more than two is relatively straightforward. The joint distribution function F(x, y) is given by

The random variables are said to be jointly continuous if there exists a function/( jc, y), called the joint probability density function, such that

Appendix A

670

The marginal distribution functions Fx (x) and FY(y) are given by and The random variables X and Y are said to be statistically independent if and only if

for all x and y. The results are similar for discrete random variables, with summation replacing integration. A.6

M O M ENTS OF TWO RANDOM VARIABLES

The covariance of X and Y is defined by

The correlation coefficient pXY is given by

where v x and a Y are the variances of X and Y , respectively. The random variables X and Y are said to be uncorrelated if pXY = 0. A.7

CO NDITIO NA L EXPECTATION

The conditional distribution of X given Y = y is defined as

The conditional distribution of Y given X = x is similarly given by Fy^yl*). For jointly continuous random variables with a joint density function fx,y(x ’ y)> the conditional probability density function of X , given Y = y, is given by

Basic Results from Probability Theory

671

where f Y(y) = dFY(y)/dy is the marginal density function of Y. Similarly,

Let E[X\Y = y] be the conditional expectation of X conditioned on Y = y. The unconditional expectation of X , given by

is related to the conditional expectation by the relation

which is symbolically written as

A.8

SUM S OF INDEPENDENT RANDOM VARIABLES

Let X h 1 < i < n, be a sequence of independent and identically distributed random variables with a common density function f x (x). Let f Y(y) denote the density function of the random variable Y given by

Then

where * is the convolution operation, given by

If the variables are nonnegative, then using Laplace transforms we have

672

Appendix A

where f Y(s) and f x (s) are the Laplace transforms of f Y(y) and f x (x), respectively. A.9

C EN TRA L LIM IT THEOREM

If X h 1 < i < n, is a sequence of independent and identically distributed random variables with mean |x and variance a 2, then, for large n,

is approximately normally distributed with mean |x and variance a 2/n.

APPENDIX B

Proofs of Results in Chapter 3

In this appendix we give the proofs for some of the results in Chapter 3. The equations cited refer to the equations of Chapter 3. B.1

PROOF OF EQUATIONS (3.19) AND (3.20)

Note that

Let

Then, conditioned on X ly the time instant for the first renewal, we have (B.l) Since the process restarts at X ly we have

673

674

Appendix B

Using this in (B .l), we have

(B2)

This is a renewal-type equation, and using the result of Chapter 3, it can be rewritten as

(B.3)

which is the same as (3.19). To prove (3.20), we need the Key Renewal Theorem, which can be stated as follows. For an integrable function h(t),

(B.4)

where

(B.5)

since Y is nonnegative. Applying the Key Renewal Theorem to (B.3) with h(t) = 1 - F(t + x), we have

Proofs of Results in Chapter 3

675

or

(B.6)

using (B.5). This the same as (3.20). B.2

PROOF OF EQUATION (3.22)

Note that

As a result,

and using (B.6) we have (B.7)

which is the same as (3.22). B.3

PROOF OF EQUATION (3.27)

Define Z, = ^ 2(/-i) + i + X 2i, i = 1, 2, . . . . These are a sequence of independent and identically distributed random variables with a distribution function H{z) given by

as Z, is a sum of two independent random variables.

Appendix B

676

By conditioning on Z 1 = z, we have (B.8) Since at time Z x the process restarts itself, we have

As a result, (B.9) Since X 2 > 0, we have (B.10) Using this in (B.9) results in (B .ll) which is the same as (3.27). B.4

PROOF OF EQUATION (3.39)

On taking the Laplace transform of (3.38), we have

where g(s), fi(s), and f(s) are the Laplace transforms of g(t), h(t), and /(f), respectively. This can be rewritten as (B.12)

Proofs of Results In C hapter 3

677

From (3.11), we have (B.13) where

(s)is the Laplace transform of m(t) = dM(t)/dt. m Using (B.13) in (B.12) results in (B.14)

On taking the inverse Laplace transform of (B.14), we have

which is Equation (3.39).

APPENDIX C

Calculation of Renewal Functions

Here we provide both tables of the renewal functions for a few important life distributions and a FORTRAN program for calculation of the values given in these tables as well as those for the renewal functions corresponding to several other distributions. The tables are given in Section C .l; the program is given in Section C.2. The distributions for which the renewal function is tabulated in Section C.l are the Weibull, gamma, and lognormal. The exponential is a special case of the first two of these. The program given in Section C.2 will calculate values of the renewal function for these distributions as well as the inverse Gaussian, truncated normal, and mixed exponential distributions.

C.1

TABLES OF THE RENEWAL FUNC TIO NS FOR THE WEIBULL, GAMMA, EXPONENTIAL, AND LOG NO RM AL DISTRIBUTIO NS

We use the notation f(t) for the density function and M(t) for the corresponding renewal function. The density functions are (for t > 0 in each case) Weibull Distribution:

Gamma Distribution:

679

680

Appendix C

where T () is the gamma function, given by

Lognormal Distribution:

The scale parameters a in these distributions are, respectively, a = X, X, and e71. In each of the following tables, the renewal function is tabulated with scale parameter equal to 1. To obtain values of the renewal functions for other values of the scale parameter, use the relationship M(i; a) = M(t/d; 1). For the Weibull and gamma distributions, p is called the shape parameter. The exponential distribution is a special case of the Weibull distribution with p = 1 and of the gamma distribution with P = 1. Tables C.l and C.2 give M(t) for the Weibull and gamma for t = 0.0(0.05)5 and values of the shape parameter that provide both increasing and decreasing failure rates. Table C.3 gives M(t) for the lognormal for t = 0.00(0.05)4.00 for several values of 02.

681

Calculation of Renewal Functions

Table C.1

Tabulated Values of the Renewal Function for the Weibull Distribution with Shape Parameter p p = 0.50

t

.00 .05 .10 .15 .20 .25 .30 .35 .40 .45 .50 .55 .60 .65 .70 .75 .80 .85 .90 .95 1.00 1.05 1.10 1.15 1.20 1.25 1.30 1.35 1.40 1.45 1.50 1.55 1.60 1.65 1.70 1.75 1.80 1.85 1.90 1.95 2.00

M(t)

t

M(t)

t

M{t)

0.0000 0.2347 0.3432 0.4296 0.5048 0.5729 0.6360 0.6952 0.7515 0.8053 0.8571 0.9072 0.9558 1.0031 1.0492 1.0943 1.1385 1.1819 1.2245 1.2664 1.3077 1.3484 1.3885 1.4281 1.4673 1.5060 1.5443 1.5822 1.6198 1.6570 1.6938 1.7304 1.7667 1.8027 1.8384 1.8738 1.9091 1.9441 1.9788 2.0134 2.0478

2.05 2.10 2.15 2.20 2.25 2.30 2.35 2.40 2.45 2.50 2.55 2.60 2.65 2.70 2.75 2.80 2.85 2.90 2.95 3.00 3.05 3.10 3.15 3.20 3.25 3.30 3.35 3.40 3.45 3.50 3.55 3.60 3.65 3.70 3.75 3.80 3.85 3.90 3.95 4.00

2.0819 2.1159 2.1497 2.1833 2.2167 2.2500 2.2832 2.3162 2.3490 2.3817 2.4143 2.4467 2.4790 2.5112 2.5432 2.5752 2.6070 2.6387 2.6703 2.7018 2.7333 2.7646 2.7958 2.8269 2.8580 2.8889 2.9198 2.9505 2.9812 3.0119 3.0424 3.0729 3.1033 3.1336 3.1638 3.1940 3.2241 3.2542 3.2841 3.3141

4.05 4.10 4.15 4.20 4.25 4.30 4.35 4.40 4.45 4.50 4.55 4.60 4.65 4.70 4.75 4.80 4.85 4.90 4.95 5.00

3.3439 3.3737 3.4035 3.4331 3.4628 3.4923 3.5218 3.5513 3.5807 3.6101 3.6394 3.6686 3.6978 3.7270 3.7561 3.7852 3.8142 3.8432 3.8721 3.9010

682

Appendix C

P = 1.0 t

.00 .05 .10 .15 .20 .25 .30 .35 .40 .45 .50 .55 .60 .65 .70 .75 .80 .85 .90 .95 1.00 1.05 1.10 1.15 1.20 1.25 1.30 1.35 1.40 1.45 1.50 1.55 1.60 1.65 1.70 1.75 1.80 1.85 1.90 1.95 2.00

M(t)

t

M(t)

t

M(t)

0.0000 0.0500 0.1000 0.1500 0.2000 0.2500 0.3000 0.3500 0.4000 0.4500 0.5000 0.5500 0.6000 0.6500 0.7000 0.7500 0.8000 0.8500 0.9000 0.9500 1.0000 1.0500 1.1000 1.1500 1.2000 1.2500 1.3000 1.3500 1.4000 1.4500 1.5000 1.5500 1.6000 1.6500 1.7000 1.7500 1.8000 1.8500 1.9000 1.9500 2.0000

2.05 2.10 2.15 2.20 2.25 2.30 2.35 2.40 2.45 2.50 2.55 2.60 2.65 2.70 2.75 2.80 2.85 2.90 2.95 3.00 3.05 3.10 3.15 3.20 3.25 3.30 3.35 3.40 3.45 3.50 3.55 3.60 3.65 3.70 3.75 3.80 3.85 3.90 3.95 4.00

2.0500 2.1000 2.1500 2.2000 2.2500 2.3000 2.3500 2.4000 2.4500 2.5000 2.5500 2.6000 2.6500 2.7000 2.7500 2.8000 2.8500 2.9000 2.9500 3.0000 3.0500 3.1000 3.1500 3.2000 3.2500 3.3000 3.3500 3.4000 3.4500 3.5000 3.5500 3.6000 3.6500 3.7000 3.7050 3.8000 3.8500 3.9000 3.9500 4.0000

4.05 4.10 4.15 4.20 4.25 4.30 4.35 4.40 4.45 4.50 4.55 4.60 4.65 4.70 4.75 4.80 4.85 4.90 4.95 5.00

4.0500 4.1000 4.1500 4.2000 4.2500 4.3000 4.3500 4.4000 4.4500 4.5000 4.5500 4.6000 4.6500 4.7000 4.7500 4.8000 4.8500 4.9000 4.9500 5.0000

683

Calculation of Renewal Functions

P = 1.5 t

.00 .05 .10 .15 .20 .25 .30 .35 .40 .45 .50 .55 .60 .65 .70 .75 .80 .85 .90 .95 1.00 1.05 1.10 1.15 1.20 1.25 1.30 1.35 1.40 1.45 1.50 1.55 1.60 1.65 1.70 1.75 1.80 1.85 1.90 1.95 2.00

M{t)

t

M(t)

t

M(t)

0.0000 0.0112 0.0314 0.0574 0.0879 0.1219 0.1591 0.1988 0.2408 0.2847 0.3303 0.3774 0.4258 0.4752 0.5256 0.5768 0.6288 0.6813 0.7343 0.7878 0.8417 0.8959 0.9502 1.0050 1.0598 1.1148 1.1699 1.2251 1.2803 1.3357 1.3910 1.4465 1.5019 1.5574 1.6128 1.6682 1.7238 1.7792 1.8347 1.8902 1.9456

2.05 2.10 2.15 2.20 2.25 2.30 2.35 2.40 2.45 2.50 2.55 2.60 2.65 2.70 2.75 2.80 2.85 2.90 2.95 3.00 3.05 3.10 3.15 3.20 3.25 3.30 3.35 3.40 3.45 3.50 3.55 3.60 3.65 3.70 3.75 3.80 3.85 3.90 3.95 4.00

2.0011 2.0565 2.1120 2.1674 2.2229 2.2783 2.3337 2.3891 2.4445 2.4999 2.5553 2.6107 2.6661 2.7215 2.7769 2.8323 2.8877 2.9431 2.9985 3.0539 3.1093 3.1646 3.2200 3.2754 3.3308 3.3862 3.4416 3.4970 3.5523 3.6077 3.6631 3.7185 3.7739 3.8293 3.8847 3.9400 3.9954 4.0508 4.1062 4.1616

4.05 4.10 4.15 4.20 4.25 4.30 4.35 4.40 4.45 4.50 4.55 4.60 4.65 4.70 4.75 4.80 4.85 4.90 4.95 5.00

4.2167 4.2724 4.3278 4.3831 4.4385 4.4939 4.5493 4.6047 4.6601 4.7155 4.7708 4.8262 4.8816 4.9370 4.9924 5.0478 5.1032 5.1586 5.2139 5.2693

Appendix C

684

0 t

.00 .05 .10 .15 .20 .25 .30 .35 .40 .45 .50 .55 .60 .65 .70 .75 .80 .85 .90 .95 1.00 1.05 1.10 1.15 1.20 1.25 1.30 1.35 1.40 1.45 1.50 1.55 1.60 1.65 1.70 1.75 1.80 1.85 1.90 1.95 2.00

M(t)

t

0.0000 0.0025 0.0100 0.0223 0.0395 0.0612 0.0874 0.1177 0.1519 0.1897 0.2308 0.2749 0.3216 0.3706 0.4216 0.4743 0.5284 0.5836 0.6397 0.6965 0.7538 0.8114 0.8692 0.9271 0.9850 1.0428 1.1005 1.1580 1.2154 1.2726 1.3296 1.3865 1.4432 1.4998 1.5563 1.6127 1.6691 1.7254 1.7817 1.8379 1.8942

2.05 2.10 2.15 2.20 2.25 2.30 2.35 2.40 2.45 2.50 2.55 2.60 2.65 2.70 2.75 2.80 2.85 2.90 2.95 3.00 3.05 3.10 3.15 3.20 3.25 3.30 3.35 3.40 3.45 3.50 3.55 3.60 3.65 3.70 3.75 3.80 3.85 3.90 3.95 4.00

=

2.0 m

1.9504 2.0066 2.0630 2.1193 2.1756 2.2320 2.2884 2.3447 2.4011 2.4575 2.5140 2.5704 2.6268 2.6833 2.7397 2.7961 2.8526 2.9090 2.9654 3.0219 3.0783 3.1347 3.1912 3.2476 3.3040 3.3604 3.4168 3.4733 3.5297 3.5861 3.6425 3.6989 3.7553 3.8118 3.8682 3.9246 3.9810 4.0374 4.0938 4.1503

t

M(t)

4.05 4.10 4.15 4.20 4.25 4.30 4.35 4.40 4.45 4.50 4.55 4.60 4.65 4.70 4.75 4.80 4.85 4.90 4.95 5.00

4.2067 4.2631 4.3195 4.3759 4.4327 4.4888 4.5452 4.6016 4.6580 4.7145 4.7709 4.8273 4.8837 4.9401 4.9966 5.0530 5.1094 5.1658 5.2222 5.2786

Calculation of Renewal Functions

685

P = 2.5 t

.00 .05 .10 .15 .20 .25 .30 .35 .40 .45 .50 .55 .60 .65 .70 .75 .80 .85 .90 .95 1.00 1.05 1.10 1.15 1.20 1.25 1.30 1.35 1.40 1.45 1.50 1.55 1.60 1.65 1.70 1.75 1.80 1.85 1.90 1.95 2.00

M(t)

t

0.0000 0.0006 0.0032 0.0087 0.0178 0.0309 0.0483 0.0704 0.0972 0.1287 0.1648 0.2053 0.2500 0.2984 0.3502 0.4048 0.4617 0.5204 0.5805 0.6413 0.7026 0.7639 0.8249 0.8854 0.9452 1.0043 1.0626 1.1201 1.1769 1.2332 1.2889 1.3444 1.3996 1.4548 1.5099 1.5652 1.6206 1.6761 1.7319 1.7879 1.8440

2.05 2.10 2.15 2.20 2.25 2.30 2.35 2.40 2.45 2.50 2.55 2.60 2.65 2.70 2.75 2.80 2.85 2.90 2.95 3.00 3.05 3.10 3.15 3.20 3.25 3.30 3.35 3.40 3.45 3.50 3.55 3.60 3.65 3.70 3.75 3.80 3.85 3.90 3.95 4.00

m

1.9004 1.9568 2.0134 2.0700 2.1266 2.1833 2.2399 2.2965 2.3531 2.4096 2.4660 2.5225 2.5788 2.6352 2.6915 2.7478 2.8041 2.8603 2.9166 2.9729 3.0292 3.0855 3.1419 3.1982 3.2545 3.3109 3.3672 3.4236 3.4800 3.5363 3.5927 3.6491 3.7055 3.7618 3.8182 3.8745 3.9309 3.9872 4.0436 4.0999

t

4.05 4.10 4.15 4.20 4.25 4.30 4.35 4.40 4.45 4.50 4.55 4.60 4.65 4.70 4.75 4.80 4.85 4.90 4.95 5.00

m

4.1563 4.2126 4.2690 4.3253 4.3817 4.4380 4.4944 4.5507 4.6071 4.6634 4.7198 4.7762 4.8325 4.8889 4.9452 5.0016 5.0579 5.1143 5.1706 5.2270

686

Appendix C

3 = 3.0 t

.00 .05 .10 .15 .20 .25 .30 .35 .40 .45 .50 .55 .60 .65 .70 .75 .80 .85 .90 .95 1.00 1.05 1.10 1.15 1.20 1.25 1.30 1.35 1.40 1.45 1.50 1.55 1.60 1.65 1.70 1.75 1.80 1.85 1.90 1.95 2.00

m

0.0000 0.0001 0.0010 0.0034 0.0080 0.0155 0.0267 0.0421 0.0622 0.0875 0.1183 0.1546 0.1965 0.2437 0.2958 0.3523 0.4125 0.4754 0.5403 0.6063 0.6724 0.7380 0.8024 0.8651 0.9259 0.9847 1.0416 1.0967 1.1504 1.2032 1.2554 1.3075 1.3597 1.4125 1.4659 1.5201 1.5751 1.6306 1.6872 1.7440 1.8012

t

M(i)

t

M(t)

2.05 2.10 2.15 2.20 2.25 2.30 2.35 2.40 2.45 2.50 2.55 2.60 2.65 2.70 2.75 2.80 2.85 2.90 2.95 3.00 3.05 3.10 3.15 3.20 3.25 3.30 3.35 3.40 3.45 3.50 3.55 3.60 3.65 3.70 3.75 3.80 3.85 3.90 3.95 4.00

1.8585 1.9159 1.9732 2.0303 2.0872 2.1438 2.2001 2.2562 2.3121 2.3678 2.4234 2.4790 2.5345 2.5900 2.6456 2.7013 2.7571 2.8129 2.8689 2.9249 2.9810 3.0371 3.0933 3.1494 3.2056 3.2617 3.3178 3.3739 3.4299 3.4859 3.5419 3.5979 3.6538 3.7098 3.7657 3.8216 3.8776 3.9335 3.9895 4.0455

4.05 4.10 4.15 4.20 4.25 4.30 4.35 4.40 4.45 4.50 4.55 4.60 4.65 4.70 4.75 4.80 4.85 4.90 4.95 5.00

4.1015 4.1574 4.2135 4.2695 4.3255 4.3815 4.4375 4.4935 4.5495 4.6055 4.6615 4.7175 4.7735 4.8295 4.8855 4.9415 4.9974 5.0534 5.1094 5.1654

Calculation of Renewal Functions

687

P = 4.0 t

.00 .05 .10 .15 .20 .25 .30 .35 .40 .45 .50 .55 .60 .65 .70 .75 .80 .85 .90 .95 1.00 1.05 1.10 1.15 1.20 1.25 1.30 1.35 1.40 1.45 1.50 1.55 1.60 1.65 1.70 1.75 1.80 1.85 1.90 1.95 2.00

M{t)

t

0.0000 0.0000 0.0001 0.0005 0.0016 0.0039 0.0081 0.0149 0.0253 0.0402 0.0606 0.0876 0.1218 0.1639 0.2143 0.2726 0.3384 0.4103 0.4868 0.5657 0.6447 0.7214 0.7940 0.8608 0.9212 0.9751 1.0234 1.0674 1.1089 1.1497 1.1914 1.2352 1.2819 1.3320 1.3854 1.4418 1.5006 1.5612 1.6229 1.6849 1.7467

2.05 2.10 2.15 2.20 2.25 2.30 2.35 2.40 2.45 2.50 2.55 2.60 2.65 2.70 2.75 2.80 2.85 2.90 2.95 3.00 3.05 3.10 3.15 3.20 3.25 3.30 3.35 3.40 3.45 3.50 3.55 3.60 3.65 3.70 3.75 3.80 3.85 3.90 3.95 4.00

m

1.8070 1.8672 1.9252 1.9817 2.0366 2.0902 2.1427 2.1945 2.2461 2.2977 2.3498 2.4025 2.4560 2.5104 2.5657 2.6216 2.6781 2.7350 2.7921 2.8491 2.9059 2.9623 3.0184 3.0740 3.1291 3.1839 3.2384 3.2927 3.3469 3.4011 3.4554 3.5099 3.5646 3.6195 3.6746 3.7299 3.7854 3.8411 3.8967 3.9524

t

M(t)

4.05 4.10 4.15 4.20 4.25 4.30 4.35 4.40 4.45 4.50 4.55 4.60 4.65 4.70 4.75 4.80 4.85 4.90 4.95 5.00

4.0080 4.0635 4.1189 4.1743 4.2294 4.2845 4.3395 4.3945 4.4496 4.5043 4.5592 4.6142 4.6692 4.7243 4.7794 4.8346 4.8899 4.9451 5.0004 5.0557

688

Appendix C

P = 5.0 t

.00 .05 .10 .15 .20 .25 .30 .35 .40 .45 .50 .55 .60 .65 .70 .75 .80 .85 .90 .95 1.00 1.05 1.10 1.15 1.20 1.25 1.30 1.35 1.40 1.45 1.50 1.55 1.60 1.65 1.70 1.75 1.80 1.85 1.90 1.95 2.00

M(t)

t

0.0000 0.0000 0.0000 0.0001 0.0003 0.0010 0.0024 0.0052 0.0102 0.0183 0.0308 0.0491 0.0748 0.1096 0.1548 0.2115 0.2798 0.3591 0.4473 0.5410 0.6358 0.7268 0.8093 0.8800 0.9373 0.9820 1.0169 1.0458 1.0727 1.1009 1.1329 1.1704 1.2138 1.2635 1.3192 1.3801 1.4453 1.5136 1.5834 1.6532 1.7216

2.05 2.10 2.15 2.20 2.25 2.30 2.35 2.40 2.45 2.50 2.55 2.60 2.65 2.70 2.75 2.80 2.85 2.90 2.95 3.00 3.05 3.10 3.15 3.20 3.25 3.30 3.35 3.40 3.45 3.50 3.55 3.60 3.65 3.70 3.75 3.80 3.85 3.90 3.95 4.00

am

1.7873 1.8493 1.9071 1.9606 2.0103 2.0570 2.1017 2.1457 2.1900 2.2358 2.2837 2.3342 2.3875 2.4434 2.5016 2.5613 2.6220 2.6830 2.7435 2.8031 2.8613 2.9179 2.9728 3.0261 3.0781 3.1290 3.1794 3.2297 3.2804 3.3317 3.3839 3.4373 3.4917 3.5471 3.6034 3.6602 3.7173 3.7743 3.8311 3.8874

t

M(t)

4.05 4.10 4.15 4.20 4.25 4.30 4.35 4.40 4.45 4.50 4.55 4.60 4.65 4.70 4.75 4.80 4.85 4.90 4.95 5.00

3.9431 3.9981 4.0525 4.1062 4.1594 4.2123 4.2651 4.3179 4.3709 4.4243 4.4781 4.5323 4.5869 4.6419 4.6972 4.7527 4.8082 4.8637 4.9190 4.9743

689

Calculation of Renewal Functions

3 t

.00 .05 .10 .15 .20 .25 .30 .35 .40 .45 .50 .55 .60 .65 .70 .75 .80 .85 .90 .95 1.00 1.05 1.10 1.15 1.20 1.25 1.30 1.35 1.40 1.45 1.50 1.55 1.60 1.65 1.70 1.75 1.80 1.85 1.90 1.95 2.00

=

6.0

M(t)

t

M(t)

t

M{t)

0.0000 0.0000 0.0000 0.0000 0.0001 0.0002 0.0007 0.0018 0.0041 0.0083 0.0155 0.0273 0.0456 0.0727 0.1110 0.1631 0.2307 0.3143 0.4125 0.5211 0.6332 0.7400 0.8331 0.9063 0.9580 0.9913 1.0124 1.0281 1.0439 1.0633 1.0881 1.1199 1.1594 1.2072 1.2635 1.3278 1.3990 1.4755 1.5549 1.6346 1.7117

2.05 2.10 2.15 2.20 2.25 2.30 2.35 2.40 2.45 2.50 2.55 2.60 2.65 2.70 2.75 2.80 2.85 2.90 2.95 3.00 3.05 3.10 3.15 3.20 3.25 3.30 3.35 3.40 3.45 3.50 3.55 3.60 3.65 3.70 3.75 3.80 3.85 3.90 3.95 4.00

1.7839 1.8492 1.9065 1.9560 1.9985 2.0360 2.0706 2.1049 2.1408 2.1802 2.2242 2.2733 2.3275 2.3864 2.4493 2.5149 2.5819 2.6491 2.7151 2.7786 2.8390 2.8956 2.9484 2.9977 3.0442 3.0890 3.1329 3.1773 3.2229 3.2707 3.3211 3.3742 3.4299 3.4879 3.5476 3.6083 3.6691 3.7295 3.7888 3.8465

4.05 4.10 4.15 4.20 4.25 4.30 4.35 4.40 4.45 4.50 4.55 4.60 4.65 4.70 4.75 4.80 4.85 4.90 4.95 5.00

3.9023 3.9561 4.0082 4.0587 4.1081 4.1570 4.2060 4.2554 4.3058 4.3574 4.4104 4.4648 4.5205 4.5771 4.6344 4.6919 4.7493 4.8061 4.8623 4.9174

Appendix C

690

p = 7.0 t

.00 .05 .10 .15 .20 .25 .30 .35 .40 .45 .50 .55 .60 .65 .70 .75 .80 .85 .90 .95 1.00 1.05 1.10 1.15 1.20 1.25 1.30 1.35 1.40 1.45 1.50 1.55 1.60 1.65 1.70 1.75 1.80 1.85 1.90 1.95 2.00

M(t)

t

M(t)

t

M(t)

0.0000 0.0000 0.0000 0.0000 0.0000 0.0001 0.0002 0.0006 0.0016 0.0037 0.0078 0.0151 0.0276 0.0478 0.0791 0.1250 0.1892 0.2743 0.3802 0.5027 0.6324 0.7557 0.8586 0.9320 0.9756 0.9974 1.0079 1.0156 1.0249 1.0382 1.0570 1.0828 1.1170 1.1609 1.2156 1.2812 1.3570 1.4411 1.5305 1.6212 1.7087

2.05 2.10 2.15 2.20 2.25 2.30 2.35 2.40 2.45 2.50 2.55 2.60 2.65 2.70 2.75 2.80 2.85 2.90 2.95 3.00 3.05 3.10 3.15 3.20 3.25 3.30 3.35 3.40 3.45 3.50 3.55 3.60 3.65 3.70 3.75 3.80 3.85 3.90 3.95 4.00

1.7888 1.8582 1.9152 1.9601 1.9949 2.0227 2.0473 2.0721 2.1001 2.1333 2.1731 2.2202 2.2747 2.3361 2.4035 2.4755 2.5500 2.6248 2.6979 2.7670 2.8306 2.8878 2.9383 2.9829 3.0228 3.0598 3.0959 3.1330 3.1728 3.2164 3.2647 3.3178 3.3755 3.4371 3.5015 3.5677 3.6341 3.6994 3.7626 3.8226

4.05 4.10 4.15 4.20 4.25 4.30 4.35 4.40 4.45 4.50 4.55 4.60 4.65 4.70 4.75 4.80 4.85 4.90 4.95 5.00

3.8790 3.9315 3.9805 4.0266 4.0706 4.1138 4.1573 4.2020 4.2489 4.2985 4.3510 4.4064 4.4643 4.5241 4.5851 4.6464 4.7072 4.7668 4.8246 4.8802

691

Calculation of Renewal Functions

Table C.2

Tabulated Values of the Renewal Function for the Gamma Distribution with Shape Parameter p P = 0.50

t

.00 .05 .10 .15 .20 .25 .30 .35 .40 .45 .50 .55 .60 .65 .70 .75 .80 .85 .90 .95 1.00 1.05 1.10 1.15 1.20 1.25 1.30 1.35 1.40 1.45 1.50 1.55 1.60 1.65 1.70 1.75 1.80 1.85 1.90 1.95 2.00

M(t)

t

M (t)

t

M(t)

0.0000 0.1988 0.3312 0.4453 0.5497 0.6477 0.7414 0.8317 0.9192 1.0046 1.0881 1.1700 1.2505 1.3297 1.4078 1.4848 1.5609 1.6360 1.7103 1.7838 1.8565 1.9286 1.9999 2.0705 2.1405 2.2099 2.2787 2.3470 2.4146 2.4817 2.5482 2.6142 2.6797 2.7447 2.8092 2.8732 2.9367 2.9997 3.0623 3.1244 3.1860

2.05 2.10 2.15 2.20 2.25 2.30 2.35 2.40 2.45 2.50 2.55 2.60 2.65 2.70 2.75 2.80 2.85 2.90 2.95 3.00 3.05 3.10 3.15 3.20 3.25 3.30 3.35 3.40 3.45 3.50 3.55 3.60 3.65 3.70 3.75 3.80 3.85 3.90 3.95 4.00

3.2472 3.3079 3.3682 3.4280 3.4875 3.5465 3.6050 3.6632 3.7209 3.7782 3.8352 3.8917 3.9478 4.0035 4.0589 4.1138 4.1684 4.2225 4.2763 4.3298 4.3828 4.4355 4.4878 4.5397 4.5913 4.6426 4.6934 4.7439 4.7941 4.8439 4.8934 4.9425 4.9913 5.0398 5.0879 5.1357 5.1832 5.2303 5.2771 5.3236

4.05 4.10 4.15 4.20 4.25 4.30 4.35 4.40 4.45 4.50 4.55 4.60 4.65 4.70 4.75 4.80 4.85 4.90 4.95 5.00

5.3697 5.4156 5.4611 5.5063 5.5512 5.5958 5.6401 5.6841 5.7278 5.7711 5.8142 5.8570 5.8995 5.9417 5.9836 6.0252 6.0665 6.1075 6.1483 6.1888

692

Appendix C

(3 = 1.00 t

.00 .05 .10 .15 .20 .25 .30 .35 .40 .45 .50 .55 .60 .65 .70 .75 .80 .85 .90 .95 1.00 1.05 1.10 1.15 1.20 1.25 1.30 1.35 1.40 1.45 1.50 1.55 1.60 1.65 1.70 1.75 1.80 1.85 1.90 1.95 2.00

M (t)

t

M(l)

t

0.0000 0.0500 0.1000 0.1500 0.2000 0.2500 0.3000 0.3500 0.4000 0.4500 0.5000 0.5500 0.6000 0.6500 0.7000 0.7500 0.8000 0.8500 0.9000 0.9500 1.0000 1.0500 1.1000 1.1500 1.2000 1.2500 1.3000 1.3500 1.4000 1.4500 1.5000 1.5500 1.6000 1.6500 1.7000 1.7500 1.8000 1.8500 1.9000 1.9500 2.0000

2.05 2.10 2.15 2.20 2.25 2.30 2.35 2.40 2.45 2.50 2.55 2.60 2.65 2.70 2.75 2.80 2.85 2.90 2.95 3.00 3.05 3.10 3.15 3.20 3.25 3.30 3.35 3.40 3.45 3.50 3.55 3.60 3.65 3.70 3.75 3.80 3.85 3.90 3.95 4.00

2.0500 2.1000 2.1500 2.2000 2.2500 2.3000 2.3500 2.4000 2.4500 2.5000 2.5500 2.6000 2.6500 2.7000 2.7500 2.8000 2.8500 2.9000 2.9500 3.0000 3.0500 3.1000 3.1500 3.2000 3.2500 3.3000 3.3500 3.4000 3.4500 3.5000 3.5500 3.6000 3.6500 3.7000 3.7500 3.8000 3.8500 3.9000 3.9500 4.0000

4.05 4.10 4.15 4.20 4.25 4.30 4.35 4.40 4.45 4.50 4.55 4.60 4.65 4.70 4.75 4.80 4.85 4.90 4.95 5.00

m

4.0500 4.1000 4.1500 4.2000 4.2500 4.3000 4.3500 4.4000 4.4500 4.5000 4.5500 4.6000 4.6500 4.7000 4.7500 4.8000 4.8500 4.9000 4.9500 5.0000

Calculation of Renewal Functions

693

3 = 1.50 t

.00 .05 .10 .15 .20 .25 .30 .35 .40 .45 .50 .55 .60 .65 .70 .75 .80 .85 .90 .95 1.00 1.05 1.10 1.15 1.20 1.25 1.30 1.35 1.40 1.45 1.50 1.55 1.60 1.65 1.70 1.75 1.80 1.85 1.90 1.95 2.00

m

0.0000 0.0087 0.0232 0.0412 0.0617 0.0841 0.1081 0.1333 0.1597 0.1868 0.2148 0.2434 0.2725 0.3021 0.3321 0.3625 0.3933 0.4243 0.4555 0.4870 0.5187 0.5505 0.5825 0.6146 0.6469 0.6792 0.7117 0.7443 0.7769 0.8096 0.8424 0.8752 0.9081 0.9410 0.9740 1.0070 1.0400 1.0731 1.1062 1.1393 1.1725

t

M(t)

t

M(t)

2.05 2.10 2.15 2.20 2.25 2.30 2.35 2.40 2.45 2.50 2.55 2.60 2.65 2.70 2.75 2.80 2.85 2.90 2.95 3.00 3.05 3.10 3.15 3.20 3.25 3.30 3.35 3.40 3.45 3.50 3.55 3.60 3.65 3.70 3.75 3.80 3.85 3.90 3.95 4.00

1.2057 1.2389 1.2721 1.3053 1.3386 1.3718 1.4051 1.4384 1.4717 1.5050 1.5383 1.5717 1.6050 1.6383 1.6717 1.7051 1.7384 1.7718 1.8052 1.8385 1.8719 1.9053 1.9387 1.9721 2.0055 2.0389 2.0723 2.1057 2.1391 2.1725 2.2060 2.2394 2.2728 2.3062 2.3396 2.3731 2.4065 2.4399 2.4734 2.5068

4.05 4.10 4.15 4.20 4.25 4.30 4.35 4.40 4.45 4.50 4.55 4.60 4.65 4.70 4.75 4.80 4.85 4.90 4.95 5.00

2.5402 2.5737 2.6071 2.6405 2.6740 2.7074 2.7409 2.7743 2.8077 2.8412 2.8746 2.9081 2.9415 2.9750 3.0084 3.0419 3.0754 3.1088 3.1423 3.1757

694

Appendix C

P = 2.00 t

.00 .05 .10 .15 .20 .25 .30 .35 .40 .45 .50 .55 .60 .65 .70 .75 .80 .85 .90 .95 1.00 1.05 1.10 1.15 1.20 1.25 1.30 1.35 1.40 1.45 1.50 1.55 1.60 1.65 1.70 1.75 1.80 1.85 1.90 1.95 2.00

M(t)

t

M(t)

t

0.0000 0.0012 0.0047 0.0102 0.0176 0.0267 0.0373 0.0492 0.0624 0.0767 0.0921 0.1083 0.1254 0.1432 0.1618 0.1809 0.2006 0.2208 0.2415 0.2625 0.2840 0.3058 0.3279 0.3502 0.3729 0.3957 0.4188 0.4420 0.4654 0.4890 0.5127 0.5365 0.5604 0.5845 0.6086 0.6328 0.6571 0.6814 0.7059 0.7303 0.7549

2.05 2.10 2.15 2.20 2.25 2.30 2.35 2.40 2.45 2.50 2.55 2.60 2.65 2.70 2.75 2.80 2.85 2.90 2.95 3.00 3.05 3.10 3.15 3.20 3.25 3.30 3.35 3.40 3.45 3.50 3.55 3.60 3.65 3.70 3.75 3.80 3.85 3.90 3.95 4.00

0.7794 0.8040 0.8287 0.8534 0.8789 0.9028 0.9276 0.9524 0.9772 1.0020 1.0269 1.0517 1.0766 1.1015 1.1264 1.1513 1.1762 1.2012 1.2261 1.2510 1.2760 1.3009 1.3259 1.3509 1.3758 1.4008 1.4258 1.4508 1.4757 1.5007 1.5257 1.5507 1.5757 1.6007 1.6257 1.6507 1.6757 1.7007 1.7257 1.7507

4.05 4.10 4.15 4.20 4.25 4.30 4.35 4.40 4.45 4.50 4.55 4.60 4.65 4.70 4.75 4.80 4.85 4.90 4.95 5.00

m

1.7757 1.8007 1.8257 1.8507 1.8757 1.9007 1.9257 1.9507 1.9757 2.0007 2.0257 2.0507 2.0757 2.1007 2.1258 2.1508 2.1758 2.2008 2.2258 2.2508

Calculation of Renewal Functions

695

0 = 2.50 t

.00 .05 .10 .15 .20 .25 .30 .35 .40 .45 .50 .55 .60 .65 .70 .75 .80 .85 .90 .95 1.00 1.05 1.10 1.15 1.20 1.25 1.30 1.35 1.40 1.45 1.50 1.55 1.60 1.65 1.70 1.75 1.80 1.85 1.90 1.95 2.00

M(t)

t

M(i)

t

M(i)

0.0000 0.0001 0.0009 0.0023 0.0046 0.0079 0.0120 0.0170 0.0230 0.0299 0.0376 0.0461 0.0555 0.0656 0.0765 0.0880 0.1002 0.1129 0.1263 0.1402 0.1545 0.1694 0.1846 0.2003 0.2164 0.2328 0.2495 0.2665 0.2837 0.3013 0.3190 0.3370 0.3551 0.3734 0.3919 0.4106 0.4293 0.4482 0.4672 0.4864 0.5056

2.05 2.10 2.15 2.20 2.25 2.30 2.35 2.40 2.45 2.50 2.55 2.60 2.65 2.70 2.75 2.80 2.85 2.90 2.95 3.00 3.05 3.10 3.15 3.20 3.25 3.30 3.35 3.40 3.45 3.50 3.55 3.60 3.65 3.70 3.75 3.80 3.85 3.90 3.95 4.00

0.5248 0.5442 0.5636 0.5831 0.6027 0.6223 0.6419 0.6616 0.6814 0.7011 0.7209 0.7407 0.7606 0.7804 0.8003 0.8202 0.8401 0.8600 0.8800 0.8999 0.9199 0.9399 0.9598 0.9798 0.9998 1.0198 1.0398 1.0598 1.0798 1.0998 1.1198 1.1398 1.1598 1.1798 1.1998 1.2198 1.2398 1.2598 1.2798 1.2998

4.05 4.10 4.15 4.20 4.25 4.30 4.35 4.40 4.45 4.50 4.55 4.60 4.65 4.70 4.75 4.80 4.85 4.90 4.95 5.00

1.3198 1.3399 1.3599 1.3799 1.3999 1.4199 1.4399 1.4599 1.4799 1.4999 1.5199 1.5399 1.5599 1.5799 1.6000 1.6200 1.6400 1.6600 1.6800 1.7000

696

Appendix C

13 = 3.00 M(t) .00 .05 .10 .15 .20 .25 .30 .35 .40 .45 .50 .55 .60 .65 .70 .75 .80 .85 .90 .95 1.00 1.05 1.10 1.15 1.20 1.25 1.30 1.35 1.40 1.45 1.50 1.55 1.60 1.65 1.70 1.75 1.80 1.85 1.90 1.95 2.00

0.0000 0.0000 0.0002 0.0005 0.0011 0.0021 0.0036 0.0055 0.0079 0.0109 0.0144 0.0185 0.0231 0.0284 0.0342 0.0406 0.0476 0.0551 0.0632 0.0718 0.0809 0.0905 0.1005 0.1111 0.1220 0.1334 0.1451 0.1572 0.1697 0.1825 0.1956 0.2090 0.2227 0.2367 0.2508 0.2652 0.2799 0.2947 0.3097 0.3248 0.3401

M(t)

M(t) 2.05 2.10 2.15 2.20 2.25 2.30 2.35 2.40 2.45 2.50 2.55 2.60 2.65 2.70 2.75 2.80 2.85 2.90 2.95 3.00 3.05 3.10 3.15 3.20 3.25 3.30 3.35 3.40 3.45 3.50 3.55 3.60 3.65 3.70 3.75 3.80 3.85 3.90 3.95 4.00

0.3556 0.3712 0.3869 0.4027 0.4186 0.4346 0.4507 0.4668 0.4831 0.4994 0.5157 0.5321 0.5486 0.5650 0.5816 0.5981 0.6147 0.6313 0.6480 0.6646 0.6813 0.6980 0.7146 0.7313 0.7481 0.7648 0.7815 0.7982 0.8150 0.8317 0.8484 0.8652 0.8819 0.8987 0.9154 0.9321 0.9489 0.9656 0.9823 0.9991

4.05 4.10 4.15 4.20 4.25 4.30 4.35 4.40 4.45 4.50 4.55 4.60 4.65 4.70 4.75 4.80 4.85 4.90 4.95 5.00

1.0158 1.0325 1.0492 1.0660 1.0827 1.0994 1.1161 1.1328 1.1495 1.1662 1.1829 1.1996 1.2163 1.2330 1.2497 1.2664 1.2831 1.2998 1.3165 1.3332

697

Calculation of Renewal Functions

3 = 4.00 t

.00 .05 .10 .15 .20 .25 .30 .35 .40 .45 .50 .55 .60 .65 .70 .75 .80 .85 .90 .95 1.00 1.05 1.10 1.15 1.20 1.25 1.30 1.35 1.40 1.45 1.50 1.55 1.60 1.65 1.70 1.75 1.80 1.85 1.90 1.95 2.00

M(t)

t

M(t)

t

M(t)

0.0000 0.0000 0.0000 0.0000 0.0001 0.0001 0.0003 0.0005 0.0008 0.0012 0.0017 0.0025 0.0034 0.0044 0.0057 0.0073 0.0091 0.0111 0.0135 0.0161 0.0190 0.0222 0.0257 0.0296 0.0338 0.0383 0.0431 0.0483 0.0538 0.0596 0.0658 0.0723 0.0791 0.0862 0.0936 0.1013 0.1093 0.1175 0.1261 0.1349 0.1440

2.05 2.10 2.15 2.20 2.25 2.30 2.35 2.40 2.45 2.50 2.55 2.60 2.65 2.70 2.75 2.80 2.85 2.90 2.95 3.00 3.05 3.10 3.15 3.20 3.25 3.30 3.35 3.40 3.45 3.50 3.55 3.60 3.65 3.70 3.75 3.80 3.85 3.90 3.95 4.00

0.1533 0.1628 0.1726 0.1826 0.1928 0.2032 0.2138 0.2246 0.2356 0.2467 0.2579 0.2693 0.2809 0.2926 0.3043 0.3162 0.3282 0.3403 0.3525 0.3647 0.3771 0.3895 0.4019 0.4144 0.4270 0.4396 0.4523 0.4649 0.4777 0.4904 0.5032 0.5159 0.5287 0.5416 0.5544 0.5672 0.5801 0.5929 0.6057 0.6186

4.05 4.10 4.15 4.20 4.25 4.30 4.35 4.40 4.45 4.50 4.55 4.60 4.65 4.70 4.75 4.80 4.85 4.90 4.95 5.00

0.6314 0.6443 0.6571 0.6699 0.6827 0.6956 0.7084 0.7212 0.7339 0.7467 0.7595 0.7722 0.7850 0.7977 0.8104 0.8231 0.8358 0.8485 0.8612 0.8739

Appendix C

698

J3 = 5.00 M(t) .00 .05 .10 .15 .20 .25 .30 .35 .40 .45 .50 .55 .60 .65 .70 .75 .80 .85 .90 .95 1.00 1.05 1.10 1.15 1.20 1.25 1.30 1.35 1.40 1.45 1.50 1.55 1.60 1.65 1.70 1.75 1.80 1.85 1.90 1.95 2.00

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0001 0.0002 0.0003 0.0004 0.0006 0.0008 0.0011 0.0014 0.0018 0.0023 0.0029 0.0037 0.0045 0.0054 0.0065 0.0077 0.0091 0.0107 0.0124 0.0142 0.0163 0.0186 0.0210 0.0237 0.0265 0.0296 0.0329 0.0364 0.0401 0.0441 0.0483 0.0527

M(t) 2.05 2.10 2.15 2.20 2.25 2.30 2.35 2.40 2.45 2.50 2.55 2.60 2.65 2.70 2.75 2.80 2.85 2.90 2.95 3.00 3.05 3.10 3.15 3.20 3.25 3.30 3.35 3.40 3.45 3.50 3.55 3.60 3.65 3.70 3.75 3.80 3.85 3.90 3.95 4.00

0.0573 0.0622 0.0673 0.0726 0.0781 0.0839 0.0899 0.0961 0.1025 0.1091 0.1159 0.1229 0.1302 0.1376 0.1452 0.1530 0.1609 0.1691 0.1774 0.1858 0.1944 0.2032 0.2121 0.2211 0.2303 0.2396 0.2490 0.2585 0.2681 0.2779 0.2877 0.2976 0.3076 0.3176 0.3278 0.3380 0.3482 0.3585 0.3689 0.3793

M(t) 4.05 4.10 4.15 4.20 4.25 4.30 4.35 4.40 4.45 4.50 4.55 4.60 4.65 4.70 4.75 4.80 4.85 4.90 4.95 5.00

0.3898 0.4003 0.4108 0.4213 0.4319 0.4425 0.4531 0.4638 0.4744 0.4851 0.4957 0.5064 0.5171 0.5277 0.5384 0.5490 0.5597 0.5703 0.5809 0.5916

Calculation of Renewal Functions

699

P = 6.00 t

.00 .05 .10 .15 .20 .25 .30 .35 .40 .45 .50 .55 .60 .65 .70 .75 .80 .85 .90 .95 1.00 1.05 1.10 1.15 1.20 1.25 1.30 1.35 1.40 1.45 1.50 1.55 1.60 1.65 1.70 1.75 1.80 1.85 1.90 1.95 2.00

M(t)

t

M(t)

t

M(t)

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0001 0.0001 0.0001 0.0002 0.0003 0.0003 0.0005 0.0006 0.0008 0.0010 0.0012 0.0015 0.0018 0.0022 0.0027 0.0032 0.0038 0.0045 0.0052 0.0060 0.0070 0.0080 0.0091 0.0104 0.0117 0.0132 0.0148 0.0166

2.05 2.10 2.15 2.20 2.25 2.30 2.35 2.40 2.45 2.50 2.55 2.60 2.65 2.70 2.75 2.80 2.85 2.90 2.95 3.00 3.05 3.10 3.15 3.20 3.25 3.30 3.35 3.40 3.45 3.50 3.55 3.60 3.65 3.70 3.75 3.80 3.85 3.90 3.95 4.00

0.0184 0.0204 0.0226 0.0249 0.0274 0.0300 0.0327 0.0357 0.0388 0.0420 0.0455 0.0490 0.0528 0.0568 0.0609 0.0651 0.0696 0.0742 0.0790 0.0840 0.0891 0.0944 0.0999 0.1055 0.1113 0.1173 0.1234 0.1297 0.1361 0.1427 0.1494 0.1562 0.1632 0.1704 0.1776 0.1850 0.1925 0.2002 0.2079 0.2158

4.05 4.10 4.15 4.20 4.25 4.30 4.35 4.40 4.45 4.50 4.55 4.60 4.65 4.70 4.75 4.80 4.85 4.90 4.95 5.00

0.2237 0.2318 0.2400 0.2482 0.2566 0.2650 0.2735 0.2821 0.2908 0.2995 0.3083 0.3171 0.3260 0.3350 0.3439 0.3530 0.3621 0.3712 0.3803 0.3895

Appendix C

700 ~

M(t) .00 .05 .10 .15 .20 .25 .30 .35 .40 .45 .50 .55 .60 .65 .70 .75 .80 .85 .90 .95 1.00 1.05 1.10 1.15 1.20 1.25 1.30 1.35 1.40 1.45 1.50 1.55 1.60 1.65 1.70 1.75 1.80 1.85 1.90 1.95 2.00

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0001 0.0001 0.0001 0.0001 0.0002 0.0003 0.0003 0.0004 0.0005 0.0006 0.0008 0.0009 0.0011 0.0013 0.0016 0.0019 0.0022 0.0026 0.0030 0.0034 0.0040 0.0045

2.05 2.10 2.15 2.20 2.25 2.30 2.35 2.40 2.45 2.50 2.55 2.60 2.65 2.70 2.75 2.80 2.85 2.90 2.95 3.00 3.05 3.10 3.15 3.20 3.25 3.30 3.35 3.40 3.45 3.50 3.55 3.60 3.65 3.70 3.75 3.80 3.85 3.90 3.95 4.00

= 7.00 M(t)

t

M(t)

0.0052 0.0059 0.0066 0.0075 0.0084 0.0094 0.0104 0.0116 0.0128 0.0142 0.0156 0.0172 0.0188 0.0206 0.0224 0.0244 0.0265 0.0287 0.0310 0.0335 0.0361 0.0388 0.0416 0.0446 0.0477 0.0510 0.0543 0.0579 0.0615 0.0653 0.0692 0.0733 0.0775 0.0818 0.0863 0.0909 0.0957 0.1006 0.1056 0.1107

4.05 4.10 4.15 4.20 4.25 4.30 4.35 4.40 4.45 4.50 4.55 4.60 4.65 4.70 4.75 4.80 4.85 4.90 4.95 5.00

0.1160 0.1214 0.1270 0.1327 0.1385 0.1444 0.1504 0.1566 0.1628 0.1692 0.1757 0.1823 0.1890 0.1958 0.2027 0.2097 0.2167 0.2240 0.2312 0.2385

701

Calculation of Renewal Functions

Table C.3 Distribution t

.00 .05 .10 .15 .20 .25 .30 .35 .40 .45 .50 .55 .60 .65 .70 .75 .80 .85 .90 .95 1.00 1.05 1.10 1.15 1.20 1.25 1.30 1.35 1.40 1.45 1.50 1.55 1.60 1.65 1.70 1.75 1.80 1.85 1.90 1.95 2.00

Tabulated Values of the Renewal Function for the Lognormal 02 = 0.10 M(t)

t

M(i)

0.0000 0.0000 0.0000 0.0000 0.0001 0.0001 0.0001 0.0005 0.0019 0.0058 0.0142 0.0294 0.0532 0.0867 0.1297 0.1816 0.2403 0.3037 0.3697 0.4359 0.5008 0.5628 0.6214 0.6760 0.7268 0.7741 0.8185 0.8610 0.9021 0.9428 0.9838 1.0255 1.0683 1.1125 1.1582 1.2052 1.2534 1.3025 1.3522 1.4022 1.4523

2.05 2.10 2.15 2.20 2.25 2.30 2.35 2.40 2.45 2.50 2.55 2.60 2.65 2.70 2.75 2.80 2.85 2.90 2.95 3.00 3.05 3.10 3.15 3.20 3.25 3.30 3.35 3.40 3.45 3.50 3.55 3.60 3.65 3.70 3.75 3.80 3.85 3.90 3.95 4.00

1.5022 1.5517 1.6007 1.6493 1.6973 1.7448 1.7920 1.8389 1.8856 1.9322 1.9789 2.0256 2.0726 2.1197 2.1670 2.2143 2.2621 2.3099 2.3577 2.4056 2.4535 2.5014 2.5492 2.5970 2.6447 2.6923 2.7398 2.7874 2.8348 2.8823 2.9297 2.9771 3.0246 3.0721 3.1196 3.1671 3.2146 3.2622 3.3098 3.3574

702

Appendix C

e2 = 0.30 t

.00 .05 .10 .15 .20 .25 .30 .35 .40 .45 .50 .55 .60 .65 .70 .75 .80 .85 .90 .95 1.00 1.05 1.10 1.15 1.20 1.25 1.30 1.35 1.40 1.45 1.50 1.55 1.60 1.65 1.70 1.75 1.80 1.85 1.90 1.95 2.00

M(f)

t

0.0000 0.0000 0.0001 0.0004 0.0017 0.0058 0.0140 0.0277 0.0472 0.0725 0.1030 0.1379 0.1762 0.2171 0.2598 0.3037 0.3482 0.3930 0.4378 0.4826 0.5271 0.5714 0.6154 0.6592 0.7029 0.7464 0.7898 0.8331 0.8763 0.9195 0.9627 1.0059 1.0490 1.0922 1.1354 1.1785 1.2217 1.2648 1.3080 1.3511 1.3943

2.05 2.10 2.15 2.20 2.25 2.30 2.35 2.40 2.45 2.50 2.55 2.60 2.65 2.70 2.75 2.80 2.85 2.90 2.95 3.00 3.05 3.10 3.15 3.20 3.25 3.30 3.35 3.40 3.45 3.50 3.55 3.60 3.65 3.70 3.75 3.80 3.85 3.90 3.95 4.00

m

1.4374 1.4806 1.5237 1.5668 1.6099 1.6531 1.6962 1.7393 1.7824 1.8255 1.8686 1.9117 1.9548 1.9978 2.0409 2.0840 2.1271 2.1702 2.2132 2.2563 2.2994 2.3425 2.3855 2.4286 2.4717 2.5147 2.5578 2.6008 2.6439 2.6870 2.7000 2.7731 2.8161 2.8592 2.9022 2.9453 2.9883 3.0314 3.0744 3.1175

703

Calculation of Renewal Functions

e2 = 0.50 t

.10 .15 .20 .25 .30 .35 .40 .45 .50 .55 .60 .65 .70 .75 .80 .85 .90 .95 1.00 1.05 1.10 1.15 1.20 1.25 1.30 1.35 1.40 1.45 1.50 1.55 1.60 1.65 1.70 1.75 1.80 1.85 1.90 1.95 2.00

M(l)

t

0.0006 0.0037 0.0115 0.0251 0.0444 0.0690 0.0980 0.1303 0.1653 0.2021 0.2403 0.2794 0.3191 0.3591 0.3994 0.4398 0.4803 0.5207 0.5611 0.6015 0.6419 0.6821 0.7224 0.7625 0.8027 0.8427 0.8828 0.9228 0.9627 1.0026 1.0425 1.0823 1.1221 1.1618 1.2016 1.2413 1.2809 1.3206 1.3602

2.05 2.10 2.15 2.20 2.25 2.30 2.35 2.40 2.45 2.50 2.55 2.60 2.65 2.70 2.75 2.80 2.85 2.90 2.95 3.00 3.05 3.10 3.15 3.20 3.25 3.30 3.35 3.40 3.45 3.50 3.55 3.60 3.65 3.70 3.75 3.80 3.85 3.90 3.95 4.00

m

1.3998 1.4393 1.4789 1.5184 1.5579 1.5974 1.6368 1.6763 1.7157 1.7551 1.7945 1.8339 1.8732 1.9126 1.9519 1.9912 2.0306 2.0698 2.1091 2.1484 2.1877 2.2269 2.2662 2.3054 2.3446 2.3838 2.4230 2.4622 2.5014 2.5406 2.5798 2.6190 2.6581 2.6973 2.7364 2.7756 2.8147 2.8539 2.8930 2.9321

704

Appendix C

e2 = 0.75 t

.10 .15 .20 .25 .30 .35 .40 .45 .50 .55 .60 .65 .70 .75 .80 .85 .90 .95 1.00 1.05 1.10 1.15 1.20 1.25 1.30 1.35 1.40 1.45 1.50 1.55 1.60 1.65 1.70 1.75 1.80 1.85 1.90 1.95 2.00

M(t)

t

M(t)

0.0040 0.0143 0.0317 0.0550 0.0828 0.1141 0.1477 0.1829 0.2192 0.2562 0.2938 0.3315 0.3695 0.4075 0.4455 0.4835 0.5215 0.5593 0.5971 0.6348 0.6724 0.7099 0.7474 0.7847 0.8219 0.8591 0.8961 0.9331 0.9700 1.0068 1.0436 1.0802 1.1169 1.1534 1.1899 1.2263 1.2627 1.2990 1.3352

2.05 2.10 2.15 2.20 2.25 2.30 2.35 2.40 2.45 2.50 2.55 2.60 2.65 2.70 2.75 2.80 2.85 2.90 2.95 3.00 3.05 3.10 3.15 3.20 3.25 3.30 3.35 3.40 3.45 3.50 3.55 3.60 3.65 3.70 3.75 3.80 3.85 3.90 3.95 4.00

1.3714 1.4076 1.4437 1.4797 1.5157 1.5517 1.5876 1.6235 1.6594 1.6952 1.7309 1.7667 1.8024 1.8381 1.8737 1.9093 1.9449 1.9804 2.0160 2.0515 2.0869 2.1224 2.1578 2.1932 2.2286 2.2639 2.2993 2.3346 2.3699 2.4052 2.4404 2.4756 2.5109 2.5461 2.5812 2,6164 2,6515 2.6867 2.7218 2.7569

Calculation of Renewal Functions

705

e2 = 1.00 t

.00 .05 .10 .15 .20 .25 .30 .15 .40 .45 .50 .55 .60 .55 .70 .75 .80 .85 .90 .95 1.00 1.05 1.10 1.15 1.20 1.25 1.30 1.35 1.40 1.45 1.50 1.55 1.60 1.65 1.70 1.75 1.80 1.85 1.90 1.95 2.00

m

0.0000 0.0030 0.0107 0.0290 0.0541 0.0838 0.1164 0.1510 0.1867 0.2231 0.2599 0.2969 0.3339 0.3709 0.4079 0.4447 0.4814 0.5179 0.5542 0.5904 0.6265 0.6623 0.6980 0.7336 0.7690 0.8043 0.8394 0.8744 0.9093 0.9440 0.9787 1.0132 1.0476 1.0819 1.1161 1.1502 1.1843 1.2182 1.2520 1.2858 1.3195

t

M(t)

2.05 2.10 2.15 2.20 2.25 2.30 2.35 2.40 2.45 2.50 2.55 2.60 2.65 2.70 2.75 2.80 2.85 2.90 2.95 3.00 3.05 3.10 3.15 3.20 3.25 3.30 3.35 3.40 3.45 3.50 3.55 3.60 3.65 3.70 3.75 3.80 3.85 3.90 3.95 4.00

1.3531 1.3866 1.4201 1.4535 1.4868 1.5201 1.5532 1.5864 1.6195 1.6525 1.6855 1.7184 1.7512 1.7840 1.8168 1.8495 1.8822 1.9148 1.9474 1.9800 2.0125 2.0449 2.0774 2.1097 2.1421 2.1744 2.2067 2.2389 2.2712 2.3033 2.3355 2.3676 2.3997 2.4318 2.4638 2.4958 2.5278 2.5598 2.5917 2.6236

706

Appendix C

e2 = i.50

t .00 .05 .10 .15 .20 .25 .30 .35 .40 .45 .50 .55 .60 .65 .70 .75 .80 .85 .90 .95 1.00 1.05 1.10 1.15 1.20 1.25 1.30 1.35 1.40 1.45 1.50 1.55 1.60 1.65 1.70 1.75 1.80 1.85 1.90 1.95 2.00

M(t)

t

0.0000

2.05 2.10 2.15 2.20 2.25 2.30 2.35 2.40 2.45 2.50 2.55 2.60 2.65 2.70 2.75 2.80 2.85 2.90 2.95 3.00 3.05 3.10 3.15 3.20 3.25 3.30 3.35 3.40 3.45 3.50 3.55 3.60 3.65 3.70 3.75 3.80 3.85 3.90 3.95 4.00

0.0136 0.0302 0.0615 0.0966 0.1334 0.1711 0.2089 0.2465 0.2840 0.3210 0.3577 0.3940 0.4300 0.4655 0.5007 0.5355 0.5700 0.6042 0.6381 0.6716 0.7050 0.7380 0.7708 0.8034 0.8357 0.8679 0.8998 0.9315 0.9631 0.9945 1.0257 1.0567 1.0876 1.1183 1.1489 1.1793 1.2097 1.2399 1.2699 1.2999

m 1.3297 1.3594 1.3890 1.4185 1.4480 1.4773 1.5065 1.5356 1.5646 1.5936 1.6225 1.6513 1.6800 1.7086 1.7372 1.7656 1.7941 1.8224 1.8507 1.8789 1.9071 1.9352 1.9632 1.9912 2.0191 2.0469 2.0748 2.1025 2.1302 2.1579 2.1855 2.2130 2.2405 2.2680 2.2954 2.3228 2.3501 2.3774 2.4046 2.4318

Calculation of Renewal Functions

707

e2 = 2.00 t

.00 .05 .10 .15 .20 .25 .30 .35 .40 .45 .50 .55 .60 .65 .70 .75 .80 .85 .90 .95 1.00 1.05 1.10 1.15 1.20 1.25 1.30 1.35 1.40 1.45 1.50 1.55 1.60 1.65 1.70 1.75 1.80 1.85 1.90 1.95 2.00

M(t)

t

MU)

0.0000 0.0257 0.0525 0.0925 0.1334 0.1742 0.2143 0.2537 0.2922 0.3299 0.3669 0.4032 0.4389 0.4739 0.5084 0.5423 0.5758 0.6088 0.6414 0.6736 0.7054 0.7368 0.7680 0.7988 0.8293 0.8596 0.8895 0.9193 0.9487 0.9780 1.0070 1.0359 1.0645 1.0929 1.1212 1.1493 1.1772 1.2049 1.2325 1.2599 1.2872

2.05 2.10 2.15 2.20 2.25 2.30 2.35 2.40 2.45 2.50 2.55 2.60 2.65 2.70 2.75 2.80 2.85 2.90 2.95 3.00 3.05 3.10 3.15 3.20 3.25 3.30 3.35 3.40 3.45 3.50 3.55 3.60 3.65 3.70 3.75 3.80 3.85 3.90 3.95 4.00

1.3144 1.3414 1.3683 1.3950 1.4216 1.4481 1.4745 1.5008 1.5270 1.5531 1.5791 1.6049 1.6307 1.6564 1.6820 1.7075 1.7329 1.7582 1.7835 1.8086 1.8337 1.8587 1.8837 1.9085 1.9333 1.9580 1.9827 2.0073 2.0318 2.0563 2.0807 2.1050 2.1293 2.1535 2.1777 2.2018 2.2259 2.2499 2.2738 2.2977

708

Appendix C

e2 = 3.00 t

.00 .05 .10 .15 .20 .25 .30 .35 .40 .45 .50 .55 .60 .65 .70 .75 .80 .85 .90 .95 1.00 1.05 1.10 1.15 1.20 1.25 1.30 1.35 1.40 1.45 1.50 1.55 1.60 1.65 1.70 1.75 1.80 1.85 1.90 1.95 2.00

M(t)

t

0.0000 0.0483 0.0956 0.1459 0.1933 0.2381 0.2805 0.3212 0.3602 0.3979 0.4343 0.4696 0.5040 0.5375 0.5702 0.6022 0.6336 0.6643 0.6945 0.7242 0.7534 0.7822 0.8105 0.8385 0.8660 0.8933 0.9202 0.9468 0.9731 0.9991 1.0248 1.0503 1.0756 1.1006 1.1254 1.1500 1.1743 1.1985 1.2225 1.2463 1.2699

2.05 2.10 2.15 2.20 2.25 2.30 2.35 2.40 2.45 2.50 2.55 2.60 2.65 2.70 2.75 2.80 2.85 2.90 2.95 3.00 3.05 3.10 3.15 3.20 3.25 3.30 3.35 3.40 3.45 3.50 3.55 3.60 3.65 3.70 3.75 3.80 3.85 3.90 3.95 4.00

m

1.2934 1.3167 1.3398 1.3628 1.3856 1.4083 1.4309 1.4533 1.4756 1.4977 1.5198 1.5417 1.5635 1.5852 1.6067 1.6282 1.6496 1.6708 1.6920 1.7131 1.7340 1.7549 1.7757 1.7964 1.8170 1.8376 1.8580 1.8784 1.8987 1.9189 1.9390 1.9591 1.9791 1.9990 2.0189 2.0387 2.0584 2.0780 2.0976 2.1172

Calculation of Renewal Functions C.2

709

RENEW AL FUNCTIO N PROGRAM

8

+

10

PROGRAM RENEW1 PA RA M ETERS = 5000) IMPLICIT DOUBLE PRECISION (A-H,0-Z) CHARACTERS FILES, FILES2 CHARACTERSO COMMENTS integer ch dimension g(n), R(0:N),F(0:N),fl(0:n) DOUBLE PRECISION LD,J,LD1,LD2,L0,L1,L2 REWIND(l) ZL= l.D-8 fl(0) = 0.0 WRITE(6,*)’ ’ WRITE(6,*) ’ PRESS 1 TO CALCULATE m(T)’ WRITE(6,*) ’ PRESS 2 TO EXIT’ WRITE(6,*)’ ’ WRITE(6,’(a,$)’)’ ENTER YOUR CHOICE:’ READ(5,*) ICH IF(ICH.EQ.2) GO TO 28 WRITE(6,*) ’ ’ WRITE(6,’(a,$)’)’ DO YOU WANT AN OUTPUT (ascii) FILE? (YES = 1 NO = 0):’ READ(5,*) ICH1 IF(IC H l.EQ .l) THEN WRITE(6,’(a,$)’)’ NAME OF OUTPUT FILE? :’ READ(5,10) FILES FORMAT(A8) OPEN(l,FILE = FILES) ENDIF WRITE(6,*)’ ’

50

WRITE(6,*) ’ PRESS 1 FOR WEIBULL DISTRIBUTION’ WRITE(6,*) ’ PRESS 2 FOR EXPONENTIAL DISTRIBUTION’ WRITE(6,*) ’ PRESS 3 FOR GAMMA DISTRIBUTION’ WRITE(6,*) ’ PRESS 4 FOR INVERSE GAUSSIAN DISTRIBUTION’ WRITE(6,*) ’ PRESS 5 FOR LOGNORMAL DISTRIBUTION’ WRITE(6,*) ’ PRESS 6 FOR TRUNCATED NORMAL DISTRIBUTION’ WRITE(6,*) ’ PRESS 7 FOR MIXTURE OF EXPONENTIALS’ WRITE(6,*) ’ ’ WRITE(6,’(a,$)’)’ ENTER YOUR CHOICE:’ READ(5,*) CH

WRITE(6,*) ’ ’ WRITE(6,’(A,$)’)’ ENTER VALUE OF STEP SIZE :’ READ(5,*) H

Appendix C

710

WRITE(6,’(A,$)’)’ ENTER VALUE OF T READ(5,*) T IF (N*H.LT.T) THEN WRITE(6,*) ’ REDUCE T AND/OR INCREASE H’ GOTO 50 ENDIF IF (CH.EQ.l) THEN WRITE (6,*) ’ F(t) = (l-EXP(-(LD*t)**AL))’ WRITE(6,*) ’ f(t) = ((AL*(t**(AL-l)))*EXP (-((t*LD)**AL)))*LD**AL’ WRITE(6,’(A,$)’)’ ENTER VALUE OF LAMBDA READ(5,*) LD WRITE(6,’(A,$)’)’ ENTER VALUE OF ALPHA READ(5,*) AL

1

ELSEIF (CH.EQ.2) THEN WRITE (6,*) ’ F(t) = l-EXP(-LD*t)’ WRITE (6,*) ’ f(t) = LD*EXP(-LD*t)’ WRITE(6,’(A,$)’)’ ENTER VALUE OF LAMBDA READ(5,*) AL ELSEIF (CH.EQ.3) THEN WRITE (6,*) ’ f(t) = (t**(AL-l))*(EXP(-t/LD))/ ((LD**AL)*Gamma)’ WRITE(6,’(A,$)’)’ ENTER VALUE OF LAMBDA READ(5,*) LD WRITE(6,’(A,$)’)’ ENTER VALUE OF ALPHA READ(5,*) AL ELSEIF (CH.EQ.4) THEN WRITE (6,*) ’ L = (LD/(2*3.1416*(t**3)))**.5’ WRITE(6,*) ’ f(t) = L*(EXP((-LD *((t-MU) **2))/ (2*MU*MU*t)))’ WRITE(6,’(A,$)’)’ ENTER VALUE OF LAMBDA READ(5,*) LD WRITE(6,’(A,$)’)’ ENTER VALUE OF MU READ(5,*) AL ELSEIF (CH.EQ.5) THEN WRITE (6,*) ’ L = (EXP(-(((LOG(t))-MU* *2)/(2*SI* *2)))’ WRITE(6,*) ’ f(t) = L/(t*SI*(2*3.1416)**.5)’ WRITE(6,’(A,$)’)’ ENTER VALUE OF SIGMA (SI) READ(5,*) LD WRITE(6,’(A,$)’)’ ENTER VALUE OF MU READ(5,*) AL ELSEIF (CH.EQ.6) THEN WRITE(6,*) ’ f(t) = EXP(-((t-MU)**2)/(2*SI**2))/(SI*(2*3.1416)**.5)’ WRITE(6,’(A,$)’)’ ENTER VALUE OF SIGMA (SI)

Calculation of Renewal Functions

READ(5,*) LD WRITE(6,’(A,$)’)’ ENTER VALUE OF MU READ(5,*) AL

25 +

27

29

IF (AL.EQ.O) THEN J = 0.5d0 GOTO 29 END IF I = 1 IF (AL.LT.O) THEN A0= -O.OldO ELSEIF (AL.GT.O) THEN A0= O.OldO ENDIF K = ABS(AL/A0 +1.5) A = dble(I-l)*A0 G(I) = dEXP(-((A-AL)**2.dO)/(2.dO*LD**2.dO)) /(LD*(2.dO*3.1416d0)* *.5d0) IF(I.EQ.K) GOTO 27 I = 1+ 1 GOTO 25 CALL INTEG(K,AO,G,S) J = S WRITE(6,*) S J = 0.5d0 + J ELSEIF (CH.EQ.7) THEN WRITE(6,’(A,$)’)’ ENTER VALUE OF LAMBDA 1 READ(5,*) LD1 WRITE(6,’(A,$)’)’ ENTER VALUE OF LAMBDA2 READ(5,*) LD2 WRITE(6,’(A,$)’)’ ENTER VALUE OF p READ(5,*) p ENDIF m = int(t/h) + 1 f(0) = 0.d0 z = 0.5d0*h IF (CH.EQ.l) THEN f05 = funl(ld,al,z) if(dabs(f05).lt.zl) f05 = 0.d0 ELSEIF (CH.EQ.2) THEN f05 = fun2(al,z) ELSEIF (CH.EQ.3) THEN f05 = fun3(ld, al,.5d0*z)*z ELSEIF (CH.EQ.4) THEN LO = (LD/(2.d0*3.1416d0*(z**3.d0)))**.5d0 f05 = fun4(10,ld,al ,.5d0*z)*z

712

Appendix C

ELSEIF (CH.EQ.5) THEN LO = (dEXP(-(((dLOG(z))-AL)**2.dO)/(2.dO*LD**2.dO))) f05 = fun5(10,ld,.5d0*z)*z ELSEIF (CH.EQ.6) THEN f05 = fun6(ld,al,j,.5d0*z)*z ELSEIF (CH.EQ.7) THEN f05 = fun7(ldl ,ld2,p,z) ENDIF do 21 i = l,m w = dble(i)*h u = (dble(i) + 0.5d0)*h v = (dble(i)-.5d0)*h q = dble(i + .25)*h

21

IF (CH.EQ.l) THEN f(i) = funl(ld,al,w) fl(i) = funl(ld,al,u) ELSEIF (CH.EQ.2) THEN f(i) = fun2(al,w) fl(i) = fun2(al,u) ELSEIF (CH.EQ.3) THEN f(i) = f(i-l) + fun3(ld,al,v)*h fl(i) = f(i) + fun3(ld,al,q)* .5d0*h ELSEIF (CH.EQ.4) THEN LI = (LD/(2.d0*3.1416d0*(v**3.d0)))**.5d0 L2 = (LD/(2.d0*3.1416d0*(w**3.d0)))**.5d0 f(i) = f(i-l) + fun4(ll,ld,al,(dble(i)-.5d0)*h)*h fl(i) = fl(i-l) + fun4(12,ld,al ,dble(i)*h)*h ELSEIF (CH.EQ.5) THEN LI = (dEXP(-(((dLOG(v))-AL)**2.dO)/(2.dO*LD**2.dO))) L2 = (dEXP(-(((dLOG(w))-AL)**2.dO)/(2.dO*LD**2.dO))) f(i) = f(i-l) + fun5(lUd,(dble(i)-.5dO)*h)*h if(dabs(f(i)).lt.zl) f(i) = 0.d0 fl(i) = fl(i-l) + fun5(12,ld,dble(i)*h)*h if(dabs(fl(i)).lt.zl) fl(i) = 0.d0 ELSEIF (CH.EQ.6) THEN f(i) = f(i-l) + fun6(ld,al,j ,(dble(i)-.5d0)*h)*h fl(i) = fl(i-l) + fun6(ld,al,j,dble(i)*h)*h ELSEIF (CH.EQ.7) THEN f(i) = fun7(ldl,ld2,p,w) f 1(i) = fun7 (ldl ,ld2 ,p ,u) ENDIF continue tem = l.d0-f05

Calculation of Renewal Functions

C

C 20 c c c C 26 C 26 26 c 31

35

28

713

write(6,*) ’ tern = ’,tern DO 31 i = 0,m if(i.eq.O) then r(i) = 0.d0 go to 31 elseif(i.eq.l) then sum = 0.d0 go to 26 elseif(i.gt.l) then sum = O.dO do 20 k = l,i-l write(6,*) ’ l ’,k,i,sum sum = sum + fl(i-k)*(r(k)-r(k-l)) write(6,*) ’ made it’,f(i),sum write(6,*) i,k,fl(i-k),r(k),r(k-l) if(i.eq.4) stop endif write(6,*) ’ at 26’ r(i) = (f(i) + sum-f05*r(i-l))/(l.d0-f05) r(i) = f(i)/tem-f05*r(i-l)/tem + sum/tem if(dabs(r(i)).lt.zl) r(i) = 0.d0 write(6,*) ’ after’,i,r(i),r(i-l),f05,tern,f(i),sum continue call output(m,h,ichl,r) go to 35 CLOSE(l) WRITE(6,*) ’ ’ WRITE(6,’(a,$)’) ’ DO YOU NEED A PRINTOUT? (YES = 1 NO = 0):’ READ(5,*) ICH2 IF (ICH2.EQ.1) THEN WRITE(6,’(a,$)’)’ NAME OF PRINTOUT FILE? :’ READ(5,’(a,$)’) FILES2 WRITE(6,’(a,$)’)’ WRITE ANY COMMENTS FOR THE FILE HERE :’ WRITE (6,*) ’ ’ READ(5,’(a$)’) COMMENTS OPEN(2,FILE = FILES2) WRITE(2,’(a,$)’) COMMENTS CALL PRIN(h ,m ,ICH2,R) CLOSE (2) ENDIF GOTO 8 stop END

Appendix C

714

C

36 37 50

SUBROUTINE PRIN(h,m,ICH,R) DOUBLE PRECISION R, H DIMENSION R(0:5000) j = int(ldO/h) + 1 DO 50 n= 0,m,4*j x = n*H IF(ICH.EQ.l) THEN WRITE(2,37) x, x + .l, x + .2, x + .3 WRITE(2,37) x, x + h, x + 2*h, x + 3*h WRITE(2,36) R(n),R(n + j),R(n + 2*j),R(n + 3*j) WRITE (2,*) endif FORMAT (4(F15.10,’ ’)) FORMAT(4(’ M(’,F4.2,’)= ’)) CONTINUE RETURN END SUBROUTINE OUTPUT(m,h,ich,R) DOUBLE PRECISION R,h DIMENSION R(0:5000)

36

37

38 50 *******

i = int(01d0/h) + l write(6,36) format(/3x,’t ’,7x, ’M(t) 7) do 50 j = 0,m y = j*h write(6,37)y,r(j) FORMAT(lx,F5.2,F10.6) if(ich,eq.l) then write(l,38)y,r(j) FORMAT(lx,F15.2,F20.6) endif continue RETURN END weibull distribution function ******* function funl(d,al,x) double precision d,al,x,funl funl = l.dO - dexp(-(x*d)**al) return end

function fun2(al,x) double precision al,x,fun2

Calculation of Renewal Functions

fun2= l.dO —dEXP(-AL*x) return end

*******

function fun3(ld,al,x) double precision ld,al,x,gm,gammln,fun3 gm = gammln(al) fun3 = (x* *(AL-1.dO)) *(dexp(-x/LD))/((LD* *AL) *dexp(gm)) return end

*******

function fun4(l,ld,al,x) double precision l,ld,al,x,fun4 fun4= L*(dEXP((-LD*((x-AL)**2.dO))/(2.dO*AL*AL*x))) return end

*******

function fun5(b,d,x) double precision b,d,x,fun5 fun5 = b/(x*d*(2.d0*3.1416d0)**.5d0) return end

*******

function fun6(d,al,j,x) double precision d,al,j,x,fun6 + *******

fun6 = dEXP(-( (x-AL) **2. d0)/(2.dO*D **2. dO)) /(D*(2.d0*3.1416d0)**.5d0)/J return end function fun7(ldl ,ld2,p,x) double precision Idl,ld2,p,x,fun7 fun7 = l.dO-p*dEXP(-ldl*x)-(l.dO-p)*dexp(-ld2*x) return end FUNCTION GAMMLN (XX) IMPLICIT DOUBLE PRECISION (A-H,0-Z) DIMENSION COF(6)

1

DATA COF,STP /76.18009173D0,-86.50532033D0 ,24.01409822D0, -1.231739516D0,.120858003D-2,-.536382D-5

715

Appendix C

716

2

,2.5066282746500/ DATA HALF,ONE,FPF/0.5D0,1.0D0,5.5D0/ X = XX-ONE TMP = X + FPF TMP = (X + HALF)*DLOG(TMP)-TMP SER = ONE

20

DO 20 J = l,6 X = X + ONE SER = SER + COF(J)/X CONTINUE GAMMLN = TMP + DLOG(STP*SER) RETURN END SUBROUTINE INTEG(N,H,G,S) DOUBLE PRECISION H,G,EVEN,ODD,S DIMENSION G(5001) IF (N.GT.4) GO TO 5 GO TO (1,2,3,4),N

1 2 3 4 5

10 20 30 40

S = 0.D00 RETURN S = H/2.D0*(G(1) + G(2)) RETURN S = H/3.D0*(G(1) + 4.D0*G(2) + G(3)) RETURN S = 3.D0/8.D0*H*(G(1) + 3.D0*(G(2) + G(3) + G(4)) RETURN M = MOD(N,2) IF (M .EO.l) TO TO 10 S = 3.D0/8.D0*H*(G(N-3) + 3.D0*(G(N-2) + G(N-l)) + G(N)) L = N-3 GO TO 20 S = 0.D00 L = N ODD = 0.D00 EVEN = 0.D00 DO 30 I = 2,L,2 EVEN = EVEN + G(I) N2 = L-2 do 40 I = 3,N2,2 ODD = ODD + G(I) S = S + H/3.D0*(G(1) + 4.D0*EVEN + 2.DO*ODD + G(L)) RETURN END

Author Index

Abel, R., 647, 651 Abramowitz, M., 77, 90, 137, 166, 176, 216, 311, 331, 365, 603 Akerlof, G., 638, 640, 650 Ascher, H., 601, 605 Agnihotri, S. R., 412, 414 Akahira, M., 568, 576 Allen, R. J., 599, 604 Amato, N. H., 161, 163, 167, 168, 213, 216, 411, 413 Andreasen, A. R., 615, 616, 648 Anderson, E. E., 161, 163, 167, 168, 213, 216, 411, 413, 618, 647, 649, 651 Ash, S. B., 617, 648

Bailey, R. J. M., 401, 413 Balaban, H. S., 61,90,276,277, 281, 285, 286, 291, 292, 297, 298, 598, 599, 601, 604, 605 Balachandran, K. R., 162, 167, 650 Balcer, Y., 162, 168, 213, 216, 577 Banks, J., 511, 513

278, 293, 603, 632, 569,

Barlow, R. E., 71,72,87,90,91,109, 112, 127-129, 502, 513, 569, 576 Bartholomew, D. J., 110, 128, 569, 576 Barton, H. R., 294, 299 Baxter, L., 113, 114, 129, 136, 150, 152, 157, 163, 166, 168, 235, 236, 267, 380, 413, 495, 513, 551, 575 Bayer, H., 600, 604 Bazovsky, I., 294, 298 Bass, F. W., 83, 90 Beall, C. W., 499, 513 Bearden, W. O., 612, 616, 628, 648 Bebuchuk, L. A., 648, 652 Bender, H. E., 411, 413 Beneke, M., 470, 472 Bergiel, B. J., 647, 651 Berk, J. H., 294, 299, 596, 603 Berke, T. M., 245,247,259-261,264, 267, 597, 603 Berger, P. D., 142, 167, 647, 651 Best, A., 615, 617, 648 Bhat, N. U., 127, 129 Bicknell, R. S., 280, 285, 287, 293, 297, 599, 604 Biedenweg, F. M., 213,217, 263,267, 411, 412, 413 717

718

Bilgen, S., 114, 129 Bleuel, W. H., 411, 413 Blischke, W. R., 86, 91, 129, 136, 150, 152, 156, 157, 163, 166168, 213, 214, 216, 235, 236, 267, 380, 413, 495, 499, 513, 528, 551, 552, 568, 574-576 Boes, D. C., 519, 574 Bogart, G. G., 9, 42 Bonner, W. J., 273, 293, 297 Bortz, J. E., 280, 285, 287, 293, 297, 599, 604 Brately, P., 511, 513 Brennan, J. R., 511, 514 Brown, J. P., 648, 652 Brown, M., 493, 513 Brown, R. G., 412, 414 Bryant, K. W., 27, 43, 411, 413, 593, 600, 603, 604 Bulgren, W. G., 511, 513

Cadotte, E. R., 628, 650 Calvo, A. B., 293, 298 Carson, J. S., 511, 513 Chakravarti, I. M., 568, 576 Chandramohan, J., 470, 471 Chandran, R., 28, 41, 43, 412, 414 Checkland, P., 41, 43 Chelson, R. C , 282-284, 297 Chiang, A., 647, 651 Chun, Y. H., 362, 365, 470, 471 Cinlar, E., 106, 109, 112, 117, 128, 151, 167 Cleroux, R., 113, 129 Cohen, A. C., 600, 604 Cohen, J., 601, 605 Coombs, C. F., 569, 576 Cooper, R., 641, 650 Cooter, R. D., 648, 652 Coppola, A., 470, 471 Courville, L., 600, 604, 642, 651 Cox, D. R., 127, 129, 150, 167 Crow, L. H., 426, 470 Cuppett, D., 293, 298, 599, 604

Author Index

Darden, W. R., 610, 648 Dardis, R., 39,43,410,413, 621, 649 Davis, D. J., 573, 577, 601, 605 Dawid, A. P., 568, 576 Day, R. C., 294, 299 Day, R. L., 616, 648 Delingonul, Z. S., 110, 114, 129 Derman, C., 601, 605 Deuermeyer, B. L., 163, 168 Devroye, L., 511, 514 Dey, K. A., 568, 576 Dhillon, B. S., 86, 87, 91, 271, 276, 293, 296, 298, 469, 471, 600, 605 Dickey, J. M., 568, 576 Dizek, S. G., 293, 298 Djamaludin, I., 441-444, 469, 471 Duane, J. T., 426, 470 Dubey, S. D., 547, 574 Dungworth, T., 30, 43

Elbright, A. H., 5, 41, 42 Ellis, D. E., 594, 603 Emmons, W., 640-642, 650 Epple, D., 640, 650 Erto, P., 567, 575, 595, 603

Farell, J., 647, 651 Feingold, H., 601,605 Ferdous, J., 568, 576 Fink, E. E., 9, 42 Fisk, G., 412, 621, 625, 647, 649 Fleig, N. G., 272, 273, 297 Flottman, W. W., 600, 604 Folse, R. O., 294, 298 Foreman, G. L., 469, 471 Fornell, C , 621, 649 Fox, B., 511, 513 Franta, W. R., 511, 513 Frees, E. W., 213, 217,263,267,549, 550, 552, 554-556, 569, 575

Author Index

Gandara, A., 61, 90, 270, 271, 296, 601, 605 Gastwirth, J., 627, 649 Gates, R. K., 280,285,287,293,297, 599, 604 Garvin, D. A., 594, 603 Gemer, J. L., 27, 43, 411, 413, 593, 600, 603, 604 Gertsbakh, I. B., 87, 91 Giblin, M. T., 136, 166 Gill, H. L., 600, 604 Gilleece, M. A., 272, 297 Glaser, A. J., 294, 299 Glaser, R. E., 87, 91 Gleser, L. J., 601, 605 Glickman, J. S., 142, 167, 647, 651 Goel, A. L., 470, 471 Goldberg, H., 469, 471 Graybill, F. A., 519, 574 Greenberg, S., 511, 513 Gregory, W. M., 271, 296 Grimlund, R. A., 632, 650 Grossman, S., 642, 650 Guglielmoni, P. B., 270, 293, 296 Guida, M., 13, 567, 575, 595, 603 Guin, L., 58, 59, 90, 247, 263, 505, 513

Hadley, G., 412, 414 Hansen, C. K., 568, 576 Hardy, C. A., 599, 604 Harris, L., 41, 43 Harrison, G., 293, 298, 599, 604 Harty, J. C., 293, 298 Harvey, D. W., 163, 168, 213, 216 Hauser, J. R., 627, 649 Hausman, W. H., 600, 604, 642, 651 Hauter, A. J., 598, 603 Heal, G., 639, 650 Heathcote, C. R., 438, 470 Hegde, G. G., 469, 471 Henderson, J. M., 82, 90 Heschel, M. S., 263, 267

719 Hill, V. L., 160, 163, 167, 168, 213, 216, 499, 503, 513, 567, 575 Hiller, G. E., 271, 296 Hjorth, U., 87, 91 Huang, C. L, 469, 471 Huberman, G., 647, 652 Hultquist, 76 Hunter, J. J., 123, 127,129, 331, 365 Hunter, L., 87, 91 Hutchinson, N. E., 412, 414

Iizuka, Y., 621, 649 Ip-Tamayo, T. C., 163, 168 Ireson, W. G., 569, 576 Isaacson, D. N., 511, 514 Isham, V., 127, 129 Iskandar, B. P., 327, 362, 365, 366, 511, 514

Jenkins, R. L., 628, 650 Johnson, N. L., 76, 87, 90, 91, 120, 129, 261, 267, 362, 365, 547, 575

Kao, E. P. C., 114, 129 Kao, J. H. K., 87, 91 Kalbfleisch, J. D., 575, 595, 603 Kalivoda, F. E., 567, 575 Kambhu, J., 647, 651 Kaplan, E. L., 552, 575 Kapur, K. C., 87, 91, 250, 267, 469, 471, 543, 574 Karmarkar, U. S., 162,167,411-414 Kelton, W. D., 485, 488, 513 Kendall, C. L., 621, 647, 649 Klause, P. J., 271, 297 Kordonsky, Kh. B., 87, 91 Kotier, P., 87, 91, 627, 649

720

Kotz, S., 76, 87, 91, 120, 129, 261, 267, 362, 365, 547, 575 Kowalski, R., 293, 298 Kruvand, D. H., 62,90,288,294,297 Kubat, P., 412, 414, 469, 471 Kubo, Y., 642, 650 Kuo, W., 469, 471

Lakey, M. J., 568, 576, 596, 603 Lamberson, L. R., 87, 91, 250, 267, 469, 471, 543, 574 Lancioni, R. A., 412, 414 Landon, E. L., 616, 628, 648, 649 Law, A. M., 485, 488, 513 Lawless, J. F., 543, 569, 574, 575, 595, 600, 603, 604 Leadbetter, M. R., I l l , 129 Lee, H. L., 470, 471 Leemis, L. M., 470, 472 Lele, M. M., 411, 413 Lie, C. H., 470, 471 Linneman, R., 28, 41, 43 Livingston, J. L., 162, 167, 632, 650 Lowerre, J. M., 161, 167 Lund, R. T., 600, 604 Lunsford, J. D., 294, 299, 596, 603 Lutz, N. A., 647, 651

Madansky, A., 569, 576 Mahajan, V., 83, 87, 90 Mahon, B. H., 401, 413 Malcolm, J. G., 469, 471 Malvern, D., 601, 605 Marner, J. W., 162, 168, 188, 204, 216 Mann, N. R., 12, 470-472, 532, 543, 574 Markowitz, O., 271, 293, 297, 298, 598, 601, 603, 605 Martz, H. F., 569, 576 Marshall, A. W., 72, 90 Marshall, C. W., 293, 298, 469, 471 Marucheck, A. S., 470, 471

Author Index

Maschmeyer, R. A., 162, 167, 632, 650 Mathews, S., 642, 650 McConalogue, D. J., 113, 129, 136, 150, 152, 157, 166, 235, 236, 267, 380, 413, 495, 513, 551, 575 McGuire, E. P., 36, 41, 43 McIntyre, L. E., 294, 299 McNeil, K., 600, 604 Meier, P., 522, 575 Menezes, M. A. J., 647, 651 Menke, W. W., 161, 167, 213, 216, 411, 413 Menzefricke, U., 569, 577 Meth, M. A., 281, 291, 292, 297 Metzer, H., 613, 648 Metzler, E. G., 294, 298 Middendorf, W. H., 2, 41, 42 Mihram, G. A., 76 Miller, R. E., 600, 604 Min, H., 412, 414 Mitra, A., 160, 163, 167, 213, 263, 267, 567, 575 Moellenberndt, R. A., 619, 649 Mood, A. M., 519, 574 Moore, E. M., 613 Moore, J., 642, 648, 650 Moorthy, K. S., 647, 651 Morgan, B. J. T., 511, 513 Morgan, F. W., 29, 30, 41, 43 Morrison, A. J., 470, 471 Moskowitz, H., 362, 365 Mundle, P. B., 568, 576 Murthy, D. N. P., 86, 87, 91, 149, 150, 159, 161, 162, 167, 213, 217, 234, 241, 263, 267, 362, 365, 366, 397, 404, 411-413, 428, 441-444, 469-472, 511, 514, 569, 576, 646, 647, 651

Nam, S. H., 213, 217, 263, 267 Nelson, W., 567, 568, 575, 576, 585, 602, 603

Author Index

Nesbit, L. D., 273,294,297,601,605 Newman, D. G., 273, 294, 297, 601, 605 Newton, D. W., 600, 604 Nguyen, D. G., 87,91,149,150,161, 162, 167, 213, 217, 234, 241, 263, 267, 397, 404, 411, 412, 469-472, 569, 576 Nixon, D. E., 294, 299, 596, 603 Nohmer, F. J., 598, 603 Nordstrom, R. D., 613, 648 Nowicki, P., 624, 647, 649

Oliver, R. L., 616, 628, 648, 650 Olkin, I., 601, 605 Ord, J. K., 519, 574 Owles, D., 41, 43 Ozbaykal, T., 110, 128

Page, N. W., 91, 646, 651 Palfrey, T., 632, 634, 650 Pandey, M., 568, 576 Park, K. S., 162, 163, 168 Park, S. K., 214, 217 Parsons, L. J., 87, 91 Patankar, J. G., 160, 163, 167, 213, 216, 263, 267, 409, 412-414, 428, 567, 569, 575, 576 Paxman, R. G., 294, 299 Perry, A., 612, 648 Perry, M., 612, 648 Peterson, R. A., 83, 87, 90 Polinsky, A. M., 648, 652 Posner, M. J., 412, 414 Priest, G. L., 25, 27, 41, 43, 636, 650 Proschan, F., 71, 87, 90, 109, 112, 128, 129, 502, 513, 569, 576, 601, 605

Quandt, R. E., 82, 90

721

Rao, C. P., 610, 648 Raviv, A., 640, 650 Reterer, B. L., 276, 277, 288, 294, 297, 298, 601, 605 Reiche, H., 469, 471 Rich, M. D., 61, 90, 270, 271, 601, 605 Reid, S., 511, 514 Reinganum, J. F., 648, 652 Ritchken, P. H., 263, 267, 470, 647, 651 Roberts, D. C., 600, 604 Robinson, J. A., 575, 595, 603 Rodin, Y. E., 91, 646, 651 Rogerson, W. P., 648, 652 Romer, T., 632, 634, 650 Ross, S. M., 96-98, 112, 116, 128, 353, 365, 493, 513 Ross, T. W., 641, 650 Rotschild, M., 647, 651 Rotz, A. O., 293, 298 Rubinfeld, D. L., 648, 652 Russ, F. A., 621, 647, 649

293, 296,

471,

127,

Sahin, L , 162,168,213,216,569,577 Sarat, A., 647, 651 Saunders, S. C., 470, 472 Schafer, R., 532, 543, 574 Scharge, L. E., 511, 513 Scheuer, E. M., 129, 136, 150, 152, 156, 157, 166, 167, 213, 214, 216, 235, 236, 267, 380, 413, 495, 513, 528, 551, 552, 569, 574-576 Schmidt, A. E., 270, 271, 296 Schmidt, B. A., 293, 298 Schultz, R. L., 87, 91 Schwartz, A., 26-28, 41,43, 641, 647, 650, 652 Serfling, R., 551, 575 Shavell, S., 647, 652 Shelton, D. K., 294, 299 Sherman, P., 41, 43

722

Shimp, T. A., 612, 648 Shmoldas, A. E., 270, 296 Shorey, R. R., 293, 298 Shuptrine, F. K., 613, 648 Simon, M., 647, 652 Singpurwalla, N. D., 362, 366, 532, 543, 574 Smith, R. L., 600, 604 Soland, R. M., 113, 129 Solomon, H., 493, 513 Speir, R. N., 600, 604 Spence, M., 640, 650 Springer, R. M., 293, 298 Steele, E. H., 622, 647, 649 Stegun, I. A., 77, 90, 137, 166, 176, 216, 311, 331, 365, 603 Stephens, M., 493, 513 Story, J. K., 294, 299 Strempke, C. W., 598, 603 Stuart, A., 519, 574 Suzuki, S., 568, 569, 576

Takeuchi, K., 568, 576 Tapiero, C. S., 412, 414, 470, 471, 647, 651 Thomas, M. U., 263, 267, 374, 413, 569, 577 Thompson, W. A., 127, 129 Thorpe, J. F., 2, 41, 42 Thyregod, P., 568, 576 Tillman, F. A., 469, 471 Toohey, E. F., 293, 298 Trimble, R. F., 60, 90, 245, 247,267, 271, 296 Trombetta, W. L., 647, 651 Truelove, A. J., 568, 576

Author Index

Udell, J. G., 618, 649 Uddin, M. B., 568, 576 Urban, G. L., 627, 649 Ursic, M. E., 624, 649

Varían, H. R., 647, 651

Waller, R. A., 569, 576 Walters, C. G., 647, 651 Wernerfeit, B., 621, 649 White, R., 293, 298 Whitin, T. M., 412, 414 Wiener, J. L., 612, 648 Wilde, L. L., 26-28,41,43,641,647, 650, 652 Wilson, B., 41, 43 Wilson, R. J., 362, 365, 366, 441444, 469, 471, 511, 514 Wilson, T. L., 647, 651 Woodruff, R. B., 628, 650 Worm, G. H., 213, 216, 569, 576 Worstell, M. R., 600, 604 Wyman, F. P., 511, 513

Yee, S. R., 162, 168 Yun, K. W., 567, 575 Yuspeh, A. R., 271, 272, 292, 296

Zaino, N. A., 245,247, 259-261, 264, 267, 597, 603 Zent, C., 39, 43, 410, 413, 621, 649

Subject Index

Advertising, 618 Air conditioners (s e e Applications) Aircraft component ( s e e Applications) Analysis: analytical, 64 computational, 64 numerical (s e e Computational) simulation, 64 Applications: aircraft components, 596 air conditioners, 594 automobiles, 595 Boeing 747 windshield, 580 gyroscope, 597 munitions, 596 Omega navigation system, 599 refrigerators, 593 Tactical navigation, 598 vehicle transmission, 594 Automobiles (s e e Applications)

BASIC, 488 BAN (s e e Estimator) Boeing 747 Windshield cations)

(s e e

Appli-

Burn-in, 64 optimal, 467 Buyer’s cost, 146 life cycle (s e e Life cycle) short term 184

Cash flow, discounted, 374 Characteristic function, 649 Caveat emptor, 5 Central limit theorem, 525 Claims, invalid, 159 Confidence coefficient, 537 Confidence interval, 477, 537 estimation, 537 Complaint behavior, 616 Consumer: activism, 1 affairs, 622 bureau, 609 decision: legal redress, 624 purchase, 612 dissatisfaction, 614 reaction, 615 education, 621 expectations, 617 723

724

[Consumer] interests, 1 movement, 1 satisfaction, 614 viewpoint (s e e u n d e r Warranty) Consumers (s e e Buyers) Cost: acquisition, 85 administration, 80 disposal, 85 handling, 81 inventory, 81 life cycle (s e e Life-cycle) maintenance, 85 operating, 85 ownership, 85 repair (s e e Repair) set up, 452 transportation, 80 Cramer-Rao bound, 523 Critical event, 619 Cumulative damage, 73

Diagnostic, 432 Design, 416 modifications, 621 optimal, 418 Discount rate, 140 Discounting factor, 374 Dispute resolution (s e e u n d e r Warranty) Distributions: Beta, 303 Beta Stacy, 120,326,336,342,343, 356, 498, 509, 510 bivariate exponential, 331 chi-Square, 541 Erlang, 109,380,381,387,391,393, 485 exponential, 67,108,135, 140, 145, 151,174,177,179,181,182, 187,190,193,194,196,200, 201,204,207,224,226,230,

Subject Index

[Distributions] 235,239,242,249,256,262, 289,371,373,377,385,443, 485,524,527,531,533,558, 560, 561, 563, 565, 709 gamma, 67,110,113, 303, 310, 312, 315,318,319,325,495,679, 691, 709 lognormal, 113, 486, 680, 709 multivariate Pareto, 121, 331 NBU, 69 NBUE, 70 NWU, 70 normal, 478, 486 NWUE, 70 revised Beta normal, 632 shifted exponential, 109 student t, 540 truncated normal, 113, 709 uniform, 303, 309, 314, 320, 485 Weibull, 67,113,114,136,145,151, 157,176,178,181,182,187, 193,194,224,226,231,235, 239,243,373,384,404,485, 496, 528, 532, 553, 559-561, 679, 681, 709 Elementary renewal theorem 112,150 Estimate, point, 518 Estimator: asymptotic variance, 532 asymptotically unbiased, 520 BAN, 525 bias, 520 consistent: mean-squared error, 522 weakly, 522 efficiency, 523 minimum variance, 525 point, 518 unbiased, 519 FASB, 519 Failure: distribution (se e bounds, 68

a lso

Distribution):

Subject Index

725

[Failure] parameter (s e e Parameter) first, 75 interaction, 74 rate, 66 bathtub shape, 67 constant, 66 decreasing, 66 DFRA, 66 increasing, 66 IFRA, 66 subsequent, 81 to inspect, 29 to warn, 29 Fraud, 26 FORTRAN, 483 Fractile, 538

Lay judge, 634 Legislative process, 636 Liability: economic impact, 30 strict, 29 Life-cycle cost, 84 buyer, 148 discounted, 153 Life-cycle profits, seller’s, 154 Likelihood equation, 529 Likelihood function, 529 Loaner, 139 Logistics, 368 Lot size, 451 optimal, 458 Lotus 1-2-3, 511

Goodness of fit, 518 GPSS, 482 Guarantee (s e e u n d e r Warranty) Gyroscope (s e e Applications)

Magnuson-Moss Act, 6 Maintenance effort, 638 Manufacture, 416 Manufacturers (s e e Suppliers) Market: equilibrium: models, 646 Nash, 636 competitive, 636 structure, 635 Markov chain, 74 Matching principle, 619 Maximum likelihood: estimator (s e e Estimators) method, 529 properties, 530 Mean time: between failures, 61, 269 guarantee, 269 to failure: estimated, 260 guaranteed, 259 to repair, 270 Merchantability, 7 Method of moments, 526 Microcomputer, 487

Hammurabi, 5 Hazard plots, 585

Incentives, 368 Infant mortality, 418 Intensity function, 97 Inventory, 408 Investment, 372 return, 372

Judicial process, 636

Lagrangian multiplier, 419 Laplace transform, 102

726

Minimal repair (s e e Repair) MINITAB, 487 Misrepresentation, 30 Model: adequate, 63 mathematical, 39, 45 simulation, 479 validation, 62 verification, 481 Modelling: Black-Box, 65 justification, 74 first failures: one-dimensional, 65 two-dimensional, 75 item sales: aggregate, 141 first purchase, 82 repeat purchase, 83 intermittent usage, 74 physically based, 65 rectification, 77 subsequent failures, 81 Modifications: ( s e e Design) Moments, 668 Money back guarantee, 223 Moral hazard, 638 Munitions (s e e Applications) NACAA, 2 Negligence, 29 theory, 618 Nonparametric, 552 Omega Navigation system (se e Applications) Omniscient mechanism, 634 Optimization, 62 Parameter estimation, 62 method of moments, 517 maximum likelihood, 517

Subject Index

Parameter estimates (s e e Estimates) Perspectives (s e e u n d e r Warranty) Point process: counting: one-dimensional, 94 two-dimensional, 95 cumulative, 116 interacting, 117 marked, 116 Poisson, 93 conditional, 99 doubly stochastic, 99 nonstationary, 97 stationary, 96 renewal, 93 one-dimensional, 94, 100, 105, 106 two-dimensional, 95, 117, 118 theory, 64 Price: indifference, 148 buyer, 155 seller, 156 noncompetitive, 609 optimal, 628 Product: choice, 614 design, 64 development, 46, 64 optimal, 413 liability, 28 misuse (s e e Usage) performance, 3 agreement, 272 quality, 24 defective, 432 nondefective, 432 recall, 368 reliability, 24 usage (s e e Usage) Products: commercial, 34 consumer durable, 34 consumer nondurable, 34 defense related, 34

727

Subject Index

[Products] industrial, 34 second hand, 84 Public policy, 637

Quality control: scheme, 434 optimal, 440 Queues, 368

Random number generator, 484 Rectification: actions, 77 duration, 79 Redundancy: active, 429 passive, 429 Refrigerators (s e e Applications) Regression analysis, 594 Reliability: allocation, 419 choice, 419 growth, 273 theory, 64 Renewal: density function, 103 function, 103 simulation, 492-494 estimators, 550, 551 integral equation, 93 analytical solution, 107 approximate solutions, 110 bounds, 107 numerical, 113 simulation, 107 two-dimensional, 120 process: one-dimensional, 93 excess life, 103 two-dimensional, 93 type equation, 115

Repair: bad as old (s e e Minimal repair) contract, 27 cost, 80 cost rate, 401 demand, 382 facility, 406 good as new, 77 imperfect, 78 limit policy, 368 minimal, 78 strategies, 138 Repairman, 368 problem, 407 Return rate, 383 Risk factor, 277

Sales: continuous, 375 rate, 375 lot, 369 Sampling distribution, 518 Seller’s costs, 133 discounted, 178 short term, 195 Seller’s profits, 192 Service: contract (s e e Contract) rate, 80 Servicing: agency, 368 third party, 410 Shock damage, 72 SIMSCRIPT, 483, 511, 513 SIMULA, 511, 513 Simulation: approach, 479 discrete, 479 event-oriented, 480 languages, 481 model (s e e Model) process oriented, 480 terminology, 479

728

SLAM II, 482 Social welfare, 635 Spares, 285 demand, 379 inventory (s e e Inventory) Stochastic process: one-dimensional, 94 two-dimensional, 94 Stopping time, 115 Systems approach, 30 System characterization, 30

Taxonomy (se e u n d e r Warranty) Technical expert, 634 Testing: accelerated, 446 presale, 416 Turnaround time, 273

UCC, 6 U-statistics, 551 Usage pattern, 74 intermittent, 158 point, 159 Utility theory, 626

Validation (s e e Model) Vehicle transmission (s e e Applications) Verification (s e e model)

Wald’s equation, 115 Wald’s Theorem, 135 Warranty accounting, 632 breach, 4

Subject Index

[Warranty] claim, 4 combination, 47 Combined (s e e Combination) comparison, 208 complex policies, 644 consumer, 45 as contractual obligation, 2 commercial, 45 costs, 36 cumulative, 58 Combination, 60 FRW, 59 PRW, 59 dispute resolution, 622 execution function, 160 extended, 608 express, 6 fleet, 259 free replacement, 9 full, 131 as guarantee, 2 historical, 5 implied: fitness, 7 merchantability, 7 as incentive, 639 as insurance, 27 integrity, 621 legislation, 609 as liability, 2 limited, 8 lump-sum, 131 lump-sum rebate, 131 maintenance contract, 47 as marketing device, 639 multicomponent, 645 multiperiod, 227 nonrenewing, 46 as normal contract, 6 one-dimensional, 47 Combination, 49 FRW, 48 PRW, 49 pro-rata, 9

Subject Index

[Warranty] PRW, 169 rebate, 169 refund, 369 reliability improvement, 60 renewing, 46 role, 3 protectional, 618 promotional, 618 reserves, 368 reserving, 367 second hand products, 646 as service contract, 2 servicing, 367 as signal, 639 simple, 47 standard, 47

729

[Warranty] storage limitation, 51 study, 36 taxonomy, 45 theories: exploitative, 25 investment, 27 signal, 26 two-dimensional, 47 combination, 57 FRW, 54 PRW, 57 viewpoint: consumer, 39 manufacturer, 39 public policy, 607 unlimited, 132

Policy Index

Nonrenewing Simple [1-Dimensional]

Policy 1: 48, 133 Policy 2: 48, 172 Policy 2a: 48, 172 Policy 2b: 49, 172 Policy 2c: 49, 172 Policy 3: 49 Policy 3a: 49 Policy 3b: 50, 222 Policy 3c: 50, 222 Policy 3d: 51, 225

Policy 10: 56, 315, 323, 341, 346 Policy 11: 56, 319, 324, 342, 349 Policy 12: 57, 350 Policy 12a: 57 Policy 12b: 57 Policy 12c: 351, 355 Policy 12d: 351, 357 Policy 12e: 352, 360 Policy 13: 57

Cumulative W arranty with Storage Lim itation

Policy 4: 51

Renewing Simple [1-Dimensional]

Policy 5: 51, 144 Policy 5a: 394 Policy 6: 52, 184 Policy 7: 52, 237 Policy 7a: 53, 228

Simple [2-Dimensional]

Policy 8: 54, 306, 322, 326, 345 Policy 9: 54, 313, 323, 335, 346

Policy Policy Policy Policy Policy

14: 15: 16: 17: 18:

59, 248 59 59, 255 60 60

Reliability Improvement

Policy 19: 61 Policy 20: 62, 284 Policy 20a: 62, 288

731