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A quantitative model for the analysis of warranty policies

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A quantitative model for the analysis of warranty policies

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Content A QUANTITATIVE MODEL FOR THE ANALYSIS OF WARRANTY POLICIES by Vickie Lee Hill A Dissertation Presented to the FACULTY OF THE GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (Business Administration) December 1983 Copyright Vickie Lee Hill 1983 UMI Number: DP22591 All rights reserved INFORMATION TO ALL USERS T he quality of this reproduction is dependent upon the quality of the copy submitted. In the unlikely event that the author did not send a com plete manuscript and there are missing pages, th e se will be noted. Also, if material had to be removed, a note will indicate the deletion. Dissertation Publishing UMI DP22591 Published by ProQ uest LLC (2014). Copyright in the Dissertation held by the Author. Microform Edition © ProQ uest LLC. All rights reserved. This work is protected against unauthorized copying under Title 17, United S tates Code ProQ uest LLC. 789 E ast Eisenhower Parkway P.O. Box 1346 Ann Arbor, Ml 4 8 1 0 6 -1 3 4 6 UNIVERSITY OF SOUTHERN CALIFORNIA THE GRADUATE SCHOOL UNIVERSITY PARK LOS ANGELES, CALIFORNIA 90089 This dissertation, w ritten by VICKIE LEE BILL under the direction of hEi. Dissertation Committee, and approved b y all its members, has been presented to and accepted b y The Graduate School, in partial fulfillm ent of re quirem ents for the degree of D O C TO R OF PH ILO SOPH Y DISSERTATION COMMITTEE i Chairperson ACKNOWLEDGEMENTS I would like to thank all the people whose help and understanding made possible the completion of this research, in particular, my dissertation committee chairman, Dr. Wallace R. Blischke, my parents, L. H. and Grace Schoonover, my husband, Joseph E. Hill, and my invaluable friend, Dr. Charles W. Beall. Also, I wish to express my appreciation to the other members of my committee, Professors J. S. Ford and C. J. Ancker, Jr. Special thanks are due as well to Dr. C. J. Pounders for all his help during my entire graduate career. TABLE OF CONTENTS Page ACKNOWLEDGMENTS................................................... ii LIST OF TABLES.................................................... v LIST OF FIGURES................................................... vi Chapter I. STATEMENT OF THE PROBLEM.................. 1 Background Warranty Models The Problem Scope and Delimitations of the Research II. REVIEW OF THE LITERATURE........................ 19 Legal and Economic Theory Accounting Marketing Behavioral Reliability Improvement Warranties Surveys and Unclassified Research Models III. DEVELOPMENT OF Step One Step Step Step Step Two Three Four F i ve THE THEORETICAL MODEL, 56 IV. DEVELOPMENT OF THE SIMULATION MODEL.. The Simulation Model Comparison of the Simulation Model to the Theoretical Models Use of the Simulation Model for Sensitivity Analysis 93 V. THE SENSITIVITY ANALYSIS: Research Design METHODOLOGY........ 104 iii Chapter Page VI. THE SENSITIVITY ANALYSIS: RESULTS............. 136 Investigation of Distributional Assumptions Investigation of Parameter Impact Secondary Investigations VII. CONCLUSIONS AND COMMENTS......................... 185 Major Conclusions Suggested Topics for Further Research BIBLIOGRAPHY........ 197 APPENDIXES.......................................................... 206 A. Detailed Flowchart For The Simulation Model.... 206 B. Example Run of The WRNTY Model...................... 210 C. Listing of WRNTY Program.............................. 213 D. Data for Sensitivity Analysis....................... 245 E. Example Regression Results........................... 272 iv LIST OF TABLES Table Page 1. Major Characteristics of Existing Warranty Valuation Models ........................ 59 2. General Notation............................................ 65 3. Existing Warranty Valuation Models Restated in General Notation........................... 67 4. Parameter Values to Reduce General Model 3.2 to Specific Single Failure Models....... 74 5. Closed Form Statement of Model 3.2 When Lifetime is Exponentially Distributed............... 76 6. Comparison of True and Simulated Mean Costs....... 102 7. Example MLE Worksheet..................................... 117 8. Example WRNTY Worksheet.................................. 121 9. Example PRMTR Worksheet.................................. 130 10. Example Regression Run................................... 134 11. Condensed Results of Sensitivity Analysis Of Distributional Assumptions................. 141 12. Over- and Underestimations Under Different Distributional Assumptions.......... 148 13. Results of Investigation of Relationship Between Warranty Cost and Lifetime Distribution Parameters................................. 153 14. Summary of Regression Results................ 164 15. Comparison of Warranty Costs at Different Levels of the Lifecycle as a Function of Warranty Length............................................ 170 v Table Page 16. Comparison of Warranty Costs at Different Levels of the Warranty Length as a Function of E(X)........................................... 178 vi LIST OF FIGURES Figure Page 1. Flowchart of the Warranty Process.................. 95 2. Non-lifetime Distribution Input Parameters......... 112 3. Input Parameters for Distributional Assumptions................................................ 123 4. Input Parameters for the Truncated Normal.......... 126 5. Failure Rate Curves For Selected Distributions With Means Equal to 1.0............... 143 vii CHAPTER I- STATEMENT OF THE PROBLEM Background In common usage, the terms warranty and guarantee are often used as synonyms. The legal distinction lies in whether a third party insurer is involved in the indemnifying relationship. A guarantee has been defined (Cochran's Law Lexicon, 1956) as, "a promise to a person to be answerable for the payment of a debt, or the performance of a duty by another, in case he should fail to perform his engagement." (emphasis added) A warranty, Cochran says, is a specific type of guarantee, "a guarantee concerning goods or land, given to a purchaser by the vendor." Since the connotation of each of these words is the compensation for damages incurred, which compensation involves some cost/benefit to the parties involved, the more specific term "warranty" will be used throughout to apply to any agreement in which an indemnifying relationship is established. For purposes of this research, a service contract shall be distinguished from a warranty by the fact that acquisition of a service contract is voluntary while a 1 warranty is attached to the product and must be included as part of the acquisition. Some military procurements specify separate pricing of warranties, but this is not the distinguishing feature of concern. It is the requirement of acquisition of both warranty and product that determines the population of potential warranty claims. It has been suggested (Gerner & Bryant, 1980) that those individuals choosing to acquire service contracts represent a special segment of the population of purchasers of a product and would thus show a different pattern of claim behavior. This research will confine itself to the analysis of warranties, as defined above, and as distinct from service contracts. Warranties exist for two primary purposes: (1) to limit liability for product or service failure, and (2) to enable the supplier to compete in the market place. The legal literature is replete with articles tracing the history and trends of product liability from its beginnings with caveat emptor (buyer beware) to the current trend toward strict liability (seller beware). Along the way the Uniform Commercial Code and the Magnuson-Moss Federal Trade Commission Improvement Act of 1975 established some rules pertaining to warranty provisions and presentation. The following references should provide an introduction to this area: Business Week, 1975; Comsumer Reports, 1975; Dean, 1977; Deutsch, 2 1980; and McKean, 1970. As liability shifted to the supplier he sought to protect himself by limiting his liability. One means of doing this was the express (written) warranty. Common warranty provisions shift some of the burden for product maintenance and proper usage onto the consumer. In return the supplier assumes some part of the cost of product repair or replacement in the event of failure. Thus, a warranty'becomes an instrument for portioning out risk and liability between consumer and supplier. With the control of liability, the supplier also achieves some measure of control over its future costs (Anderson, 1973). The second, and equally important, reason for warranties is their value as a marketing tool. Competition in the market place has been found to have the effect of maintaining some similarity in warranty provisions for competing products (Bryant & Gerner, 1978). Warranties are also seen as a signaling device which can be used to influence consumers’ perceptions of the quality and reliability of a product (Fisk, 1970). Extensive research has been, and is being, done on the behaviorial aspects of warranty policy. The behavior of consumers with respect to warranty claims during the warranty period is one of the areas of recent interest (Gerner & Bryant, 1980). 3 Warranty Structures Warranties are either implied or written. Implied warranties are rights, created by state law, which come automatically with every product. A supplier may avoid being bound by implied warranties by stating in writing that no warranty at all is given. However, if a supplier gives a written warranty, the implied warranties are necessarily included. The most common of these implied warranties is "merchantability,” meaning that a product is fit for the ordinary uses of that product, i.e., a toaster must toast or the consumer has the legal right to redress. This has been upheld in most state courts under the Uniform Commercial Code (Fisk, 1973). Written warranties may be either "full" or "limited.” These words were used in the Magnuson-Moss Warranty Act of 1975. A pamphlet from the Federal Trade Commission explaining this act gives the following explanation of the terms "full” and "limited" in the context of written warranties. Full Warranty: The label "FULL” on a warranty means all this: A defective product will be fixed (or replaced) including removal and rein stallation if necessary. It will be fixed within a reasonable time after you complain. You will not have to do anything un reasonable to get warranty service. (Such as ship a piano to the factory.) The warranty is good for anyone who owns the product during the warranty period. If the product can’t be fixed (or hasn’t been after a reasonable number of tries), you get your choice of a new one or your money back. A FULL warranty doesn't have to cover the whole product. It may cover only part of the product, like the picture tube of a TV. Or it may leave out some parts, like tires on a car. Limited Warranty. A warranty is "LIMITED" if it gives you anything less than what a full warranty gives. For example a LIMITED warranty may: Cover only parts, not labor. Allow only a pro-rata refund or credit. (You get a smaller refund or credit the longer you had the product.) Require you to return a heavy product to the store for service. Cover only the first purchaser. Charge for handling. The concern of this research is with written warranties and their specific provisions rather than with the very general protection allowed by implied warranties. Blischke and Scheuer (1979) divide written warranties into four distinct groups based on certain aspects of their provisions. This classification scheme makes use of certain major provisions identifiable in virtually all warranties, while recognizing that individual differences exist in the fine print. 5 Free-replacement warranty. The consumer incurs a one-time fixed cost to secure the benefits of a product or service for some specified period of time. The supplier assumes the responsibility for any repair or replacement necessary during the period of the warranty. Expiration of the warranty shifts the burden for repair/replacement costs onto the consumer. Typical of this type of warranty are new car warranties and many appliance warranties (Blisckhe & Scheuer, 1979). This is equivalent to the consumer receiving a total lump sum rebate. This will be referred to from now on as the lump sum rebate to distinguish it from the pro rata form of rebate. Pro rata warranty. The pro rata warranty . differs from the free-replacement warranty in that replacements are not free but may be obtained at a discount based on the amount of the warranty period which remains unexpired. The consumer receives a credit toward the replacement item and the replacement item carries its own warranty so that a new warranty period is begun with its purchase. This reduces to the free-replacement warranty in the case where a 100% credit is given and the new item does not carry its own warranty, but assumes the unexpired portion of the original warranty. Theoretically, the terms of the pro r‘ \ ration could be either linear or nonlinear, although virtually all warranties of this type now specify linear pro ration (Blischke & Scheuer, 1979). This type of 6 warranty is generally offered on nonrepairable items such as automobile tires and batteries. Combination free-replacement, pro rata warranty. The usual combination is a free replacement policy until some specified time when coverage changes to pro rata for the duration of the warranty period. Blischke and Scheuer (1979) contend that if analysis of the pro rata warranty is sufficiently general, ”. . . (including nonlinear as well as linear proration), the combination warranty will be dealt with as well." (p. 5) Consumer and commercial warranties differ primarily in terms of duration and the precision with which provisions are spelled-out. However, most warranties, both consumer and commercial, fall into one of the preceding categories. The final category has been used in commercial applications (it was originally developed by the airlines) but it is most commonly seen in military acquisitions. It is not included in the general model due to its distinctly different structure and the fact that it is virtually never seen in consumer warranties. Reliability improvement warranty. The reliability improvement warranty (RIW) is characterized by a fixed price contract including both support (repair/replacement of failed items) and improvement of the reliability of a system during a specified time period 7 (Schmoldas, 1977). The RIW supercedes the failure-free warranty (FFW) which was basically a maintenance contract for a certain period of time after equipment delivery. The FFW provided financial incentives for improved reliability by reducing returns for repair (Schmidt, 1977). The RIW provides contractural incentives to improve reliability through requirements that product improvement modifications be undertaken by the supplier at his cost for the duration of the warranty. The financial incentives are still present, encouraging suppliers to design and build high reliability into their products since support costs incurred by the supplier under an RIW will have a direct effect upon his profit. One way of measuring improvements made under an RIW is to include a Mean-Time-Before-Failure (MTBF) guarantee clause. This feature specifies that an operational MTBF is to be achieved and maintained. It also requires the supplier to perform engineering analysis of failed items and to make design changes required to achieve specified MTBF rates. These changes are to be made at the supplier's cost. The MTBF clause is perceived by some to put the supplier in "double jeopardy." This opinion is not held by all experts on RIW (Newman & Nesbitt, 1978). An RIW may also include a guaranteed turnaround 8 time (TAT) which provides for either consignment spares or a liquidated damages payment (Newman & Nesbitt, 1978). One of the questions arising in the application of RIWs is duration of the warranty. Concern centers around duration of the initial RIW period, terms of any extension of this period, and any special modifications needed in the RIW to maintain the supplier’s motivation to improve product reliability toward the end of the warranty period when it may begin to slacken (Springer, 1977). The RIW is a relatively new type of warranty. The government is exploring its possibilities and pitfalls through the application of RIWs to some of its major acquisitions. RIW programs vary considerably in many respects, and the sharing of risk between the supplier and the consumer depends on specific contract requirements. Originally developed by the airlines to improve the reliability of avionics components of commercial aircraft, RIWs are used mainly in the marketing of high technology electronic industrial products (Springer, 1977). Warranty Models Analysis of warranties has been approached from various directions. The section on models in Chapter II provides a discussion of a variety of the models proposed. Some of these models contributed to the current research through the. suggestion of additional factors which may 9 bear on the value of a warranty from either the supplier’s or consumer's point of view, while others formed the foundation upon which the new model was built. These models have been developed for various purposes including explanation of behavior, exploration of risk-sharing, determination of optimal warranty policies, and analysis of warranty provisions with respect to valuation measures such as monetary cost or utility. It is this last class of models which is of interest. Determination of the expected cost of a particular warranty is important for a number of reasons, not the least of which is the requirement of the Financial Accounting Standards Board that after-sale costs associated with warranties be accrued by a charge to income in the period of sale (Amato, Anderson & Harvey, 1976). The most important reason for determining the cost of a warranty is the need to price the warranty. By offering a warranty, a-supplier is assuming an obligation to the consumer. There are costs associated with the fulfillment■of this obligation (repair or replacement of items which fail, consequential damages, etc.). Since the supplier expects to make a profit, he must recover his costs. The more nearly determinable warranty costs are, the more accurately suppliers may make decisions about product pricing and product quality through design and manufacturing specifications. 10 Many considerations go into the determination of warranty provisions (length of the warranty, amount of protection, exclusions from coverage), and the cost of the provisions is only one aspect of the decision. The models which provide normative information about optimal amounts and durations of coverage provide a useful input to the decision-making process, but by no means the only input. The decision .maker must also consider competition in the marketplace and the signals his warranty provisions provide to prospective consumers. Finally, having considered all relevant factors, the supplier adopts a warranty policy. How much is it going to cost? From the consumer's perspective, the concept of life cycle costing recognizes that costs of operations, maintenance and repair, and disposal should all be included in an acquisition decision (Lund, 1978). If the warranty on an item reduces repair costs, then it has a value to the consumer. When the consumer perceives a warranty to have value, it follows that an unwarrantied item should be priced lower than a warrantied item. An objective determination of the expected monetary value of a warranty would enable consumers to make more rational acquisition decisions. Warranty value, however, is only one piece of the information necessary to employ a life cycle cost analysis. This research is concerned only with that portion of the model related to valuation of 11 warranties. For a complete discussion of the life cycle cost model see Prohaska (1978). The Problem The problem to be addressed by this research is the lack of a general model flexible enough to allow determination of the value of a wide range of different warranty policies. The major limitations of the existing models are: (a) they are specific to a particular warranty application or to a limited number of warranty provisions, e.g., applicable only to the first failure of a product or structured to accept only pro rata warranty provisions; (b) the assumptions made are too restrictive; and (c) only very limited sensitivity analysis has been done to investigate any model’s response to Changes in its assumptions. A more complete discussion of these limitations follows. The following assumptions are restrictive to a model purporting to represent warranty provisions realistically. While individual models have relaxed one to two of these assumptions, no single model has relaxed more than two of them at a time, and some have never been investigated. The Lifetime Distribution When specified, the lifetime distribution of the item is almost always assumed to be exponential. Only one 12 model uses the gamma distribution (Glickman & Berger, 1976) and one model is demonstrated numerically with data from a normal distribution (Blischke & Scheuer, 1975). The exponential, while it is a convenient distribution for computation, is by no means the only, or even the best, distribution for modeling lifetimes. Some distributions postulated to. be more representative of actual phenomena are the gamma, Weibull, and truncated normal. The Pro Ration Scheme Pro rations figured as a function of the remaining proportion of the warranty period are assumed to be linear by all authors except Bazovsky (19,68). While in practice all warranties with a pro rata provision specify linear pro ration, a non-linear provision would be possible theoretically, and could be used to model a combination warranty, as noted above. Failure Mode Half of the models assume only single failures. Heschel, Brown, and Glickman and Berger expand this assumption to deal with multiple failures during a product's lifetime. Blischke and Scheuer introduce renewal theory to model these multiple failures. They note that a further extension to deal with multiple modes of failure through sources of failure that are not independent is possible, although it has not been done in the context of warranty analysis. Mamer (1981) also uses 13 renewal theory and, in an earlier paper (1980), suggests a diffusion model to allow for description of degrees of failure. His is the only paper to suggest this in the context of warranty analysis. Repair Cost If failure always results in replacement it is relatively easy to assign a constant cost to the failure of an item. However, if failure results in replacement sometimes and repair sometimes, and if differing degrees of repair are necessary, the cost of a failure is certainly not a constant. It must vary with the degree of the failure. Some of the models treat repair cost as a constant and others recognize it as a variable. A general model should be able to accomodate both eventualities. Validity of Claims Most models ignore claim validation as an issue. Only Amato, Anderson and Harvey (1976) recognize that not all claims may be made, and that those actually made may not all be valid. Some cost must attach to the administration of a warranty policy, and the various processes that lead to settling of a claim must be recognized. For instance, not only may the consumer be in error in submitting a claim, but the supplier may be in error in validating or denying the claim. It should also be recognized that the cost of claim validation is not necessarily a constant. 14 The Time Value of Money Under warranty provisions with long duration, costs may be incurred at some distance in time from the original purchase. The fact that present value theory is applicable to this situation was noted in six of the papers reviewed; Amato and Anderson (1976), Amato, Anderson and Harvey (1976), Brown (1974), Blischke and Scheuer (1975), Mamer (1980), and Patankar and Worm ( 1981). The sensitivity analyses reported are brief numerical examples which deal superficially with costs, usage rates, discount rates, life cycle lengths, and warranty duration (Blischke & Scheuer, 1975; Glickman & Berger, 1976; and Mamer, 1980). Only Mamer (1980) varies the lifetime distribution in his numerical example, and his example is quite brief. No serious investigation has been made into the sensitivity of costs or benefits to assumptions about either the form or the parameters of the lifetime distribution. Certainly these assumptions and estimates need to be explored as well as some of the other assumptions mentioned, taken both singly and in combination. A purely practical problem that makes comparison among models difficult is the diversity of notation used by the various authors. One of the contributions of this research is a common notation in which to express the 15 various proposed models. Scope and Delimitatios of the Research The objective of this research was to develop a general model capable of generating the monetary cost associated with a warranty of specific provisions. It was desired to make the model general enough to analyze any of the three types of consumer warranties; lump sum rebate, pro rata rebate, and combination. In addition, the model should incorporate factors for claims behavior of both the consumer and supplier. The final model should then be able to describe the vast majority of existing warranties through the inclusion or exclusion of various components of its functional statement. A second, equally-important objective was to perform a sensitivity analysis to address the major limitation of existing research, the lack of sensitivity analysis information. Such an analysis should investigate the impact on warranty costs of changes in the assumptions and provisions of the warranty model. Of particular interest was the lifetime distribution assumption. Assuming ,an exponential lifetime distribution greatly simplifies calculations, but is there a significant loss of accuracy in predicting warranty costs when this is not the appropriate distribution to be used? As another example, often the MTBF of a product’s life distribution is difficult, if not impossible, to determine. For this 16 reason, the sensitivity analysis includes the variation of this parameter to determine the impact of estimation errors on predicted warranty cost. Excluded Areas of Research The RIWs are not included in the general model developed by this research. They are virtually never seen in consumer usage, and their provisions and objectives are quite different. Also, there is no discussion of the estimation of the parameters to be used in the model. There is a vast body of literature on the development of estimations of MTBF and the form of life distributions, some of which was noted in-the above sections of this chapter. The Final Result There are three end products of this research; (1) a statement of the' model which specifies warranty cost to be a function of a number of factors including constant parameters and random variables; (2) a computer simulation model which yields a simulated warranty cost for a specified set of provisions and assumptions; and (3) a set of simulation results from which conclusions about the sensitivity of the model to distributional assumptions and parameter estimates can be drawn. Chapters III, IV and V detail the steps toward achieving these goals. Chapter III addresses the theoretical development of the general functional 17 statement of the model, Chapter IV explains the simulation model, and Chapter V details the research design for the sensitivity analysis. Chapter VI shows the results of the sensitivity analysis, and Chapter VII supplies comments upon these results and areas for further research. 18 CHAPTER II REVIEW OF THE LITERATURE With certain exceptions, which shall be noted, the literature on the subject of warranties falls into six general categories. These will be treated individually with special emphasis on the category which encompasses actual modeling of warranties as functions of stochastic processes. The other areas, while, in most cases, not actually proposing models of warranty provisions contribute to the model developed by this research through the suggestion of components which should be included, or through provision of justification for the pursuit of the stated goal. They provide the background and context in which the model, as a tool, must be viewed. This does not purport to be a complete survey of all of the literature on every aspect of warranties. It is intended, rather, to provide a source of introduction to those areas where research is being conducted which includes consideration of warranty provisions. An annotated bibliography of some of the literature of warranty analysis is presented by Blischke 19 and Scheuer (1977). Legal and Economic Theory Dean (1977) provides an outline of the evolution of products liability from its beginnings with the concept of caveat emptor (buyer beware) which prevailed until the mid 1950's when the Uniform Commercial Code was adopted (which created implied warranties and limited the effects of disclaimers) through the evolution in the courts to "strict liability." A comprehensive review of the legal history and economic aspects of products liability was written by McKean (1970) for the Joint Committee of the American Association of Law'Schools and the American Economic Association. His article reviews the literature on products liability and the economic implications of products liabilty. His research was commented upon by two economists (Buchanan, 1970; Dorfman, 1970) and three law professors (Calabresi & Bass, 1970; Gilmore, 1970) at the request of the Joint Committee. McKean's response to his commentators is available in the Edited Transcript of the Joint Conference on Products Liability (1970). The enactment of the Magnuson-Moss Warranty-Federal Trade Commission Improvement Act in 1975 changed the face of warranty law. A brief discussion of the events leading up to the Act and a delineation of the actual provisons of the Act appeared in Consumer Reports 20 (1975). The early months of the Act were confusing for manufacturers because implementing regulations were not available to spell out the intention of the Act. Articles on this subject appeared in Business Week (1975) and the Los Angeles Times (1975). An interesting critique of the strict liability concept currenly prevalent appeared in the 1980 Proceedings of the Annual Reliability and Maintainability Symposium. The author, Irving Deutsch, points out instances of accidents caused by overprotection, and concludes that a shift of some, though not all, of the responsibility back to the consumer is advisable. He maintains that such a shift would reduce accidents, cause the cost of goods and services to drop, and make the consumer more selective in purchasing. Accounting Another area subject to strict regulation is that of accounting. Such regulation with respect to warranties appeared in the form of "Statement of Financial Accounting Standards No. 5: Accounting for Contingencies" which was released and became effective for fiscal years beginning on or after July 1, 1975. The Statement requires that the after-sale costs associated with warranties be accrued by a charge to income in the period of sale when it is probable that a liability has been incurred and the amount of future costs is reasonably estimable. Menke (1969) 21 whose paper will be discussed in a later section appears to have been the first to propose an actual model for determining warranty reserves, but the topic of warranty reserves had been addressed by Heck in 1963, at which time he noted that the existence of warranties had been almost completely ignored in accounting literature up to that time. Amato and Anderson (1976) proposed an extension to Menke’s model which involved acknowledgement of the time value of money. Amato, Anderson and Harvey (1976) propose their own model, which will be discussed in a later section, and develop the "theoretical bases for warranty cost estimation;” A further d-isussion of the purely theoretical cnsiderations of warranty cost recognition is undertaken by Moellenbendt (1977). Marketing As early as I960, the Federal Trade Commission (FTC) was concerned about the marketing of warranties. It issued Guides Against Deceptive Advertising of Guarantees. Eight years later, in 1968, it was announced that a task force was to be formed to investigate product warranties and service repairs of major home appliances. As part of this effort, the FTC was to oversee a study of warranties. A 1968 article in Sales Management notes the reactions of certain appliance manufacturers to this probe. The general reaction appeared to be to broaden warranties and publicize them. 22 Fisk (1970) notes the growing dissatisfaction of consumers with warranties offered by manufacturers and distributors and suggests that firms use warranties as a marketing tool. His article proposes certain guidelines for establishing -warranty policies that could be a positive force in the marketing strategy developed by a company. Another, later, article (Kendall & Russ, 1975) also supplied guidelines for formulating a warranty policy which could enhance the marketing effort. This article is based, in part, on a survey of manufacturers of consumer packaged goods. An interesting comment at the close of the article suggests that the warranty is not only an attribute of the product which can add to its marketability, but may also provide data (through claims experience) which, ". . . can supplement marketing research data as an aid in executive decisions, and the cost (of retaining claims data) is low compared to most marketing research studies." p.43 Behavioral Several studies have been conducted to investigate how consumers perceive warranties. A 1976 study (Wilkes & Wilcox) was conducted to determine whom the consumer holds responsible for product performance and warranty support and to identify specific consumer concerns regarding warranties. The conclusions were that retailers were expected to be responsible for product problems and that 23 consumers have generally negative feelings about product quality and complexity. Warranties were criticized for technical complexity and deceptive wording. Suppliers were faulted for not calling warranties to the consumer's attention. The literature in the area centers on consumer perceptions of warranties and on the behavior of consumers with respect to use of products and demand for repairs. It is generally accepted that the warranty acts as a signaling device to indicate product quality and reliability to the consumer. Courville and Hausman (1979) incorporate consumer perceived reliability as part of their model for expected utility. This, and the following reference, will be discussed in a later section. Gerner and Bryant (1980) model the consumer’s behavior during the warranty period. A 1979 article in the Journal of Marketing Research (Darden & Rao) explored an attitudinal question to see if consumers’ attitudes toward warranties would influence their purchasing behavior. The conclusions were directed at applications of marketing techniques, but suggested that consumers’ attitudes toward warranties were not a significant characteristic upon which to segment a market. 24 Reliability Improvement War ranties The literature on Reliability Improvement Warranties (RIWs) falls into three general categories; discussions of the theory and evolution of RIWs, descriptions of individual procurements with RIW provisions, and presentation of economic models of risks and costs associated with RIWs. Much of the research on RIWs comes from ARINC research Corporation, and the majority of the applications are military procurements. In the first category, a good review of the history of RIWs is presented by Schmidt (1977). He tracks the evolution of the RIW and its implementation by the Department of Defense. Springer (1977) discusses the aspect of duration decisions for RIWs, giving guidelines for what the purchaser must consider in establishing warranty duration. A general discussion of the early experiences with the then "failure free warranty" can be found in the Proceedings of the Failure Free Warranty Seminar held in 1973. Markowitz (1971) introduced the failure free warranty concept and contrasted it with the "normal method of commercial overhaul." His paper is quite thorough and includes flow charts of the overhaul processes being contrasted. He also proposes three dimensions for failure classification to serve the needs of relating life cycle costs to failures in service. The designer/supplier perspective on the failure free warranty 25 is presented by Harty (1971), while a clear exposition of the fundamentals of the FFW/RIW from the buyer’s point of view is available in Markowitz's paper (1975). As RIWs became more common, authors began to investigate their impact and alternative forms. Springer (1977) considered cost sharing incentive pricing as an alternative to the firm fixed price method in current use as a way of sharing the risk between buyer and supplier. His conclusions indicate that, ". . . traditional cost sharing with shallow share ratios may tend to dilute the contractor’s motivation to improve product reliability and control costs." p. 391 He points out that a contractor who is relatively confident of the reliability of his product would be indifferent between the two pricing methods, while a contractor who is not confident of pre-production reliability estimates will find neither option desirable, since the contractor's cost increases exponentially at very low MTBF. Bonner (1977) raised the question of an RIW's impact on a contractor’s risk under then-current product liability law. He concludes that, since liability is absolute (in a practical sense) anyway, no aspect of a warranty can increase the contractor's liability. The liability in question is that for injury caused by a defect in the supplier’s product. Day and McIntyre (1978) address the requirements 26 for data included in RIW programs. They suggest a reporting system for RIW data that covers collection, analysis and summarization of the available information. By 1978 the Air Force had five major applications of RIW in force. Newman and Nesbitt (1978) summarize the status of these five applications, identify some results achieved with the use of RIWs, and express some concerns about the concept. Their analysis includes putting a dollar value on the commitment to RIWs by the Air Force. At that time the Air Force had spent $65 million on its RIW program. Many papers detailing the provisions of specific RIW procurements are avaiable. Among these papers are the following; Feder and Kowalski (1975) detail the RIW terms and conditions for the U. S. Army’s CONUS NAV VOR/ILS Radios procurement, Balaban and Nohmer (1975) present the case history of the development of the warranty provisions for the Air Force’s ARN-XXX Tactical Navigation (TACAN) set procurement. In 1967 the first Department of Defense FFW/RIW contract was entered into by the U.S. Navy Aviation- Supply Office (ASO) (Harty, 1971; Markowitz, 1975). It was followed in 1972 by an ASO contract with Abex Corporation (Markowitz, 1976). The Army’s Lightweight Doppler Navigation System procurement solicitation RIW terms are described in a paper by Kowalski and White (1977). The F-16 RIW program was 27 considered to be the most complex warranty procurement to date in 1979. A discussion of this program by Balaban, Cuppett and Harrison (1979) incudes the objectives and provisions of the contract as well as the major issues which had not yet been fully resolved. Bayer and Speir (1978) provide a comparison of Douglas Aircraft Company’s commercial aircraft warranty coverage as it has evolved over the years. They explain warranty terms in some detail for the DC-8, DC-9, and DC-10. This is one of the few non-military applications of RIW. The third area of RIW literature, modeling of the economic aspects of RIWs, will be covered in the section on Models. Surveys and Unclassified Research As noted in the introduction to this literature review, some references do not fit into the six established categories. These have been grouped in the present section. The Center for Policy Alternatives at the Massachusetts Institute of Technology published a four-volUme research report titled Consumer Durables: Warranties, Service Contracts and Alternatives (1978). This very comprehensive report covers policy alternatives (Vol. I), the description, pricing and legal aspects of warranties and service contracts (Vol. II), surveys of 28 consumer experiences with warranties and service contracts (Vol. Ill), and modeling of life cycle costs, warranties, and multiple failures of appliances (Vol. IV). Although the report restricts itself to specific appliances such as televisions and refrigerators, the theories developed are central to any study of warranty models. In addition to this report, the Center for Policy Alternatives published proceedings of a conference on Research for Consumer Policy held in March of 1978. One of the conference presentations (Bryant & Gerner) summarizes the work then being done in economic research on warranties and service contracts at the Center. The vast majority of research, at this time, was being conducted under the auspices of the Center. Blischke and Scheuer (1979) have compiled a comprehensive survey of the literature on warranty models published through 1978. Their presentation includes both consumer and military type warranties, and their bibliography covers the history, some legal aspects, and policy considerations associated with warranty usage as well as the aforementioned models. This survey is the only collection of economic warranty models by the various authors who have published on the subject. Blischke and Scheuer suggest a notation for the expression of such economic models, and proceed to restate various published models in the common notation. This technique makes 29 possible the comparison of various models. This paper forms the inspiration for the current research, since it theorizes that a very general model for warranty valuation could be developed. Fisk (1973) states that most warranties are components in a complex interacting system. He applies a systems analysis approach to illustrate the dynamics of the warranty system. He uses automobiles in his example, but notes that the system for appliances differs little from that for automobiles. Fisk uses a matrix representaion to describe the stimulus and response interactions between consumers, dealers, manufacturers, and the government. This approach to the problem is unique. Models The mathematical models of warranty policies and behaviors come from all of the categories mentioned above. These models can be generally classified into three groups. Those that deal with reliability improvement warranties, those that deal with the economic analysis of the other three types of warranties, and those that deal with some aspect of the warranty process other than the estimation of the cost or value of a warranty. The second group is of primary interest for this research, although the last group provides some interesting insights into the working of the warranty process as a dynamic system. The 30 models which have been published in the three areas will be reviewed in generally chronological order. Models of Reliability Improvement Warranties An early paper in this area (Bazovsky, 1968) develops a scheme for appraising MTBF warranties by determining their effect on the overall system life cycle costs. He considers both linear and non-linear remuneration functions in determination of costs. Markowitz (1971) provides an assessment of the state of the art in life cycle costing which references many reports which develop models representing mathematical relationships between costs and the environments in which procurements exist. He uses actual military procurement data to illustrate experiences with existing life cycle costing contracts. A warranty life cycle cost model was developed by Balaban and Retterer (1973) to assist in determining whether a warranty is economically attractive and, if it is, the best warranty period. The authors introduce a generic model for life cycle cost. Life cycle costs over (0,T) = (number of units bought x price per unit) + (expected number of failures over (0,T) x average cost per failure) + (maintenance support costs over (0,T)). The generic model is extended to the case of a no-warranty procurement, giving consideration to reliability modification, initial and recurring support cost and 31 amortization. The result is (the "o" superscript is used to denote a no warranty procurement) LCC°t = N°C°pA W W . r° i r0 + L MOD + ° DMU + C° A + ISU T„ • W + G RSUTW (2-1) where LCC° = life cycle costs over T W (0,T^) for a no-warranty procurement Ttt = calendar time in months N° = number of units purchased Cp = purchase price per unit At = amortization factor for (0,T ) W = T^/expected equipment life d°MOD = expected amortized costs of reliability modification C°~wIT = direct user maintenance costs DMU G°ISU =: support costs G°RSU = mont^ly recurring support costs A further extension to the case of life cycle cost with warranty yields the following model (the "1" superscript is used to denote a procurement with warranty) 32 LCC1™, = N1C1pAT W W + MOD+C DMC^ R(TW )(1+P) + C DMU + C ISU AT w + C RSUTW (2.2) where C^dj^c ~ contractor direct warranty repair costs R(T^) = risk factor contractor applies to costs for a warranty period of T /12 months = (1+r) W where 0<r<l P = contractor fee = contrator costs for modif ication, MOD discounted and amortized All other symbols represent the same factors as for the no-warranty case except that the numerical values will generally be different. Balaban and Retterer conclude (1974) that a properly constituted and applied warranty can yield significant reliability and life cycle cost benefits, and they advise broader use of warranties in military avionics procurement. They state that further development of the 33 model presented in both their 1973 and 1974 research is required, but that the life cycle cost approach for assessing the economic value of a warranty is necessary and possible. Gates, Bortz, and Bicknell (1976) developed three models which view RIWs from the standpoint of profit to the supplier. Their ’’simplified" profit model for the basic RIW is, P = W - Cf -[(QTUtw )/(MTBFa)] Cr - I (MTBF ) - D (2.3) 3 t where P = profit (loss) under the RIW W = fixed price paid to the contractor for the warranty cf = fixed costs to the contractor associated with the warranty q t = total number of systems to be delivered u = usage rate in operating time per calendar time fcw = duration of warranty period MTBF a = achieved MTBF (average over the RIW period) C r = cost to the contractor per unit repair I (MTBF )= cost of improvement actions to achieve MTBF 34 'D = damages for not meeting the turnaround time requirement Their model for an RIW with spares shipment guarantee is, P = W - Cf - (QTUtw )/(MTBFa) (C ) - I(MTBF ) - D (2.4) x a. o where, in addition to previous definitions, Dg = damages for stockouts at the bonded storage over the warranty period Finally, they propose a model for an RIW with MTBF guarantee, P - W - Cf - [(QTUtw)/(MTBFa)] (Cr ) - I(MTBFa) - CgSc - Dt (2.5) where Cg = cost per consignment spare Sq = number of consignment spares required Suggestions for calculation of the various component parts of the models are made in the paper. The authors also discuss reliability growth under RIWs and the probable responses of a profit-motivated contractor to alternative forms of the RIW. They conclude that as certain RIW provisions shift the risk to the contractor, the price of the warranty will rise, with the greatest shift in the case of the guaranteed MTBF option. Markowitz and Giordano (undated report) develop a model for determining cost escalation adjustments within fixed price RIW/FFW contracts. They state that the basic 35 premise of adjustment is to maintain the risk exposure for both parties to the contract as a constant agreed upon in the original contract. This is accomplished by defining the contractor’s future economic uncertainty to specified limits. This escalation procedure is offered as an alternative to current fixed rate/non-composite index inflation clauses. While this model does not propose to provide a cost or assign a value to a warranty, and is thus not central to the current research, it does give insight into the costs and risks involved in RIWs. of risk evaluation in RIWs by looking at contractor profit as the major financial variable. The model they use is a simplified one which considers only those factors significant with respect to pricing risks in the authors’ opinions. The model is Balaban and Retterer (1977) address this question WC FC + A x TOH x CR (2.6) where WC contractor warranty cost over the warranty period FC contractor fixed costs associated with the warranty TOH total operate hours of all warranted equipment over the warranty period average failure rate of all warranted equipment over the warranty period 36 CR = average contractor cost to repair a unit returned for warranty service The warranty price model used for their subsequent analysis of profit variance as a measurement of risk is WP = (FC-j + XD x T0Ho Jj D D X CRg)FW (2.7) where the subscript B denotes "bid" value for pricing. The factor FW is the loading applied to expected costs to yield the fee or profit. Therefore, FW will generally be greater than 1.0, and its value may very well be influenced by the contractor’s view of his risks at the time of pricing. Balaban and Meth conclude, in a subsequent presentation of the above models (1978), that achieved MTBF is the key variable in determining contractor risk under warranty, and that equipment MTBF under warranty can be controlled to provide contractors a reasonable profit expectation and that increased risk of loss is balanced to some extent by the greater opportunity for profit and by long-term benefits. Chelson (1978) develops a model for evaluating the economic incentive for a contractor to implement an engineering change proposal (ECP). This is another model that does not bear directly on the present research but is of peripheral interest in the investigation of costs under specific RIW options. 37 Anderson (1979) explores the idea that it is possible for a warranty to provide a counter-productive result, i.e., lower reliability at high cost. While his model is not, again, one of central interest, it does provide an additional look at some of the costs involved in the repair process. He postulates that there is some optimal level of reliability determined for the supplier by the tradeoff of repair costs and reliability improvement costs, and also some optimal level of reliability defined as an operational requirement for the item (probability of successfully completing a deployment). Depending upon the relationship of these two reliabilities, the warranty may or may not be advisable. Models of Pro Rata, Free-replacement, and Combination Warranties One of the earliest papers to deal with the subject of the value of warranty provisions (called "quality" in this paper and included in the general category of reliability and services offered with the sale) is Dorfman and Steiner (1954). They find the profit maximizing equilibrium in the joint optimization of quality and price through analysis of the demand curve's relationship to a curve expressing the rate at which sales increase in response to increases in quality and concommitant increases in average cost. Their conclusion is that the general level of quality in any market depends 38 on the sensitivity of consumers to quality variation and price variation as well as the effect on average cost of quality changes. The first actual model of the value of a warranty is presented by Lowerre (1968). He seeks to develop a model for the determination of warranty reserves as a percentage of the selling price of the item. His is a rather simplistic model based on the binomial distribution. He finds the expected cost of a warranty to be $ = M(l+k)$cp (2.8) where M = the production quantity in the program covered by the warranty (l+k)$ = cost to replace a unit under warranty, k = a non-negative number p = constant probability each unit has of being a defect Lowerre also uses the binomial model to determine probabilities of exceeding or falling short of warranty reserve needs. Menke (1969) is also interested in calculating warranty claims costs for the purposes of estimating warranty reserves. He assumes exponentially distributed lifetimes, and develops models for both pro rata warranties and lump sum rebate warranties. The model 39 which yields total warranty reserve funds for a pro rata warranty i s R = c(l - (m/w)(l - exp-w/m))N (2.9) where R = total warranty reserve cost for lot size N c = constant unit product price, including warranty cost w = duration of warranty period m = product MTBF (mean time before failure) N = product lot size for warranty reserve determination, typically one year's production for planning purposes The lump sum rebate model for total warranty reserve funds i s R' ' =' Nc(l/(1 - exp-w/m) - m/w)(l - exp-w/m) = Nkc(I-e-W/m) (2.10) where double primed variables refer to lump sum calculations k = proportion of the unit cost to be returned as a lump sum rebate Amato and Anderson (1976) extend Menke’s warranty reserve model by including components to recognize the time value of money. Their total pro rata cost of claims i s 40 R = ((Nc)/(1 + m ln( l + 0 + ( } ) ) ) ) ( 1 - (m/w) x (l/(l+ra In ( 1 + 0 + < t > ) ) ) x (1 - (l + e + d>)~We~w/m)) (2.11) where • J f * R = real present value of the costs associated with the warranty claims on a product lot size N N =• product lot size for warranty reserve determination c = constant unit product price, including warranty cost t = time of product failure w = warranty duration m = product mean time before failure (MTBF) C(t) = c(l - t/w) the customer rebate given for valid claims at any time t within the warranty period 0 = the opportunity nominal rate of return which could be earned through the investment of warranty reserves 4> = expected change per period in the general price level Menke’s response (1976) to Amato and Anderson's extension of his model (1976) questions whether their treatment of inflation is really needed. He suggests that if w is so long or 4> so large that the purchasing power of 41 the price premium is significantly eroded, the warranty rebate offered by the vendor would be based on the replacement price at the time of failure rather than on the purchase price. Heschel’s model (1971) is, again, a rather simple one, but his is the first model to recognize multiple failures. The model looks at warranties from the consumer’s point of view. He assumes exponentially distributed lifetimes and a combination full-replacement, pro rata warranty. The expected total cost of repairs, including warranty effects, for the item’s life is E(total cost) = (At^)C(((exp(-Atj) - exp(-At2)) /(A(t2-t1))) exp(-Atg)) (2.12) where A = the equipment constant failure rate per unit time t^ = time at which full compensation stops and pro rata compensation begins t2 = time at which pro rata compensation stops, no compensation after this time t^ = equipment useful life (life cycle) c = average repair cost In 1973, Oi modeled demand for a risky product as a function of the cost of the product plus the cost of 42 damages caused by product failure. His analysis allocates costs to attain some specified utility to the good units obtained, and uses this "full price" concept to reach an insurance premium value. He expands his analysis to results of consumer versus producer liability and aggregate accident costs, and concludes that product decisions between more and less risky products hinge on full prices, developed through reallocating the cost of bad items replaced to obtain specified utility, to the good items. price or cost of a warranty but with the necessary conditions for optimality of the amount and duration of a warranty. He takes the consumer's point of view and optimizes the warranty terms by minimization of the consumer's utility loss function. He concludes that the optimal warranty for perfect competition and costless transactions is not a lifetime, money-back warranty. This model also recognizes multiple failures; the expected discounted cost of the entire future stream of replacements is Brown (1974) is not directly concerned with the V(R,G) = (1/(1 - Eexp(-pt)))K(R,G) (2.13) where rG e f(t) dt Eexp(-pt) G the largest t for which f(t) > 0 (the life cycle of the product) 43 p = discount rate K(R,G) = expected total discounted cost of replacing the asset Blischke and Scheuer (1975) begin a new line of investigation by applying renewal theory to the analysis of warranty policies as a means of modeling multiple failures during the life cycle of a product. They develop models for the consumer’s indifference price (the price at which the consumer is indifferent between a warrantied item and an unwarrantied item) for free-replacement and pro rata warranties. In the case of the free-replacement warranty the consumer's indifference price is (the ”1” subscript denotes the free-replacement warranty) Cx** = (UK(1+M(L))(1+M(W1))) /(U(1+M(W1))+L.) (2.14) where N =N(t) = n um ber of replacements of the item made in the time interval (0,t) M = M(t) = E(N(t)) (the "renewal function") U = E(X) the expected lifetime of the item K = cost of new item L = life cycle of the item = length of the free-replacement warranty In the case of the pro rata warranty, the model is (the "2" subscript denotes the pro rata warranty) 44 C2* = KW2/(uw + W2F(W2)) (2.15) where W„ = length of the pro rata warranty f W f 2 yT7 = x dF(x) (the partial expectation 2 0-' of X) X = lifetime of the item F(.) = cumulative distribution function of X F(t) = 1 - F(t) The authors note that by slight changes in their models the seller' s indifference price may be obtained. Thus, they serve both points of view. Blischke and Scheuer extend their analysis to a consideration of multiple independent failure modes, and conclude their paper with a numerical example of their model using observed data that appeared to fit a normal distribution. Blischke and Scheuer extended their work in a 1977 paper which concentrated on non-parametrie methods of solution.' Where the earlier paper had assumed knowledge of the cumulative distribution function for the item’s time-to-failure, the 1977 paper considers estimation of the distribution function from incomplete data using non-parametric methods. Current research by the above authors (1981) goes further into the use of renewal theory in the analysis of warranties and provides tables of values for the renewal 45 functions of several important life distributions. Amato, Anderson and Harvey (1976) develop a model whose components describe, as a function of time, what the authors call rebate and non-rebate costs. The model is then used to measure resulting costs of a warranty rebate program and to assign these costs to separable accounting periods. This paper, for the first time, considers the issue of invalid and unfiled claims, and, in addition, recognizes the costs associated with administration of a warranty program. For these reasons the paper contributes much to the development of theoretical bases for warranty cost evaluation. However, the paper does not specify the form of any of the functional relationships. Collecting the pieces of the theoretical model developed yields the present value of rebate and evaluation costs for any quantity of product demanded at time x, which quantity is given by some function Q(x). The resulting model is ft* c t t ( l-0)R(t)Q(x)f(t)dt CKt* * + ’ (l-TT)3R(t)Q(x)f (t)dt (2.16) where c per unit present value of the rebate costs associated with a particular quantity of sales Q(x) + evaluation costs 46 = time when consumers would cease to seek redress from warranties even though they may hold a valid claim = proportion of valid claims = proportion of valid claims rejected = the present value of per unit warranty rebates made at time t = quantity of product demanded at time T = failure density function = proportion of invalid claims accepted = failure density function associated with invalid claims = per unit inspection and evaluation costs Glickman and Berger (1976) also raised an innovative point with their proposed model. Rather than assume the usual exponential life distribution they employed an integer valued gamma distribution which led them to the use of the Poisson distribution to determine the expected number of failures during the warranty period. They used their results in determining the joint optimization of selling price of a product and the duration of its warranty. Their model states the unit profit to the seller to be p - (v + cp(t)) (2.17) where p = unit price 47 * TT 0 'R(t) Q(t) f (t) 3 * f (t) v = unit variable cost c = constant cost of repair under warranty p(t) = expected number of repairs to be performed on a unit which is protected by warranty for a length of time t Yun and Kalivoda (1977) investigate further the theory proposed by Amato, Anderson and Harvey (1976) that human error is involved in the warranty claims process. Their model is designed to estimate failure rates recognizing possible human errors by both customers and the product manufacturer in determination of failures. The model provides a correction factor to the customers’ perceived failure rate to obtain the true failure rate. This model does not actually provide any cost or benefit analysis of a warranty policy, but does extend the theory necessary for incorporating a component involving claims behavior and claims validation into the general model proposed by this research. Prohaska, Briggs, DeWolf, and Lund (1978) develop an extensive life cycle costing model. Their research is primarily concerned with consumer applicances such as television sets and refrigerators, but the model is general enough to have broad applicability. The life cycle cost model includes factors other than warranty cost/benefit values, but these are not part of the present analysis. The expected warranty cost calculation is part 48 of the acquisition cost and also appears as part of the service cost computation. Their model is of only peripheral interest to the current research, since it does not include the stochastic component central to a predictive model. The portion of the model which deals with expected warranty cost yields the following. L I WV.(m,a) * WF.(m,a) a = l 3 3 * DF(a) * (1 + POVH) AW . (m) J wher e AW .(m) J WV .(m, a): J WF . J DF( a ) POVH expected warranty cost in constant dollars for the jth service type cost element cost per service incident weighted for service incidence frequency of jth service type cost element for year of age (a) and model year (m) in constant dollars warranty factor. A 1 indicates coverage by warranty in year of age (a) and model year (m) for jth service type cost element. A 0 indicates no coverage, discount factor fractional increment to account for extra administrative overhead year of age (2.18) 49 m .= model.year L = retention life in years j ;=- service type cost elements (consumer responsible, general guarantor responsible, specific guarantor responsible, parts cost, labor cost, other costs) In an extension of the Blischke and Scheuer (1975) research, Mamer (1980) specifically addresses the question of adverse selection by proposing that consumer’s usage rates differ, and could be modeled by some theoretic continuous function. This introduces an additional factor which may account for differing failure rate experiences by consumers. Mamer also suggests the use of a diffusion process to model the state of a product over time when the product's state may be described by a continuous variable. This is the only paper to provide any sort of sensitivity analysis to the assumptions of life distributions. However, in a later paper (Mamer, 1981) he substantially changes the model used in this analysis. This subsequent paper (Mamer, 1981) presents a model which supercedes his earlier model, and, using renewal theory, corrects the pro rata total cost model developed by Blischke and Scheuer (1975) to eliminate an overestimation in the cost of ownership during the product’s lifetime. The corrected version of the Blischke and Scheuer model is the one shown above as 2.15. For the 50 case of the free-replacement warranty, the author develops exact, closed form expressions for average cost and profit in the long run case only. Mamer states his model from both the consumer and supplier points of view. The expected cost of ownership under a pro rata warranty of a product during its lifetime is ) = q2 + (q2m(T)/W) ((1 - F(W))W +nw ) + (q2/W)(pw - F(W)W) (2.19) = time at which consumer will no longer replace item (life cycle) = total costs of replacement during period (0,T) = unit cost to the consumer = expected number of failures in product life cycle = length of the warranty period = distribution function of product lifetime fW = xdF(x) 0J The expected total profit to the seller under a pro rata warranty is 51 E(TCpR(T) where T TCpR(T) <*2 m(T) W F ( . ) UW E(ttpr(T) ) = (q2 —qx) - q2F(W) + q2uy/W + (q2m(T)/W) ((1 - F(W))W - Wq1/q2+yw) (2.20) where TTp^(T) = total profit to seller from supplying items during period (0,T) q^ = unit cost to the supplier Patankar and Worm (1981) are involved in a totally different aspect of the modeling of warranties from Blischke and Scheuer and Mamer. Their research is an extension of the Menke (1969), Amato and Anderson (1976) approach to estimation of warranty reserves. However, they propose something entirely new; the estimation of prediction intervals for warranty reserves. Through use of the Central Limit Theorem, the authors state that the present value of warranty reserves is normally distributed, and, thus, a prediction interval may be calculated for the estimated present value of warranty reserves. They extend their investigation into the area of cash flow by looking at the expected total warranty cost (or cash flow) for N items for a specified time period. Patankar and Worm use a continuous discounting procedure rather than the discrete procedure used by Amato and Anderson (1976). In all other respects, their basic model is virtually the same. The expected present value 52 of the rebate paid per unit is given by fW c(l - t/W) exp(-( 0 + 4>) t ) 0J (l/m)exp(-t/m)dt PV(r) = (2. 21) where W warranty length c = constant unit product price, including warranty cost t = time of product failure, distributed exponentially 9 the firm's discount rate ( J ) expected change per period in the general price level m = product mean time before failure (MTBF) Models of Other Aspects of the Warranty Process Mann and Saunders (1969) and Mann (1976) deal with the determination of warranty duration. In both cases, the failure time is assumed to be modeled by a two-parameter Weibull distribution with neither parameter known. The warranty period is to be determined so as to assure a specified probability that no failures will occur before the expiration of the warranty period. This warranty period is to be calculated from a small sample of life lengths. The expression for the warranty period i s then derived as a funcion of the ordered observations • In a totally different area of investigation, Bryant and Gerner (1978) use hedonic price analysis to 53 estimate the implicit market prices of the warranty provisions of refrigerators. Their results suggest that it is in the consumer's own interest to bear the risk of failure himself. They specify the particular conditions when this is true, namely when implicit market prices of warranty coverage are high because their supply prices are high. Courvile and Hausman (1979) view warranty analysis from yet another point of view. Their model introduces the consumer's perceived reliability of the product as part of the consumer's expected utility of the product. The authors deal explicitly with the case when consumer information regarding product liability is imperfect. They investigate the relationship between the supply of reliability and voluntary liability (warranty). They conclude that the market system provides a solution and that regulation may not be necessary. Indeed, they state that the supply of additional information may produce adverse effects. Gerner and Bryant (1980), in their continuing research in the area of warranties on consumer appliances, examine consumer behavior within the warranty period. They conclude that consumers respond to economic variables, particularly the value of female's time, in their demand, for warranty provisions. An interesting conclusion is that consumers who are more likely to own 54 comprehensive warranties are also more likely to use the item more and are more likely to experience repair. This research implies that consumers have differing usage rates which may impact the failure rate of the item. This issue was specifically addressed by Mamer (1980) in the previously discussed paper. 55 CHAPTER III DEVELOPMENT OF THE THEORETICAL MODEL The objective of this research was to build a general model for the quantitative analysis of warranty policies. As work progressed, it became clear that there would really be two models; a theoretical model which would enumerate the variables of which warranty cost is a function, and a working model which would actually generate an estimated cost from a set of warranty parameters and assumed product lifetime information. This chapter describes the steps taken in developing the theoretical model and the decision to use simulation for the working model. Chapter IV discusses the simulation m'odel in depth. There were five major steps in the development of the theoretical model: (1) the characteristics of the existing models were analyzed and compared, (2) a general notation was developed and the existing models were restated in the general notation and, where necessary, altered to provide the expected warranty cost from the supplier’s point of view as opposed to the consumer's, 56 (3) a general theoretical model which incorporated all of the features of the existing single failure models was stated in the new notation, and parameters were developed which would reduce the general model to any of the existing single failure models, (4) the general model was stated in closed form for the exponential lifetime case and each of the parameter settings from step 3 was checked for agreement of final results with the original existing model, (5) the general theoretical model was extended to the case of recognition of multiple failures. The results of each of these steps will now be shown and discussed. Step One The first step taken was to analyze and compare the existing models. Most earlier authors model single failures. They, each in their differing ways, calculate the expected cost of the first failure of the item. None of these models addresses subsequent failures or their costs.’ This is a shortcoming which other models seek to remedy. The authors who tackle the problem of modeling multiple failures use diverse methods, the results of which do not agree. This second group of models counts and sums up the results of multiple failures during the life cycle of an item, i.e., until such time as the item will no longer be repaired or replaced upon failure. Once one considers multiple failures, the question of whether 57 or not the warranty renews upon repair or replacement becomes central. The difference in policy has a direct impact upon the mathematics of the modeling process. The difficulties which arise will be discussed in a later section of this chapter. Table 1 is a chronological listing of the major characteristics of each of the models studied. Models which are conceptual in nature rather than mathematical (Dorfman and Steiner, 1954; Oi, 1973; Prohaska, Briggs, DeWolf, and Lund, 1978; and Yun and, Kalivoda, 1977) will be omitted from discussion. The table has been constructed with headings for the major characteristics of comparison between models. Multiple Failures The first, and most important, distinguishing feature of these models is whether they count only the first failure of an item or all failures occurring in the life cycle. This summing up of a stream of costs over time is a characteristic of five of the ten models (Blischke and Scheuer, 1975; Brown, 1974; Glickman and Berger, 1976; Heschel, 1971; Mamer, 1981). Warranty Renews The question of whether or not the warranty renews upon the repair or replacement of a covered item is germane only to these multiple failure cases. Two of the multiple failure models limit coverage to nonrenewing 58 Table 1 Major Characteristics of Existing Warranty Valuation Models Sales After Multiple Warranty Warranty Lifetime Discount Rebate Author(s) Date Failures Renews Expiration Distribution Function . Function Viewpoint Sensitivity Lowerre 1968 no no no Bernoulli failures none lump sum supplier no Menke 1969 no no no exponential none lump sum pro rata supplier no Heschel 1971 yes no no exponential none combination consumer no Broun 1974 yes yes yes unspecified continuous unspecified supplier consumer no Blischke & Scheuer 1975 yes yes yes unspecified continuous lump sum pro rata supplier consumer limited Amato & Anderson 1976 no no no exponential discrete pro rata supplier no Glickman & Berger 1976 yes no no gamma none lump sum supplier limited Oi vO Table 1— Continued fluthor(s) Date Multiple Failures Warranty Renews Sales After Warranty Expiration Lifetime Distribution Discount Function Rebate Function Viewpoint Sensitivity Amato, Anderson & Harvey 1976 no no no unspecified unspecified pro rata supplier no Patankar & Worm 1981 no no no exponential continuous pro rata supplier no Mamer 1981 yes yes yes unspecified none lump sum pro rata supplier consumer no on o warranties (Glickman and Berger, 1976; Heschel, 1971), and one discusses only renewing warranties (Brown, 1974). The Mamer models are virtually identical to those of Blischke and Scheuer since Mamer was building on their work. The primary thrust of these models is toward the renewing warranty, although the case of nonrenewing warranties is addressed in both papers. Sales After Warranty Expiration Because of the mathematics used in some of the models with renewing warranty processes the authors adopted the assumption that sales will occur after the end of the warranty period. That is, if you are a supplier you assume that whether or not an item is under warranty the consumer will return to you to replace it until the end of its life cycle. This assumption may or may not be realistic, but it is relaxed 'in the models proposed by this research. This feature will be discussed with the mathematics of renewing warranties later in this chapter. Lifetime Distribution By far the most popular lifetime distribution is the exponential. This is because of its ease of calculation. Of those authors who specify a distribution, only Lowerre (1968) and Glickman and Berger (1976) depart from the exponential Lowerre using a Bernoulli process to generate failures, and Glickman and Berger extending the concept of the exponential life to a gamma distribution. 61 Other authors do not specify a form for their lifetime distribution but leave their models stated in generalized functional form. Discounting Function The earliest three models do not include discounting of future cash flows, but later models use a variety of ways of recognizing the time value of money. In fact, the Amato and Anderson paper (1976) modifies Menke's model (1969) only by adding discounting. Patankar and Worm (1981) take this same model a step further by using a continuous rather than a discrete discounting method. Brown (1974) actually uses a specific discounting formula to achieve a statement of his multiple failure model in terms of a Laplace transform of-the renewal function. There are two distinct formulas for discounting used in these models and both have been incorporated into the theoretical and the simulation models of this research. Rebate Function Only one author, Heschel (1971), uses a combination of the lump sum and pro rata rebates. All other authors treat only one form of rebate in a given model. Whenever pro rata rebates are specified the function is assumed to be linear. All three cases, lump sum rebates, pro rata rebates, and combinations of the 62 two can be modeled by both the theoretical and simulation models proposed by this paper. Viewpoint The cost of a warranty may be calculated from either the supplier’s point of view or the consumer's. A warranty results in an obligation to repair or replace an item, and thereby a cost to the supplier, and also in a potential savings or benefit to the consumer through being protected against loss of the use of an item. Only one author, Heschel (1971), deals exclusively with the consumer’s side. Three papers provide models for both the consumer and the supplier. The majority of the models provide the supplier’s cost exclusively. Further comparisons of these models in this chapter will require that all address the same question. For this reason, Heschel’s model has been restated to provide the supplier's cost and, where both viewpoints were available, the model addressing the supplier’s cost has been used. The current theoretical and simulation models provide the supplier's cost. Sensitivity The question of how sensitive the model is to its assumptions is central to this research. Only two papers, Blischke and Scheuer (1975) and Glickman and Berger (1976), provide any sensitivity analysis, and in both cases it is limited. The simulation model of this 63 research provides the data for an extensive examination of the results of changing certain assumptions about the model or certain of its parameters. Step Two The second step in this research was to develop a general notation in which to state all models. This general notation was necessary to a comparison of the various models’ component parts. A listing of the notation developed appears in Table 2. Also necessary to a comparison of the different models was a common point of view. To this end, all models were restated in terms of the supplier’s expected cost as noted above. Table 3 is a listing of the existing models restated in the general notation to provide the supplier's expected cost for offering a warranty of specified length and terms. Single Failure Models Lump sum rebate functions. Lowerre’s model and Menke's model are identical. Pro rata rebate functions. Menke, Amato and Anderson, and Patankar and Worm differ only by the inclusion or form of the discounting function. Amato, Anderson and Harvey introduce several new components in their model, but its basic structure is still Menke's. 64 Table 2 H E(H.) E(H(W)) E(H(L)) L W R C T f(.) f*(.) y m( . ) S(.) t( 1) q( t) General Notation = cost of offering a warranty, includes repair or replacement cost and claim validation cost t h = expected cost to supplier of i failure of an item offered with warranty = expected cost to supplier of all failures of an item during the warranty period = expected cost to supplier of all failures of an item during the life cycle = life cycle of an item (point at which it will no longer be replaced or repaired) chosen so that F(L)=1.0 = length of warranty = supplier's cost of a failure, may be a function of time, R(t) = consumer's cost for an item before pro-ration or consumer's cost of a failure = random lifetime of an item (a nonnegative random variable) = probability density function of T = lifetime distribution of items experiencing invalid claims = E(T) = MTTF = MTBF •W t f ( t ) d t = renewal function = Laplace transform of renewal function = time at which combination warranty switches from lump sum to pro rata = pro-ration function 65 Table 2— Continued g(t) = discounting function K = administration and claim validation cost a = proportion of valid claims made a* = proportion of invalid claims made 0 = proportion of valid claims rejected 0* = proportion of invalid claims rejected 4) = optimal rate of return on invested funds 6 = expected change per period in the general price level 66 Table 3 Existing Warranty Valuation Models Restated In General Notation Single Failure Models Lowerre ECHj) = R f(t) dt O' Menke' ECHj) = R f(t) dt Cr Menke f W ECHO = {R - C(t/W)} f(t) dt o-' Amato & Anderson E(H -r x) = {R - C(t/W)} (l+<j> + 6) r f(t) dt n J Patankar & Worm Amato, Ander son & Harvey E(HX) = ECH^ ^{R - G (t /W)} e t^ + <5> f(t) dt a(1-0){R-Cq(t)}g(t) f(t) dt a*(l-e*){R-Cq(t)}g(t) f*(t). dt a K f(t) dt + a* K f*(t) dt pro rata model lump sum model 67 Table 3— Continued Multiple Failure Models Glickman & Berger Heschel Brown Blischke & Scheuer' E {H(W)} = {m(W)} R E {H(W ) } = {m( W) } { t{n) R f(t) dt O-' + l i l t(i y R - C {(t-t(l))/(W-t(l)> } f(t) dt} u E (H(L) } = { l + m(L) > { [ R( t ) e C + 6 ) f ( t ) dt cr L(R-C) g-tC'N-6) f(t) dt ■t (ct>+5) lit + K e f(t) dt } E{H(L)} = {m(L)}{CF(W) - C(uw/W) + R - C} Blischke ^ & Scheuer Mainer a + {CF(W) - C(yw/W)} E (H(W)} = {m(W)} R same as Blischke & Scheuer' Mamer same as Blischke & Scheuer pro rata model lump sum model 68 Multiple Failure Models Nonrenewing warranties3 The Blischke and Scheuer (1975) lump sum model and the Glickman and Berger (1976) and Heschel (1971) models differ from other multiple failure models because their warranties are nonrenewing. Therefore, their models pay off only for failures during the initial warranty period, while other models must look at all failures in the life cycle since the warranty does renew and, thus, the end of the initial warranty period is not an absolute cut-off point for benefits. This difference in assumptions leads to different mathematical models. Renewing warranties. In the pro rata case,Blischke and Scheuer, in their research, and Mamer, in his, propose the same model, as noted in Chapter II, but their model is different from Brown’s, since they use renewal theory to develop their model while Brown does not . It appears that there is a consensus among authors of single failure models. However, the multiple failure case has proven to be much more complicated. Additional assumptions must be made about warranty provisions and consumer behavior, and these assumptions directly affect the form the model takes. 69 Step Three Thirdly, a general model was developed which incorporated all of the features of the existing single failure, or first-failure, models. The result is components. 1. The -first is the expected discounted ( g ( t) ) cost (R - Cq(t)) of the first "real" failure at time t. This cost being■the difference between the supplier's cost of a failure and the consumer's cost of a failure. The consumer will incur a cost when the pro ration provision of a warranty is being modeled. The probability of this first failure is generated by the lifetime distribution f(t). The cost is modified by the probability that the failure will be claimed (a) and the claim validated (1 - 0). 2. The second is the expected discounted cost of the first "perceived" failure modified by the probability 0 a (1 - 0) [R - C(q(t))] g(t) f(t) dt + 0 a* (1 - 0*) [R - C(q(t))] g(t) f*(t) dt + a K g(t) f(t) dt 0 rW + a* K g(t) f*(t) dt 0J The model is designed to generate the expected cost, (3.1) E(H^), of the first failure of an item, based on the assumption that any item can be classified as either failed or good. This cost is the sum of four cost 70 that the invalid claim will be made and validated. The asterisks on the parameters of this component indicate that a lifetime distribution different from that of the first component applies. This modified lifetime distribution is the distribution describing the time to a perceived failure when a real failure has not occurred. It may be one of the same family as the lifetime distribution of valid claims (real failures) with different parameters, or it may take a totally different f orm. 3. The third component is the expected discounted (g(T)) cost (K) of claim validation for valid claims arising from "real" failures. Again, this cost is modified by the probability that a claim will be made. 4. The fourth component is the expected discounted cost of claim validation for invalid claims arising from "perceived" failures modified by the probability that a claim will be made. The only difference between the first and second components and the third and fourth components is the recognition of a modified lifetime distribution designated by the asterisk. If it is felt that claims will no longer be made at some time preceeding W, the upper limit of integration may be changed to t* to reflect this decreased period of jeopardy. This was suggested by Amato, Anderson and 71 Harvey (1976) as a way of dealing with consumer claims behavior. The current set up of the model makes this the supplier's expected cost; however, by changing the signs on the R and C parameters of the model (supplier's and consumer's costs respectively) it could yield an expected profit instead. This would be of interest in the case where sales were assumed after the end of the warranty period, W. In which case, an additional component would be needed to account for sales after W. The limits of integration on this new component would be W to L, and the cost would be R-C, since the consumer would have to pay full price to replace an item after the end of the warranty period. Further adjustment in the rebate component of the model would allow the calculation of consumer's cost rather than supplier's cost. This would entail replacement of the supplier's cost, R, with the consumer's cost, Cq(t), and deletion of the claims validation cost, K. As stated, the model defines expected cost to the supplier of offering a warranty to be primarily a function of the lifetime distribution of the item, modified by a discounting function, a pro ration function, and claim filing and validating probabilities. The model at this point does not purport to deal with the multiple failure case . 72 Each of the single failure models currently existing can be fit into this model through the use of the parameters developed for this purpose. A listing of the appropriate parameter values for each of these models appears in Table 4. The parameterized model is E(H ) = Q' t-a 2 * - ( 1 ) <x ( 1-0) {R - Ca I m w ,(t) { i }} 1 W - a - t ( l ) X {a„(l + 4)+5) + a , (e t^ +<5>)} f(t) dt , hl t-a„ t ( 1 ) + a* ( 1 - 0 * ) {R - C . 1 ( t ) { i }} n I 1 ' J W-a2t ( l ) X {a,(l + d)+5)“t + a, (e"t(4)+<5)) } f*(t) dt + K {a^l + Q+S)'* + a^(e( t^+S)))} f(t) dt 0 + J^a* K (a3(l + < t , + 5) 1 + a4(e( f*(t) dt (3.2) One of the multiple failure models, that of Heschel, also fits into the framework of the general model with only the addition of a multiplier term to account for failures subsequent to the first. Heschel’s model is, therefore, included in Table 4. 73 Table 4 Parameter Values to Reduce General Model 3.2 To Specific Single Failure Models Model a e al a 2 a3 a4 < t > 6 a* 6* K Lowerre 1 0 0 0 1 0 0 0 0 0 0 Menke3 1 0 1 0 1 0 0 0 0 0 0 Menke^ 1 0 0 0 1 0 0 0 0 0 0 Amato & Anderson 1 0 1 0 1 0 < j ) 6 0 0 0 Amato, Anderson & Harvey a 0 lc 0 1 0 < t > 0 a* 0* K Patankar & Worm 1 0 1 0 0 1 $ 6 0 0 0 Heschel*^ 1 0 1 1 1 0 0 0 0 0 0 apro rata model ^lump sum model Q pro-ration function is unspecified in model, but linear pro-ration is assumed in authors1 example ^Heschel's model also fits the general model if a multiplier term m(L) is appended to account for multiple failures 74 The f*(t) distribution is only used by Amato, Anderson and Harvey (1976). Therefore, components two and four will not appear for any other models. They are zeroed out by setting a* to 0. Amato, Anderson and Harvey's model is the only single failure model to include claim filing and validation behavior. In all other cases a and 9 are, respectively, 1 and 0. Parameter a^ is set to 1 to indicate linear pro ration in the rebate function and to 0 to indicate that the rebate is a lump sum. Parameter a2 is 0 in all cases except Heschel’s combination warranty where it is 1 to indicate that at time t(l) the warranty shifts from lump sum rebate to pro rata. Parameters a„ and a, select one of the two ■ 3 4 discounting functions, discrete or continuous, respectively. When 4>, the optimal rate of return on invested funds, and 6, the expected change per period in the general price level, are both set to 0, the model reduces to the no discounting mode. Step Four Subsequently, the general model was stated in closed form for the case of the exponential lifetime distribution. The resulting model is shown in Table 5. Each of the parameter settings listed in Table 4 was then worked through and the results were found to 75 Table 5 Closed Form Statement Of Model 3.2 When Lifetime Is Exponentially Distributed l_<e1/u(l+<H5)rW 1-e (1^ +(t ,+ 6)w E(H.) = (a (l-0 )) ({R{a«(--------------------------------) +--a4(— -----------------------) ) ) l+)jlog( 1+4>+< 5) y(1 / v j + < b +6 ) a,C a2t ( l ) ( e 1/u( H * + 6 ) r a2t ( I ) ( e 1/u( l+*+S) ) ' a2l ( 1} - t— U,C- + W-(a2t(l)) ^ 1 + yl og( l+(J>+6) y( l/y+log( l+<J>+6)) ^ W(e1 / y ( l + d)+6))_W ( e 1 / y ( l + d > + 6 )rW _ . _ 2) i+iog(l+<|)+6) y ( l / y+1 o g ( 1 + < j)+ 6)) a 2t ( l ) ( l / u + < t ) + 6 ) e ' ^ 1//u + <l>+6)a2t ( 1 ) e ~0/y+<i>+6 ) a 2t ( 1 ^ + a , ( ■ 2 + : 2 y(l/y+4»+6) y(l/y+<j)+6) W(l/jj+4,+ 6 ) e " (1/w+<t>+6)W e _(1/u+<1)+6)W y(l/y+<|)+6)^ y( 1 / y+4>+ 6) Table 5— Continued 77 l + log( 1 + ( J > + 6) l/u#+log( l + (J>+6)) Table 5--Continued 78 Table 5— Continued 79 agree with the original model. No other distributions were used at this point because of the impossibility of obtaining closed form expressions for some required functions for lifetime distributions other than the exponential. This is a shortcoming overcome by the use of simulation. Step Five It is intuitively appealing to move from a single failure model to a multiple failure model through the use of a multiplier term such as the expected number of failures in a specified time period. This is the method Heschel (1971) used, and the reason his model can be included in Table 4 as fitting into the framework of 3.2. The multiple failure models of Brown (1974), Glickman and Berger (1976), Blischke and Scheuer (1975), and Mamer (1981) do not reduce to 3.2, nor do they reduce to any common, framework, with the exception of the Blischke and Scheuer model and the Mamer model which are virtually identical, since Mamer1s paper is.a correction to the Blischke and Scheuer paper, see Chapter II section on Models. Since Glickman and Berger and Heschel are the only authors in this group that do not use a renewing warranty, it is not surprising that their models take a different mathematical form. Brown uses a discounting scheme to arrive at his final model for multiple failures which 80 includes a Laplace transform of the renewal function rather than the renewal function itself. This is in contrast to the models of Blischke and Scheuer and Mamer. These models use the renewal,function directly, having been derived through renewal theory. The differences between renewing and nonrenewing warranties and lump sum versus pro rata rebate schemes lead to differences in the multiple failure models which can be used to describe them. Nonrenewing, Lump Sum Warranties In the simplest case of the nonrenewing, lump sum warranty, without discounting, the cost would be R each time the item failed while covered by the initial warranty. The expected number of failures in the warranty rW period, m(W) = (1 + m(W-t)) f(t) dt, could be 0j obtained through the use of the renewal function appropriate to the lifetime distribution. This yields the simple model E(H(W)) = R m(W) (3.3) Glickman and Berger (1976) use this approach, although they do not state it in terms of renewal theory, but rather note that failures of items with exponential lifetimes constitute a Poisson process. They use this fact to state their multiplier term as a summation of Poisson probabilities. If we simply ignore failures after the warranty 81 E(H) = E(H(W)|t1-t) = period, not assuming that they will result in new sales, Blischke and Scheuer1s (1975) and Mamer's (1981) renewal theory approach will yield 3.3. If 0 if t > W R if t < W and we condition on the time of the first failure 0 if t > W R- + E(H(W-t)) if t < W where E(H) = expected cost of a failure E(H(W)) = expected cost of all failures over the warranty period = supplier’s constant cost of a failure = warranty length = random lifetime of an item (a nonnegative random variable) R W T then -W E(H(W)) f = [R + E(H(W-t))] f(t) dt A J w 0 = R 0 rW 0 f(t) d(t) E(H(W-t)) f(t) dt rW = R F(W) + R F(W-t) dm(x) = R F(W) + R (m(W) - F(W)) = R m ( W ). 82 Attempting to modify the new single failure general model, 3.1, to also model multiple failures does not result in the above expression. Model 3.1 calculates the expected, undiscounted, cost of the first failure under a lump sum rebate policy as fW E(H.) = R f(t) dt 0j Simply appending the multiplier term m(w) to this cost term does not yield 3.3, but fW E(H(W)) = m(W) ( R f(t) dt) = m(W) RF(W) (3.4) 0J which agrees with the lump sum portion of Heschel's (1971) combination model. He also approaches the problem with a multiplicative model. His rationale is that, Given a sum S of independent and identically distributed random variables with mean in , the number rx in the sum being a discrete random variable, then from Johnson, . . . E(S) = mE(n) where E(S) = expected value or mean of S E(n) = expected value or mean of n. His reference is to Johnson, N.I., "A Proof of Wald’s Theorem on Cumulative Sums," Annals of mathematical Statistics, Volumn 30, 1959. This rationale supports 3.3, but not 3.4, since the expected cost is R not RF(W). Both approaches lead to the conclusion that simply modifying the new single failure general model, 3.1, by a multiplier term will not yield a multiple failure model for a nonrenewing lump sum warranty. 83 Nonrenewing, Pro Rata Warranties Extending the use of a multiplicative model to the case of a nonrenewing, pro rata warranty yields a much more complicated expected cost term and the conclusion that a multiplier term is not appropriate. Now we are faced with a cost which is not a constant. The cost of a failure, R(t), depends upon when it happens. Since the warranty does not renew, the expected cost of failures subsequent to the first would not be the same as the first, nor equal to each other. E(H ) = 0 W R(t) f(t) dt but W E(H.) - j R(t) f(t) dt where ^ E(H^) = the expected cost of the i failure t. = time of the i*"*1 failure x Thus, the expected cost of all failures during the warranty period would be 00 . y 1 R(t) f(t) dt (3.5) i = 0 t . x Heschel's model (1971) is the only one to include a term for pro ration in a nonrenewing warranty. His model uses the same expected cost for all failures during the period covered by the pro rata warranty. For the reason stated above, his model is incorrect. 84 Renewing, Pro Rata Warranties Unlike the nonrenewing warranty, in the case of the renewing warranty, neglecting discounting, the expected cost of the first and all subsequent failures remains the same E(H ) = 1 0 W R(t) f(t) dt (3.6) since each failure renews the warranty. Every failure is treated as the first failure. The problem then lies with the multiplier term. If one tries to count the number of failures in the warranty period the result can not be more than one, since each failure renews the process. One must then consider the stream of failures and renewals through time. At what point does one stop counting? The life cycle concept is useful here in putting a bound on this renewal process. One counts failures up to some arbitrary point in time which one declares to be the end of the life cycle, beyond which date failures will no longer result in renewals. This raises the question of what happens if a failure occurs outside a warranty period. If we consider the case where a new sale is not assumed, we have an expected cost for any failure in or out of warranty of • W f L : E(H ) = R(t) f(t) dt + 0 f(t) dt (3.7) 0J Wj Authors who have used renewal theory have adopted the assumption of a repurchase of the item even if it fails 85 outside the warranty period, which yields an expected cost per failure of Since C is presumably larger than R, this line of reasoning could well lead to a resulting profit rather than a cost. From a practical viewpoint, it seems unreasonable to assume a continuing stream of sales stemming from every initial sale. One solution to this problem is to combine 3.7 and 3.8 by modifying the after warranty sales by a probability term, 8 , representing the likelihood of a follow-on sale. The resulting expected cost per failure is E(H1) = R(t) f(t) dt + 3(R - C) f(t) dt (3.9) If we assume linear pro ration for q(t), i.e., q(t) = t/W, and apply renewal theory, the result is as follows. L E( Hx ) R(t) f(t) dt + (R - C) f(t) dt (3.8) 0 If E(H1) 0 e(R-c) R-C(t/W) if L < t if W < t 1 L if 0 < t < W and we condition on the time of the first failure 0 E(H(L)|t1-t) =■ 8(R-C) + E(H(L-t)) 8(R-C) + E(H(L-t)) if W < t < L R-C(t/W) + E(H(L-t)) if 0 < t £ W if L < t 86 then E(H(L)) 0 W (R-C(t/W)) + E(H(L-t) ) f(t) dt •L 3(R-C) + E(H(L-t)) f(t) dt W; •W t W + 0 W' 0 R f(t) dt - 0 L $R f(t) dt - W L E(H(L-t)) f(t) dt C(t/W) f(t) dt 3C f(t) dt RF(¥) - Cuw/W + BR(F(L)-F(W)) L 3C(F(L)-F(W)) + 0J RF(W ) - Cyw/W + 3RF(L-t) - 3RF(W) - 3CF(L-t) + 3CF(W) dm(t) = RF(W) - cyy/W + SRF(L) - 3RF(W) - 3CF(L) + 3CF(W) + RF(W)m(L) - (Cyw/W) m(L) + 3R(M(L)-F(L)) - 3RF(W)m(L) - 3C(m(L) - F(L)) + 3CF(W)m(L) = (l+m(L)) (RF(W) - cyw/W - 3RF(W) + 3CF(W) + 3R - 3C) - (3R-3C) (3.10) where E(Hj) = expected cost of a failure E(H(L)) = expected cost of all failures occurring in the life cycle 87 R supplier's cost of a failure C consumer's cost of a failure W = warranty length L life cycle length (L is chosen so that F(L) = 1.0 T = random lifetime of an item (a non negative random variable) t f ( t ) d t 0J fW j t f(t) It would appear that in the renewing warranty case the multiple failure model arrived at through renewal theory ought to be obtainable by appending a multiplier term m(L) = the expected number of failures in the life cycle, to model 3.9, which is recognizable as a modified version of 3.1. This change would result in the following If we again assume linear pro ration for q(t), then 3.11 yields where all notation is as above, which differs from 3.10 by the lack of the term E(H(L)) = m(L) ( (R - Cq(t)) f(t) dt 0 rL + B(R - C) f(t) dt) WJ (3.11) E(H) = m(L) (RF(W) * - Cyir/W w + BR - 3C - BRF(W) + BCF(W)) (3.12) + (RF(W) - Cyw/W - 3RF(W) + BCF(W)) (3.13) Setting B = 1.0 , the case of assumed sales 88 resulting from failures after expiration of the warranty, reduces 3.13 to + (CF(W) - Cuy/W) (3.14) while setting 3 = 0 , the case of no sales resulting from failures after the warranty period, reduces 3.13 to + (RF(W) - Cuw/W) (3.15) The term 3.14 will always be negative, but depending upon the relative values of R and C, 3.15 could yield a positive value. In either case, 3.11 is incorrect, since the use of that multiplier term causes an alteration in the distribution of time between failures due to conditioning on the number of failures in the period (0,L) . Brown (1974) derives a multiplicative model without the use of renewal theory or the problem encountered by 3.11 through the use of the discounting function 8(t) = 'e-'C*+«> (3.16) and the concept of a continuing stream of failures and renewals to arrive at an expression which happens to be a Laplace transform of a renewal function (m(L)) (see model 2.13 in Chapter II). He could only arrive at this expression by using the specific discounting function 3.16. His model then is a multiplicative one, but his multiplier term is unique. A simplified version of his model restated to the supplier’s point of view is 89 E(H) = (m(L) + 1) ( f R (t) e~t((1)+6) f(t) dt nJ •W I 0 + [ (R - C) e_t(cl)+<5) f(t) dt (3.17) WJ This model also circumvents a major problem encountered by all multiple failure models, discounting. Discounting A problem arising when multiple failures are modeled by any method is discounting. Each time there is a failure it is necessary to discount back to the time of the initial purchase. Returning to our earlier simplest case of the nonrenewing, lump sum warranty, when we discount R it is no longer a constant. Instead of a cost of R each time the item fails, we have a cost of R g(t) = R (1 + c | > + 6) t or R (e for each failure at a time t. Clearly, we no longer have an expression comparable to 3.3, since, when R is a function of time, it can not be removed from the renewal equation for solution. The same problem would, of course, exist in the nonrenewing and renewing, pro rata warranty cases. Brown's (1974) double use of the discounting function in the multiplier term as well as the cost term would appear to solve the problem of discounting back to the initial' sale. However, this expression of the model, as a Laplace transform of the renewal function, depends 90 upon the use of the specific discounting function g(t) = e-t(*+S). Limiting Assumptions All of the multiple failure models noted in this section have three assumptions in common; these assumptions are: 1. The lifetime distribution of replacement or repaired items is identical to the lifetime distribution of the original item. 2. Repair or replacement is instantaneous. 3. All items are either failed or good, and there is only one mode of failure. Note that this assumption could be relaxed through the use of a lifetime distribution such as the gamma, which would allow the modeling of a situation where the failure of an item is the result of the failure of component parts, each of which has an identical exponential lifetime distribution. The Decision To Use Simulation It is clear that moving into the area of multiple failure models greatly complicates the search for one general theoretical model. Mamer has proven that a simple multiplier term is not sufficient, and yet his renewal theory approach, since it can not adequately include nonrenewing warranties nor discounting, is not a satisfactory alternative. Brown's model is limited by the requirement for the use of a particular discounting 91 function and also cannot include nonrenewing warranties. These difficulties can be overcome through the use of a simulation model rather than a theoretical or purely mathematical one. Simulation also allows the flexibility to investigate many desired lifetime distributions, under both renewing and nonrenewing warranty policies, to generate the data necessary for a sensitivity analysis of expected cost to various distributional assumptions. For these reasons, it was decided to build a working computer simulation model in addition to the theoretical model. The development of the simulation model is the subject of the next chapter. 92 CHAPTER IV DEVELOPMENT OF THE SIMULATION MODEL Simulation is a technique, usually utilizing a computer, which evaluates a model numerically over a time period of interest. A simulation generates data with which to estimate the desired true characteristics of the model. Simulation models are useful for studying real-world systems which are too complex to be evaluated analytically. They also allow the estimation and comparison of system performances under different configurations and the study of a system with a long time frame in compressed time (Law & Kelton, 1982). The major drawback of simulation as a technique for evaluating warranty costs is that it produces only estimates of the model's true result for a particular set of input parameters. If a valid analytic model could be developed, it would be prefereable. However, because of the complexity of the mathematical expressions describing warranty models and the impossibility of solving the expressions derived for various of the lifetime distributions, simulation estimates are the best 93 alternative. Given a sufficiently large number of independent runs of the simulation model, the results should be entirely adequate to draw comparisons between the results achieved when the warranty model’s parameters are changed. The Simulation Model The technique used herein is called Monte Carlo simulation. It involves the generation of random numbers to simulate real-world events. In the case of warranty analysis, the random numbers represent item lifetimes. By generating these random numbers from specified distributions we can simulate the failure data for items with lifetime distributions of interest. As mentioned above, we are also able to investigate the behavior of very long-lived items for which it is difficult to collect actual data. The first step in building a simulation model is to construct a flowchart of the system to be modeled. There are various levels of detail that may appear. The flowchart in Figure 1 is a simplified version showing the basic dynamics of the warranty process. A more detailed flowchart appears in Appendix A. The warranty begins when the item is placed in service. Nothing happens until a failure is perceived. Such a failure may be real (generated by a lifetime distribution, f(t)) or merely perceived (from a lifetime 94 NO YES NO YES YES YES YES Renewing Warranty s Fail-\ ure Within New ^ Purchase ^ Fail- ^ ure Within Warranty Begins W Set Perceived Failure Service Item Placed Rebate Hade Replaced Figure,!. Flowchart of the Warranty Process 95 distribution, f*(t)). The model assumes that the item will not be replaced if the failure occurs after the end of the life cycle and a claim will not be pursued if the failure occurs outside of the warranty period. In the latter case, a new purchase may or may not be made (corresponds to a 3 probability of a new sale after W in model 3.10). If the failure has occurred during the warranty period, a claim may or may not be made (a probability of a claim from model 3.1), and validated (1-9, model 3.1). If the claim is both made and validated, the rebate function determines the results of the claim, and the form of the warranty (renewing or nonrenewing) determines whether a new warranty period begins or the original warranty period continues. The cycle starts over and continues until one of the paths leads to the box, "Item not replaced." The simulation model encompasses both the single or first failure case (model 3.1) and the multiple failure case (models 3.3, 3.5, and 3.10). It is capable of discounting as well, which allows it to escape that limitation of the multiple failure models. Output From the Simulation Model The model provides an estimated cost to the supplier of servicing his warranty, based on the input values, which is the mean of n simulated costs arising from n initial sales of an item (replications). That is, 96 the model starts with an initial sale, generates a random number simulating the time until the first failure, it calculates the cost to the supplier of that failure, and, depending upon the time of that failure and the provisions of the warranty, it stores that cost or adds to it the cost of all subsequent failures over the life cycle of the item and then stores the accumulated costs. Then it begins again with a new item. When it has done this n times it calculates the mean and standard deviation of the n simulated costs for this particular set of inputs, and displays them. Naturally, the larger n is the less error variance to be expected, however, as n increases so does the running time of the simulation. This expected cost may be either positive or negative, depending upon whether the profit from follow-on sales of new items resulting when an item fails after the expiration of its warranty exceeds the rebate cost. A negative cost indicates a net profit. Inputs to the Simulation Model In order to produce ah estimated cost for a warranty of particular specifications, the simulation model must have input to it the characteristics of the item's lifetime distribution and the particulars of the warranty provisions. The necessary inputs to the model are : 1. the form of the lifetime distribution, 97 2 . the parameters of the lifetime distribution, 3. the length of the life cycle of the item, 4. the length of the warranty, and if it is a combination warranty, the length of the first phase of the warranty, 5. whether or not the warranty renews upon failure, 6 . the form of the rebate function, 7. the seller's and buyer’s costs and claim validation costs (if included), 8 . the probabilities of a claim being made, a claim being validated, and a new purchse being made if the failure occurs after warranty expiration, 9. the form and rate of the discounting function (if included), and 1 0. the number of replications of the simulation desired (initial sales of an item). The simulation model is limited only by the functions programmed into it. To keep the initial model from becoming too complex, certain aspects of the warranty process were simplified. The lifetime distribution function, f(t), of the item has been limited to three choices; the gamma, Weibull, and truncated normal distributions. The gamma 98 and Weibull distributions, through variations in the value of their shape parameters, can represent increasing failure rates, decreasing failure rates, and constant failure rates (through reduction to the exponential distribution). These distributions, along with the truncated normal distribution, have been found to fit failure data for a wide variety of products (Barlow & Proschan, 1975). The lifetime distribution, f*(t), describing perceived (not real) failures has been omitted, thereby also deleting any need for estimates of asterisked probabilities. This reduces the model to one where only real failures result in claims rather than one which includes claims based on perceived failures not generated by the item's true lifetime distribution. The rebate functions have been limited to either lump sum or a linear pro ration. Virtually all authors of consumer-type warranty models have assumed linear pro ration where pro rata rebates are used or, at least, in their numerical examples. This function could be liberalized, but for purposes of comparison of varying lifetime distributions and parameter estimates it would only be a complicating factor. It has also been assumed, in the model, that repair or replacement is instantaneous. Again, this would only be a complicating factor if relaxed. 99 The choice of discounting functions is limited to the two previously introduced, but there is no limit on the discount rate that may be used. The maximum number of replications available is 1000. Programming Considerations of the Simulation Model At the heart of any Monte Carlo simulation model is the random number generator. The model of this research uses four of them. The three which generate the gamma, Weibull and truncated normal distributed random numbers are as described in Law and Kelton (1982), and are as good as the underlying uniform distribution generator on which they are based, since they contain no approximations. The uniform random number generator is as described by Lewis, Goodman, and Miller (1969). It is documented and tested in the aforementioned reference. The simulation program is written in FORTRAN and runs on the DEC20 computer at the University of Hawaii, Manoa Campus, Computer Center. It is menu driven; an example run of the program appears in Appendix B. The program listing is in Appendix C. Comparison of the Simulation Model to the Theoretical Models As a means of validating the simulation model, the theoretical models 3.1, 3.3, and 3.10 were used to calculate expected costs for a specified set of values. 100 Those values were: 1 . lifetime distribution: exponential 2 . E ( X ) : 1.0 3. L: 20.0 (where applicable) 4. W: 0.75 5. t ( 1) : 0.10 (where applicable) 6, rebate lump sum combination or linear pro ration 7. R: 10.0 8 . C: 15.0 9. K: 0 (where applicable) 10. a : 1.0 (where applicable) 11. 9 : 0 (where applicable) 12. 13. 3 : discount: 1.0 (where applicable) e ^ ^ (where applicable) 14. (<J>+ 5) : 0.10 (where applicable) Use of the exponential distribution for the lifetime distribution allows expression of the models in closed form. Table 5 in Chapter III shows the closed form expression of Model 3. 1 when the lifetime distribution i s exponential. The closed form expression of Model 3.3 for an exponential lifetime i s E [H (W) ] = RW/y 101 The closed form expression of Model 3.10 with an exponential lifetime is E[H(L)] = [1 + L/y] [R - Re W/w + 3Re~W/y - Cu/W + Ce_W^W + (Cu/W)e_W^W - 3Ce~W/y] - [3R - BC] The results were compared against the estimated costs generated by the simulation model, with its input parameters set accordingly, using 1000 replications. The model's output is the run shown in Appendix B. These costs are shown in Table 6 . Table 6 Comparison of True and Simulated Mean Costs Analytical Model 3.1: pro rata combination lump sum True Expected Cost 1.797512 2.1847763 5.10695 Simulation Model Mean Cost Std. Dev;, of Cost 1.83 3.532118 2.26 4.003891 5.29 4.821291 p Value for Difference 0.7718 0.5552 0.2302 3.3: 7.5 7.14 8.584524 0.1836 3.10: -6.6060 -7.32 22.42999 0.3124 It appears that the simulated figures are quite close to the exact figures calculated by the analytical models. In fact, the p values show that the smallest level of significance which would cause rejection of any 102 of the null hypotheses of no difference is 0.1837, and the rest are much larger. These p values were found from the standard normal distribution. This distribution applies because the means available from the simulation model are the sample means calculated from large samples, thus enabling the use of the Central Limit Theorem to declare that they come from normally distributed populations of sample means, although the distributions of the individual costs are not known. Use of the Simulation Model for Sensitivity Analysis In order to analyze the sensitivity of expected warranty costs to variations in lifetime distribution form and parameters, data is needed. The simulation model can provide that data. By simply changing the input values to the simulation model, one can generate expected costs for any set of parameters desired, within the constraints of the programmed model. These costs can then be compared to see where differences are significant and conclusions can be drawn about the sensitivity involved. The research design for the sensitivity analysis is the subject of the next chapter. 103 CHAPTER V THE SENSITIVITY ANALYSIS: METHODOLOGY The models developed in Chapters III and IV provide an expected cost to the supplier of offering a warranty of particular terms on an item with a specified lifetime distribution. This expected cost is a function of the supplier's and consumer's costs of a failure, the rebate terms of the warranty, the discounting method and rate, and the lifetime distribution of the item. If any of these component parts are changed, one would expect the result to be a change in expected cost. For instance, going from a lump sum rebate to a pro rated rebate, everything else remaining constant, should lower the supplier's cost, since he is shifting part of the cost of a failure onto the consumer. Similarly, an increase in the discount rate, all else constant, should lower the supplier's expected cost, since a future dollar is worth less at a higher discount rate. These results seem obvious, but what about changes in the lifetime distribution? Suppose one chooses to use the exponential distribution to model the lifetime of a 104 particular item when the true lifetime distribution is normal. Will this cause a substantial over- or underestimation of the expected cost of the warranty 7 Also of interest is the result of misestimating the parameters of the lifetime distribution even if the form is known. Estimates of these parameters must often be made from incomplete failure data; not all failures are reported and/or the lifetime of the item is sufficiently long that many unfailed items are still in service. It would be helpful to the suppplier to know the consequences of such misestimates. No author, to.date, has undertaken to answer these questions. The purpose of this sensitivity analysis is to investigate whether changes in the form or parameters of the lifetime distribution effect significant changes in the expected cost of a warranty. Research Design There are two distinct questions to be answered by the sensitivity analysis: 1. Does the form of the lifetime distribution significantly affect the predicted expected cost of a warranty? 2. What effect do errors in the estimation of the lifetime distribution parameters have on the predicted expected cost of a warranty ? The data required to answer the first of these two 105 questions differs from the data required for analysis of the second. The simulation model introduced in Chapter IV provides the necessary data for both cases, but only when given the proper sets of input values. The selection of these input values and the analysis of the data resulting will be discussed separately for each of the two primary investigations undertaken. Also of secondary interest is the investigation of the effect on the cost of a warranty of changes in the ratios of W to L and W to E(X). The data for this enquiry is generated as a by-product of the investigation of the two primary questions. It is to be remembered that the data collection and analysis described in this chapter and the one to follow are part of the search for a general model which will quantify the costs associated with offering a warranty of specified terms on a product with particular characteristics. Many models have been proposed, and most of these models have found it necessary to make some simplifying assumptions about the lifetime distribution of the product involved. The theoretical models developed in Chapter III can only be stated in closed mathematical form in the case where an exponential lifetime is assumed. However, not all items experience the constant failure rate associated with the exponential distribution. The purpose of this sensitivity analysis is to investigate the effects of changes in the assumptions underlying the 106 theoretical models on the expected costs generated by the models. The data generation processes will now be explained followed by examples of how data points were obtained . Comparison of Expected Costs Under Different Distributional Assumptions In order to compare expected costs under different distributional assumptions it is necessary to equate the distributions to be used with respect to their mean and variance. It is not possible to select mean and variance values for the gamma and Weibull distributions that are equal to those of the exponential without having the gamma and Weibull distributions degenerate into distributions identical to the exponential against which they are being compared. This raises the problem of how to select the parameters for input to the simulation model so that it will generate expected cost data based on lifetimes of a different form but with comparable parameters. In real-world situations, estimation of the form and parameters of lifetime distributions is usually made from failure data (generally incomplete). This same approach is possible through simulation. The problem of input parameter values was solved by using a simulation model to generate failure data from a specified distribution with particular parameters. 107 The same random number generators used in the warranty cost simulation (WRNTY) model were used. First a set of random numbers was generated from a particular probability distribution with specified parameters (these numbers simulate failure data, lifetimes of items), for instance, the gamma distribution with shape parameter A = 1.5 and scale parameter B = 0.33. Then the maximum likelihood estimates of each of the parameters of the generating distribution and three other distributions were calculated from these sample data. There are six distributions from which original failure data can be generated. The desired theoretical mean lifetime of an item determines the input parameters for the distribution which generates the sample data. The desired theoretical mean lifetime used in this study was 1.0 , and the sample size was set at 1,000 simulated lifetimes. The distributions are: 1. the exponential distribution 2 . the gamma distribution with an increasing failure rate (IFR) 3. the gamma distribution with a decreasing failure rate (DFR) 4. the Weibull distribution with an IFR 5. the Weibull distribution with a DFR 108 6 . the truncated normal distribution For each of the six sets of failure data, the maximum likelihood estimators of the parameters of the exponential, truncated normal, gamma, and Weibull distributions were calculated. The results constitute input parameters to the WRNTY model. generate the simulated expected cost to the supplier of offering a warranty of particular terms based on each of the fitted lifetime distributions listed above. to the simulation model were selected to eliminate terms such as a (the probability of a valid claim being made) and 0 (the probability of a claim not being validated) which were mere modifiers of the final expected cost. The profile of the simulation model used for data generation for this portion of the sensitivity analysis is: 1. lifetime The next step was to use the WRNTY model to Setup of the simulation model. The fixed inputs distribution: variable 2. E(X): 1.0 3. L: (life cycle) 2 W, 20 W 4. W: (warranty length) 0.75, 1.0, 1.25 5. t(l): (length of first phase of a combi nation warranty) 0 109 6. warranty type: renewing, pro rata nonrenewing, lump sum 7. R: (supplier’s cost) 10 8 . C: (consumer's cost) 1.2R, 1.6R 9. K: (administrative costs) 0 10. a : (probability that a claim is made) 1.0 11. 0 : (probability that a claim is not validated) 0 12. 0 : (probability of sales resulting from failures after the end of the warranty) 0 , 1.0 13. discounting: none 14. re plications: 100 Since we are interested only in comparisons between costs and not in the actual costs generated, the values set for L, W, R, and C are immaterial. It is their ratios which are important. This simplified model is ei ther a renewing, pro rata warranty or a nonrenewing, lump sum warranty, where all claims are made and validated, there is no discounting, and new sales may or may not occur after the expiration of the warranty period. 110 Figure 2 is a tree diagram of the non-lifetime distribution input parameters to the simulation model. The form of the lifetime distribution is varied at each of the levels of the life cycle, L, the warranty length, W, the consumer’s cost, C, and each of the two forms of warranty type for both assumptions about follow-on sales, to yield the results for the following comparisons: 1. exponential vs. gamma Weibull truncated normal 2. gamma (IFR) vs. exponential Weibull truncated normal 3. gamma (DFR) vs. exponential Weibull truncated normal 4. Weibull (IFR) vs. exponential gamma truncated normal 5. Weibull (DFR) vs. exponential gamma truncated normal 111 0=1«2R L=2ll/ , n C=1.6R W=0.75 0=1.2R “XI L=20li) u 0=1.SR □ 0=1.2R L=2ld . " O 0=1.BR SALES W=1.0 0=1.2R u AFTER l i J L=20U u 0=1.BR ' .T3 0=1.2R L=2lii U 0=1.BR Ui=1 .25 L I 0=1.2R L=20U D 0=1.BR PRO RATA 0=1.2R ID L=2U L) 0=1.BR li/=0.75 0=1,2R " D L=20lil LJ 0=1,6R U 0=1.2R L=2UI U 0=1,6R IMO SALES td=1.0 0=1.2R □ L=20UJ n 0=1.6R u 0=1.2R L=2li/ □ .0=1.6R lil=1.25 0=1.2R L J L=20UJ U 0=1.BR U W=0.75 L I LUP1P SUM U=1 .0 L i L)=1.25 o Figure 2. Non-lifetime Distribution Input Parameters 112 6 . truncated normal vs. exponential gamma Weibull The first distribution listed is the generating distribution used to obtain the failure data (true distribution). The three distributions which are compared against it are those of different form fitted to the failure data generated by the first distribution (fitted distributions) . The mean and standard deviation of the simulated expected cost for each combination of WRNTY model input parameters are then analyzed. A significance level of 0.05 was selected for all hypothesis tests. The hypotheses to be tested are: V WF = U0F Ha : yF ^ U0F where: Up = the mean cost when the parameters of the fitted distributions are used, e.g., input truncated normal distri bution with MU = 1.013840 and SG = 1.011606 113 Uqp = the mean cost when the parameters of the fitted version of the originating distribution are used, e.g., input distri bution exponential with B = 1.013840 There are eighteen tests to be performed for each set of WRNTY simulation model inputs exclusive of distribution type and distribution parameters (e.g., warranty type, W, L, etc.). For each of the six distributions used to generate failure data (exponential, truncated normal, gamma dfr, gamma ifr, Weibull dfr, Weibull ifr), four simulated expected costs were generated based on the form and fitted parameters of the exponential, truncated normal, gamma, and Weibull distributions. Each of the simulated expected costs resulting from the input of the fitted parameters of each of the distributions not used to generate the sample data (fitted distributions) is hypothesized to be equal to the simulated expected cost resulting from the input of the generating distribution’s fitted parameters (the true form of the lifetime distribution). If this hypothesis is rejected one can conclude that the form of the lifetime distribution has a significant effect on the magnitude of the expected cost of the warranty. 114 Example of data collection for comparison of expected costs under different distributional assumptions. The following is an example of how data was generated to answer the first of the two questions posed at the beginning of this chapter - does the form of the lifetime distribution significantly affect the predicted expected cost of a warranty? The first two steps need be performed only once for each desired mean lifetime. The resulting parameter estimates then become input for all subsequent runs of the WRNTY model program which does the actual generation of data for analysis. 1. A theoretical mean lifetime for the item to be warrantied is chosen, such as 1.0 . 2. Six sets of sample data are simulated, selecting, in turn, each of the distributions listed below as generating distribution for the sample. The A (shape parameter) values for the gamma and Weibull distributions were selected to insure that both the decreasing and increasing failure rate forms of those distributions would be represented. It was decided to set the standard deviation (SG) of the truncated normal to one-third of its mean to keep that distribution as nearly normal in shape as possible. After fixing A and SG, the rest of the parameters used to generate the sample data were determined by the selection of the desired theoretical 115 mean lifetime, i.e., selecting a desired mean of 1.0 forces the exponential parameter B to be input as 1.0. Distribution Parameter s exponential B = 1.0 truncated normal MU= 1.0, SG= 0.3333 gamma DFR A = 0.5, B = 2.0 gamma IFR A = 1.5, B = 0.6667 Weibull DFR A = 0.5, B = 0.5 Weibull IFR A = 1.5, B = 1.1077 Table 7 is an example of the worksheet used to collect the maximum likelihood estimates of the various distributions' parameters for input to the WRNTY simulation program. The purpose of these first two steps was to obtain input parameters for the WRNTY model which would allow the comparison of simulated costs, varying only the form of the distribution and not the expected lifetime. By using the same sample data to which to fit each distribution, the results can be compared without the confounding effects of differing parameters. We know the lifetime distributions are comparable because they are based on exactly the same sample data. 3. The third step, to which the first two have led, is to actually generate the data for analysis. The data consist' of simulated expected costs to the supplier of servicing his warranty committments over the length of 116 Table 7 Example PILE Worksheet THEORETICAL MEAN =1.0 EXPONENTIAL GENERATOR GENERATING DISTRIBUTION exponen. MLE FDR PARAMETERS OF FITTED DISTRIBUTION gamma Ueibull tr normal 0 = 1 D = 1 D = 2 D = 3 D = 4 B = 1 B = 1.013840 A = 1.0673010 A = 1.032567 MU = 1.013840 B = 0.9499104 B = 1.027135 SG = 1.011606 TR NORMAL GENERATOR GENERATING DISTRIBUTION tr normal MLE FOR PARAMETERS OF FITTED DISTRIBUTION exponen gamma Ueibull D = 4 D = 4 D = 1 D = 2 D = 3 MU = 1 MU = 0.9951098 B = 0.9951098 A = 7.290287 A = 3.290662 SG = 0.3333 SG = 0.3309437 B = 0.136498 B = 1.106672 117 Table 7— Continued GAMMA DFR GENERATOR MLE FOR PARAMETERS OF FITTED DISTRIBUTION GENERATING -------------------- DISTRIBUTION gamma exponen lileibull tr normal 0 = 2 . D = 2 D = 1 D = 3 D = 4 A = 0.5 A = 0.4810658 B = 0.9569498 A = 0.613596 MU = 0.9569498 B = 2.0 . B = 1.989228 B = 0.674543 SG = 1.41617 GAMMA IFR GENERATOR MLE FOR PARAMETERS OF FITTED DISTRIBUTION GENERATING ------------------- - DISTRIBUTION gamma exponen bJeibull tr normal D = 2 D = 2 D = 1 D = 3 D = 4 A = 1.5 A = 1.464641 B = 0.9826341 A = 1.252239 MU = 0.9826341 B = 0.6667 B = 0.6709043 B = 1.056545 SG = 0.8078971 118 Table 7— Continued WEIBULL DFR GENERATOR MLE FOR PARAMETERS OF FITTED DISTRIBUTION GENERATING -------------------- DISTRIBUTION Weibull exponen gamma tr normal D = 3 D = 3 D = 1 D = 2 D = 4 A = 0.5 A = 0.4919837 B = 1.022308 A = 0.354235 MU = 1.022308 B = 0.5 B = 0.5007594 B = 2.8B5962 SG = 2.234603 a-###*#####*##*# WEIBULL IFR GENERATOR MLE FOR PARAMETERS OF FITTED DISTRIBUTION DISTRIBUTION Weibull exponen gamma tr normal D = 3 D = 3 D = 1 D = 2 D = 4 A = 1.5 A = 1.516497 B = 0.9826432 A = 1.974693 MU = 0.9826432 B = 1.1077 B = 1.090317 B = 0.497618 SG = 0.6638593 119 the warranty or over the life cycle of the product, depending on how the model is set up, as well as the simulated standard deviations of these costs. These costs are based upon certain assumptions about the form and parameters of the lifetime distribution, the warranty terms and the costs involved in claim satisfaction. Carrying our example further, the fitted parameters shown in’Table 7 were used as input to the WRNTY model to obtain the results shown in Table 8 . The warranty modeled is a renewing, pro rata warranty without discounting. The consumer's cost represents a 20% markup from the supplier's costs, and follow-on sales if an item fails after the warranty period are not assumed. Figure 3 is a tree diagram illustrating the input parameters used in this example as a subset of the full set of input parameters used in the final data generation. 120 Table 8 Example WRNTY Worksheet FITTED DISTRIBUTION mut DISTRIBUTION exponen. gamma Weibull tr normal exponential! • • • • • • mean = 4.08 * • 4.81 5.00 : 1.38 std. dev. = 7.01 BO • • 6.2006 ! 8.9856 : 3.1287 Z = • • 0.78 • • 0.81 ! -3.51 P = : 0.4354 • 0.4180 ! 0.0004 tr normal! • • : • • mean = 0.37 5.13 : 0.12 • • 0.30 • std. dev. = 1.6677 7.1980 • • 1.2691 • • 1.3150 • ■ Z = 6.44 : -1.19 : -0.33 : P = 0.0000 5 0.2340 : 0.7414 : gamma DFR: : : : mean = 10.14 6.02 • • 14.46 ! 1.03 std. dev. = 12.1675 7.8648 • • • 17.0718 ! 2.6289 Z = -2.84 • : 2.06 ! -7.32 P = 0.0046 : : 0.0394 ! 0.0000 gamma IRF: : • « • « mean = 3.67 4.81 ; • 2.77 ! 1.19 std. dev. = 5.5916 8.9744 : • « 4.6748 : 2.9144 Z = 1.08 • : -1.24 1 -3.94 P = 0.2802 • * • 0.2150 : 0.0000 * 121 Table 8— Continued Ibull DFR: » • • • mean = 14.76 4.23 : 14.01 • • ! 1.29 std. dev. = 18.4701 6.0551 • 16.3792 • : 3.2239 Z = -5.42 • -0.30 • • s-7.18 P = 0.0000 : 0.9760 : : 0.0000 Weibull IFF?! : : : mean = 1.93 4.91 : 2.54 : ; 1.15 std. dev. = 4.2330 7.0952 : 4.5299 : : 3.2367 Z = 3.61 ! 0.98 : s-1.46 P = 0.0004 : 0.3270 : : 0.1442 122 GAMMA 3 < m c n rt H- 00 05 r+ H* O 3 3j H- 00 3 3 n) CO H 3 >3 3 rt 13 05 3 3 g (0 rt n> 3 05 Hi O 3 « h > ‘ V) rt 3 H1 cr 3 rt H- O 3 05 EXPONENTIAL TRUNCATEn EXPONENTIAL NEIBULL PRO RATA NO SALES C=1.2R AFTER 1 1 1 SALES C=1.6R AFTER 1 1 1 ' liM.25 iAMMA lilEIBULL IFR K 5 U 5 4. The hypotheses to be tested at the 0.05 level of significance are: H : u u o trnorra exp H : u 4 - U a trnorm exp H : u y o gamma exp H : u ^ U a gamma exp o Weibull exp H : Wtt • u i ^ W a Weibull exp Table 8 shows the test statistics calculated from the sample data (simulated mean cost and standard deviation of cost) and the resulting p-values. The analysis of these results will be presented in the next chapter. Investigation of Expected Costs at Different Parameter Levels of a Fixed Lifetime Distribution The generation of the data necessary to investigate the second question stated in the Research Design section of this chapter is straightforward. In order to investigate what effect errors in the estimation of the lifetime distribution's parameters have on the predicted expected cost of a warranty, expected costs are generated with the WRNTY simulation program holding the form of the lifetime distribution constant while selecting various levels of the distribution's parameters for input to the simulation model. 124 The distributions1 input parameters were set singly and in combination (where a distribution has two parameters) at five levels: 75%, 90%, 100%, 110% and 125% of the true parameter value. A simulated expected warranty cost was then generated at each level. Figure 4 illustrates the parameters input to the WRNTY simulation program for the truncated normal distribution as a subset of the full set of input parameters used in the final data generation. Although there is no theoretical reason to assume a linear relationship between the lifetime distributions’ parameters and the simulated expected costs on the basis of the form of the cost model, preliminary scatterplots of the data exhibited sufficiently strong linear characteristics to justify the use of least squares linear regression techniques; for analysis of the data. It is necessary to realize that the regression equations obtained are.dependent upon the sample data used, and therefore do not provide general statements about the amount of change to be expected in the mean cost for specified changes in the lifetime distributions' parameters for all warranties. With the possible violations of the model’s assumptions in mind, the correlation coefficients will be allowed to speak for themselves. Each of the six lifetime distributions’ 75 percent 125 u :XPONENTIAL PRO RATA W O SALES AFTER U 50* 0 .3 6 6 6 SALES GAPW fl IFR UigXBULLOFR ’igure 4. Input Parameters for the Truncated Normal 126 through 110 percent parameter values were used as the independent variables in regression analyses with the simulated expected costs generated at each level as the dependent variable. The resulting regression equations indicate how, for this data, expected costs change with changes in the input parameters. The correlation coefficients indicate the strength of the linear relationship between lifetime distributions’ parameters and expected warranty costs within this data. The regression program used was QSTAT which runs on the Hewlett-Packard 3000 at the School of Business of the University of Hawaii. A significance level of 0.05 was chosen for tests of the contributions of the independent variables as well as the test of the strength of the overall model. The assumptions of linearity and homoscedasticity were checked by examination of plots of residual values, while the correlation coefficients of the independent variables with each other were used to rule out autocorrelation. Setup of the simulation model. The fixed inputs to the simulation model were, again, selected to eliminate terms which were modifiers of the final expected cost. The profile of the simulation model used for data generation for this portion of the sensitivity analysis i s : 127 1 . li f e time distribution: exponential, gamma IFR, gamma DFR, Weibull IFR, Weibull DFR, truncated norma1 2 . par ame te r s : variable 3. E(X): 1.0 4. L: 2 W, 20 W 5. W: 0.75, 1.0, 1.25 6 . t ( 1) : 0 7. warranty type: renewing, pro rata nonrenewing, lump sum 8 . R: 10 9. C: 1.2R, 1.6R 10. K: 0 11. a : 1.0 12. 0 : 0 13. 8 : 0 , 1.0 14. discounting: none 15. replications: 100 Since we are still interested only in comparisons between costs and not in the actual costs generated, the values set for L, W, R, and C are immaterial. It is their ratios which are important. This is still the simplified model of a renewing, pro rata warranty or a nonrenewing, lump sum warranty where all claims are made and validated. 128 There is no discounting, and new sales may or may not occur after the expiration of the warranty period. The parameters of the six lifetime distributions were set at each of the five levels noted previously for every -combination of WRNTY model inputs specified in the profile to generate data for the regression analyses. The mean and standard deviation of the simulated expected cost for each combination of the WRNTY model input parameters were then collected for input to the QSTAT program, and linear regression equations and correlation coefficients were obtained. The analysis of the resulting data is the subject of the next chapter. Example of data collection for investigation of expected costs under different parameter estimations. The following example shows how data was generated to answer the second of the two questions posed at the beginning of this chapter - what effect do changes in the estimates of the parameter values of the lifetime distribution have on the predicted expected cost of a warranty? The first step was to run the WRNTY simulation program with the various distributions and their 75 percent to 110 percent parameters for all combinations of warranty type input parameters and collect the resulting simulated expected cost data on a worksheet like the example shown in Table 9. This example represents a renewing, pro rata warranty with no sales occuring if an 129 Table 9 Example PRMTR Worksheet TRUNCATED NORMAL 755? 0.25 90# 0.30 SG 1005? 0.3333 1105? 0.3666 125$ 0.4166 755? 0.75 905? 0.90 100# 1.0 1105? 1.10 1255? 1.25 mean= 3.54 sd= 3.8548 4.26 5.7319 3.85 4.9413 4.22 5.7302 5.29 6.8875 mean= 1.07 sd= 2.5402 2.37 3.7853 1.94 3.8344 2.05 3.5705 1.90 3.7466 mean= 0.26 sd= 1.5887 0.60 ** 1.07 ** 0.82 #*_____ 2.4152 #* 2.6604 ** 2.3663 : 3.4B03 1.58 mean= 0.04 sd= 1.0807 0.25 1.6202 0.44 1.6630 0.95 2.2462 0.80 1.9350 mean= -0.00 sd= 0.3788 - 0.02 0.9165 -0.08 0.5938 0.56 1.8212 0.31 1.5829 130 item fails after the expiration of the warranty period and a 20 percent markup in price from supplier to consumer. Figure 4 illustrates where this example data fits into the overall data collection. Data points were generated by running the WRNTY model with all inputs held constant except for variations in the parameters of the lifetime distribution, MU and SG for this example. Thus, the inputs which produced the ”75% value" in Table 9 were the same as those which produced the "100% value" with the exceptions that MU = 75% of 1.0, or 0.75, and SG = 75% of 0.3333, or 0.25. .Similar modifications in the input values of MU and SG were made to produce the other example values shown. The analysis of this data is accomplished by the use of the linear regression subprogram in the QSTAT .program mentioned above. Using the simulated expected costs as the dependent variable and the parameter values as the independent variables results in an equation describing the mathematical relationship between these expected warranty costs and the 75 percent to 110 percent parameters of this truncated normal lifetime distribution. From this equation the effect, on cost, of changes in lifetime distribution parameter values can be inferred, with the reservations noted at the beginning of this chapter concerning the dependency of the results on the sample data and violations of the assumptions of the 131 linear regression and correlation models. The preceding example data in Table 9 was as : input to the regression model Y X(l) X(2) 3. 54 0. 75 0.2500 4.26 0. 75 0.3000 3.85 0.75 0.3333 4.22 0.75 0.3666 5.29 0.75 0.4166 1 .07 0.90 0.2500 2.37 0.90 0.3000 1.94 0.90 0.3333 2.05 0.90 0.3666 1.90 0.90 0.4166 0 . 26 1.00 0.2500 0 . 60 1.00 0.3000 1.07 1.00 0.3333 0.82 1 . 00 0.3666 1. 58 1 . 00 0.4166 0.04 1 . 10 0.2500 0.25 1.10 0.3000 0.44 1.10 0.3333 0.95 1.10 0.3666 0.80 1.10 0.4166 -0.00 1.25 0.2500 -0.02 1. 25 0.3000 132 -0.08 1.25 0.3333 0.56 1.25 0.3666 0.31. 1.25 0.4166 The results of the regression analysis were checked to establish'whether or- not a significant (at the 0.05 level of significance) linear relationship existed, and each independent variable was tested for its contribution to the explanation of variation in costs (significant at the 0.05 level). An examination of the correlation between the independent variables and a plot of the residuals was made. Table 10 shows the results of the QSTAT analysis of the above data from Table 9. Finally, the correlation coefficient and the appropriate model of the mathematical relationship between warranty cost and lifetime distribution parameter values were collected. The presentation of the results and the conclusions drawn from them are the subject of the next chapter. 133 Table 10 Example Regression Run DEPENDENT VARIABLE: COLUMN 1 (EXPECTED COST) INDEPENDENT VARIABLES: COLUMNS 2, 3 (MU, SG) DEP VAR 1 2 3 1 1.000 -.862 .332 2 -.862 1.000 .DOO 3 .332 .000 1.000 IV BETA/SE RAW B / SE 2 2 r uiith . . . COL t P<= sr pr Y IV 2— -.8622 -4.82621 -10.576 .001 .7434 .8356 -.8622 .0000 .082 .456338 3— .3323 5.58245 4.076 .001 .1104 .4302 .3323 .0000 .082 1.36968 INTERCEPT 3.89478 For each t, df = 22 MULTIPLE OVERALL DF R R-squared F num den P<= .9240 .853EI 64.231 2 22 .001 RESIDUALS VS. PREDICTED EXPECTED COST (Y) 1.0 I:::: ;:I + + + + 2 + + + + + -1.0 I: -1 :I: 0 134 Table 10— Continued VALUES IN COL. 1 (EXPECTED COST) VS. VALUES IN COL. 2 (NU) 7.5 I::::I::::I::::I::::I::::I::::I::::I::::I::::I::::I 5.0 2.5 + 2 2 2 .0 2 2 + + 2 + + 3 -2.5 .5 1 1.5 2 2.5 3 VALUES IN COL. 1 (EXPECTED COST) VS. VALUES IN COL. 3 (SG) 7.5 I : : : I : : : I : : : : I 5.0 -2.5 I: .2 .25 .3 .35 .4 .45 135 CHAPTER VI THE SENSITIVITY ANALYSIS: RESULTS The two major questions addressed by this sensitivity analysis were: (1) does the form of the lifetime distribution significantly affect the predicted expected cost of a warranty, and (2) what effect do errors in the estimation of the lifetime distribution parameters have on the predicted expected cost of a warranty? The answer to the first question is yes, for certain distributions. Notably, if the lifetime distribution is assumed to be exponential when in fact it is normal, gamma ifr, or Weibull, warranty costs may be significantly over- or underestimated, while, if the lifetime distribution is assumed to be gamma or Weibull, these distributions will not result in substantial misestimations of cost for the case where the true lifetime distribution is exponential, gamma, Weibull or normal. The truncated normal distribution, however, resulted in misestimations of cost when it was assumed to fit data that were actually exponentially distributed, distributed as gamma, either dfr or ifr, or Weibull dfr. 136 The detailed presentation of these results is the subject of the second section of this chapter. The answer to the second question is that changes in the lifetime distributions' parameters have substantial effects on costs in most cases. Strong inverse relationships exist, within this sample data, between warranty cost and the exponential parameter B, the truncated normal parameter mu, the A and B parameters of both the decreasing and increasing failure rate cases of the gamma distribution, and the B parametr of the Weibull distribution wih either a decreasing or an increasing failure rate. A further discussion of these results can be found in the third section of this chapter. The motivation for this sensitivity analysis was presented in the final section of Chapter I. It is reviewed as each question is discussed separately in the next two sections of this chapter. The generation of the data which led to the conclusions stated in this chapter was explained in detail in Chapter V. It also will be briefly reviewed in the following two sections. The final section of this chapter deals with the answers to questions of secondary interest noted in Chapter V. Investigation of Distributional Assumptions In Chapter I, it was noted that the assumption of an exponentially distributed lifetime greatly simplifies the calculation of warranty costs with mathematical 137 models. In Chapter III, the fact that closed form expressions of mathematical models for warranty costs are only possible using the exponential distribution for the lifetime was mentioned. Chapter III also highlighted, in Table 1, how many of the authors of mathematical warranty models have assumed an exponential lifetime when stating their models. The question which naturally arises is whether there is a significant loss of accuracy (as measured by the difference between the expected cost under the true distribution and the expected cost under the assumed distributon) in cost prediction attendant upon this assumption of an exponentially distributed lifetime. To answer the question, expected costs were generated from each of five lifetime distributions (truncated normal, gamma dfr, gamma ifr, Weibull dfr, and Weibull ifr). Then five expected costs were generated under the assumption that the lifetime was exponentially distributed. In order to make the assumed exponential distribution comparable in each case with the true distribution against which it was being compared, its parameter was estimated from the same sample failure data which provided the estimates of the parameters of the true distribution. Selection of the parameters of these various distributions was discussed fully in Chapter V. The mean costs were compared, testing to see if the cost generated under the exponential lifetime assumption was significantly different from each 138 of the five costs generated from non-exponential di stributions. It should be remembered through all analysis of these results that references to the gamma and Weibull distributions' decreasing and increasing failure rate types are specific to only one case representative of each type, these cases being shape parameter A is 0.5 (decreasing failure rate case) and 1.5 (increasing failure rate case). For this reason, these results may not be generalizable to all decreasing or increasing failure rate forms of these two distributions. To carry the question about the accuracy of assumed distributions further, the same test for equality of means was conducted on each of the other three distributional assumptions (truncated normal, gamma, and Weibull). It was unknown if the non-distributional parameters of the warranty (i.e., L/W, R/C) would affect the results of these comparisons. Therefore, these non-distributional parameters were varied over different levels, and multiple tests were made on each distributional assumption. There were a total of 27 tests, representing 27 different warranty cases, done for each of the four assumed distributions. Figure 2, in Chapter V, is a tree diagram of the parameter values which led to these 27 different cases. The data exhibited the same patterns for all 139 levels of the non-distributional parameters. For example, neither the ratio of L to W nor the form of the rebate function, lump sum versus pro rata, had any effect upon the closeness of estimated costs to true costs under different distributional assumptions. Table 11 shows the condensed results of this entire section of the sensitivity analysis. The data used to compile Table 11 are in Appendix D. Table 11 contains the relative frequencies with which p-values less than 0.05 were obtained for the sample data in each of the classifications, where relative frequency is calculated by ( number of tests resulting in a p-value < 0.05)/27. These p-values represent the probability of obtaining a sample result at least as extreme as the one calculated from the data, from the sampling distribution resulting from the null hypothesis of equality between the two mean costs. The null hypothesis is that the mean cost generated under the assumed distribution is equal to the mean cost generated by the true distribution. A two tailed test was•performed at the 0.05 level of significance. The percentage values in Table 11 are the proportion of tests, out of the total of 27, which resulted in rejection of this null hypothesis, thus, leading to the conclusion that the assumed distribution generated a cost significantly different from the true cost. 140 Table 11 Condensed Results of Sensitivity Analysis Of Distributional Assumptions Percentage of p-values < 0.05 Assumed Distribution True Distribution Exponential Truncated Normal Gamma Weibull Exponential 100# 11# 15# Truncated Normal 100# 19# 4# Gamma DFR 89# 100# 15# Gamma IFR 52# 93# —4 C O Weibull DFR 100# 100# 37# Weibull IFR 93# 74# C D ■ 2 * 141 The sample results in Table 11 support the conclusion that, for these sample data, the assumption of an exponentially distributed lifetime results in estimated warranty costs significantly different from the true warranty cost (the cost to be expected under the true lifetime distribution form) when the true lifetime distribution is truncated normal, Weibull, or gamma with a decreasing failure rate. However, the fact that the assumption of an exponential lifetime frequently yields an expected cost not significantly different from the expected cost generated by a gamma distributed lifetime with an increasing failure rate is not to be ignored. These results are not unexpected in light of an examination of the theoretical expectations associated with a study of the characteristics of the lifetime distributions involved. Figure 5 shows the failure rate curves for the exponential, normal, gamma and Weibull distributions with means of 1.0, which is the mean lifetime used in this study. The conditional failure rate at time t for a specified distribution is defined as r(t) = f(t)/(l - F(t)) wher e r(t) is the conditional failure rate at time t f(t) is the density function of the specified life distribution 142 Failure Rate 4.0 3.5 - Weibull A=1.5 3.0 2.5 _ 2.0 - ^ normal SG=0.3333 exponential t mt 0.5 Weibull Time Failure Rate Curves for Selected Distributions Figure 5 With Means Equal to 1.0 143 F(t) is the distribution function of the specified life distribution A look at the failure rate curves for the particular distributions used in this study (Figure 5) reveals that the gamma distribution for both the, decreasing and increasing failure rate cases more closely approximates the constant failure rate of the exponential distribution than do the Weibull or normal distributions’ failure rate curves. Further analysis of Table 11 shows that the assumption of a truncated normal distribution also results in a high incidence of misestimation of cost for all of the true distributions. The relative frequency of misestimation is lowest when the true lifetime distribution is Weibull with an increasing failure rate. Again, comparison of the failure rate curves for these distributions reveals that the normal distribution has an increasing failure rate, and, therefore, would be expected to provide the best approximation when the true distribution was also one with an increasing failure rate. Additionally, the particular Weibull distribution used in this study to represent the increasing failure rate case of the Weibull appears closer to the normal distribution in a plot of its density function than does the corresponding gamma distribution selected to represent the gamma with an increasing failure rate. This result is, of 144 course, specific to the gamma and Weibull distributions with shape parameter A = 1.5. Both the gamma and Weibull distributions, in either the decreasing failure rate cast or the increasing failure rate case, appear to yield close approximations to the true cost for any of the true distributions studied. This is not surprising, since, as distributions with shape parameters, they are more flexible in their ability to adapt to other distributional forms. It is to be expected that they would both yield good results when used as approximations to a true exponential distribution, since both the gamma and Weibull distributions reduce to the exponential distribution when their shape parameters are equal to 1.0. We see that the Weibull provides a better cost estimate than the. gamma when the true distribution is truncated normal. Further examination of the failure rate curves for the gamma and Weibull distributions indicates that it is to be expected that the gamma would provide better cost estimates compared to a true Weibull distribution for the increasing failure rate case than for the decreasing failure rate case. This is supported by the results shown in Table 11. In conclusion, it appears that the choice of the exponential distribution as an assumed lifetime distribution results in such a high probability of an over- or underestimated cost that it is not justified. 145 The Weibull would be a better all-purpose distribution for assumption because of its greater flexibility which should result in closer approximations to true costs. However, the Weibull distribution is not as tractable mathematically as the exponential distribution and would not allow closed form expressions of mathematical warranty cost models. These results should be applied with the caution that only one case of the decreasing failure rate, shape parameter A = 0.5, and one case of the increasing failure rate, shape parameter A = 1.5, for each of the gamma and Weibull distributions were studied. A study of the failure rate curves indicates, that as these shape parameters approach 1.0, the exponential distribution should yield cost estimates closer to the true costs, and the converse would also be true. Having concluded that the exponential lifetime distribution will result in misestimation of true costs, the direction of those misestimations is of interest. Does it consistently over- or underestimate? Theory indicates that this will depend on the form of the true distribution to which the exponential assumption is being compared. It is to be expected that the exponential distribution, with its constant failure rate, would overestimate costs when the true distribution is of the increasing failue rate type and underestimate costs when 146 the true distribution is of the decreasing failure rate type. Table 12 displays the results of comparisons of approximated costs, when the various distributions are assumed for lifetimes, with true costs generated by the truncated normal, gamma ifr, gamma dfr, Weibull ifr, and Weibull dfr distributions. Again, the percentage figures represent the relative frequency with which a particular result was found. A value of 100% means that all of the 27 comparisons of costs generated by the assumed distribution under- or overestimated the cost generated by the true distribution. Again, the sample results support the theory. The exponential distributional assumption results in consistent overestimations for the truncated normal, gamma ifr, and Weibull ifr distributions, while it yields consistent underestimates in the cases where the true distribution is gamma dfr or Weibull dfr. It is interesting to note that the truncated normal, as an assumed distribution, always results in the underestimation of warranty cost. Underestimation is to be expected in cases where the truncated normal distribution is used as an approximation to a distribution with a constant or decreasing failure rate, however, the underestimation shown here for the increasing failure rate cases may be specific to the particular value of the shape parameter selected to represent the increasing failure rate case for the gamma and Weibull distributions. It is 147 Table 12 Over- and Underestimations Under Different Distributional Assumptions Percentage of Over- and Underestimates Assumed Distribution True Distribution Exponential Truncated Normal Gamma Weibull Exponential 100$ Under 7$ Under 93$ Over 67$ Under 33$ Over Truncated Normal 100$ Over 44$ Under 56$' Over 55$ Under 45$ Over Gamma DFR 100$ Under 100$ Under 26$ Under 74$ Over Gamma IFR 4$ Under 96$ Over 100$ Under 63$ Under 37$ Over Weibull DFR 100$ Under 100$ Under 89$ Under 11$ Over Weibull IFR 100$ Over 100$ Under 30$ Under 70$ Over 148 possible that the approximation would improve as the value of the shape parameter increases, particularly in the case when the Weibull is the true distribution. The next section addresses the second question posed in this sensitivity analysis: what effect do errors in the estimation of the lifetime distribution's parameters have on the expected cost of a warranty? Investigation of Parameter Impact In order to use any mathematical or simulation model of warranty costs, it is necessary to input the parameter values of the lifetime distribution in some form, even if only to state the mean of the distribution. In practice, these parameters are usually estimated from incomplete failure data. Much has been published regarding the estimation of these lifetime distribution parameters. It is not the purpose of this study to discuss the estimation technique, but to investigate the effects of errors in the estimations as they affect estimated warranty costs. To provide the data necessary for this investigation, warranty costs were generated from each of six lifetime distributions (exponential, truncated normal, gamma with decreasing failure rate, gamma with increasing failure rate, Weibull with decreasing failure rate, and Weibull with increasing failure rate) at varying levels of their parameters’ values. Again, it should be noted that the class of decreasing failure rate 149 distributions for the gamma and Weibull is represented by setting the shape parameter A to 0.5, and the class of increasing failure rate distributions by setting the shape parameter A to 1.5. Various other non-distributional inputs to the warranty model (i.e., W/L and R/C) were also considered, since their impact on the relatonship between warranty cost and parameter value was unknown. The levels of these non-distributional parameters are shown in Figure 2, in Chapter V. They are the same as those used in the first part of this sensitivity analysis. A total of 27 sets of warranty costs were generated for each combination of parameter values under each of the six distributions. Figure 4, in Chapter V, shows an example of the settings for parameter levels for one of the six distributions used. Chapter V presents a complete discussion of the non-distributional parameter levels used and the combinations of lifetime distribution parameters selected. The first step in analyzing this data was to plot the warranty costs generated at each of the different lifetime distribution parameter level combinations. Each of the parameters of the lifetime distributions was set at five levels: 25% under the true value, 10% under the true value, the true value itself, 10% above the true value, and 25% above the true value. Since the exponential distribution is a single parameter distribution, this resulted in 27 plots of only five points. However, each 150 of the other five distributions has two parameters, which results in 25 data points for plotting. The exponential is the only case for which a closed form statement of the model is obtainable. Analysis of this mathematical statement of the model reveals that the relationship between warranty cost and the exponential parameter B is not linear. Furthermore, there is no reason to believe that the relationship between warranty cost and the parameters of any of the other lifetime distributions used is linear. However, examination of the plots of the sample data showed that, over this range of values of the distributions’ parameters, the variables did exhibit a nearly linear relationship. Consequently, a least-squares linear regression line was fit to each set of sample data. The resulting high R-square (coefficient of determination) values support the assumption that deviation from linearity is small in this sample data. Plots of the residual values for these regressions do exhibit some indication of an underlying nonlinear relationship for all of the exponential cases, but this is not the case for the 2 other five lifetime distributions. When smaller R values were obtained, examination of plots of the 2 variables involved showed that the lower R was indicating a lack of any recognizable form of relationship between the variables, not merely a lack of linearity. The assumption of homoscedasticity which underlies true 151 linear regression analysis is violated to some extent in most of these regressions, since the variance of warranty costs seems to increase along with the mean cost. No inferential techniques were used on these sample regression results, and extreme caution should be exercised in generalizing from this sample to a larger population. These regression results are specific to the distributional and non-distributionl parameters used in this study. The intent of this analysis is not to provide a predictive model for all cases, but to indicate how misestimations in lifetime distribution parameters can result in over- or underestimation of warranty costs, which may be significant in any given situation. Pursuant to this objective, Table 13 shows the relative contribution of each of the lifetime 2 distributions’ parameters to the overall R value for each of the 27 regressions, as well as the slopes associated with each of these parameters. In the case of the two-parameter distributions, the sum of these two 2 R values for each set of parameters within a 2 specified distribution equals the total R for the multiple regression. Since the independent variables are 2 independent of each other, these are also the R values that would be observed in a simple regression using only the single parameter as the independent variable. As mentioned, no tests of significance were performed on this 152 Table 13 Results of Investigation of Relationship Between Warranty Cost and Lifetime Distribution Parameters Lifetime Distribution Parameters Warranty Case Exponential B Truncated Normal mu SG R* SLOPE R* SLOPE Ra SLOPE I i im n Qi im M n n r o n o i i i i n o _ _ _ _ _ _ _ _ _ L-Ultt(-> O U III9 l\IUI IL c l IcUJ-LI l y Warranty = 0.75 0.86 - 13.00 , 0.89 - 9.50 0.04 5.98 Warranty =1.0 0.94 - 12.86 0,91 - 13.46 0,00 0.37 Warranty = 1.25 0.77 - 11.79 0.94 - 14.68 0.00 - 1.01 Pro Rata, Renewing Sales After W, Beta =1.0 Warranty = 0.75 Lifecycle = 2W C = 1.2R 0.70 - 8.14 0.63 - 4.56 0.17 7.04 C = 1.6R 0,46 - 4.20 0.04 - 0.80 0.48 8.66 Lifecycle = 20W C = 1.2R 0.B9 - 82.49 0.66 - 38.43 0.14 52.52 C = 1.6R 0.86 - 45.26 0.01 - 4.01 0.52 79.83 Warranty =1.00 Lifecycle = 2W C = 1.2R 0.79 - 14.29 0.88 - 10.39 0.02 4.51 C = 1.6R 0.84 - 14.16 0.65 - 7.80 0.14 11.02 Lifecycle = 20W C = 1.2R 0.97 -133.87 0.87 -118.45 0.03 64.94 ro CD • o tr CO • T— I I CJ - 89.07 0.75 - 87.88 0.09 91.94 Warranty =1.25 Lifecycle = 2W C = 1.2R 0.94 - 18.56 0.93 - 20.01 0.01 5.78 C = 1.6R 0.91 - 16.96 0.85 - 15.85 0.04 10.39 Lifecycle = 20W C = 1.2R 0.98 -185.60 0.94 -209.66 0.00 42.59 C = 1.6R 0.92 -144.74 0.92 -194.62 0.01 69.58 153 Table 13— Continued Lifetime Distribution Parameters T runcated Exponential ________Normal_______ Warranty Case B MU 5G Ra SLOPE Ra SLOPE Ra SLOPE Pro Rata, Renewing -------------------------------------------- No Sales After W, Beta = 0 Warranty = 0.75 Lifecycle = C = 1.2R 2W 0.76 5.56 0.54 - 2.75 0.10 3.54 C = 1.6R 0.90 - 6.79 0.20 0.87 0.14 2.15 Lifecycle = C = 1.2R 20W 0.52 8.37 0.63 - 3.06 0.13 4.18 C = 1.6R 0.69 - 5.25 0.02 0.34 0.04 1.24 Warranty =1.00 Lifecycle = C = 1.2R 2W 0.93 15.77 0.79 - 7.97 0.04 5.61 C = 1.6R 0.59 - 9.47 0.11 - 1.19 0.31 6.00 Lifecycle = C = 1.2R 20W 0.87 15.31 0.74 - 16.51 0.00 3.50 C = 1.6R 0.84 - 12.19 0.06 1.92 0.46 15.62 Warranty = 1.25 Lifecycle = C = 1.2R 2W 0.93 17.22 0.86 - 17.56 0.01 5.90 C = 1.6R 0.80 - 17.45 0.61 - 10.27 0.07 10.69 Lifecycle = C = 1.2R 20W 0.63 25.43 0.65 - 77.36 0.04 -55.94 C = 1.6R 0.85 - 25.03 0.39 - 23.92 0.03 20.80 154 Table 13— Continued Lifetime Distribution Parameters Gamma DFR____ A B Warranty Case R2 Lump Sum, Nonrenewing ---- Warranty = 0.75 0.62 Warranty = 1.00 0.52 Warranty =1.25 0.61 Pro Rata, Renewing Sales After W, Beta =1.0 Warranty = 0.75 Lifetcycle = 2W C = 1.2R 0.74 C = 1.6R 0.69 Lifecycle = 20W C = 1.2R 0.61 C = 1.6R 0.69 Warranty = 1.00 Lifecycle = 2W C = 1.2R 0.65 C = 1.6R 0.67 Lifecycle = 20W C = 1.2R 0.60 C = 1.6R 0.66 Warranty = 1.25 Lifecycle = 2W C = 1.2R 0.59 C = 1.6R 0.64 Lifecycle = 20W C = 1.2R 0.59 C = 1.6R 0.64 SLOPE R2 SLOPE - 31.12 0.23 - 4.74 - 38.47 0.35 - 7.87 - 45.26 0.26 - 7.41 - 42.22 0.13 - 4.50 - 42.92 0.18 - 5.55 -250.75 0.31 - 44.88 -260.78 0.21 - 35.86 - 59.25 0.23 - 8.77 - 49.56 0.22 - 7.12 -392.81 0.31 - 70.95 -389.97 0.25 - 59.60 - 76.48 0.26 - 12.76 - 64.75 0.25 - 10.14 -503.37. 0.34 - 95.36 -481.21 0.28 - 79.66 155 Table 13-—-Continued Warranty Case Lifetime Distribution Parameters Gamma DFR A B R2 SLOPE R2 SLOPE Pro Rata, Renewing No Sales After Id, Beta = 0 Warranty = 0.75 Lifecycle = 2W C = 1.2R O.BB - 41.86 0.22 - 6.08 C = 1.6R 0.7B - 38.16 0.13 - 3.98 Lifecycle = 20W C = 1.2R 0.6B - 46.44 0.26 - 7.28 C = 1,BR 0.71 - 38.91 0.13 - 4.10 Warranty = 1.00 Lifecycle = 2W C = 1.2R 0.66 - 48.88 0.18 - 6.38 C = 1.6R 0.69 - 47.97 0.22 - 6.80 Lifecycle = 20W C = 1.2R 0.59 - 55.57 0.27 - 9.31 C = 1.6R 0.72 - 67.29 0.17 - 8.13 Warranty =1.25 Lifecycle = 2W C = 1.2R 0.62 - 55.44 0.29 - 9.54 C = 1.BR 0.64 - 57.80 0.29 - 9.84 Lifecycle = 20W C = 1.2R 0.49 - 70.94 0.36 - 15.22 C = 1.BR 0.62 - 77.96 0.24 12.26 156 Table 13— Continued Lifetime Distribution Parameters Gamma IFR A B R’ Lump Sum, Nonrenewing ---- Warranty = 0.75 0.B2 Warranty =1.0 0.58 Warranty = 1.25 0.58 Pro Rata, Renewing Sales After W, Beta = 1.0 Warranty = 0.75 Lifecycle = 2W C = 1.2R 0.61 C = 1.6R‘ 0.B9 Lifecycle = 20W C = 1.2R 0.59 C = 1.BR 0.6B Warranty =1.0 Lifecycle = 2W C = 1.2R 0.56 C = 1.BR 0.65 Lifecycle = 20W C = 1.2R 0.54 C = 1.BR 0.B0 Warranty =1.25 Lifecycle = 2W C = 1.2R 0.43 C = 1.6R 0.54 Lifecycle = 20W C = 1.2R 0.55 C = 1.6R 0.5B SLOPE Rs SLOPE - 7.44 - 8.76 - 10.85 0.27 0.34 0.30 - 10.92 - 15.20 - 17.66 - 8.42 7.10 0.27 0.13 - 12.63 - 6.81 - 67.04 - 53.76 0.29 0.14 -105.77 - 54.91 - 11,58 - 11.63 0.31 0.23 - 19.28 - 15.64 -108.84 - 95.81 0.35 0.26 -198.13 -143.76 - 13.70 - 14.62 0.43 0.33 - 31.11 - 25.88 -147.91 -138.72 0.37 0.34 -275.19 -241.89 157 Table 13— Continued Lifetime Distribution Parameters Gamma IFR A B R2 SLOPE R2 SLOPE Pro Rata, Reneu/ing ---------------------------- No Sales After W, Beta = 0 Warranty = 0.75 Lifecycle = C = 1.2R 2W 0.59 - 8.33 0.25 12.20 C = 1.6R 0.5B - 5.47 0.26 - 8.42 Lifecycle = C = 1.2R 20W 0.55 - 9.32 0.23 13.71 C = 1.6R 0.51 - 6.24 0.24 - 9.63 Warranty =1.0 Lifecycle = C = 1.2R 2W 0.62 - 10.66 0.31 16.90 C = 1.6R 0.57 - 9.43 0.24 - 13.63 Lifecycle = C = 1.2R 20W 0.53 - 15.98 0.31 27.38 C = 1.6R 0.53 - 13.60 0.22 - 19.96 Warranty = 1.25 Lifecycle = 2W C = 1.2R 0.50 - 14.60 0.38 28.66 C = 1.BR 0.59 - 14.14 0.28 - 22.18 Lifecycle = C = 1.2R 20W 0.47 - 28.75 0.34 55.20 C = 1.BR 0.39 - 20.55 0.39 - 45.81 158 Table 13— Continued Weibull DFR Warranty Case A B Rs SLOPE Ra SLOPE Lump Sum, IMonreneuing Warranty = 0.75 0.00 0.68 0.55 - 18.88 Warranty =1.00 0.02 - 5.08 0.60 - 25.81 Warranty = 1.25 0.05 7.49 0.50 - 24.65 Pro Rata, Renewing Sales After W, Beta = 1.0 Warranty = 0.75 Lifecycle = 2W C = 1.2R 0.04 8.46 0.42 - 27.89 C = 1.6R 0.05 - 6.88 0.51 - 22.63 Lifecycle = 20W C = 1.2R 0.32 169.26 0.58 -227.60 C = 1.BR 0.19 90.98 0.68 -170.65 Warranty =1.00 Lifecycle = 2W C = 1.2R 0.02 7.39 0.66 - 41.84 C = 1.BR 0.00 2.80 0.83 - 45.75 Lifecycle = 20W C = 1.2R 0.42 302.79 0.50 -332.14 C = 1.6R 0.27 179.37 0.64 -274.46 Warranty =1.25 Lifecycle = 2W C = 1.2R 0.04 14.16 0.67 - 54.44 C = 1.BR 0.03 8.91 0.72 - 42.15 Lifecycle = 20W C = 1.2R 0.51 442.08 0.44 -409.14 C = 1.6R 0.37 335.48 0.52 -396.21 159 Table 13— Continued Lifetime Distribution Parameters Weibull DFR Warranty Case A B Ra SLOPE R° SLOPE Pro Rata, Renewing ---------------------------- No Sales After W, Beta = 0 Warranty = 0.75 Lifecycle = C = 1.2R 2W 0.02 - 5.42 0.B4 - 31.42 C = 1.BR 0.01 - 3.36 0.38 - 23.98 Lifecycle = C = 1.2R 20W 0.06 10.46 0.B6 - 36.26 C = 1.BR 0.00 0.B0 0.75 - 34.07 Warranty = 1.00 Lifecycle = C = 1.2R 2W D.12 14.98 0.59 - 32.63 C = 1.6R 0.02 6.69 0.79 - 43.14 Lifecycle = C = 1.2R 20W 0.05 15.05 0.65 - 53.86 C = 1.6R 0.15 : 19.93 0.61 - 40.28 Warranty •= 1.25 Lifecycle = C = 1.2R 2W 0.05 13.75 0.63 - 47.94 C = 1.6R 0.02 7.70 0.71 - 45.09 Lifecycle = C = 1,2R 20W 0.35 53.83 0.47 - 62.82 C = 1.6R 0.27 46.44 0.58 - 67.98 160 Table 13— Continued Lifetime Distribution Parameters Warranty Case Weibull IFR A B R2 SLOPE R2 SLOPE 0.11 _ 1.88 0.75 - 6.59 0.11 - 2.36 0.71 - 8.24 0.05 - 2.15 0.81 - 11.99 ,0 0.31 3.66 0.41 - 5.84 0.63 - 3.28 0.13 - 2.03 0.27 _ 24.71 0.66 - 52.13 0.79 - 37.58 0.12 - 20.26 0.17 4.05 0.66 - 10.88 0.40 - 4.64 0.43 - 6.48 0.13 _ 28.44 0.81 - 97.72 0.36 — 39.99 0.54 - 65.70 0.06 2.99 0.82 - 14.53 0.11 - 3.71 0.79 - 13.40 0.04 _ 22.38 0.90 -153.62 0.11 - 37.17 0.80 -133.88 Lump Sum, IMonreneuiing Warranty = 0.75 Warranty =1.0 Warranty =1.25 Pro Rata, Renewing Sales After W, Beta = Warranty = 0.75 Lifecycle = 2W C = 1.2R C = 1.6R Lifecycle C = 1.2R C = 1.BR Warranty =1.0 Lifecycle C = 1.2R C = 1.BR Lifecycle = 20W C = 12.R C = 1.6R Warranty =1.25 Lifecycle = 2W C = 1.2R C = 1.6R Lifecycle = 20W C = 1.2R C = 1.6R 20W 2W 161 Table 13— Continued Warranty Case Lifetime A Distribution Parameters Weibull IFR B R2 SLOPE Ra SLOPE Pro Rata, Renewing IMo Sales After W, Beta = 0 Warranty = 0.75 Lifecycle = 2W C = -1.2R 0,21 2.70 0.56 - 5.95 C = 1.BR 0.64 2.72 0.18 - 1.94 Lifecycle = 20W C = 1.2R 0.29 2.98 0.56 - 5.63 C = 1.6R 0.43 2.83 0.31 - 3.28 Warranty =1.0 Lifecycle = 2W C = 1.2R 0.23 4.33 0.57 - 9.19 C = 1.6R 0.39 3.68 0.40 - 5.02 Lifecycle = 20W C = 1.2R 0.03 2.12 0.79 - 15.10 C = 1.BR 0.28 3.99 0.54 - 7.47 Warranty =1.25 Lifecycle = 2W C = 1.2R 0.18 5.14 0.72 - 13.80 C = 1.6R 0.20 3.78 0.62 - 8.90 Lifecycle = 20W C = 1.2R 0.00 0.55 0.81 - 33.16 C = 1.6R 0.07 4.29 0.74 - 19.26 162 sample data, because of the expectation of the violation of the assumptions of linear regression. However, a certain strength in the linear relationship between most of the lifetime distributions' parameters and warranty 2 cost can be inferred from the consistently high R values.- Table 14 summarizes the characteristics of the 2 R values for each of the distributions. Means and standard deviations, as descriptive terms only, of the 2 R values obtained for each parameter over the 27 different warranty cases were calculated from the 27 2 values of R shown in Table 13 for each parameter. There does not appear to be a pattern to the high and low 2 R values over the 27 different warranty cases, which 2 leads to the conclusion that variations in R values are random in this sample, and not due to changes in the non-distributional parameters of the warranty. The exponential parameter B exhibits the highest 2 mean R , however, the sample size used in the 2 calculations of these R values for the exponential parameter B was only five. The truncated normal parameter, MU, exhibits a 2 relatively high mean R value, but it shows a 2 considerably larger standard deviation of R values than any of the other parameters, except the Weibull ifr's A parameter. The sample size for these regressions, and for all other two-parameter distributions' regressions, 163 Table 14 Summary of Regression Results Lifetime Distribution Parameter Mean R2 Standard Deviation of R2 High R2 Low R2 exponential B 0.82 0.14 0.98 0.46 truncated normal MU SG 0.61 0.11 0.32 0.15 0.94 0.52 0.01 0.00 gamma dfr A 0.64 0.06 0.76 0.49 B 0.24 0.06 0.35 0.13 gamma ifr A 0.56 0.06 0.69 0.39 B 0.29 0.07 0.43 0.13 Weibull dfr A 0.13 0.15 0.51 0.00 B 0.60 0.11 0.B3 0.38 Weibull ifr A 0.24 0.60 0.79 0.00 B 0.20 0.22 0.82 0.12 164 was 25. The truncated normal's standard deviation, 2 parameter SG, shows the lowest average R value, and 2 the standard deviation of these R values is larger than the mean, indicating considerable variation in the 2 R values obtained when warranty cost was regressed on the truncated normal prameter SG. In both the gamma dfr and ifr cases, the shape 2 parameter A has a higher mean R value than the scale 2 parameter B. All the standard deviations of R values for all of the gamma parameters are the smallest for this study, which indicates that there is relatively less 2 variability in the R values when warranty cost is regressed against any of the gamma parameters than is seen when warranty cost is regressed against any of the other lifetime distributions* parameters. In the case of the Weibull distribution with a decreasing failure rate, the scale parameter B has a 2 considerably larger mean R value than the shape parameter A, and a relatively small standard deviation of 2 R values. However, for the rest of the Weibull 2 parameters, mean R values are low, and the standard 2 deviations of R values accompanying them are 2 consistently larger than the mean R values. The best and worst (those showing highest and 2 lowest overall R ) of the 27 regressions for each of the six lifetime distributions are included in Appendix E. 165 In several instances, there was more than one "best" or "worst" case regression. The particular regressions chosen are representative of these multiple "best" or "worst" regression results. It is to be expected that increases in the lifetime distribution parameters which affect mean life (mu, A, B) would decrease warranty cost. This inverse relationship is evident in the sample data, with the notable exception of the parameter A for the decreasing failure rate case of the Weibull distribution, which exhibits a primarily positive relationship with cost. Any conclusions drawn from this deviation from the expected pattern should be tempered by the fact that the 2 contribution to total R for this variable is generally quite low. Also, this represents only one case of a Weibull distribution with a decreasing failure rate (A = 0.5), and the results might be quite different for other values of this shape parameter. While the mean of the truncated normal distribution, MU, behaves as expected, there appears to be little relationship between its standard deviation, SG, and warranty cost. Where a relationship is exhibited, through a relatively high 2 R value, that relationship is positive. To summarize, the sample data exhibit an indication of relatively strong inverse relationships between warranty cost and the exponential parameter B, the 166 truncated normal parameter MU, the gamma parameters A and B, and the Weibull parameter B. Again, it should be stressed that these results should not be generalized to all dfr and ifr cases of the gamma and Weibull distributions. In conclusion, the sample data indicate that errors in the estimation of certain lifetime distribution parameters can cause substantial errors in the estimation of warranty cost. As an example, for the sample data, a unit change in the exponential parameter B caused a change in estimated warranty cost, measured by the slope of the regression line varying from $4.18 per unit change in B to as much as $185.60 per unit change in B, depending upon the magnitude of the true expected cost. The mean slope, over the 27 exponential regressions, is -$35.51, with a standard deviation of the slope value of $47.30. The regression runs for the two extreme cases mentioned here are in Appendix E. The other parameters studied exhibit a similar variability in their slope values. Much of this variability in the slope values appears to be explainable by the variability in expected cost levels for different sets of non-distributional warranty parameters, particularly the assumption of sales occurring when items fail after the expiration of the warranty perod and the ratio of lifecycle length to warranty length. The observed effect of changes in two of these 167 non-distributional warranty parameters is the subject of the next section. Secondary Investigations An additional topic, of secondary interest to this study, was noted in the introductory section of Chapter V. The question raised was: what effect, if any, do the non-distributional warranty parameters involving the ratios of warranty length to lifecycle length and warranty length to expected lifetime have on the expected cost of a warranty? These non-distributional warranty parameters (¥, L, E(X)) are shown with the other non-distributional warranty parameters used in this study in Figure 2 in Chapter V. Two basic types of warranties were investigated; the lump sum rebate, nonrenewing warranty and the pro rata rebate, renewing warranty. Three levels of warranty length were used corresponding to 75% of the mean lifetime, 100% of the mean lifetime, and 125% of the mean lifetime. For each of these warranty lengths, in the pro rata case, lifecycle length was set at either twice warranty length or 20 times warranty length, and rebate costs were based on either a 20% or 60% markup from supplier’s cost to consumer's cost. Additionally, two cases of consumer behavior were considered. Setting the beta input parameter of the simulation model to 1.0 guaranteed that, if an item failed after the expiration of the warranty period, the consumer would purchase another 168 at full price. This was introduced in Chapter III as the probability of a sale occurring when an item fails after the expiration of the warranty period. Setting beta to 0 guarantees the opposite; follow-on sales are not assumed. A thorough investigation of the effects of all of these parameters would require more levels of each parameter to be studied. However, the data generated to address the two major questions of this sensitivity analysis provides some insight into how warranty cost is affected by changes in the ratios of warranty length (W) to lifecycle length (L) and warranty length to expected lifetime of an item (E(X)). A comparison of costs at the different levels of the ratios of W to L and W to E(X) for each of the six distributions indicates that, as one might suspect, costs tend to increase as the warranty period increases relative to the expected lifetime, and differences between costs generated when warranty length is short relative to lifecycle versus costs generated when warranty length is long relative to lifecycle are substantial only when follow-on sales are assumed (the probability of a new item being purchased when a failure occurs after the expiration of the warranty period, Beta, = 1.0). Table 15 shows the expected warranty costs and the standard deviations of warranty cost from Appendix D as they compare for the two levels of the ratio of lifecycle length to warranty length over the six different 169 Table 15 Comparison of Warranty Costs at Different Levels of the Lifecycle as a Function of Warranty Length Lifetime exponential L = 2W L = 20W Distribution truncated normal L = 2W L = 20W No Sales After W, Beta = 0 Warranty = 0.75 C = 1.2R mean cost std. dev. of cost 4.08 7.02 4.61 8.45 0.37 1.67 0.27 1.94 C = 1.6R mean cost 3.36 3.16 -0.29 -0.14 std. dev. of cost 6.32 7.52 2.01 1.71 Warranty =1.0 C = 1.2R mean cost 6.93 6.13 0.83 1.11 std. dev. of cost 9.06 8.45 2.28 2.67 C = 1.5R mean cost 5.48 5.79 CD o • 1 -1.73 std. dev. of cost 8.91 8.86 2.92 3.76 Warranty = 1.25 C = 1.2R mean cost std. dev. of cost 12.01 12.45 14.62 17.88 3.04 4.22 4.73 5.83 C = 1.6R mean cost 8.27 9.99 -0.98 -3.49 std. dev. of cost 11.79 14.76 3.70 7.09 170 Table 15— Continued Lifetime exponential L = 2W L = 20W Distribution truncated normal L = 2W L = 20W Sales After W, Beta = Warranty = 0.75 C = 1.2R mean cost std. dev. of = 1.0 cost 5.33 7.78 24.16 22.80 -1.12 1.76 -18.72 6.49 C = 1.6R mean cost 1.86 -14.27 -4.80 -73.66 std. dev. of cost 7.54 17.65 2.85 6.88 Warranty =1.0 C = 1.2R mean cost 6.93 53.34 -0.14 -7.18 std. dev. of cost 9.75 28.96 3.32 10.81 C = 1.6R mean cost 4.30 2.31 -4.76 -74.00 std. dev. of cost 10.95 24.75 3.00 8.99 Warranty = 1.25 C = 1.2R mean cost 12.63 86.00 3.14 22.89 std. dev. of cost 13.00 37.61 4.79 14.64 C = 1.6R mean cost 6.27 21.63 -3.45 -53.64 std. dev. of cost 10.82 30.06 4.64 12.13 171 Table 15---Continued Lifetime Distribution gamma dfr gamma ifr ' 3 CM I ! L = 20W L = 2W L = 20W No Sales After Ui, Beta = 0 Warranty = 0.75 C = 1.2R mean cost std. dev. of cost 10.14 12.17 12.71 14.90 3.67 5.59 3.83 5.99 C = 1.6R mean cost std. dev. of cost B.28 9.10 9.25 14.78 2.88 5.55 0.95 4.92 Warranty = 1.0 C = 1.2R mean cost std. dev. of cost 10.43 14.38 16.25 19.22 5.28 7.06 6.26 8.92 C = 1.BR mean cost std. dev. of cost 12.10 15.17 14.04 20.57 1.93 5.39 4.01 6.30 Warranty =1.25 C = 1.2R mean cost std. dev. of cost 19.30 17.99 22.83 24.77 7.76 7.90 9.02 10.12 C = 1.6R mean cost std. dev. of cost 13.65 16.56 17.76 19.79 5.83 8.44 7.16 10.67 172 Table 15— Continued gamma L = 2W Lifetime dfr L = 20W Distribution gamma L = 2W ifr L-= 20W Sales After W, Beta = 1.0 Warranty = 0.75 C = 1.2R mean cost std. dev. of cost 10.69 13.86 59.17 35.58 3.88 6.70 14.45 17.44 C = 1.6R mean cost 8.33 22.91 -1.29 -33.95 std. dev. of cost 11.02 28.75 4.74 14.74 Warranty =1.0 C = 1.2R mean cost 14.51 92.65 6.69 40.01 std. dev. of cost 14.31 46.26 8.96 26.55 C = 1.6R mean cost 10.00 45.98 0.70 -13.74 std. dev. of cost 17.45 37.44 6.11 21.72 Warranty = 1.25 C = 1.2R mean cost 16.B2 128.74 8.11 70.69 std. dev. of cost 17.04 55.05 8.81 28.87 C = 1.6R mean cost 17.78 80.68 3.87 3.78 std. dev. of cost 20.83 51.79 9.22 25.58 173 Table 15— Continued Lifetime Distribution Weibull dfr Weibull ifr L = 2W L = 20W L = 2W L = 20W No Sales After Id, Beta = 0 Warranty = 0.75 C = 1.2R mean cost std. dev. of cost 14.76 18.47 14.05 16.98 1.93 4.23 3.07 4.75 C = 1.6R mean cost std. dev. of cost 16.28 -19.59 13.05 15.58 0.95 3.56 0.66 3.78 Warranty = 1.0 C = 1.2R mean cost std. dev. of cost 19.70 21.64 23.44 26.74 5.16 7.77 5.54 7.19 C = 1.6R mean cost std. dev. of cost 17.95 21.59 23.09 26.17 1.87 4.84 1.31 5.95 Warranty = 1.25 C = 1.2R mean cost std. dev. of cost 23.44 23.20 29.81 32.35 6.09 7.38 9.90 12.46 ‘ C = 1.6R mean cost std. dev. of cost 21.82 20.37 22.71 23.81 2.72 6.21 4.27 8.96 174 Table 15— Continued Lifetime Distribution Weibull dfr Weibull ifr L = 2W L = 20W L = 2W L = 20W Sales After W, Beta =1.0 Warranty = 0.75 C = 1.2R mean cost std. dev. of cost 16.59 19.65 83.76 53.22 2.52 5.28 5.94 13.55 C = 1.6R mean cost std. dev. of cost 15.95 19.6? 57.17 45.48 -1.16 4.49 -42.09 13.13 Warranty =1.0 C = 1.2R mean cost 25.7? 119.42 3.56 24.50 std. dev/, of cost 22.01 69.66 6.13 18.71 C = 1.6R mean cost 19.48 79.25 -0.76 -32.70 std. dev. of cost 21.37 56.12 5.56 15.26 Warranty = 1.25 C = 1.2R mean cost 24.13 153.60 6.40 48.20 std. dev. of cost 25.56 ■62.98 7.65 22.30 C= 1.6R mean cost 23.61 119.51 0.26 -12.84 std. dev. of cost 28.23 79.70 6.21 24.86 175 distributions and the other non-distributional parameters included in the study. Costs may be compared on the basis of holding all model inputs constant except the value of lifecycle length, which varies from twice the warranty length to 20 times the warrant length. For all six lifetime distributions, the major factor seems to be the assumption of follow-on sales. When the probability of sales resulting from failures after the expiration of the warranty (sales after W) is 0, there appears to be little difference in the expected warranty cost whether the lifecycle is long or short compared to the warranty length. However, when the probability of sales after W is 1.0, the difference between the two costs generated with lifecycle equal to twice warranty length and lifecycle equal to twenty times warranty length, respectively, is much larger. The direction of the observed differences in this sample data varies, apparently randomly, over the other non-distributional parameters. Only for the decreasing failure rate case of the gamma distribution does the relationship between increases in lifecycle length relative to warranty length and increases in warranty cost appear monotonic. In conclusion, changes in lifecycle length relative to warranty length may affect the warranty cost in either a decreasing or an increasing manner; however, 176 those changes in warranty cost are observed to be of less magnitude when follow-on sales are not assumed. Data for the comparison of expected warranty costs at different levels of the warranty length relative to the mean of the lifetime distribution are shown in Table 16, comprised of data from Appendix D. Again, costs are comparable over different lifetime distributions and non-distributional warranty parameters when these variables are held constant and only the length of the warranty as a function of the expected lifetime of the item is varied. The warranty length was set at 75% of the expected lifetime, 100% of the expected lifetime, and 125% of the expected lifetime. Out of the 53 sets of costs shown in Table 16, only six do not exhibit a monotonically increasing relationship between warranty cost and length of warranty as a function of expected lifetime. This is not surprising in simulation data, and the effect of the length of the warranty relative to the lifecycle is also apparent from the magnitude of the changes noted. The percentage change in expected cost is not constant, and shows no particular pattern relative to the non-distributional warranty parameters. Further investigation of the impact of either of these ratios on the expected cost of a warranty would require the collection of additional data with this specific purpose in mind. The sample data available from Table 16 Comparison of Warranty Costs At Different Levels of the Warranty Length as a Function of E(X) Exponential Lifetime Distribution W = 0.75E(X) W = 1.0E(X) W=1.25E(X) Pro Rata, Renewing Warranty Lifecycle = 2W C = 1.2R mean cost std. dev. of cost A. 08 7.02 No Sales After W, Beta = 0 6.93 12.01 9.06 12.45 C = 1.6R mean cost 3.36 5.48 8.27 std. dev. of cost 6.32 8.91 11.79 Lifecycle= 2DW C = 1.2R mean cost 4.61 6.13 14.62 std. dev. of cost 8.45 • - 8.45 17.88 C = 1.6R mean cost 3.16 5.77 9.99 ; std. dev. of cost ' 7.52 8.86 14.76 Lump Sum, Nonrenewing Warranty mean cost 7.20 9.90 13.50 std. dev. of cost 8.65 9.48 11.84 Pro Rata, Renewing Warranty Sales After W, Beta =1.0 Lifecycle = 2W C = 1.2R mean cost 5.33 6.93 12.63 std. dev. of cost 7.78 9.75 13.00 C= 1.6R mean cost 1.86 4.30 6.27 std. dev. of cost 7.54 10.95 10.82 Lifecycle = 20W C = 1.2R mean cost 24.16 53.34 ' 86.00 std. dev. of cost 22.80 28.96 37.61 C = 1.6R mean cost -14.27 2.31 21.65 std. dev. of cost 17.65 24.75 30.06 178 Table 16— Continued Truncated Normal Lifetime Distribution W = 0.75E(X) W = 1. 0E (X) W = 1.25E(X) Pro Rata, Renewing Warranty No Sales After W, Beta = 0 Lifecycle = 2 W C = 1.2R mean cost 0.37 0.83 3.04 std. dev. of cost 1.67 2.28 4.22 C = 1.6R mean cost -0.29 -1.06 -0.98 std. dev. of cost 2.01 2.92 3.70 Lifecycle = 20W C = 1.2R mean cost 0.27 1.11 4.73 std. dev. of cost 1.94 2.67 5.85 C = 1.6R mean cost -0.14 -1.73 -3.49 std. dev. of cost 1.71 3.76 7.D9 Lump Sum, Nonrenewing Warranty mean cost 3.00 5.60 9.20 std. dev. of cost 4.82 4.99 4.42 Pro Rata, Renewing Warranty Sales After W, Beta= 1.0 Lifecycle = 2W C = 1.2R mean cost -1.12 -0.14 3.14 std. dev. of cost 1.76 3.32 4.79 C = 1.6R mean cost -4.80 -4.76 -3.45 std. dev. of cost 2.85 3.00 4.64 Lifecycle = 20W C = 1.2R mean cost -18.72 -7.18 22.89 std. dev. of cost 6.49 10.81 14.64 C= 1.6R mean cost -73.66 -74.00 -53.64 std. dev. of cost 6.88 8.99 12.13 179 Table 16— Continued Gamma DFR Lifetime Distribution W = 0.75E(X) W = 1. 0E (X) W = 1.25E(X) Pro Rata, Renewing Warranty No Sales After W, Beta = 0 Lifecycle = 2W C = 1.2R mean cost 10.14 10.43 19.30 std. dev. of cost 12.17 14.38 17.99 C = 1.6R mean cost 8.28 12.10 13.65 std. dev. of cost 9.10 15.17 16.56 Lifecycle = 20W C = 1.2R mean cost 12.71 16.25 22.83 std. dev. of cost 14.90 19.22 24.77 C = 1.6R mean cost 9.25 14.04 17.76 std. dev. of cost 14.78 20.57 19.79 Lump Sum, Nonrenewing Warranty mean cost 12.50 14.30 17.90 std. dev. of cost 14.10 14.37 17.25 Pro Rata, Renewing Warranty Sales i After W, Beta= 1.0 Lifecycle = 2W C = 1.2R mean cost 10.69 14.51 16.82 std. dev. of cost 13.86 14.31 17.04 C = 1.6R mean cost 8.33 10.00 17.78 std. dev. of cost 11.02 12.45 20.83 Lifecycle = 20W C = 1.2R mean cost 59.17 92.65 128.74 std. dev. of cost 35.58 46.26 55.05 C = 1.6R mean cost 22.91 45.98 80.68 std. dev. of cost 28.75 37.44 51.79 180 Table 16— Continued Gamma IFR Lifetime Distribution W = 0.75E(X) W = 1.OE(X) W = 1.25E(X) Pro Rata, Renewing Warranty No Sales After W, Beta = □ Lifecycle = 2 1 1 1 C = 1.2R mean cost 3.67 5.28 7.76 std. dev. of cost 5.59 7.06 7.90 C = 1.6R mean cost 2.88 1.93 5.83 std. dev. of cost 5.55 5.39 8.44 Lifecycle = 20U C = 1.2R mean cost 3.83 6.26 9.02 std. dev. of cost 5.99 8.92 10.12 C = 1.6R mean cost 0.95 4.01 7.16 std. dev. of cost 4.92 6.30 10.67 Lump Sum, Nonrenewing Warranty mean cost 7.10 9.70 12.10 std. dev. of cost 7.00 9.04 10.47 Pro Rata, Renewing Warranty Sales After W, Beta= 1.0 Lifecycle = 2W C = 1.2R mean cost 3.88 6.69 8.11 std. dev. of cost 6.70 8.96 8.81 C = 1.6R mean cost -1.29 0.70 3.87 std. dev. of cost 4.74 6.11 9.22 Lifecycle - 20W C = 1.2R mean cost 14.45 40.01 70.69 std. dev. of cost 17.44 26.55 28.87 C = 1.6R mean cost -33.95 -13.74 3.78 std. dev. of cost 14.74 21.72 25.58 181 Table 16— Continued Weibull W = 0.75E(X) DFR Lifetime Distribution W = 1.OE(X) W = 1.25E(X) Pro Rata, Renewing Warranty No Sales After W, Beta = □ Lifecycle = 2W C = 1.2R mean cost 14.76 19.70 23.44 std. dev. of cost 18.47 21.64 23.20 C = 1.6R mean cost 16.28 17.95 21.82 std. dev. of cost 19.59 21.59 20.37 Lifecycle = 20W C = 1.2R mean cost 14.05 23.44 29.81 std. dev. of cost 16.98 26.74 32.35 C = 1.6R mean cost 13.05 23.09 22.71 std. dev. of cost 15.58 26.17 23.81 Lump Sum, Nonrenewing Warranty mean cost 15.60 19.20 22.70 std. dev. of cost 15.66 18.51 20.74 Pro Rata, Renewing Warranty Sales After W, Beta =1.0 Lifecycle = 2W C = 1.2R mean cost 16.59 25.77 24.13 std. dev. of cost 19.65 22.01 25.56 C = 1.6R mean cost 15.95 19.48 23.61 std. dev. of cost 19.67 21.37 28.23 Lifecycle = 20W C = 1.2R mean cost 83.76 119.42 153.60 std. dev. of cost 53.22 69.66 62.98 C= 2.6R mean cost 57.17 79.25 119.51 std. dev. of cost 45.48 56.12 79.70 182 Table 16— Continued Weibull IFR Lifetime Distribution W = 0.75E(X) W = 1. 0E ( X) W = 1.25E(X) Pro Rata, Renewing Warranty No Sales After W, Beta = 0 Lifecycle = 2W C = 1.2R mean cost 1.93 5.16 6.09 std. dev. of cost 4.23 7.77 7.38 C= 1.BR mean cost 0.95 1.87 2.72 std. dev. of cost 3.56 4.84 6.21 Lifecycle = 20W C = 1.2R mean cost 3.07 5.54 9.90 std. dev. of cost 4.75 7.19 12.46 C = 1.6R mean cost 0.66 1.31 4.27 std. dev. of cost 3.78 5.95 8.96 Lump Sum, Nonrenewing Warranty mean cost 4.90 7.30 8.80 std. dev. of cost 6.11 7.89 8.79 Pro Rata, Renewing Warranty Sales After W, Beta =1.0 Lifecycle = 2W C = 1.2R mean cost 2.52 3.56 6.40 std. dev. of cost 5.28 6.13 7.65 C = 1.BR mean cost -1.16 -0.76 0.26 std. dev. of cost 4.49 5.56 6.21 Lifecycle = 20W C = 1.2R mean cost 5.94 24.50 48.2D std. dev. of cost 13.55 18.71 22.30 C = 1.6R mean cost -42.09 -32.70 -12.84 std. dev. of cost 13.13 15.26 24.86 183 this study indicate, however, that these ratios do have some impact on expected warranty costs. The conclusions drawn from the results of the theoretical modeling presented in Chapter III and from the results of the sensitivity analysis presented in this chapter are discussed in Chapter VII in relation to suggested areas for further research. 184 CHAPTER VII CONCLUSIONS AND COMMENTS The topic of this research, the quantitative analysis of warranty policies, is of practical as well as academic interest. The introduction to this study presented in Chapter I noted several motivations for the development of a quantitative model for the prediction of warranty costs. Among these reasons were the need to price the warranty to be offered with an item and the requirement to account for after-sale costs associated with warranties by a charge to income in the period of the sale. Because of the practical nature of the topic, input was sought early in the study from two executives at a large Southern California aerospace company. Prior to interviews with the Director of Product Support Advanced Development and the Manager of Warranty-Product Support, the following list of factors, which were hypothesized to contribute to the cost of a warranty program, was developed and presented to both men as a basis for discussion. 185 1. , reliability of item measured through MTBF, lifetime distribution, and/or renewal function 2 . probability of correct assessment of reliability 3. length of the warranty 4. warranty provisions specifying the rebate to be full replacement, pro rata replacement, or repair of an item 5. cost of repair or replacement of an item 6 . cost of validating claims 7. fixed costs of warranty administration 8 . probability of a claim being made 9. probability of a claim being valid 10. probability of a claim being correctly validated 1 1. warehousing costs of replacement parts 12. fixed costs of repair facilities 13. transportation costs for items or repair crews 14. costs of consequential damages 15. probability of consequential damages As a result of the interviews with these two executives, the following conclusions were reached. This company does not appear to be using engineering test data to arrive at item reliability. . Rather, historical failures are used in forecasting future warranty experience. The exception to this practice is the item of 186 such new technology that no history is available. In the case of new technology items, the test data is utilized. The Manager of Warranty-Product Support thinks that reliability as a component of warranty cost is actually being considered when he includes prior experience with a given aircraft in his forecast. He did say that if he were dealing with a "simple" line of products he would consider reliability of the item directly. This company does have a group which collects reliability data from suppliers and "runs the numbers" which they supply to this manager's group during the writing of a new warranty. Much of the reliability data available on aircraft parts comes from the company's suppliers, although they do have their own testing department. Accounting for warranty reserves is done at a very high level and with a cloak of some secrecy. This manager's forecasts are one input to the process. He admits that his forecasts are not particularly accurate as to the timing of warranty expenses, but claims that the magnitude is reasonably close. The major components of the manager's forecasts are: 1. An analysis of the major costs of known defects. 2. Dollar cost per aircraft in prior years plus an escalation factor. 187 3. Estimation of the difference in cost for a new design as opposed to an existing design. 4. Dates of deliveries and estimated dollar value per aircraft (for timing of claims costs). 5. Terms and conditions of new contracts compared to existing contracts. One gets the feeling that the forecasts are very "seat of the pants." Administrative costs are charged to overhead, and do not appear in the manager's forecasts. As overhead, these costs become part of the burden cost of the product. According to this manager, 90% of the cost of a warranty is design cost. By this, he seems to mean warranting that the parts will do what they were designed to do. If there is a major design flaw, service bulletins are issued which describe the flaw and explain how to fix it. Historical data on repair costs on specific aircraft are easily available. They are, in fact, used in his forecasts. He noted that transportation could be a significant part of a given total repair cost. The company does not address consequential damages in its warranty administration, and such damages are actually excluded in all of their written warranties. The manager thought that the notion of some probability being attached to the making of a claim was an interesting point. He noted that it depends on to whom you are selling. Some of his customers will not claim 188 while others will file claims for things not covered by the warranty. Each claim is adjusted individually, and in some cases invalid claims are settled. He does consider this in his cost analysis, but finds it difficult to estimate. The Director of Product Support Advanced Development is not actually involved in warranty administration, but in his capacity as Director of Product Support he interacts closely with the Manger of Warranty-Product Support, and appears to have given some thought to the subject of warranties. His primary concern was the difference between actual and intended use. The manager had touched on this subject when he made a comment about claims being dependent upon whom your customer is. The director suggested a case in which departure from intended use could have a significant quantitative impact on one’s assessment of costs. He pointed out that it is necessary to consider the physical environment and the skill of the user as well as the entire range of possible uses to which the product might be put. This area is similar to some of the considerations of liability for implied warranty. The director noted that the concept of discounting a stream of costs seemed strange to him, but would indeed yield an estimate for warranty reserves. One final, and interesting, point made by the 189 director was that there is a risk involved in assessing the reliability of an item. Since most measures of reliability (i.e., MTBF or renewal function) are based on incomplete data and unknown distributions, the assignment of a level of reliability is actually an estimate. As a consequence of this conversation, this factor was added to the list of factors discussed with the Manger of Warranty-Product Support. Some of the factors mentioned by these two executives were specific to their particular product, but most of the factors they agreed upon are in the theoretical and simulation models developed in Chapters III and IV. Notably, factors were included to account for the probabilities attached to consumer claims behavior and supplier claims validation behavior in the form of assessments of the probabilities associated with the placing of a claim, the validity of that claim, and the likelihood of its being validated by the supplier. A subject about which they expressed concern, the estimation of some measure of reliability through MTBF or a renewal function, was addressed in the sensitivity analysis of Chapters V and VI, and found to be substantially important to the estimation of warranty costs. At least in the case of these two executives, the models developed in this research would appear to address their informational needs. Their access to the input values for any of these 190 models, however, seems to be limited. This last point represents the major flaw in the models for quantitative analysis presented by this study, namely, they require many input parameters, some of which may be relatively easy to obtain, such as the supplier’s and consumer's costs, while others are incalculable, such as the probability of invalid claims being received. Any warranty costs generated by these models will be only as good as the estimates of the distributional and non-distributional parameters input to the models. Major Conclusions The theoretical portion of this research resulted in the conclusion that no one mathematical model could be stated which would account for both nonrenewing and renewing warranties and discount costs in the renewing case as well. As a result of this conclusion, a computer simulation model was built which was capable of modeling either renewing or nonrenewing warranties with lump sum or linear pro rata rebate functions. Discounting can be done in either the renewing or nonrenewing case. Supplier and consumer claims behavior in the validating and placing of claims is accounted for by the use of probabilities input to describe these behaviors. Invalid claims are dealt with through the use of a modified lifetime distribution, if desired. Costs may be generated under the lifecycle concept in which the consumer is assumed to purchase a new 191 item when the old item fails after the expiration of the warranty period, or without this assumption, which results in an end to the renewal process when any lifetime exceeds the warranty period. Through the variety of its inputs the simulation model is flexible enough to address any of the concepts introduced in the search for a theoretical model. It does, however, share the flaw of being dependent upon those inputs for the goodness of its warranty cost estimates. The computer simulation model allowed the investigation of two areas of sensitivity analysis never fully explored before, sensitivity of warranty costs to the form of the lifetime distribution, and sensitivity of warranty costs to estimations of the warranty parameters. This second question specifically addresses the point raised in the interviews discussed at the beginning of this chapter about estimation of reliability measures. The result of the investigation of the first of these two areas was that the model is sensitive to the form of the distribution when certain forms of lifetime distributions are being used. The primary result of interest was that the exponential distribution does not provide good estimates of warranty costs when being used to approximate the truncated normal distribution, the Weibull distribution, or the gamma distribution with shape parameter equal to 0.5. Its estimation abilities were 192 better for the case of the gamma distribution with shape parameter equal to 1.5, but not much. This result was tempered by the fact that only one decreasing failure rate and one increasing failure rate case of each of the gamma and Weibull distributions was considered. However, the result is significant in light of the frequency with which the exponential distribution is used in the literature on mathematical warranty cost models because of its ease of computation. The second area explored under the sensitivity analysis was the misestimation of lifetime distribution parameters. It was found that the warranty cost model is very sensitive to changes in certain parameters of the lifetime distributions studied, particularly the exponential parameter B, the truncated normal parameter MU, the Weibull parameter B, and both the A and B parameters of the gamma distribution. This result presents a good argument for allocating sufficient resources to ensure the accurate estimation of the lifetime distribution of the item to be warrantied. These conclusions, while interesting in themselves, raise certain other questions which might be answered by further research into this area of sensitivity analysis . 193 Suggested Topics For Further Research Additional research which might be done with the existing data from this study would include using analysis of variance techniques on the cost data generated through varying the parameters of the distributions. Such analysis would provide another look at this data beyond the regression results obtained here. Although the assumptions of analysis of variance are probably not satisfied in this application, it could be used as a first exploratory technique to yield a measure of the effect, on cost, of varying the lifetime distribution’s parameter values. The technique should provide a means of investigating interactions of the factors as well as main effects. A second type of analysis which might be pursued with the existing data is additional plots of the costs versus the values of the lifetime distributions’ parameters. The current plots, for the two-parameter distributions, show all 25 data points generated over the five levels of each of the distributions* parameters. This obscures any interaction effects, between the two parameters, which might become evident if the individual points could be identified by their values on the second parameter which does not appear on the axes of the graph. Other topics for further research would involve the generation of additional simulation data. Of 194 particular interest in this area is the form of the distribution of the simulated costs under different distributional and non-distributional assumptions. One of the conclusions of this research was that the exponential distribution did not provide a good approximation to the gamma and Weibull distributions for either their dfr or ifr cases as represented here by shape parameter A set to 0.5 and 1.5. It was noted that the approximation should improve as the value of A approaches the exponential equivalent 1.0. A further investigation of this theorized improvement could be undertaken by generating more simulated expected costs at varying levels of the shape parameter A. There are many non-distributional parameters used in the simulation model, such as, a (the probability of a claim being made), 0 (the probability of a claim not being validated), and 3 (the probability of a new sale occurring when a failure occurs after the expiration of the warranty period). 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Reliability Improvement Warranty (Tech. Rep. ASO-TEE-2-77). Philadelphia: Aviation Supply Office, 1977. Markowitz, 0., & Giordano, J. C. Avionics Escalation Composite Index, Multiyear Composite Clauses Adjustment with no Exchange of Dollars (Tech. Rep.) Philadelphia: Aviation Supply Office, undated. McKean, R. N. Products liability, trends and implications. University of Chicago Law Review, 1970, 38, 3-63. Menke, W. W. Determination of warranty reserves. Management Science, 1969, 15, 542-549. Menke, W. W. Comments on Amato and Anderson’s paper, 'Determination of Warranty Reserves: an Extension’. Management Science, 1976, 22, 1395-1396. Metzler, E. G. Forcing functions integrate r & m into design-DOD TACAN procurement policy on reliability and maintainability. Proceedings 1974 Annual Reliability and Maintainability Symposium, 1974, 52-55. Moellenberndt, R. A. A critical look at warranty cost recognition. National Public Accountant, 1977, 2 2, 18-22. Neter, J., & Wasserman, W. Applied Linear Statistical Models. Illinois: Richard D. Irwin, Inc., 1974. New warranty law under attack by business - unfair, confusing. Advertising Age, 1975, 46, 1; 37. Newman, D. G., & Nesbitt, L. D. USAF experience with RIW. Proceedings 1978 Annual Reliability and Maintainability Symposium, 1978, 55-61. Oi. W. Y. The economics of product safety. The Bell Journal of Economics and Management Science, 1973, 4_, 3-28. Patankar, J. G., & Worm, G. H. Prediction intervals for warranty reserves and cash flows. Management Science, 1981, 17, 237-241. Perry, M. Service contract compared to warranty as a means to reduce consumer’s risk. Journal of Retailing, 1976, 51, 33-40; 90. 203 Proceedings, Failure Free Warranty Seminar. Philadelphia: U.S. Navy, Aviation Supply Office, 1973. Produce warranties: Congress Lends a Helping Hand. Consumer Reports, 1973, 40, 164-165. Prohaska, J. T., Briggs, W. G., DeWolf, J. B., & Lund, R. T. Life-cycle costs, concepts, considerations and modeling. In R. T. Lund (Ed.), Consumer Durables: Warranties, Service Contracts and Alternatives (Vol. 4). Cambridge, Mass.: Center for Policy Alternatives, Massachusetts Institute of Technology, 1978. Retterer, B. L. Considerations for effective warranty application (Tech. Rep. 6107-1467). Annapolis, Md.: ARINC Research Corp., 1976. Revett, J. Reliability warranty concept growing. Industrial Marketing, 1976, 61, 6-7. Schmidtt, A. E. A view of the evolution of the reliability improvement warrnaty (RIW). Fort Belvoir, Va.: Defense Systems Management School, 1977. Schmoldas, J. D. Improvement of weapon systems reliability through reliability improvement warranties (Tech. Rep.). Fort Belvoir, Va.: Defense Systems Management School, 1977. Shorey, R. R. Factors in balancing government and contractors risk with warranties. Proceedings 1976 Annual Reliability and Maintainability Symposium, 1976, 366-368. Southwick, A. F., Jr. Products liability: a broadening concept. Management Review, 1966, 21-25. Spengler, J. J. The economics of safety. Law and Contemporary Problems, 1968, 33, 619-638. Springer, R. M., Jr. Risks and benefits in reliability warranties. Journal of Purchasing and Materials Management, 1977, JJ3, 3-13. Springer, R. M., Jr. RIW with cost sharing. Proceedings 1977 Annual Reliability and Maintainability Symposium, 1977, 391-395. Stansbury, J. W. Source selection and contracting approach to life cycle cost management. Defense Management Journal. 1976, _12» 19-22. 204 Udell, J. G., & Anderson, E. E. The product warranty as an element of competitive strategy. Journal of Marketing, 1968, _3_2, 1-8. Warranties: there ought to be a law. Washington, D.C.: U.S. Government Printing Office (undated brochure). Warranty programs attacked. Merchandising Weekly, 1972, 104, 14. Weaver, P. The warranty dilemma deepens. Los Angeles Times, September 12, 1975, Part IV, p. 6. Weinstein, A. Product safety: dimension for consumer policy. In W. M. Denny and R. t. Lund (Eds.), Research for Consumer Policy. Cambridge, Mass.: Center for Policy Research, Massachusetts Institute of Technology, 1978. Yun, K. W., & Kalivoda, F. E. Model for an estimation of the warranty return rate. Proceedings 1977 Annual Reliability and Maintainability Symposium, 1977, 31-37 205 APPENDIX A DETAILED FLOWCHART FOR THE SIMULATION MODEL START Y a s length of lifetime from prob. dist. for real & unreal failures real or unreal failures INPUT PARAMETERS GENERATE UNIFORM RANDOM NUMBERS CALCULATE Y, Y* 0*0 L, H I , T, R, C, K, a, 6 , 6 , a*, 0 « , 0 , 6 , n, rebate function, lifetime distribution for lifetime, probability claim, probability of sales after l i ) , probability of claim being validated unreal failures recognized JltL Y* Y* a = a t * e = e* CALCULATE Y s* X *X + Y u s0 from prob. dist. for real f ailures time of failure relative to initial sale cost of failure so far failure beyond lifecycle length item will not be replaced 206 process ends process ends NO NO item under warranty YES NO YES YES purchase after claim cost NO process ends YES counts purchases outside W costs accumulated cost of failure NO YES combo warranty pro rata cost WARRANTY .RENEWS YES YES NO RESET WARRANTY costs .accumulated YES costs accumulated costs accumulated 207 total discounted cost total discounted cost for sigma claim process ends NO YES COUNT FAILURES DISCOUNTED 208 total discounted cost total number of failures total squared cost number of purchases after t i l NO YES MEAN COST iTD. DEV. COST, CALCULATE P T E AN AND STD. DEV. OF COSTS RESET Z, Cf T . RESET X, F 209 APPENDIX B EXAMPLE RUN OF WRNTY MODEL D FAILURE DISTRIBUTION EXPON B BETA 1 .00000 WR WARRANTY RENEWAL TYPE RENEW WF FAILURES COVERED FIRST AL ALPHA (PROB. OF CLAIM) 1.00000 BT BETA (PROB. OF SALES AFTER W) 1.00000 W WARRANTY PERIOD 0. 750 L LIFETIME 20.000 PB PAYBACK FUNCTION COMBO T1 T1 FOR COMBO WARRANTY 0.0 R SELLER’S COST 10.00000 C BUYER’S COST 15.00000 TH THETA (PROB. OF NOT VAL. CLAIM) 0.0 DF DISCOUNT FUNCTION 2 PH PHI 0.0 DL DELTA 0.0 NR NUMBER OF REPLICATIONS 100 K ADMINISTRATIVE COST 0.0 SK SKIP CODE OFF DB DEBUG LEVEL 0 ENTER REQUEST SK SKIP CODE CHANGED TO: ON ENTER REQUEST NR ENTER NUMBER OF REPLICATIONS ? 1000 ENTER REQUEST GO FOR 1000 REPLICATIONS: DISCOUNTED MEAN TOTAL COST = -7.32 STD DEV OF TOTAL COST = 22.42999 210 ENTER REQUEST WR WARRANTY RENEWAL TYPE CHANGED TO: NONREN ENTER REQUEST WF NUMBER OF FAILURES COVERED BY WARRANTY CHANGED TO: MULTIPLE ENTER REQUEST BT ENTER BETAC (PROB. OF SALES AFTER W) ? 0 ENTER REQUEST PB PAYBACK FUNCTION CHANGED TO: CONSTANT ENTER REQUEST GO FOR 1000 REPLICATIONS: DISCOUNTED MEAN TOTAL COST = 7.14 STD DEV OF TOTAL COST = 8.584524 ENTER REQUEST WF NUMBER OF FAILURES COVERED BY WARRANTY CHANGED TO: FIRST ENTER REQUEST PH •ENTER PHI (AS A DECIMAL FRACTION) ? 0.10 ENTER REQUEST GO FOR 1000 REPLICATIONS: DISCOUNTED MEAN TOTAL COST = 5.29 STD DEV OF TOTAL COST = 4.821291 ENTER REQUEST PB PAYBACK FUNCTION CHANGED TO: COMBO 211 ENTER REQUEST T1 ENTER T1 FOR COMBO WARRANTY ? 0 . 10 ENTER REQUEST GO FOR 1000 REPLICATIONS: DISCOUNTED MEAN TOTAL COST = 2.26 STD DEV OF TOTAL COST = 4.00389 ENTER REQUEST T1 ENTER T1 FOR COMBO WARRANTY ? 0 ENTER REQUEST GO FOR 1000 REPLICATIONS: DISCOUNTED MEAN TOTAL COST - 1.83 STD DEV OF TOTAL COST = 3.532118 ENTER REQUEST Q * ■»# # all p a u * * * * * 212 APPENDIX C LISTING OF WRNTY PROGRAM 00001 00002 C: 00003 C; COPYRIGHT 1983, V. L. HILL 00004 nnnnc Cs p • •; UUUuD LI IJ 00008 C: 00007 BLOCK DATA 00008 nnnnn C: p. . . u u u u y LS 3 : 00010 Ci 00011 REAL*8 DTYPE, WRTYPE, WFTYPE, PBFTYP 00012 INTEGER*4 CHGCOD. 00013 C: 00014 COMMON /LITRAL/. DTYPE(4), WRTYPE(2), WFTYPE(2), 00015 + LGQ, LSKIP(2), LQUIT, CHGC0D(24), 00016 + PBFTYP(2) 00017 C: 00D18 C: DISTRIBUTION TYPES 00019 C: 00020 DATA DTYPE / ’EXPON’, ’GAMMA', 'WE [BULL*, ’TRNORM’ / 00021 C: 00022 C: WARRANTY RENEWING/NONRENREWING 00023 C: 00024 DATA WRTYPE / 'RENEW’, ’NONREN* / 00025 C: 00026 C: FIRST/MULTIPLE FAILURES 00027 C: 00028 DATA WFTYPE / ’FIRST’, ’MULTIPLE’ / 00029 C: 00030 C: PAYBACK FUNCTION TYPE 00031 C: 00032 DATA PBFTYP / ’CONSTANT*, ’COMBO* / 00033 C: 00034 C: ”G0" CODE 00035 C: 00038 DATA LGO / ’GO’ / 00037 C: 00038 C: "SKIP” CODE 213 00039 00040 00041 00042 0G043 00044 00045 00046 00047 00048 00049 00050 00051 00052 00053 00054 00055 00056 00057 00058 00059 00060 00061 00062 00063 00064 00065 00006 00067 00068 00069 00070 00075 00076 00077 00078 00079 00080 00081 00082 00083 00084 00085 00086 00087 00088 00089 C: C: C: C: C: G: C: C: C: C: C: C: C: C: C: C: C: C: C: C; C: C: C: C: C: C; C: C: C: C: C: C: DATA LSKIP / 'ON1, 'OFF' / "QUIT" CODE DATA LQUIT / 'Qf / CHANGE CODES: D - CHANGE WR UIF A B MU SG SK AL BT W L T1 R C TH PB PH DL DF NR K DB CHANGE CHANGE CHANGE CHANGE CHANGE CHANGE CHANGE CHANGE CHANGE CHANGE CHANGE CHANGE CHANGE CHANGE CHANGE CHANGE CHANGE CHANGE CHANGE CHANGE CHANGE CHANGE DISTRIBUTION TYPE WARRANTY RENEWAL TYPE NUMBER OF FAILURES COVERED BY WARRANTY ALPHA FOR GAMMA OR WEIBULL BETA FOR GAMMA, WEIBULL, OR EXPONENTIAL MU FOR TRUNCATED NORMAL SIGMA FOR TRUNCATED NORMAL SKIP CODE ALPHA (PROB. OF CLAIM) BETA (PROB. OF SALES AFTER W) W L T1 FOR COMBINATION WARRANTY R (SELLER'S COST) C (BUYER'S COST) THETA (PROB. OF NOT VALIDATING CLAIM) PAYBACK FUNCTION PHI DELTA discount function NUMBER OF REPLICATIONS ADMINISTRATIVE COST DEBUG LEVEL 00071 DATA CHGCOD / 'O', 'WR', 'WF1, 'A', 'B', 'MU’, 00072 + 'SG', 'SK', 'AL', 'BT', 'W, 'L' 00073 + 'T1', *R', 'C', 'TH', 'PB', 'PH' 00074 + *DLf, *DF', 'NR', 'K1, 'DB', t / END REAL*8 DTYPE, WRTYPE, WFTYPE, PBFTYP INTEGER*4 CHGCOD L0GICAL*1 LGOFLG, LQFLG COMMON /LITRAL/ DTYPE(4), WRTYPE(2), WFTYPE(2), + LGO, LSKIP(2), LQUIT, CHGC0D(24), + PBFTYP(2) COMMON /DEFVAL/ IDTYPE, IWRTYP, IWFTYP, IPBF, ALPHA, BETA, XMU, SIGMA, ALPHAC, BETAC, W, XL, T1, R, C, THETA, PHI, DELTA, IDF, NR, XK, LSK, LDB 214 00090 C: 00091 OPEN (UNIT=6, DEUICE='TTY') 00092 C: 00093 C: INITIALIZE RANDOM NUMBER SEEDS 00094 C: 00095 ISEED1 = 12345 00096 ISEED2 = 98765 00097 ISEED3 = 24680 00098 C: 00099 C: 00100 C: SET DEFAULT VALUES 00101 C: 00102 CALL DEFSET 00103 C: 00104 C: PRINT OUT MAIN MENU IF NOT SKIPPING 00105 C: 00106 10 CONTINUE 00107 IF (LSK .EQ. LSKIP(1)) GO TO 20 00108 CALL MENU 00109 0: 00110 C: PROCESS USER INPUTS 00111 C j 00112 20 CONTINUE 00113 C: 00114 CALL UPDATE (LGOFLG, LQFLG) 00115 C: 00116 C: CHECK GO FLAG 00117 C: 00118 IF (LGOFLG) GO TO 100 00119 C: 00120 C: CHECK QUIT FLAG 00121 C: 00122 IF (LQFLG) GO TO 9900 00123 Ci 00124 Ci IF NEITHER FLAG SET, PROCESS NEXT INPUT 00125 Ci 00123 GO TO 10 00127 u! 00128 Ci RUN ONE SIMULATION 00129 Zz 00130 100 CONTINUE 00131 CALL SIMCTL (ISEED1, ISEED2, ISEED3) 00132 C: 00133 GO TO 10 00134 C: 00135 C: FINISH 00136 C: 00137 9900 CONTINUE 00138 WRITE (6,9100) 00139 9100 FORMAT (' ***** ALL PAU ***»*’) 00140 C: 00141 C: 00142 CLOSE (UNIT=6, DEVICE='TTY’) 00143 C: 00144 STOP 00145 END 00146 SUBROUTINE UPDATE (LGOFLG, LQFLG) 00147 C: 00148 00149 r* • ■ C: 00150 C: THIS SUBROUTINE READS USER INPUTS AND UPDATES THE 00151 C: CURRENT SETTINGS FOR EACH SELECTION 00152 C: 00153 00154 C* •1 C: 00155 REAL*8 DTYPE, WRTYPE, WFTYPE, PBFTYP 00156 INTEGER*4 CHGCOD 00157 0: 00158 L0GICAL*1 LGOFLG, LQFLG 00159 Cs 00160 CDMMON /LITRAL/ DTYPE(4), WRTYPE(2), WFTYPE(2), 00161 + LGO, LSKIP(2), LQUIT, CHGC0D(24), 00162 + PBFTYP(2) 00163 C: 00164 COMMON /DEFVAL/ IDTYPE, IWRTYP, IWFTYP, IPBF, 00165 + ALPHA, BETA, XMU, SIGMA, ALPHAC, 00166 + BETAC, LI, XL, T1, R, C, THETA, 00167 + PHI, DELTA, IDF, NR, XK, LSK, 00168 + LDB 00169 C: 00170 C: INITIALIZE GO FLAG AND QUIT FLAG 00171 C: 00172 LGOFLG = .FALSE. 00173 LQFLG = .FALSE. 00174 C: 00175 C: INITIALIZE MAXIMUM ALLOWABLE CONSECUTIVE ERRORS 00176 C: 00177 DATA MAXERR / 5 / 00178 C: 00179 C: INITIALIZE MAXIMUM NUMBER OF DEBUG LEVELS 00180 C: 00181 DATA MAXDBG / 2 / 00182 C: 00183 C: INITIALIZE CONSECUTIVE ERROR COUNT 00184 C: 00185 NERR = 0 00186 C: 00187 C: SET CURRENT NUMBER OF CHANGE CODES TO BE SEARCHED 00188 C: 00189 NCC = 23 00190 C: 00191 C: REQUEST USER INPUT 216 □□192 C: □0193 WRITE (6,9000) □□194 9D00 FORMAT (' ', ’ENTER REQUEST') 00195 C: □□196 C: READ USER INPUT □0197 C; □0198 5 CONTINUE 00199 C: 00200 READ (5,9002) ICC 00201 9002 FORMAT (A2) 00202 IF (ICC .NE. LGO) GO TO 10 00203 LGOFLG = .TRUE. 00204 GO TO 9900 00205 C: 00206 10 CONTINUE 00207 C: 00208 IF (ICC .NE. LQUIT) GO TO 20 00209 LQFLG = .TRUE. 00210 GO TO 9900 00211 C: 00212 20 CONTINUE 00213 C: 00214 C: CHECK FOR CHANGE CODE MATCH 00215 C: 00216 DO 30 1=1, NCC 00217 ISAVE = I 00218 IF (ICC .EQ. CHGCOD(I)) GO TO 40 00219 30 CONTINUE 00220 C: 00221 C: NORMAL EXIT FROM LOOP MEANS NO HALID CHANGE CODE 00222 C: WAS FOUND 00223 C: 00224 WRITE (6, 9005) ICC 00225 9005 FORMAT. (' ', 'CHANGE CODE ’, A2, 'IS NOT VALID' / 00226 f ' ', 'REENTER REQUEST' ) 00227 NERR = NERR + 1 00228 C: 00229 C: IF TJO MANY CONSECUTIVE ERRORS, THEN QUIT 00230 C: 00231 IF (NERR .LE. MAXERR) GO TO 5 00232 LQFLG = .TRUE. 00233 GO TO 9900 00234 C: 00235 C: PROCESS CHANGE REQUEST 00236 C: 00237 40 CONTINUE 00233 C: 00239 GO TO (100, 200,. 300, 400, 500, 600, 700, 800, 00240 f 900, 1000, 1100, 1200, 1300, 1400, 1503, 1500, 00241 f 1700, 1800, 1900, 2000, 2100, 2200, 2300), ISAVE 00242 C: □0243 WRITE (6, 9010)i ISAVE 00244 9010 FORMAT (’ ‘PROGRAM ERROR 1 IN UPDATE, ISAVE = ', 13 00245 + ’ ' + » ’QUITTING1) 00246 LQFLG = .TRUE. 00247 GO TO 9900 00248 C: 00249 C: PROCESS CHANGE CODE 1: CHANGE DISTRIBUTION TYPE 00250 C; 00251 100 CONTINUE 00252 WRITE (6,9015) 00253 9015 FORMAT (' ', 'ENTER DISTRIBUTION TYPE’ / 00254 1 \ •1-EXPON 2-GAMMA 3-WEI8ULL 4-TRNORM’) 00255 110 CONTINUE 0025S READ (5, ■*) ITEMP 00257 IF ((ITEMP .GE. 1) .AND. (ITEMP .LE. 4)) GO TO 120 00258 WRITE (6,9020) ITEMP 00259 9020 FORMAT (' ’, 14, ' IS NOT 1, 2, 3, OR 4 - REENTER11) 00260 GO TO 110 00261 120 CONTINUE 00262 IDTYPE = ITEMP 00263 GO TO 9900 00264 C; 00265 C: PROCESS CHANGE CODE 2: CHANGE WARRANTY RENEWAL TYPE 00266 C: 00267 200 CONTINUE 00268 IWRTYP = 3 - IWRTYP 00269 WRITE (6,9025) WRTYPE(IWRTYP) 00270 9025 FORMAT (’ ', •WARRANTY RENEWAL TYPE CHANGED TO: ’, 00271 Hv A8 ) 00272 GO TO 9900 00273 C: 00274 C: PROCESS CHANGE CODE 3: CHANGE NUMBER OF FAILURES COVERED 00275 C: BY WARRANTY 00276 C: 00277 300 CONTINUE 00278 IWFTYP = 3 - IWFTYP 00279 WRITE (6,9030) WFTYPE(IWFTYP) 00280 9030 FORMAT (' •NUMBER OF FAILURES COVERED BY WARRANTY', 00281 h ' CHANGED TO: A8) 00282 GO TO 9900 00283 C: 00234 C: PROCESS CHANGE CODE 4: CHANGE ALPHA FOR DISTRIBUTION 00285 C: 00286 400 CONTINUE 00287 WRITE (6,9035) 00288 9035 FORMAT (' ’, ’ENTER ALPHA FOR DISTRIBUTION’) 00289 410 CONTINUE 00290 READ (5,*) TEMP 00291 IF (TEMP .GT.. 0.) GO TO 420 00292 WRITE (6,9065) TEMP 00293 9040 FORMAT (' ’, G13.7, ' IS NOT BETWEEN 0 AND 1' / 218 □0294 * ‘REENTER’ ) 00295 GO TO 410 00296 420 CONTINUE 00297 ALPHA = TEMP 00293 GO TO 9900 00299 C: 00300 C: PROCESS CHANGE CODE 5: CHANGE BETA FOR DISTRIBUTION 00301 C: 00302 500 CONTINUE 00303 WRITE (6,9045) 00304 9045 FORMAT (' ', ’ENTER BETA FOR DISTRIBUTION’) 00305 510 CONTINUE 00306 READ (5,*) TEMP 00307 IF (TEMP .GT 0.) GO TO 520 00308 WRITE (6,9065) TEMP 00309 GO TO 510 00310 520 CONTINUE 00311 BETA = TEMP 00312 GO TO 9900 00313 C; 00314 C: PROCESS CHANGE CODE 6: CHANGE MU FOR DISTRIBUTION 0031 5 C: 00316 600 CONTINUE 00317 WRITE (6,9055) 00318 9055 FORMAT (1 1, ’ENTER MU FOR DISTRIBUTION’) 003'! 9 READ (5,*) XMU 00320 GO TO 9900 00321 C: 00322 C: PROCESS CHANGE CODE 7: CHANGE SIGMA FOR DISTRIBUTION 00323 C: 00324 700 CONTINUE 00325 WRITE (6,9060) 00326 9060 FORMAT (' ’, ’ENTER SIGMA FOR DISTRIBUTION’) 00327 710 CONTINUE 00328 READ (5,*) TEMP 00329 IF (TEMP .GE. 0.) GO TO 720 00330 WRITE (6,9065) TEMP 00331 9065 FORMAT (’ ’, G13.7, * MUST BE POSITIVE’ / 00332 ’REENTER’ ) 00333 GO TO 710 00334 720 CONTINUE 00335 SIGMA = TEMP 00336 GO TO 9900 00337 C: 00338 ■ C: PROCESS CHANGE CODE 8: CHANGE SKIP CODE 00339 C: 00340 800 CONTINUE 00341 IF (LSK .EQ. LSKIP(1)) GO TO 810 00342 LSK = LSKIP(1) 00343 GO TO 820 00344 810 CONTINUE 219 00345 LSK = LSKIP(2) 00346 820 CONTINUE 00347 WRITE (6,9070) LSK 0034B 9070 FORMAT (' V S K I P CODE CHANGED TO: ’, A4) 00349 GO TO 9900 00350 C: 00351 C: PROCESS CHANGE CODE 9: CHANGE ALPHAC (PROB. OF CLAIM) 00352 C: 00353 900 CONTINUE 00354 WRITE (6,9075) 00355 9075 FORMAT (’ ', ’ENTER ALPHAC (PROB. OF CLAIM)’) 00356 910 -CONTINUE 00357 READ (5,*) TEMP 00358 IF ((TEMP .GE. 0.) .AND. (TEMP .LE. 1.)) GO TO 920 00359 WRITE (6,9040) TEMP 00360 GO TO 910 00.361 920 CONTINUE 00362 ALPHAC = TEMP 00333 GO TO 9900 00364 C: 00365 C: PROCESS CHANGE CODE 10: CHANGE BETAC 00366 0: (PROB. OF SALES AFTER W) 00367 C: 00368 1000 CONTINUE 00369 WRITE (6,9080) 00370 9080 FORMAT (' ’, 'ENTER BETAC (PROB. OF SALES AFTER W)') 00371 1010 CONTINUE 00372 READ (5,*) TEMP 00373 IF ((TEMP .GE. 0.) .AND. (TEMP .LE. 1.)) GO TO 1020 00374 WRITE (6,9040) TEMP 00375 GO TO 1010 00376 1020 CONTINUE 00377 BETAC = TEMP 00378 GO TO 9900 00379 C: 00380 C: PROCESS CHANGE CODE 11: CHANGE W (WARRANTY LENGTH) 00381 C: 00382 1100 CONTINUE 00383 WRITE (6,9085) 00.384 9085 FORMAT (' ’, ’ENTER W (WARRANTY LENGTH)') 00385 1110 CONTINUE 00336 READ (5,*) TEMP D0387 IF ((TEMP .GT. 0.) .AND. (TEMP .LE. XL)) GO TO 1120 00388 WRITE (6,9095) TEMP, XL 00389 9095 FORMAT (’ G13.7, ’ MUST BE BETWEEN 0 AND ’, 0.0390 Hh G13.7 / ’ ’, ’REENTER’ ) 0039-1 GO TO 1110 00392 1120 CONTINUE 0039.3 W = TEMP 00394 GO TO 9900 00395 C: 220 00396 C: PROCESS CHANGE CODE 12: CHANGE L (LIFETIME) 00397 C: 00398 1200 CONTINUE 00399 WRITE (6,9090) 00400 9090 FORMAT (' ', ’ENTER L (LIFETIME)*) 00401 1210 CONTINUE 00402 READ (5,*) TEMP 00403 IF (TEMP .GT,. 0.) GO TO 1220 00404 WRITE (6,9065) TEMP 00405 GO TO 1210 00406 1220 CONTINUE 00407 XL = TEMP 00408 GO TO 9900 00409 C: 00410 C: PROCESS CHANGE CODE 13: CHANGE T1 FOR COMBO WARRANTY 00411 C: 00412 1300 CONTINUE 00413 WRITE (6,9100) 00414 9100 FORMAT (' ', 'ENTER T1 FOR COMBO WARRANTY') 00415 1310 CONTINUE 00416 READ (5,*) TEMP 00417 IF ((TEMP .GE. 0.) .AND. (TEMP .LT. W)) GO TO 1320 00418 WRITE (6,9095) TEMP, W 00419 GO TO 1310 00420 1320 CONTINUE 00421 T1 = TEMP 00422 GO TO 9900 00423 C: 00424 C: PROCESS CHANGE CODE 14: CHANGE R (SELLER’S COST) 00425 C: 00426 1400 CONTINUE 00427 WRITE (6,9105) 00428 9105 FORMAT’(' ', 'ENTER R (SELLER"S COST)') 00429 1410 CONTINUE 00430 READ (5,*) TEMP 00431 IF (TEMP .GE. 0.) GO TO 1420 00432 WRITE (6,9065) TEMP 00433 GO TO 1410 00434 1420 CONTINUE 00435 R = TEMP 00436 GO TO 9900 00437 C? 0043B C: PROCESS CHANGE CODE 15: CHANGE C (BUYER'S COST) 00439 C: 0044D 1500 CONTINUE 00441 WRITE (6,9110) 00442 9110 FORMAT (* 'ENTER C (BUYER"S COST)') 00443 1510 CONTINUE 00444 READ (5,*) TEMP 00445 IF (TEMP .GE. 0.) GO TO 1520 00446 WRITE (6,9065) TEMP 221 00447 GO TO 1510 00448 1520 CONTINUE 00449 C = TEMP 00450 GO TO 9900 00451 C: 00452 C: PROCESS CHANGE CODE 16: CHANGE THETA 00453 C: (PROB. OF NOT VALIDATING A CLAIM) 00454 C: 00455 1600 CONTINUE 0045B WRITE (6,9115) 00457 9115 FORMAT (' 'ENTER THETA (PROB. OF NOT VALIDATING', 0D458 h 1 A CLAIM)' ) 00459 1610 CONTINUE 00460 READ (5,*) TEMP 00461 IF ((TEMP .GE. 0.) .AND. (TEMP .LE. 1.)) GO TO 1620 00462 WRITE (6,9040) TEMP 00463 GO TO 1610 00464 1620 CONTINUE 00465 THETA = TEMP 00465 GO TO 9900 D0467 C: 00468 C: PROCESS CHANGE CODE 17: CHANGE PAYBACK FUNCTION 00469 0: 00470 1700 CONTINUE 00471 IPBF = 3 - IPBF 00472 WRITE (6,9120) PBFTYP(IPBF) 00473 9120 FORMAT (' ', 'PAYBACK FUNCTION CHANGED TO: ', A8) 00474 GO TO 9900 00475 C: 00476 C: PROCESS CHANGE CODE 18: CHANGE PHI 00477 C: 00478 1800 CONTINUE 00479 WRITE (6,9125) 00480 9125 FORMAT (' ', 'ENTER PHI (AS A DECIMAL FRACTION)' ) 00481 1810 CONTINUE 00482 READ (5,*) TEMP 00483 IF ((TEMP .GE. 0.) .AND. (TEMP .LE. 1.)) GO TO 1820 00484 WRITE (6,9040) TEMP 00485 GO TO 1810 00486 1820 CONTINUE 00487 PHI = TEMP 00488 GO TO 9900 00483 C: 00490 C: PROCESS CHANGE CODE 19: CHANGE DELTA 00491 C: 00492 1900 CONTINUE 00493 WRITE (6,9130) 00494 9130 FORMAT (' ', 'ENTER DELTA (AS A DECIMAL FRACTION)' ) 00495 1910 CONTINUE 00496 READ (5,*) TEMP 00497 IF ((TEMP .GE. 0.) .AND. (TEMP .LE. 1.)) GO TO 1920 222 00498 WRITE (6,9040) TEMP 00490 GO TO 1910 00500 1920 CONTINUE 005131 DELTA = TEMP 00502 GO TO 9900 00503 C: 00504 Cs. PROCESS CHANGE CODE 20: CHANGE DISCOUNT FUNCTION 00505 C: 00506 2000 CONTINUE 00507 WRITE (6,9135) 00508 9135 FORMAT (' ', ’ENTER DISCOUNT FUNCTION TYPE: 1, 2, OR 3’ ) 00509 2010 CONTINUE 00510 READ (5, *) ITEMP 00511 IF ((ITEMP .GE. 1) .AND. (ITEMP .LE. 3)) GO TO 2020 00512 WRITE (6,9140) ITEMP 00513 9140 FORMAT (' ', 14, ’ IS NOT 1, 2, OR 3 - REENTER1) 00514 GO TO 2010 00515 2020 CONTINUE 00516 IDF = ITEMP 00517 GO TO 9900 00518 G: 00519 C: PROCESS CHANGE CODE 21: CHANGE NR (NUMBER OF REPLICATIONS) 00520 0: 00521 2100 CONTINUE 00522 WRITE (6,9145) 00523 9145 FORMAT (' ', 'ENTER NUMBER OF REPLICATIONS') 00524 2110 CONTINUE 00525 READ (5,*) ITEMP 00526 IF (ITEMP .GT. 0) GO TO 2120 00527 WRITE (6,9150) ITEMP 00528 9150 FORMAT (' ', 18, ’ MUST BE POSITIVE' / 00529 H 'REENTER' ) 00530 GO TO 2110 00531 2120 CONTINUE 00532 NR = ITEMP 00533 GO TO 9900 00534 C: 00535 C; PROCESS CHANGE CODE 22: CHANGE K (ADMINISTRATIVE COST) 00536 C: 00537 2200 CONTINUE 00538 WRITE (6,9155) 00539 9155 FORMAT (' 1, 'ENTER K (ADMIN. COST)') 00540 2210 CONTINUE 00541 READ (5,*) TEMP 00542 IF (TEMP .GE. 0.) GO TO 2220 00543 WRITE (6,9065) TEMP 00544 GO TO 2210 00545 2220 CONTINUE 00546 Xi< = TEMP 00547 GO TO 9900 00548 C: 223 00549 00550 00551 00552 00553 00554 00555 00556 00557 00558 00559 00560 00561 00562 00563 00564 00565 00566 00567 00568 00569 00570 00571 00572 00573 00574 00575 00573 00577 00578 00579 00580 00581 00582 00583 00584 00585 00586 00587 00588 00589 00590 00591 00592 00593 00594 00595 00596 00597 00598 00599 C: PROCESS CHANGE CODE 23: CHANGE DEBUG LEVEL C: 2300 CONTINUE WRITE (6,9160) 9160 FORMAT (1 ’ENTER DEBUG LEVEL' ) 2310 CONTINUE READ (5, *) ITEMP IF ((ITEMP .GE. 0) .AND. (ITEMP .LE. MAXDBG)) GO TO 2320 WRITE (6,9165) ITEMP, MAXDBG 9165 FORMAT (’ ', 14, ' IS NOT BETWEEN 0 AND ', 12, + 1 - REENTER*) GO TO 2310 232D CONTINUE LDB = ITEMP GO TO 9900 C: C: INPUT PROCESSING COMPLETE C: 9900 CONTINUE RETURN END SUBROUTINE DEFSET C: C C C C C C C: C: C: C: Cs C: C: THIS SUBROUTINE SETS THE DEFAULT VALUES FOR ITEMS IN H E MAIN MENU REAL*8 DTYPE, WRTYPE, WFTYPE, PBFTYP INTEGERS CHGCOD COMMON /LITRAL/ DTYPE(4), WRTYPE(2), WFTYPE(2), LGO, LSKIP(2), LQUIT, CHGC0D(24), PBFTYP(2) COMMON /DEFVAL/ IDTYPE, IWRTYP, IWFTYP, IPBF, ALPHA, BETA, XMU, SIGMA, ALPHAC, BETAC, W, XL, T1, R, C, THETA, PHI, DELTA, IDF, NR, XK, LSK, LDB SET DISTRIBUTION TYPE CODE TO "EXPONENTIAL" IDTYPE = 1 SET WARRANTY RENEWAL TYPE TO "RENEWING" IWRTYP = 1 + + 224 00600 C: SET WARRANTY FAILURE TYPE TJ "FIRST" 00601 C: 00602 IWFTYP = 1 00603 C: 00604 C: SET PAYBACK FUNCTION TYPE TO "CONSTANT" 00605 C: 00606 IPBF = 1 00607 0: 00608 C: INITIALIZE ALPHA, BETA FOR DISTRIBUTIONS 00609 Cs 00610 ALPHA = 1. 00611 BETA = 1. 00612 C: 00613 C: INITIALIZE MU AND SIGMA FOR TRUNCATED NORMAL 00614 C: 00615 XMU = 0. 00616 SIGMA = 1. 00617 C: 00618 C: INITIALIZE ALPHA (PROBABILITY OF A CLAIM) 00619 C: 00620 ALPHAC = 1. 00621 C: 00622 C: INITIALIZE BETA (PROBABILITY OF SALES AFTER W) 00623 C: 00624 BETAC = 0. 00625 C: 00626 C: INITIALIZE W (WARRANTY DURATION) 00627 C: 00628 W = 1. 00629 C: 00630 C: INITIALIZE LIFETIME 00631 C: 00632 XL = •20. 00633 Cs 00634 C: INITIALIZE PAYBACK FUNCTION TYPE TO "CONSTANT" 00635 C: 00636 IPBF = 1 00637 C: 00638 C: INITIALIZE T1 FOR COMBINATION WARRANTY 00639 C: (T1 = 0 SAME AS PURE PRO-RATA WARRANTY) 00640 C: 00641 T1 = 0. 00642 C: 00643 C: INITIALIZE SELLER'S COST 00644 0: 00645 R = 1. 00646 C: 00647 C: INITIALIZE BUYER'S COST 00648 C: 00649 • o I I u 00650 C: 225 00651 C: INITIALIZE THETA (PROBABILITY OF NOT VALIDATING A CLAIM) 00652 C: 00653 THETA = 0. 00654 C: 00655 C; INITIALIZE DISCOUNT FUNCTION TO TYPE 1 00656 C: 00657 IDF = 1 00658 C: 00659 C; INITIALIZE VALUES OF PHI AND DELTA FOR DISCOUNT 00660 C: FUNCTION 00661 C: 00662 PHI = 0. 00663 DELTA = 0. 00664 C: 00665 C: INITIALIZE NUMBER OF REPLICATIONS 00666 C: 00667 NR = 100 00668 0: 00669 C: INITIALIZE ADMINISTRATIVE COSTS 00670 C: 00671 XK = 0. 00672 C: 00673 C: INITIALIZE SKIP CODE TO "OFF” 00674 C: 00675 LSK = LSKIP(2) 00676 C: 00677 C: INITIALIZE DEBUG LEVEL 00678 r* 00679 LDB = 0 00630 C: 00681 RETURN 00682 END 00683 SUBROUTINE MENU 00684 0: 00685 00686 r* • C; 00687 C: THIS SUBROUTINE PRINTS THE MAIN MENU TOGETHER WITH 00638 0: CURRENT SETTINGS FOR EACH SELECTION 00689 C: 00690 00691 r* • C: 00692 REAL*8 DIYPE, LIRTYPE, WFTYPE, PBFTYP 00693 INTEGER*4 CHGCOD 00694 C: 00695 COMMON /LITRAL/ DTYPE(4), WRTYPE(2), WFTYPE(2), 00696 + LGO, LSKIP(2), LQUIT, CHGC0D(24), 00697 + PBFTYP(2) 00698 0: 00699 COMMON /DEFVAL/ IDTYPE, IWRTYP, IWFTYP, IPBF, 00700 + ALPHA, BETA, XMU, SIGMA, ALPHAC, 00701 + BETAC, W, XL, T1, R, C, THETA, 226 00702 + PHI, DELTA, IDF, NR, XK, LSK, 00703 + LDB 00704 C: 00705 WRITE (6,9000) CHGC00{1), DTYPE(IDTYPE) 00706 9000 FORMAT (* A1, 1X, 00707 + FAILURE DISTRIBUTION1, T45, AS) 00708 C: 00709 IF (IDTYPE .EQ. 4) GO TO 20 00710 IF (IDTYPE .EQ. 1) GO TO 10 00711 C: 00712 WRITE (6,9005) CHGC0D(4), ALPHA 00713 9005 FORMAT (' ’, A1, 1X, 00714 + ALPHA*, T45, F8.5) 00715 C: 00716 10 CONTINUE 00717 WRITE (6,9010) CHGC0D(5), BETA 00718 9010 FORMAT (’ *, A1, 1X, 00719 + *: BETA', T45, F8.5) 00720 GO TO 30 00721 C: 00722 20 CONTINUE 00723 WRITE (6,9015) CHGC0D(6), XMU, 00724 + CHGC0Q(7), SIGMA 00725 9015 FORMAT (* ’, A2, *: MU', T45, F8.5 / 00726 + * ', A2, SIGMA*, T45, F8.5 ) 00727 C: 00728 30 CONTINUE 00729 C: 00730 WRITE (6,9020) CHGC0D(2), WRTYPE(IWRTYP) 00731 9020 FORMAT (’ ’, A2, *: WARRANTY RENEWAL TYPE', 00732 + T45, A8) 00733 C: 00734 WRITE (6,9025) CHGC0D(3), WFTYPE(IWFTYP) 00735 9025 FORMAT (* ’, A2, *: FAILURES COVERED*, 00736 + T45, A8) 00737 C: 00738 WRITE (6,9030) CHGC0D(9), ALPHAC, 00739 + CHGC0D(10), BETAC 00740 9030 FORMAT (’ ', A2, ALPHA (PROB, OF CLAIM)*, 00741 + T45, F8.5 / 00742 + * *, A2, 00743 + *: BETA (PROB, OF SALES AFTER W)', 00744 + T45, F8.5 ) 00745 C: 00746 WRITE (6,9035) CHGC0D(11), W, CHGC0D(12), XL 00747 9035 FORMAT (* ’, A1, 1X, *: WARRANTY PERIOD1, 00748 + T45, F8.3 / 00749 + * ', A1, 1X, LIFETIME*, T45, F8.3 ) 00750 C:^ 00751 WRITE (6,9040) CHGC0D(17), PBFTYP(IPBF) 00752 9040 FORMAT (' *, A2, PAYBACK FUNCTION*, T45, A8) 227 00753 C: 00754 IF (IPBF .NE. 2) GO TO 40 00755 WRITE (6,9045) CHGC0D(13), T1 00755 9045 FORMAT (' ', A2, ': T1 FOR COMBO WARRANTY’, 00757 + T45, F8.3) 00758 C; 00759 40 CONTINUE 00760 C: 00761 WRITE (6,9050) CHGC0D(14), R, 00762 + CHGC0D(15), C 00763 9050 FORMAT (' ', A1, 1X, SELLER1'S COST1, 00764 T45, F8.5 / 00765 A1, 1X, BUYER1’S COST1, 00766 + T45, F8.5 ) 00767 C; 00768 WRITE (6,9055) CHGC0D(16), THETA 00763 9055 FORMAT (' ', A2, 00770 + THETA (PROB. OF NOT VAL. CLAIM)’, 00771 + T45, F8.5 ) 00772 C: 00773 WRITE (6,9060) CHGCQD(20), IDF, 00774 + CHGC0D(18), PHI, 00775 + CHGC0D(19), DELTA 00776 9060 FORMAT (' f, A2, ’: DISCOUNT FUNCTION’, T45, 11 / 00777 ’ *, A2, ’: PHI', T45, F8.5 / 00778 A2, DELTA', T45, F8.5 ) 00779 C: 00780 WRITE (6,9065) CHGC0D(21), NR 00781 9065 FORMAT (' ', A2, ': NUMBER OF REPLICATIONS', T45, 15) 00782 C: 00783 WRITE (6,9070) CHGC0D(22), XK 00784 9070 FORMAT (' \ A2, ADMINISTRATIVE COST1, T45, F8.5) 00785 C: 00786 WRITE (6,9075) CHGC0D(8), LSK 00787 9075 FORMAT (’ ', A2, SKIP CODE', T45, A4) 00788 ' C: 00789 WRITE (6,9080) CHGC0D(23), LDB 00790 9080 FORMAT (' ', A2, ': DEBUG LEVEL', T45, 11) 00791 C: 00792 RETURN 00793 END 00794 SUBROUTINE SIMCTL (ISEED'l, ISEED2, ISEED3) 00795 00796 C: L:::::::::::::::::::: 00797 C: 00798 C: THIS SUBROUTINE CONTROLS ALL ITERATIONS OF THE SIMULATION 00799 C: FOR A SINGLE COMBINATION OF INPUT PARAMETERS. IT THEN 00800 C: GETS THE SAMPLE MEAN AND SAMPLE STANDARD DEVIATION OF THE 00801 C: DISCOUNTED TOTAL COST. 00802 00803 C: 228 00804 C: 00805 REAL*8 DTYPE, liJRTYPE, WFT/PE, PBFTYP 00806 INTEGERS CHGCOD 00807 C: 00808 LOGICAL^ LGOFLG, LQFLG 00809 C; 00810 COMMON /LITRAL/ DTYPE(4), WRTYPE(2), WFTYPE(2), 00811 + LGO, LSKIP(2), LQUIT, CHGC0D(24), 00812 + PBFTYP(2) 00813 C: 00814 COMMON /DEFVAL/ IDTYPE, IliJRTYP, IWFTYP, IPBF, 00815 + ALPHA, BETA, XMU, SIGMA, ALPHAC, 00816 + BETAC, W, XL, T1, R, C, THETA, 00817 + PHI, DELTA, IDF, NR, Xi<, LSK, 00818 + LDB 00819 C: 00820 DIMENSION DTCOST(IOOQ) 00821 C: 00822 DO 1000 N = 1, NR 00823 C: 00824 C: PERFORM A THE NTH SINGLE REPLICATION 00825 C: 00826 CALL SIM (DTCOST(N), ISEED1, ISEED2, ISEED3) 00827 C: 00828 1000l CONTINUE 00829 C: 00830 C: GET MEAN AND STANDARD DEVIATION OF THE SAMPLES 00831 Cs DTCOST(N); N = 1, NR 00832 C; 00833 CALL STD (DTCOST, NR, DTAVG, DTSTD) 00834 C: 00835 WRITE (6,9000) NR, DTAVG, DTSTD 00836 9000 FORMAT (* ’, ’FOR ’, 14, ' REPLICATIONS:’ / 00837 + ’ 5X, ’DISCOUNTED MEAN TOTAL COST = F12.2 / 00838 + ' ’, 5X, ’ STD DEV OF TOTAL COST = ’, G13.7 /// ) 00839 RETURN 00840 END 00841 SUBROUTINE SIM (DISCST, ISEED1, ISEED2, I5EED3) 00842 C: 00843 00844 C* • * * C: 00845 C: THIS SUBROUTINE CONTROLS A SINGLE REPLICATION OF THE 00846 C: WARRANTY SIMULATION. IT RETURNS DISCST, THE DISCOUNTED 00847 C: TOTAL COST FOR THIS REPLICATION. 00848 nno/iQ C: P • • • ■ uubay 00850 C: 00851 REAL*8 DTYPE, WRTYPE, WFTYPE, PBFTYP 00852 INTEGER*4 CHGCOD 00853 C: 00854 LOGICAL^I LGOFLG, LQFLG 229 00855 Cs 00856 COMMON /LITRAL/ DTYPE(4), WRTYPE(2), WFTYPE(2), 00857 + LGO, LSKIP(2), LQUIT, CHGC0D(24), 00858 + PBFTYP(2) 00859 C: 00860 COMMON /DEFVAL/ IDTYPE, IWRTYP, IWFTYP, IPBF, 00861 + ALPHA, BETA, XMU, SIGMA, ALPHAC, 0D862 + BETAC, W, XL, T1, R, C, THETA, 00863 + PHI, DELTA, IDF, NR, XK, LSK, 00864 + LDB 00865 G: 00866 C: T(I) = ABSOLUTE TIME OF THE ITH FAILURE 00867 C: DT(I) = TIME BETWEEN THE (1-1)TH AND ITH FAILURE 00868 C: D •H /-S I I —I -a 00869 C: COST(l) = NONDISCOUNTED COST OF THE ITH FAILURE 00870 C: DCOST(I) = DISCOUNTED COST OF THE ITHE FAILURE 00871 C: 00872 DIMENSION T(1000), DT(1000), COST(IOOO), DCOST(IOOO) 00873 C: 00874 C: GENERATE A SEQUENCE OF FAILURES IN THE TIME INTERVAL (0,XL). 00875 C: NF IS RETURNED AS THE NUMBER OF FAILURES IN THE: INTERVAL.. 00876 Cr 00877 CALL GEN (T, DT, NF, ISEED1, ISEED2) 00878 Cs. 00879 C: COMPUTE NONDISCOUNTED COSTS UNDER THE TERMS OF THE WARRANTY 00880 C: 00881 CALL WARR (T, DT, NF, COST, ISEED3) 00882 C: 00883 C: DISCOUNT THE SERIES OF CASH FLOWS 00884 C: 00885 XMEAN = 0. 00886 CALL DISCNT (COST, T, NF, IDF, PHI, DELTA, 00887 + XMEAN, DCOST, LDB) 00888 C: 00889 Cs COMPUTE the: TOTAL DISCOUNTED COST 00890 C: 00891 DISCST = TCOST (DCOST, NF) 00892 C: 00893 RETURN 00894 END 00895 SUBROUTINE GEN (T, DT, NF, ISEED1, ISEED2) 00896 Ci 00897 L:: 00898 C: 00899 Cs THIS SUBROUTINE GENERATES A SEQUENCE OF FAILURES OVER 00900 C: THE TIME INTERVAL (0,XL) 00901 Cs 00902 Cs INPUTS: 00903 Cs 00904 Cs ISEED1, ISEED2 = RANDOM INTEGER SEEDS 00905 Cs 230 □0906 C: OUTPUTS: □0907 C: D09D8 C: NF = NUMBER OF FAILURES IN THE INTERVAL (0,XL) 00909 C: 00910 C: T (I) = ABSOLUTE TIME OF THE ITH FAILURE; 00911 C: I = 1, 2, NF 00912 C: 00913 C: DT(I) = TIME BETWEEN THE (1-1)TH AND ITH 00914 C: FAILURE; I = 1, 2, ..., NF; 00915 C: DT(1) = T(1) 00916 nnoi * 7 C: P ■ ■ uuy 1 ( Li i 00918 C: 00919 REAL*8 DTYPE, WRTYPE, WFTYPE, PBFTYP 00920 INTEGER*4 CHGCOD 00921 C: 00922 LOGICAL*'! LGOFLG, LQFLG 00923 C: 00924 COMMON /LlfRAL/ DTYPE(4), WRTYPE(2), WFTYPE(2), 00925 + LGO, LSKIP(2), LQUIT, CHGC0D(24), 00926 + PBFTYP(2) 00927 C: 00928 COMMON /DEFVAL/ IDTYPE, IWRTYP, IWFTYP, IPBF, 00929 + ALPHA, BETA, XMU, SIGMA, ALPHAC, 00930 + BETAC, W, XL, T1, R, C, THETA, 00931 + PHI, DELTA, IDF, NR, XK, LSK, 00932 + LDB 00933 C: 00934 DIMENSION T(1000), DT(1000) 00935 Cr □0936 C: MAXIMUM NUMBER OF FAILURES TO BE GENERATED 00937 C: 00938 DATA NFMAX / 1000 / 00939 C: 00940 C: ZERO OUT ALL VALUES PRIOR TO GENERATING FAILURES 00941 C: 00942 DO 100 1=1, NFMAX 00943 T(I> = 0. 00944 DT(I) = 0. 00945 100 CONTINUE 00946 NF = 0 00947 C: 00948 DO 1000 1=1, NFMAX 00949 NF = I 00950 C: 00951 C: DETERMINE FAILURE DISTRIBUTION TYPE 00952 C: 00953 GO TO (210, 220, 230, 240), IDTYPE 00954 C: 00955 WRITE (6,9000) NF, IDTYPE 00956 9000 FORMAT (' ', 'PROGRAM ERROR 1 IN GEN, 231 □0957 + ’NF = 15, ’ IDTYPE = ’, 15) 00958 GO TO 9800 00959 C: 00960 C: GENERATE AN EXPONENTIAL INTERFAILURE TIME 00961 C: 00962 210 CONTINUE 00963 DT(l) = EXPON (BETA, ISEED1) 00964 GO TO 500 00965 C: 00966 C: GENERATE A GAMMA INTERFAILURE TIME 00967 C: 00968 220 CONTINUE 00969 DT(I) = RGAMMA (ALPHA, BETA, I5EED1, ISEED2) 00970 GO TO 500 00971 C: 00972 C: GENERATE A UEIBULL INTERFAILURE TIME 00973 C: 0D974 230 CONTINUE 00975 DT(I) = UEIBUL (ALPHA, BETA, ISEED1, ISEED2) 00976 GO TO 500 00977 C: 00978 Cs IF I IS ODD, GENERATE A PAIR OF TRUNCATED NORMAL 00979 C: INTERFAILURE TIMES. IF I IS EVEN, USE ONE OF THE 00980 C: INTERFAILURE TIMES GENERATED ON THE PREVIOUS 00981 C: ITERATION. 00982 C: 00983 240 CONTINUE 00984 IF (M0D(I,2) .EQ. 0) GO TO 500 00985 IP1 = 1 + 1 00986 CALL TRNORM (XMU, SIGMA, ISEED1, ISEED2, 00987 + DT(I), DT(IP1)) 00988 C: 00989 C: COMPUTE ABSOLUTE TIME BASED ON INTERFAILURE TIME 00990 C: 00991 500 CONTINUE 00992 IF (I .EQ. 1) GO TO 510 00993 T(I) = T(1—1) + DT(l) 00994 GO TO 600 00995 C: 00996 5T0 CONTINUE 00997 T(I) = DT(I) 00998 C: 00999 600 CONTINUE 01000 C: 01001 C: CHECK TO SEE IF MOST RECENT FAILURE OCCURS AFTER 01002 C: TIME XL. 01003 C: 01004 IF (T(I) .LE. XL) GO TO 1000 01005 —I M I ! O • 01006 DT(l) = 0. 01007 NF = NF - 1 232 □1008 GO TO 9800 □1009 C: 01010 1000 CONTINUE 01011 C: 01012 C: ALL DONE, CHECK FOR DEBUG PRINTOUT 01013 C: 01014 9800 CONTINUE 01015 IF (LDB .EQ. 0) GO TO 9900 01016 C: 01017 WRITE (6, 9010) LDB, NF 01018 9010 FORMAT (’ ', ’DEBUG LEVEL ', 11, ' IN GEN, NF = ’, 15) 01019 C: 01020 WRITE (6, 9020) LDB, (DT(I), 1=1,NF) 01021 9020 FORMAT (' ’, ’DEBUG LEVEL ', 11, ’ IN GEN, DT =' / 01022 H h 200(1 ', 5G15.7 /)) 01023 C: 01D24 WRITE (6, 9030) LDB, (T(l), 1=1,NF) 01025 9030 FORMAT (' ’DEBUG LEVEL 11, ' IN GEN, T =’ / 01026 H h 200(1 ’, 5G15.7 /)) 01027 C: 01028 9900 CONTINUE 01029 C: 01030 RETURN 01031 END 01032 SUBROUTINE WARR (T, DT, NF, COST, ISEED3) 01033 Cs 01034 r. .. . . L:::;: 01035 C: 01036 C: THIS SUBROUTINE COMPUTES THE COST TO THE SELLER OF EACH 01037 C: OF THE FAILURES OCCURRING AT THE TIMES GIVEN BY T(l). 01038 C: 01039 C: INPUTS: 01040 C: 01041 C: NF = NUMBER OF FAILURES IN THE INTERVAL (0,XL) 01042 Cs 01043 C: T(I) = ABSOLUTE TIME OF THE ITH FAILURE; 01044 Cs I = 1, 2, ..., NF 01045 C: 01046 C: DT(I) = TIME BETWEEN THE (1-1)TH AND ITH 01047 C: FAILURE; I = 1, 2, ..., NF; 01048 C: DT(1) = T(1) 01049 Cs 01050 C: OUTPUTS: 01051 C: 01052 C: COST(I) = NONDISCOUNTED COST (TO THE SELLER) OF 01053 C: THE ITH FAILURE, BASED ON THE WARRANTY 01054 C: CONDITIONS 01055 n*i ncc C: p _ . . . , U 1 Uhb l ::::: 01057 Cs 01058 REAL*8 DTYPE, WRTYPE, WFTYPE, PBFTYP 233 □1059 INTEGER** CHGCOD 01060 C: 01061 L0GICAL*1 LGOFLG, LQFLG 01062 C: 01063 COMMON /LITRAL/ DTYPE(4), WRTYPE(2), WFTYPE(2), 01064 + LGO, LSKIP(2), LQUIT, CHGC00(24) 01065 + PBFTYP(2) 01066 C: 01067 COMMON /DEFVAL/ IDTYPE, IWRTYP, IWFTYP, IPBF, 01068 + ALPHA, BETA, XMU, SIGMA, ALPHAC 01069 + BETAC, W, XL, T1, R, C, THETA 01070 + PHI, DELTA, IDF, NR, XK, LSK, 01071 + LDB 01072 C: 01073 DIMENSION T(1000), DT(1000), COST(lOOO) 01074 C: 01075 C: MAXIMUM NUMBER OF COSTS TO BE COMPUTED 01076 C: 01077 DATA NFMAX / 1000 / 01078 C: 01079 C: ZERO OUT ALL VALUES PRIOR TO GENERATING FAILURES 01080 C: 01081 DO 100 1=1, NFMAX 01082 c o s t(i ) = a. 01083 100 CONTINUE 01084 C: 01085 C: CHECK WARRANTY RENEWAL TYPE 01086 C: 1: RENEWING 01087 C: 2: NDNRENEWING 01088 C: 01 089 IF (IWRTYP .EQ. 1) GO TO 1000 01090 C: 01091 C: ELSE, WARRANTY TYPE IS 2: NONRENEWING 01092 C: 01093 C: CHECK WARRANTY FAILURE MODE 01094 C: 1: FIRST 01095 C: 2: MULTIPLE 01096 C: 01097 IF (IWFTYP .EQ. 2) GO TO 200 01098 C: 01099 C: WARRANTY IS NONRENEWING, FIRST FAILURE 01100 C: GET COST OF FIRST FAILURE ONLY 01101 C: 01102 C: CHECK TO SEE IF FAILURE WAS UNDER WARRANTY 01103 C: 01104 IF (T(1) .GE. W) GO TO 2000 01105 C: 01106 C: FAILURE IS UNDER WARRANTY 01107 C: 01108 C0ST(1) = PAYBAK (T(1), R, C, T1, W, IPBF, LDB) 01109 GO TO 2000 234 □1110 C: 01111 C: WARRANTY IS NONRENEWING, MULTIPLE FAILURE 01112 C: 01113 200 CONTINUE 01114 DO 300 I = 1, NF 01115 C: 01116 C: IS FAILURE NOT COVERED BY WARRANTY? 01117 C: 01118 IF (T(I) .GE. W) GO TO 2000 01119 Cs 01120 C: FAILURE IS COVERED 01121 C: 01122 COST(I) = PAYBAK (T(l), R, C, T1, W, IPBF, LDB) 01123 Cs 01124 300 CONTINUE 01125 C; 01126 C: NORMAL EXIT FROM LOOP IMPLIES THAT ALL FAILURES 01127 C: WERE COVERED BY WARRANTY 01128 Cs 01129 GO TO 2000 01130 Cs 01131 Cs WARRANTY IS RENEWING 01132 Cs 01133 1000 CONTINUE 01134 Cs 01135 DO 1500 I = 1, NF 01136 Cs 01137 Cs IS FAILURE DURING THE WARRANTY PERIOD? 01138 Cs 01139 IF (DT(I) .GT. W) GO TO 1100 01140 Cs 01141 Cs YES, COMPUTE COST OF WARRANTY 01142 Cs 01143 COST(I) = PAYBAK (DT(I), R, C, T1, W, IPBF, 01144 GO TO 1500 01145 C: 01146 Cs NO, COMPUTE "COST" OF NEW PURCHASE 01147 Cs 01148 1100 CONTINUE 01149 Q = RANDOM (ISEED3) 01150 IF (Q .GT. BETAC) GO TO 2000 01151 COST(I) = R - C 01152 Cs 01153 1500 CONTINUE 01154 Cs 01155 Cs ALL DONE, CHECK FOR DEBUG PRINTOUT 01156 Cs 01157 2000 CONTINUE 01158 IF (LDB .EQ. 0) GO TO 9900 01159 Cs 01160 WRITE (6, 9010) LDB, NF 235 □11 61 901□ FORMAT (’ ', 'DEBUG LEVELV 12, ' IN WARR, NF =', 15) □1162 C: □1163 WRITE (6, 9020) LDB, (COST(l), 1=1,NF) □1164 9020 FORMAT (* *, ’DEBUG LEVEL’, 12, 'IN WARR, COST =’ / D1165 i - 200 (’ ’, SG15.7 /)) D1166 C: □1167 9900 CONTINUE 01168 RETURN □1169 END □1170 FUNCTION PAYBAK (DELT, R, C, T1, W, IPBF, LDB) □1171 C: 01172 r* • m m m , L::::: □1173 C: □1174 C: THIS FUNCTION COMPUTES THE WARRANTY COST GIVEN; □1175 C: 01176 Cs DELT = THE TIME THAT THE WARRANTY HAS BEEN IN FORCE □1177 C: R = THE SELLER'S COST 01178 C: C = THE BUYER'S COST 01179 C: T1 = "BREAKPOINT" IN A COMBINATION WARRANTY 01180 C: (IF APPLICABLE) 01181 C: W = WARRANTY LENGTH □1182 fV1 A O ~ 2 C: U1 1 0*5 L::::; □1184 C: □1185 C: □1186 C; PROGRAM CHECK 01187 IF (DELT .LE. W) GO TO 10 01188 WRITE (6,9000) DELT, W □1189 9D00 FORMAT (' ', 'PROGRAM ERROR 1 IN PAYBAK' / □1190 Hh ' ', 'DELT = ', G15.7, 5X, ’W = ’, G15.7 //) □1191 GO TO 9800 □1192 10 CONTINUE □1193 C: CHECK WARRANTY PAYBACK TYPE □1194 C: 1: CONSTANT □1195 C: 2: COMBINATION WITH T1 = TIME FROM BEGINNING OF □1196 C: WARRANTY TO SWITCH FROM CONSTANT TO PRO-RATA □1197 C: D1198 IF (IPBF .EQ. 1) GO TO 1000 □1199 C: □1200 C: WARRANTY TYPE IS "COMBO" □1201 C: □1202 C: CHECK TO SEE IF DURING THE CONSTANT OR PRO-RATA PHASE □1203 C: 01204 IF (DELT .LE. T1) GO TO 1000 □1205 C: 01206 C: DETERMINE PRO-RATA COST □1207 C: □12D8 P = R - C * (DELT-T1) / (W-T1) □1209 GO TO 5000 01210 C: □1211 C: PAYBACK IS CONSTANT 236 □1212 C: □1213 1000 CONTINUE 01214 P = R 01215 C: 01216 5000 CONTINUE 01217 PAYBAK = P 01218 C: 01219 C: ALL DONE, CHECK FOR DEBUG PRINTOUT 01220 C: 01221 9800 CONTINUE 01222 IF (LDB .LE. 1) GO TO 9900 01223 C: 01224 WRITE (6, 9010) LDB, DELT, R, C, T1, W, P 01225 9010 FORMAT (' ’, 'DEBUG LEVEL', 12, ’ IN PAYBAK’ / 01226 i t » 5X, 'DELT = ', G15.7, 5X, 'R = ’, G15.7 / 01227 t ? » 5X, 'C = ’, G15.7, 5X, ’T1 = ’, G15.7 / 01228 » t » 5X, 'W = ’, G15.7, 5X, ’P = ’, G15.7 //) 01229 C: 01230 9900 CONTINUE 01231 RETURN 01232 END 01233 SUBROUTINE DISCNT (COST, T, NF, IDF, PHI, DELTA, 01234 XMEAN, DCOST, LDB) 01235 C: 01236 01237 p..... C: 01238 Cs THIS SUBROUTINE DISCOUNTS A SERIES OF CASH FLOWS 01239 Cs 01240 C: INPUTS; 01241 C: 01242 C: NF = THE NUMBER OF COSTS IN THE SEQUENCE 01243 C: 01244 C: COST(I) = THE ITH CASH FLOW IN THE SEQUENCE; 01245 C: 1 = 1, 2.... . NF 01246 C: 01247 Cs T(I) = THE TIME OF THE ITH CASH FLOW; 01248 Cs I = 1, 2, ..., NF 01249 C: 01250 Cs PHI, DELTA = DISCOUNT FUNCTION PARAMETERS 01251 Cs 01252 Cs XMEAN = PARAMETER FOR DISCOUNT FUNCTION 3 01253 Cs 01254 Cs IDF = 1, 2, OR 3; THE DISCOUNT FUNCTION TYPE 01255 Cs 01256 Cs OUTPUTS 01257 Cs 01258 Cs DCOST(I) = THE PRESENT VALUE OF THE ITH CASH FLOW; 01259 C: I = 1, 2, ..., NF — BASED ON THE 01260 Cs GIVEN DISCOUNT FUNCTION TYPE 01261 01262 Cs 237 01263 C: 01264 DIMENSION COST(1000), T(1000), DCOST(IOOO) 01265 C: 01266 C: SET 1 MAXIMUM NUMBER OF CASH FLOWS 01267 C: 01268 DATA NFMAX / 1000 / 01269 C: 01270 C: SAVE THE CONSTANT PHI + DELTA 01271 C: 01272 Q = PHI + DELTA 01273 C: 01274 C: INITIALIZE ALL DISCOUNTED FLOWS TO ZERO 01275 C: 01276 DO 10 1=1, NFMAX 01277 DCOST(I) = Q. 01278 10 CONTINUE 01279 C: 01280 DO 1000 I = 1, NF 01281 TIME = T(I) 01282 C: 01283 C: DETERMINE DISCOUNT FUNCTION TYPE 01284 C: 01285 GO TO (100, 200, 300), IDF 01 286 C: 01 287 C: PROGRAM ERROR TRAP 01288 C: 01289 WRITE (6,9000) IDF 01 290 9000 FORMAT (' ’, ’PROGRAM ERROR 1 IN DISCNT, 01 291 GO TO 9800 01 292 C: 01293 Cs DISCOUNT FUNCTION 1 01294 C: 01295 100 CONTINUE 01296 DF = 1. / ((1.+Q)**TIME) 01297 GO TO 500 01298 C: 01299 C: DISCOUNT FUNCTION 2 01300 C: 01301 200 CONTINUE 01 302 Q2 = -TIME * Q 01303 DF = EXP (Q2) 01304 GO TO 500 01305 C: 01306 C: DISCOUNT FUNCTION 3 01307 C: 01308 300 CONTINUE 01309 Q2 = -XMEAN * Q 01310 Q3 = 1. - EXP (Q2) 01311 DF = 1. / Q3 01312 GO TO 500 01313 C: 238 01314 C: DISCOUNT THIS CASH FLOW 01315 C: 01316 500 CONTINUE 01 317 DCOST(I) = DF * COST(l) 01318 C: 01319 1000 CONTINUE 01320 C: 01321 C: ALL DONE, CHECK FOR DEBUG PRINTOUT 01322 C: 01323 9800 CONTINUE 01324 IF (LDB .EQ. 0) GO TO 9900 01325 C: 01326 URITE (6,9010) LDB, NF, (DC0ST(l),I=1,NF) 01327 9010 FORMAT (' ’, ’DEBUG LEVEL1, 12, ’ IN DISCNT, NF =’, 15 / 01328 + 200 (’ ’, 5G15.7 /)) 01329 C: 01330 9900 CONTINUE 01331 RETURN 01332 END 01333 FUNCTION TCOST (DCOST, NF) 01334 ni C: U I JJJ 01 336 C: 01337 C: THIS FUNCTION FINDS THE SUM OF DCOST(l); 01338 C: I = 1, 2, ..., NF 01339 C: U 1 J4U 01341 C: 01342 DIMENSION DCOST(1000) 01343 C: 01344 SUMT = 0. 01345 C: Q1346 DO 100 I = 1, NF 01347 SUMT = SUMT + DCOST(I) 01348 100 CONTINUE 01349 C: 01350 TCOST = SUMT 01351 RETURN 01352 END 01353 FUNCTION RANDOM (I) 01354 DOUBLE PRECISION A, P, X 01355 C: 01356 DATA A, P, C / 01357 1I 16B07.0DO, 2147483647.ODO, 4.656613E-10 / 01358 C: 01359 X = I 01360 X = A' * X 01361 X = DMOD(X,P) + 0.5 01362 I = X 01363 R = X 01364 R = C * R 239 01365 RANDOM = R 01366 RETURN 01367 END 01 368 SUBROUTINE RNORM (XMU, SIGMA, X, ISEED1, ISEED2) 01369 C: 01370 DIMENSION X(2) 01 371 DATA TUiOPI / 6.283185307179586 / 01372 C: 01373 U1 = RAND0M(ISEED1) 01374 U2 = RAND0M(ISEED2) 01375 A = SQRT(-2.*AL0G(U1)) 01376 TH = TUIOPI * U2 01377 B = COS(TH) 01378 C = SIN(TH) 01 379 Q1 = A * B 01380 02 = A * C 01381 X(1) = Q1 * SIGMA + XMU 01382 X(2) = Q2 * SIGMA + XMU 01383 C: 01384 RETURN 01385 END 01386 FUNCTION EXPON (BETA, ISEED) 01387 C: 01388 L::: 01389 C: EXPON GENERATES EXPONENTIALLY DISTRIBUTED RANDOM 01390 C: VARIATES 01391 C: 01392 C: INPUTS: 01393 C: BETA = MEAN OF THE EXPONENTIAL DISTRIBUTION 01394 C: ISEED = A RANDOM INTEGER SEED 01395 C: 01396 C: OUTPUT: 01397 C: EXPON = AN EXPO(BETA) RANDOM VARIATE 01398 ni "znn C: n. . . U I obb LS S * 01400 C: 01401 U = RANDOM(ISEED) 01402 EXPON = -BETA * ALOG(U) 01403 RETURN 01404 END 01405 FUNCTION RGAMMA (ALPHA, BETA, ISEED1, ISEED2) 01406 n*i /«no C: U 1 4U f l : :: 01408 C: 01409 C: RGAMMA GENERATES GAMMA DISTRIBUTED RANDOM VARIATES 01410 C: 01411 C: INPUTS: 01412 C: ALPHA, BETA = THE PARAMETERS DF THE GAMMA 01413 C: DISTRIBUTION 01414 C: ISEED1, ISEED2 = RANDOM INTEGER SEEDS 01415 C: 240 01416 C: OUTPUT: 01417 C: GANNA = A GANNA(ALPHA,BETA) DISTRIBUTED RANDON 01 418 C: VARIATE 01419 C: 01420 C: MOTES: 01421 C: NEAN: 01422 C: ALPHA * BETA 01423 C: VARIANCE: 01424 C: ALPHA * (BETA ** 2) 01425 Cs 01426 p. 4 L:: 01427 C: 01428 DATA E / 2.7182818284 / 01429 DATA XLN4 / 1.3862944 / 01430 DATA THETA / 4.5 / 01431 DATA D / 2.504774 / 01432 C: 01433 C: FIRST, GENERATE A GANNA (ALPHA,1) RANDON VARIATE 01434 C: 01435 1 CONTINUE 01436 IF (ALPHA .NE. 1.0) GO TO 10 01437 X = EXPON (1.0, ISEED1) 01438 GO TO 5000 01439 C: 01440 10 CONTINUE 01441 IF (ALPHA .GT. 1.0) GO TO 1000 01442 C: 01443 C: CASE: 0 < ALPHA < 1 01444 C: 01445 C: 01446 B = (E + ALPHA) / E 01447 C: 01448 C: STEP 1 01449 C: 01450 110 CONTINUE 01451 C: 01452 U1 = RANDON (ISEED1) 01453 P = B # U1 01454 IF (P .GT. 1.0) GO TO 130 01455 C: 01456 C: STEP 2 01457 C: 01458 120 CONTINUE 01459 C: 01460 Y = P ** (1./ALPHA) 01461 U2 = RANDON (ISEED2) 01462 IF (U2 .GT. EXP(-Y)) GO TO 110 01463 X = Y 01464 GO TO 5000 01465 C: 01466 C: STEP 3 241 □1467 C: □1468 130 CONTINUE □1469 C: 01470 Y = -ALDG ((B - P) / ALPHA) □1471 U2 = RANDOM (ISEED2) 01472 IF (U2 .GT. Y**(ALPHA-1.)) GO TO 110 01473 X = Y 01474 GO TO 5000 01475 C: 01476 C: CASE: ALPHA > 1 01477 C: 01478 1000 CONTINUE 01479 C: 01480 A = 1. / SQRT (2.*ALPHA - 1.) 01481 B = ALPHA - XLN4 01482 Q = ALPHA + 1. / A 01483 C: 01484 C: STEP 1 01485 C: 01486 1 010 CONTINUE 01487 C: 01488 U1 = RANDOM (ISEED1) 01489 U2 = RANDOM (ISEED2) 01490 C: 01491 C: STEP 2 01492 C: 01493 1020 CONTINUE 01494 C: 01495 V = A * ALOG (U1 / (1. - U1)) 01496 Y = ALPHA * EXP(V) 01497 Z = U1 * U1 * U2 01498 U = B + Q*V-Y 01499 C: 01500 C: STEP 3 01501 C: 01502 1 030 CONTINUE 01503 C: 01504 IF (111 + D - THETA*Z .LT. 0.) GO TO 1040 01505 X = Y 01506 GO TO 5000 01507 C: 01508 C: STEP 4 01509 C: 01510 1040 CONTINUE 01511 C: 01512 IF (W .LT. ALOG (Z) ) GO TO 1010 01513 X = Y 01514 C: 01515 C: CONVERT FROM GAMMA(ALPHA,1) TO GAMMA(ALPHA,BETA) 01516 C: 01 517 5000 CONTINUE 242 01518 CD CO CD CONTINUE 01519 C: 01520 RGAMMA = BETA * X 01521 RETURN 01522 END 01523 FUNCTION WEIBUL- (ALPHA, BETA, I SEED) 01524 n-i c . n c . C: U 1 01526 C: 01527 C: GENERATES RANDOM VARIATES THAT ARE WEIBULL DISTRIBUTED WITH 01528 C: PARAMETERS, ALPHA AND BETA 01529 C: 01530 C: INPUTS: 01531 0: ALPHA, BETA = PARAMETERS OF THE WEIBULL DISTRIBUTION 01532 C: ISEED = INTEGER RANDOM NUMBER SEED 01533 C: 01534 C: OUTPUT: 01535 C: UEIBUL = A WEIBULL(ALPHA,BETA) RANDOM VARIATE 01536 C: 01537 C: NOTES: 01538 C: MEAN: 01539 C: 01540 C: BETA 01541 C: GAMMA (1/ALPHA) 01542 C: ALPHA 01543 ni ra/ i C: U I OH4 01545 C: 01546 U = RANDOM (ISEED) 01547 UEIBUL = BETA * ((-ALOG(U)) *# (1./ALPHA)) 01548 RETURN 01549 END 01550 SUBROUTINE STD (X, N, XBAR, S) 01551 DIMENSION X(1) 01552 SUMX = 0. 01553 SUMXSQ = 0. 01554 DO 20 I = 1, N 01555 SUMX = SUMX + X(I) 01556 SUMXSQ = SUMXSQ + X(l)*X(l) 01557 20 CONTINUE 01558 FN = FLOAT(N) 01559 FN1 = FLOAT(N-I) 01560 XBAR = SUMX / FN 01561 IF (N .EQ. 1) GO TO 30 01562 SSQ = (SUMXSQ - FN*XBAR*XBAR) / FN1 01563 S = SQRT(SSQ) 01564 GO TO 40 01565 30 CONTINUE 01566 S = 99999.99 01567 40 CONTINUE 01568 RETURN 243 □1569 01570 01571 m >=179 C: END SUBROUTINE TRNORM (XMU, SIGMA, ISEED1, ISEED2, X1, X2) Ul J(«' 01 573 C: 01 574 C: THIS SUBROUTINE COMPUTES TWO TRUNCATED NORMAL RANDOM 01575 C: VARIATES. 01576 C: 01577 C: INPUTS: 01578 C: XMU = THE MEAN OF THE NORMAL DISTRIBUTION 01579 C: (BEFORE TRUNCATION) 01580 C: SIGMA = THE STANDARD DEVIATION OF THE NORMAL 01581 C: DISTRIBUTION (BEFORE TRUNCATION) 01582 C: ISEED1, ISEED2 = INTEGER RANDOM SEEDS 01583 C: 01584 C: OUTPUTS: 01585 C: X1, X2 = TWO INDEPENDENT RANDOM VARIATES FROM THE 01586 C: TRUNCATED NORMAL DISTRIBUTION 01587 C: 01588 C: NOTE: 01589 C: ALL VALUES OF X1, X2 .LE. ZERO ARE DISCARDED; HENCE, 01590 Cs TRUNCATION IS OVER ALL ZERO AND NEGATIVE VALUES. 01591 C: U I 01593 C: 01594 DIMENSION X(2) 01595 C: 01596 10 CONTINUE 01597 C: 01598 CALL RNORM (XMU, SIGMA, X, 2, ISEED1, ISEED2) 01599 C: 01600 X1 = X(1) 01601 X2 = X(2) 01602 C: 01603 IF ((X1 .LE. 0.) .OR. (X2 .LE. 0.)) GO TO 10 01604 RETURN 01605 END 244 APPENDIX D DATA FOR SENSITIVITY ANALYSIS LUMP SUM, NONRENEWING WARRANTIES Warranty Length = 0.75 Assumed Distribution True Distribution exponen tial gamma truncated Weibull normal exponential mean cost = 7.20 std. dev/. = 8.BB 7.40 7.99 7.00 8.35 3.20 5.66 truncated normal mean cost = 3.00 std. dev. = 4.82 7.20 8.54 2.70 4.46 2.80 4.51 gamma dfr mean cost = 12,50 8.40 std. dev. = 14.10 8.25 14.30 14.44 1.70 3.78 gamma ifr mean cost = 7.10 std. dev. = 7.00 6.40 7.98 6.00 7.25 3.00 5.22 Weibull dfr mean cost = 15.60 std. dev. = 15.66 6.70 8.29 15.70 16.53 1.90 4.65 Weibull ifr mean cost = 4.90 9.30 5.80 std. dev. = 6.11 9.35 6.06 3.90 5.84 245 LUWP SUm, NONRENEWING WARRANTIES Warranty Length =1.0 Assumed Distribution True exponen - truncated Distribution tial gamma Weibull normal exponential - ■ mean cost = 9.90 8.90 10.90 4.00 std. dev. = 9.48 8.86 11.81 5.68 truncated normal mean cost = 5.60 10.50 5.40 4.60 st.d dev. = 4.99 9.99 5.21 5.40 gamma dfr mean cost = 14.30 11.00 16.80 3.40 std. dev. = 14.37 11.15 17.34 5.54 gamma ifr mean cost = 9.70 10.80 8.50 4.80 std. dev. = 9.04 9.81 8.80 6.11 Weibull dfr mean cost = 19.20 9.70 17.50 3.50 std. dev. = 18.51 9.48 17.60 5.57 Weibull ifr mean cost = 7.30 10.10 7.50 5.70 std. dev. = 7.90 10.30 7.96 6.40 246 LUMP SUM, NONRENEWING WARRANTIES Warranty Length = 1.25 Assumed Distribution True exponen- truncated Distribution tial gamma Weibull normal exponential mean cost = 13.50 10.90 10.60 7.10 std. dev. =11.84 . 9.11 9.72 7.29 truncated normal mean cost = 9.20 14.70 8.10 8.70 std. dev. = 4.42 14.10 5.63 5.06 gamma dfr mean cost = 17.90 13.70 17.60 5.00 std. dev. = 17.25 11.43 17.47 6.89 gamma ifr mean cost = 12.10 13.40 10.60 7.80 std. dev. = 10.47 11.74 10.23 7.05 Weibull dfr mean cost = 22.70 12.80 18.80 4.20 std. dev. = 20.74 13.03 18.71 6.06 Weibull ifr mean cost = 8.80 13.00 9.30 8.00 std. dev. = 8.79 10.96 8.79 6.96 247 PRO RATA, RENEWING WARRANTIES Sales After W Lifecycle = 2U Warranty = 0.75 C = 1.2R Assumed Distribution exponen- truncated tial gamma Weibull normal True Distribution exponential mean cost = 5.33 B.05 4.92 0.63 std. dev. = 7.78 8.01 7.37 3.42 truncated normal mean cost = -1.12 5.21 -0.91 -1.81 std. dev. = 1.76 7.39 1.66 1.86 gamma dfr mean cost = 10.69 6.23 14.66 1.14 std. dev. = 13.86 8.15 14.61 3.77 gamma ifr mean cost = 3.88 4.21 2.88 0.67 std. dev. = 6.70 7.12 5.18 3.87 Weibull dfr mean cost = 16.59 3.98 15.64 -0.09 std. dev. = 19.65 7.37 18.48 1.98 Weibull ifr mean cost = 2.52 5.42 1.87 0.91 std. dev. = 5.28 7.03 4.48 4.21 248 PRO RATA, RENEWING WARRANTIES Sales After W Lifecycle = 2W Warranty = 0.75 C = 1.6R Assumed Distribution exponen- truncated tial gamma . Weibull normal True Distribution exponential mean cost = 1.86 1.35 0.49 -1.01 std. dev. = 7.54 7.29 6.75 4.51 truncated normal mean cost = -4.80 1.44 -5.37 -5.08 std. dev. = 2.85 6.56 2.37 2.47 gamma dfr mean cost = 8.33 2.75 11.32 -1.73 std. dev.= 11.02 8.57 13.12 3.95 gamma ifr mean cost = -1.29 2.12 0.34 -2.20 std.dev. = 4.74 7.00 5.60 4.32 Weibull dfr mean cost = 15.95 1.23 12.15 -1.27 std. dev. = 19.67 7.18 16.55 3.02 Weibull ifr mean cost = -1.16 0.63 -1.91 -2.58 std. dev. = 4.49 8.49 4.64 4.18 249 PRO RATA, RENEWING WARRANTIES Sales After W Lifecycle = 20W Warranty = 0.75 C = 1.2R Assumed Distribution exponen- truncated tial gamma Weibull normal True Distribution exponential mean cost = 24.16 17.86 25.91 -3.23 std. dev. = 21.20 17.94 21.79 9.80 truncated normal mean cost = -18.72 27.95 -18.55 -17.87 std. dev. = 6.49 21.53 5.50 5.69 gamma dfr mean cost = 59.17 27.02 68.61 -3.80 std. dev. = 35.58 23.29 43.56 9.01 gamma ifr mean cost = 14.45 25.98 14.35 -3.37 std. dev. =17.44 21.90 16.70 9.99 Weibull dfr mean cost = 83.76 24.04 63.38 -4.24 std. dev. = 53.22 23.07 44.85 7.62 Weibull ifr mean cost = 5.94 26.27 5.83 -6.04 std. dev. = 13.55 22.50 12.59 10.35 250 PRO RATA, RENEWING WARRANTIES Sales After W Lifecycle = 20W Warranty = 0.75 C = 1.6R Assumed Distribution exponen- truncated tial gamma Weibull normal True Distribution exponential mean cost = -14.27 -19.04 -17.38 -41.13 std. dev. = 17.65 17.02 18.33 10.51 truncated normal mean cost = -73.66 -13.40 -74.70 -73.39 std. dev. = 6.88 20.62 7.24 6.85 gamma dfr mean cost = 22.91 -16.06 26.81 -35.49 std. dev. = 28.75 18.25 30.71 8.72 gamma ifr mean cost = -33.95 -15.08 -28.71 -46.08 std. dev. = 14.74 17.69 15.88 10.93 Weibull dfr mean cost = 57.17 -16.37 34.39 -27.21 std. dev. = 45.48 18.18 33.72 8.84 Weibull ifr mean cost = -42.09 -15.29 -41.94 -52.60 std. dev. = 13.13 20.59 11.95 10.31 251 PRO RATA, RENEWING WARRANTIES Sales After W Lifecycle = 2W Warranty = 1.00 C = 1.2R Assumed Distribution True exponen- truncated Distribution tial gamma Weibull normal exponential mean cost = 6.93 7.32 7.39 1.95 std. dev. = 9.75 9.82 9.19 6.28 truncated normal mean cost = -0.14 6.75 0.66 -0.04 std. dev. = 3.32 9.68 3.34 2.87 gamma dfr mean cost = 14.51 8.52 19.90 1.92 std. dev. = 14.31 8.83 19.39 6.16 gamma ifr mean cost = 6.69 8.01 5.45 2,16 std. dev. = 8.96 10.09 7.39 5.53 Weibull dfr mean cost = 25.77 7.86 18.05 1.14 std. dev. = 22.01 9.13 19.13 3.58 Weibull ifr mean cost = 3.56 9.05 4.13 1.41 std. dev. = 6.13 9.74 4.58 5.13 252 PRO RATA, RENEWING WARRANTIES Sales After W Lifecycle = 2W Warranty = 1.00 C = 1.6R Assumed Distribution True exponen _ truncated Distribution tial gamma Weibull normal exponential mean cost = 4,30 3.61 5.27 -1.64 std. dev. = 10.95 7.17 11.85 5.26 truncated normal mean cost = -4.76 4.33 -4.41 -5.06 std. dev. = 3.00 9.17 3.38 2.86 gamma dfr mean cost = 10.00 6.88 10.88 -1.94 std. dev. = 12.45 10.32 13.51 5.09 gamma ifr mean cost = 0.70 4.18 1.98 -1.90 std. dev. = 6.11 10.00 8.76 4.47 Weibull dfr mean cost = 19.48 3.51 13.56 -0.70 std. dev. = 21.37 8.99 17.20 4.88 Weibull ifr mean cost = -0.76 3.28 0.25 -2.26 std. dev. = 5.56 8.48 6.96 5.55 253 PRO RATA, RENEWING WARRANTIES Sales After W Lifecycle = 20W Warranty =1.00 C = 1.2R Assumed Distribution exponen- truncated tial gamma Weibull normal True Distribution exponential mean cost = 53.34 46.98 48.70 1.93 std. dev. = 28.96 26.49 27.08 14.25 truncated normal mean cost = -7.18 50.30 -3.21 -5.41 std. dev. = 10.81 27.28 10.01 10.85 gamma dfr mean cost = 92.65 55.60 91.80 2.69 std. dev. = 46.26 33.60 48.31 12.55 gamma ifr mean cost = 40.01 56.16 36.01 6.86 std. dev. = 26.55 ' 37.44 24.94 14.41 Weibull dfr mean cost = 119.42 48.27 88.15 -3.51 std. dev. = 69.66 27.71 43.98 8.87 Weibull ifr mean cost = 24.50 54.64 30.74 10.06 std. dev. = 18.71 34.05 20.98 14.69 254 PRO RATA, RENEWING WARRANTIES Sales After W Lifecycle = 20W Warranty =1.00 C = 1.6R Assumed Distribution exponen- truncated tial gamma Weibull normal True Distribution exponential ' mean cost = 2.31 0.10 -1.01 -42.BA std. dev/. = 24.75 25.27 24.08 16.67 truncated normal mean cost = -74.00 4.90 -69.34 -72.81 std. dev. = 8.99 26.67 8.85 9.93 gamma dfr mean cost = 45.98 4.34 54.50 -39.68 std. dev. = 37.44 28.19 37.33 12.66 gamma ifr mean cost = -13.74 4.37 -15.85 -47.24 std. dev. = 21.72 24.44 19.45 12.35 Weibull dfr mean cost = 79.25 4.44 61.07 -34.61 std. dev. = 56.12 27.60 49.66 10.01 Weibull ifr mean cost = -32.70 8.22 -30.08 -49.34 std. dev. = 15.26 27.17 18.21 14.19 255 PRO RATA, RENEWING WARRANTIES Sales After W Lifecycle = 2W Warranty =1.25 C = 1.2R True Distribution Assumed Distribution exponen tial gamma truncated Weibull normal exponential mean cost = 12.63 std. dev. = 13.00 10.58 11.49 10.48 9.97 3.27 5.23 truncated normal mean cost = 3.14 12.17 2.78 3.07 std. dev. = 4.79 12.31 4.52 4.51 gamma dfr mean cost = 16.82 11.66 std. dev. = 17.04 10.76 21.51 22.25 2.43 6.03 gamma ifr mean cost = 8.11 std. dev. = 8.81 11.54 13.24 8.02 10.38 4.23 7.62 Weibull dfr mean cost = 24.13 12.88 19.56 std. dev. = 25.56 12.27 23.31 2.08 4.86 Weibull ifr mean cost = 6.40 std. dev. = 7.65 11.71 12.81 7.62 8.02 3.B9 5.66 256 PRO RATA, RENEWING WARRANTIES Sales After W Lifecycle = 2W Warranty = 1.25 C = 1.6R Assumed Distribution True exponen- truncated Distribution tial gamma Weibull normal exponential mean cost = 6.27 std. dev. = 10.82 truncated normal mean cost = -3.45 std. dev. = 4.64 gamma dfr mean cost = 17.78 std. dev. = 20.83 gamma ifr mean cost = 3.87 std. dev. = 7.22 Weibull dfr mean cost = 23.61 std. dev. = 28.23 Weibull ifr mean cost = 0.26 std. dev. = 6.21 257 7.13 10.90 8.68 12 61 8.72 11.39 6.89 11.02 6.04 11.13 6.09 10.38 - 2.88 4.18 19.67 20.58 3.16 8.69 5.55 10.85 -3.58 4.89 14.24 18.22 3.44 7.61 -1.04 6.18 -0.48 5.65 -1.32 5.97 -9.97 4.74 -1.52 5.34 PRO RATA, RENEWING WARRANTIES Sales After W Lifecycle = 20W Warranty = 1.25 C = 1.2R True Distribution Assumed Distribution- exponen tial gamma truncated Weibull normal exponential mean cost = 86.00 std. dev/. = 37.61 74.08 34.37 75.21 30.57 21.28 18.26 truncated normal mean cost = 22.89 86.85 25.87 22.65 std. dev/. = 14.64 38.41 12.96 14.93 gamma dfr mean cost = 128.74 85.19 std. del/. = 55.05 33.57 134.23 57.51 8.98 17.75 gamma ifr mean cost = 70.69 81.31 std. dev/. = 28.87 35.72 68.87 29.19 26.59 17.18 Weibull dfr mean cost = 153.60 73.79 136.21 std. dev/. = 62.98 35.22 62.65 2.26 11.47 Weibull ifr mean cost = 48.20 89.15 61.56 std. dev. = 22.31 37.68 26.80 29.00 18.20 258 PRO RATA, RENEWING WARRANTIES Sales After W Lifecycle = 20W Warranty = 1.25 C = 1.6R Assumed Distribution exponen- truncated tial gamma Weibull normal True Distribution exponential mean cost = 21.65 19.38 23.42 -39.18 std. dew. = 30.06 30.66 38.53 17.67 truncated normal mean cost = -53.64 28.14 -45.39 -50.20 std. dew. = 12.13 28.47 12.81 15.50 gamma dfr mean cost = 80.68 32.18 100.73 -36.39 std. dev. = 51.79 32.73 56.54 15.01 gamma ifr mean cost = 3.78 30.48 11.80 -37.24. std. dev. = 25.58 32.03 30.21 17.17 Weibull dfr mean cost = 119.51 24.52 93.86 -34.94 std. dev. = 74.70 38.52 58.98 11.67 Weibull ifr mean cost = -12.84 29.82 -2.71 -32.87 std. dev. = 24.86 31.52 21.71 17.98 259 PRO RATA, RENEWING WARRANTIES No Sales After W Lifecycle = 2W Warranty = 0.75 C = 1.2R Assumed Distribution exponen- truncated tial gamma Weibull normal True Distribution exponential mean cost = 4.08 4.81 5.00 1.38 std. dev. = 7.02 6.20 8.98 3.13 truncated normal mean cost = 0.37 5.13 0.12 0.30 std. dev. = 1.67 7.20 1.27 1.31 gamma dfr mean cost =10.14 6.02 14.46 1.03 std. dev. = 12.17 7.86 17.07 2.63 gamma ifr mean cost = 3.67 4.81 2.77 1.19 std. dev. = 5.59 8.97 4.67 2.91 Weibull dfr mean cost = 14.76 4.23 14.01 1.29 std. dev. = 18.47 6.05 16.38 3.22 Weibull ifr mean cost = 1.93 4.91 2.54 1.15 std. dev. = 4.23 7.09 4.53 3.24 260 PRO RATA, RENEWING WARRANTIES No Sales After W Lifecycle = 2W Warranty = 0.75 C = 1.BR Assumed Distribution exponen- truncated tial gamma Weibull normal True Distribution exponential mean cost = 3.36 2.91 2.93 0.39 std. dev. = 6.32 7.59 5.19 2.93 truncated normal mean cost = -0.29 2.45 -0.94 -0.52 std. dev. = 2.01 5.38 2.16 1.79 gamma dfr mean cost = 8.28 4.52 9.18 0.77 std. dev. = 9.10 9.52 11.94 3.31 gamma ifr mean cost = 2.88 3.05 1.48 0.91 std. dev. = 5.55 5.09 4.15 3.46 Weibull dfr mean cost = 16.28 2.76 9.27 0.76 std. dev. = 19.59 5.63 12.28 2.76 Weibull ifr mean cost = 0.95 2.97 0.37 0.06 std. dev. = 3.56 6.71 3.31 2.59 261 PRO RATA, RENEWING WARRANTIES No Sales After U Lifecycle = 20W Warranty = 0.75 C = 1.2R Assumed Distribution exponen- truncated tial gamma Weibull normal True Distribution exponential mean cost = 4.61 5.06 5.15 1.90 std. dev. = 8.45 7.86 8.21 4.27 truncated normal mean cost = 0.27 6.72 0.24 0.87 std. dev. = 1.94 10.01 ' 1.32 2.35 gamma dfr mean cost = 12.71 4.82 15.32 1.66 std. dev. = 14.90 7.39 17.48 4.32 gamma ifr mean cost = 3.83 7.87 3.02 1.82 std. dev. = 5.99 12.78 5.01 3.44 Weibull dfr mean cost = 14.05 4.48 14.47 0.86 ' std. dev. = 16.98 7.76 17.80 2.48 Weibull ifr mean cost = 3.07 5.21 2.77 1.20 std. dev. = 4.75 8.35 5.18 2.87 262 PRO RATA, RENEWING WARRANTIES No Sales After W Lifecycle = 20W Warranty = 0.75 C = 1.6R Assumed Distribution exponen- truncated tial gamma Weibull normal True Distribution exponential mean cost = 3.16 2.94 5.09 0.13 std. dev. = 7.53 5.69 7.09 2.39 truncated normal mean cost = -0.14 ‘2.97 -0.59 -0.50 std. dev. = 1.71 6.47 1.76 2.20 gamma dfr mean cost = 9.25 4.55 11.12 0.26 std. dev. = 14.78 8.74 13.86 3.49 gamma ifr mean cost = 0.95 2.86 2.08 0.78 std. dev. = 4.92 6.13 4.55 3.09 Weibull dfr mean cost = 13.05 2.77 11.52 0.66 std. dev. = 15.58 5.60 14.95 2.30 Weibull ifr mean cost = 0.66 3.58 1.25 0.37 std. dev. = 3.77 8.93 5.00 2.47 263 PRO RATA, RENEWING WARRANTIES No Sales After W Lifecycle = 2W Warranty =1.00 C = 1.2R Assumed Distribution True exponen- truncated Distribution tial gamma Weibull normal exponential mean cost = 6.93 7.93 6.11 2.12 std. dev. = 9.06 9.17 9.34 3.58 truncated normal mean cost = 0.83 7.61 1.29 0.91 std. dev. = 2.28 10.49 2.35 2.71 gamma dfr mean cost = 10.43 8.47 14.80 1.72 std. dev. = 13.38 10.36 14.64 3.78 gamma ifr mean cost = 5.28 6.09 6.35 2.53 std. dev. = 7.06 9.20 8.31 5.21 Weibull dfr mean cost = 19.70 6.79 15.16 1.72 std. dev. = 21.64 8.48 17.32 3.66 Weibull ifr mean cost = 5.16 8.52 5.04 2.73 std. dev. = 7.77 11.00 7.22 4.19 264 PRO RATA, RENEWING WARRANTIES No Sales After W Lifecycle = 2W Warranty =1.00 C = 1.6R True Distribution Assumed Distribution exponen tial gamma truncated Weibull normal exponential mean cost =5.48 std. dev. = 8.91 6.28 8.75 5.01 8.72 0.69 3.79 truncated normal mean cost = -1.06 std. dev. = 2.92 4.85 8.11 -1.64 3.08 -1.27 2.87 gamma dfr mean cost = 12.10 5.68 std. dev. = 15.17 9.68 12.61 14.43 1.67 4.41 gamma ifr mean cost = 1.92 std. dev. = 5.39 4.62 8.02 1.99 4.54 0.21 2.70 Weibull dfr mean cost = 17.94 std. dev. = 21.59 3.88 6.59 12.43 15.19 0.63 2.99 Weibull ifr mean cost = 1.87 std. dev. = 4.84 4.99 7.37 2.28 5.71 0.33 4.01 265 PRO RATA, RENEWING WARRANTIES No Sales After W Lifecycle = 20W Warranty = 1.00 C = 1.2R •Assumed Distribution exponen- truncated tial gamma Weibull normal True Distribution exponential mean cost = 6.13 9.7B 6.32 2.09 std. dev. = 8.44 12.47 9.28 4.24 truncated normal mean cost =1.11 8.53 1.74 1.24 std. dev. = 2.67 10.09 3.03 3.04 gamma dfr mean cost = 16.25 6.69 15.55 1.73 std. dev. = 19.22 10.85 17.74 4.11 gamma ifr mean cost = 6.26 8.62 5.80 2.48 std. dev. = 8.92 10.98 9.36 4.39 Weibull dfr mean cost = 23.44 9.31 21.47 1.45 std. dev. = 25.74 11.88 26.24 2.99 Weibull ifr mean cost = 5.54 7.42 5.43 2.24 std. dev. = 7.19 12.36 8.51 4.13 266 PRO RATA, RENEWING WARRANTIES No Sales After W Lifecycle = 20W Warranty = 1.00 C = 1.6R Assumed Distribution True Distribution exponen tial gamma truncated Weibull normal exponential mean cost = 5.79 std. dev. = 8.86 5.58 9.74 5.53 8.81 0.56 3.90 truncated normal mean cost = -1.73 7.33 -2.12 -1.62 std. dev. = 3.76 10.79 4.51 4.06 gamma dfr mean cost =14.04 5.89 std. dev. = 20.57 11.11 12.64 16.82 1.14 3.22 gamma ifr mean cost = 4 . 0 1 6.96 std. dev. = 6.30 10.06 2.24 5.48 1.06 4.43 Weibull dfr mean cost = 23.09 std. dev. = 25.17 4.54 8.32 14.93 20.02 0.63 3.16 Weibull ifr mean cost = 1.31 7.19 2.61 std. dev. = 5.95 11.03 4.92 0.64 3.68 267 PRO RATA, RENEWING WARRANTIES No Sales After W Lifecycle = 2W Warranty =1.25 C = 1.2R Assumed Distribution True exponen- truncated Distribution tial gamma Weibull normal exponential mean cost = 12.01 std. dev/. = 12.45 truncated normal mean cost = 3.04 std. dew. = 4.22 gamma dfr mean cost = 19.30 std. dev. = 17.99 gamma ifr mean cost = 7.76 std. dev. = 7.90 Weibull dfr mean cost = 23.44 std. dev. = 23.20 Weibull ifr mean cost = 6.09 std. dev. = 7.37 268 10.63 11.56 10.68 12.07 10.56 12.98 11.08 12.27 11.17 10.95 9.57 12.00 3.25 4.26 23.41 23.86 8.57 8.66 8.69 9.39 2.81 3.54 16.29 18.69 8.97 9.83 2.90 5.76 2.40 4.99 4.87 5.91 1.21 2.78 3.73 5.55 PRO RATA, RENEWING WARRANTIES No Sales After W Lifecycle = 2W Warranty =1.25 C = 1.6R Assumed Distribution exponen- truncated tial gamma Weibull normal True Distribution exponential mean cost = 8.27 6.80 5.08 1.69 std. dev. =11.79 9.64 7.20 5.01 truncated normal mean cost = -0.98 7.19 -0.86 -1.74 std. dev. = 3.70 10.58 2.98 3.87 gamma dfr mean cost = 13.65 7.95 14.85 0.51 std. dev. = 16.56 12.04 16.87 3.48 gamma ifr mean cost = 5.B3 8.79 8.28 1.32 std. dev. = 8.44 12.79 9.24 5.40 Weibull dfr mean cost = 21.82 8.06 18.13 1.16 std. dev. = 20.37 9.66 18.86 4.23 Weibull ifr mean cost = 2.72 8.41 2.67 0.87 std. dev. = 6.21 10.33 5.85 5.66 269 PRO RATA, RENEWING WARRANTIES No Sales After W Lifecycle = 20W Warranty =1.25 C = 1.2R Assumed Distribution exponen- truncated tial gamma Weibull normal True Distribution exponential mean cost = 14.62 10.76 8.07 2.84 std. dev. = 17.88 12.49 9.37 5.60 truncated normal mean cost =4.73 13.17 5.82 6.09 std. dev. = 5.85 14.35 7.20 8.14 gamma dfr mean cost = 22.83 14.83 17.94 3,20 std. dev. = 24.77 15.12 21.55 5.94 gamma ifr mean cost = 9.02' 11.14 10.60 4.20 std. dev. =10.12 13.98 12.51 6.50 Weibull dfr mean cost = 29.81 14.14 31.90 2.28 std. dev. = 32.35 15.04 36.00 4.69 Weibull ifr mean cost = 9.90 ’ 15.27 8.62 5.61 std. dev. = 12.46 20.94 12.11 7.99 270 PRO RATA, RENEWING WARRANTIES No Sales After W Lifecycle = 20W Warranty =1.25 C = 1.6R Assumed Distribution exponen- truncated tial gamma Weibull normal True Distribution exponential mean cost = 9.99 9.06 7.63 1.83 std. dev. = 14.76 14.66 9.95 5.70 truncated normal mean cost = -3.49 8.82 -2.64 -3.49 std. dev. = 7.09 11.91 5.69 6.13 gamma dfr mean cost =17.76 9.06 19.01 2.07 std. dev. = 19.79 12.53 22.63 5.74 gamma ifr mean cost = 7.16 8.53 6.18 1.81 std. dev. = 10.67 13.69 11.89 5.48 Weibull dfr mean cost = 22.71 8.62 19.13 1.18 std. dev. = 23.81 13.29 23.59 3.77 Weibull ifr mean cost = 4.27 9.37 6.54 1.75 std. dev. = 8.96 13.37 11.75 5.50 271 APPENDIX £ EXAMPLE REGRESSION RESULTS EXPONENTIAL - "BEST CASE" PRO RATA, RENEWING SALES AFTER W WARRANTY =1.25 LIFECYCLE = 20W IV BETA/SE RAW B / SE t • P<= B— -.9881 -185.604 -11.148 .002 .089 16.6494 INTERCEPT 271.76 For each t, df = 3 MULTIPLE OVERALL DF R R-squared F num den P<= .9881 .9764 124.273 1 3 .002 C = 1.2R 272 EXPONENTIAL - "BEST CASE" PRO RATA, RENEWING SALES AFTER U l WARRANTY =1.25 LIFECYCLE = 20W C = 1.2R RESIDUALS 10 + + -10 I::::I::::I::::I::::I::::I::::I::::I::::I::::Is:::I □ 50 100 150 200 250 PREDICTED COST (Y) 2.73 EXPONENTIAL - "BEST CASE" PRO RATA, RENEWING SALES AFTER W WARRANTY =1.25 LIFECYCLE = 20W C = 1.2R ■ COST 200 I:s s s i : s ssis st siss s sis sssl 150 100 50 0 I::::I::::I:r::I::::I::::I::::I::::I::::I::::I::::I .5 1 1.5 2 2.5 3 EXPONENTIAL B 274 EXPONENTIAL - "WORST CASE" PRO RATA, RENEWING SALES AFTER W WARRANTY = 0.75 LIFECYCLE = 2W 1 1 / BETA/SE RAW B / SE t P<= B— -.6762 -4.19655 -1.590 .210 .425 2.6398 INTERCEPT 5.66055 For each t, df = 3 MULTIPLE OVERALL DF R R-squared F num den P<= .6762 .4572 2.527 1 3 .210 RESIDUALS 2 I : : ; I : 1 . -1 -2 I::::I::::I::::I::::I::::I::::I::::I::::I::::I::::I 0 1 2 3 A 4 5 PREDICTED COST (Y) C = 1.6R 275 EXPONENTIAL - "WORST CASE" PRO RATA, RENEWING SALES AFTER W WARRANTY = 0.75 LIFECYCLE = 2W C = 1.6R COST 0 Is:: iI:::tI:s::I::::I::::I::s:I::::I::::Iis:sI::s:I .5 1 1.5 2 2.5 3 EXPONENTIAL B 276 TRUNCATED NORMAL - "BEST CASE" LUMP SUM, NONRENEWING WARRANTY =1.25 IV BETA/SE RAW B / SE 2 2 r with . . . t P<= sr pr Y IV MU- -.9689 -14.6828 -10.432 .001 .938? .9392 -.9689 .0000 .053 .796572 SG- -.0223 -1.01432 -.424 .675 .0005 .0081 -.0223 .0000 .053 2.390B7 INTERCEPT 23.565 For each t df = 22 MULTIPLE OVERALL DF R R-squared F num den P<- .9691 .9392 169.967 2 22 .001 RESIDUALS O 1 /£ J + 1 + + + + 4* + + + + + 0 + + + + + 3 + + + + -1 + ■ ■ T i : 1 4 6 8 10 A 12 14 PREDICTED COST (Y) 277 TRUNCATED NORMAL - "BEST CASE" LUMP SUM, NONRENEUJING WARRANTY =1.25 COST 20 s s i s : I 15 + 2 2 10 : 4 + 2 2 + + 3 + + 4 0 .5 1 1.5 2 2.5 3 TRUNCATED NORMAL MU 278 TRUNCATED NORMAL - "BEST CASE" LUMP SUM, NONRENEltJING WARRANTY =1.25 COST 20 15 10 + £ + + + + 2 0 .2 .25 .3 .35 .4 .45 TRUNCATED NORMAL SG 279 TRUNCATED NORMAL - "WORST CASE" PRO RATA, RENEWING NO SALES AFTER W WARRANTY = 0.75 LIFECYCLE = 20W C = 1.6R IV BETA/SII RAW 0 / SE 2 2 r with • • • t P<= sr pr Y IV MU- ' .1605 .206 .34 .436931 .778 .445 .0258 .0268 .1605 .0000 SG- .1957 .206 1.24429 1.31158 .949 .353 .0383 .0393 .1957 .0000 INTERCEPT -1.16752 For each t, df = 22 MULTIPLE OVERALL DF R R-squared F num den P<= .2531 .0640 .753 2 22 .483 RESIDUALS 1.0 I : : : I : : : : I j j:I: .5 + + + + + + + + + + ++ + + + + + + + 4- + + -.7 -.6 -.5 -* 4 ^-.3 -.2 PREDICTED COST (Y) 280 TRUNCATED NORMAL - "WORST CASE" PRO RATA, RENEWING NO SALES AFTER W WARRANTY = 0.75 LIFECYCLE = 20W C = 1.6R COST .5 I : s i : s i : s sis s s si -1.0 + + + + + + + + + 3 2 + + + + 2 -1.5 .5 1 1.5 2 2.5 3 TRUNCATED NORMAL MU 281 TRUNCATED NORMAL - "WORST CASE" PRO RATA, RENEWING NO SALES AFTER W WARRANTY = 0.75 LIFECYCLE = 20W C = 1,6R COST .5 I::::I::::I::::I::::I::::I::::I::::I::::I::::I::::I : + s + .0 : 2 s + : 2 : + 2 : 2 2 + + -.5 : + + : + : + + + : + -1.0 : + : + .2 .25 .3 .35 .4 .45 TRUNCATED NORMAL SG 282 GANNA DFR - "BEST CASE" PRO RATA, RENEWING NO SALES AFTER U WARRANTY =1.25 LIFECYCLE = 2U C - 1.6R IV BETA/SE RAW B / SE 2 2 r with . . . t P<= sr pr Y IV A— -.7970 -57.8014 -14,146 ,001 .6353 .9009 -.7970 .0000 .056 4.0861B B— -.5430 -9.84483 -9.63? .001 .2949 .8085 -.5430 .0000 .056 1.02154 INTERCEPT 64.1384 For each t df = 22 MULTIPLE OVERALL DF R R-squared F num den P<- .9644 9302 146.488 2 22 .001 RESIDUALS ‘ n D 1 : :I J«U + 2.5 + + + + + + .0 + 4- + + + + + 4- + + +4- 4 * + + -2.5 4- + n . -X 0 10 20 30 A 40 50 PREDICTED COST (Y) 2 8 3 GAMMA DFR - "BEST CASE" PRO RATA, RENEWING NO SALES AFTER W WARRANTY = 1.25 LIFECYCLE = 2U COST 40 I: 30 20 10 + + + + + + + + + + 2 + + 0 I::::I::::I::::I::::I::::I::::I::::I::::I::::I::::I .2 .4 .6 .8 1 1.2 GAMMA DFR A C = 1.6R 284 GAMMA DFR - "BEST CASE" PRO RATA, RENEWING NO SALES AFTER W WARRANTY = 1.25 LIFECYCLE = 2W COST 40 I : : : I : : s : I : : : I 30 + + + 20 : + + + + + + + + + + + + + 10 : + + + 2 + + 0 I::::I::::I::::I::::I:::!l::::I::::I::::I::::I::::I 1 1.5 2 2.5 3 3.5 GAMMA DFR B C = 1.BR 285 GAMMA DFR - "WORST CASE" NO SALES AFTER W PRO RATA, RENEWING WARRANTY =1.0 LIFECYCLE = 2W C = 1.2R IV BETA/SE RAW B / SE A— -.8105 -48.88 .086 5.20813 B— -.4232 -6.38069 .086 1.30203 INTERCEPT 51.2534 For each t, df = 22 MULTIPLE OVERALL R R-squared F .9143 .8359 56.050 2 2 P<= sr pr r with . . . Y IV -9.385 .001 .6569 .8002 -.8105 .0000 -4.901 .001 .1791 .5219 -.4232 .0000 DF num den P<= 2 22 .001 RESIDUALS 15 I : : : : I : t : : I : j : : I : s s s i s 10 -5 I: + + + + + + + ++ 2+ + + 10 15 A 20 PREDICTED COST (Y) :: I 25 286 GANNA DFR - "WORST CASE" PRO RATA, RENEWING NO SALES AFTER W WARRANTY =1.0 LIFECYCLE = 2W C = 1.2R • COST 40 I s s i s : :sis::s i s : : I : 30 20 10 2 + 4 3 + + + 2 + 2 2 4 0 I::::I::::I::::I::::I::::I::::I::::I::::I::::I::::I .2 .4 .6 .8 1 1.2 GANNA DFR A 287 GANNA DFR - "WORST CASE" PRO RATA, RENEWING NO SALES AFTER W WARRANTY = 1,0 LIFECYCLE = 2W C = 1,2R COST 40 I::;:I:;::I::::Ii 30 20 10 2 2 + + + + + 2 + GANNA DFR B 288 GAMMA' IFR - "BEST CASE" PRO RATA, RENEWING NO SALES AFTER l i i WARRANTY = 1.0 LIFECYCLE = 2W C = 1.2R IV BETA/SE RAW B / SE 2 2 r u/ith • • • t P<= sr pr Y IV A— -.7870 .058 -10.6608 .780631 -13.657 .001 .6193 .8945 -.7870 .0000 B— -.5546 -16.9002 -9.624 .001 .3076 .8081 -.5546 .0000 .058 1.756 INTERCEPT 33.6053 For each t, df = 22 MULTIPLE OVERALL DF R R-squared F num den P<= .9628 .9269 139.565 2 22 .001 RESIDUALS 2 I::::I::::I::::I::::I::::I::::I::::I::::I::::I::::I + + ++ + + + + + 4* + + 2 + + + + + -5 0 5 10 A 15 20 ■PREDICTED COST (Y) 289 GAMMA IFR - "BEST CASE” PRO RATA, RENEWING NO SALES AFTER l i J WARRANTY = 1.0 LIFECYCLE = 2W COST 20 I : : : : X : 15 4 - + 10 : + + + + 2 + + + + + 1 1.2 1.4 1.6 1.8 GAITO IFR A 2 + + C = 1.2R 290 GAMMA IFR - "BEST CASE" PRO RATA, RENEWING NO SALES AFTER W WARRANTY = 1.0 LIFECYCLE = 2W C = 1.2R COST 20 I : : : I : : : : I : : : I : t : : I : s : I 15 + + “ h 10 : + + + + ■ + 2 + .+ + + + + + 2 + + 0 I::::I::::I::::I::::I::::I::::I::::I::::I::::I::::I .A .5 .6 .7 .B .9 GAMMA IFR B 291 NO SALES AFTER 1 1 1 GAWIA IFR - "WORST CASE" PRO RATA, RENEWING WARRANTY =1.0 LIFECYCLE = 20W C = 1.6R 2 2 r with • • • P<= sr pr Y IV ,001 .5286 .6822 -.7271 .0000 ,001 .2251 .4776 -.4745 .0000 IV BETA/SE RAW B / SE A— -.7271 -13.5977 .106 1.97872 B— -.4745 -19.9614 .106 4.45104 INTERCEPT 38.5496 For each t, df = 22 MULTIPLE OVERALL R R-squared F .8682 ' .7537 33.668 DF num den P<= 2 22 .001 RESIDUALS 15 I::::I::::I::::I::::I::::I::::I: 10 + + + +2 + + ++++ + +++ + + 2 -5 I: -5 :I::::I::::I::::I::::I::::I::::I 5 10 A 15 20 PREDICTED COST (Y) 292 GAMMA IFR - "WORST CASE" PRO RATA, RENEWING NO SALES AFTER W WARRANTY = 0.75 LIFECYCLE = 20W C = 1.6R COST 30 I : : : I : : : I : t : : I : : : : I 20 10 2 + + 3 + -10 I::::I::::I::::I::::I::::I::::I::::I::::I::::I::::I 1 1.2 1.4 1.6 1.8 2 GAMMA IFR A 293 GAMMA IFR - "WORST CASE" PRO RATA, RENEWING NO SALES AFTER W WARRANTY =0.75 LIFECYCLE = 20W C = 1.6R COST 30 I : : : I : : : : I 20 10 -10 Is .4 .5 + + 2 2 + + 2 + .6 .7 .8 GAMMA IFR B .9 294 WEIBULl DFR - "BEST CASE" PRO RATA,RENEWING SALES AFTER W WARRANTY =1.25 LIFECYCLE = 20W C = 1.2R IV BETA/SE RAW B / SE t P<= 2 sr 2 pr r with Y • • • IV A— .7135 .050 442.08 30.9023 14.306 .001 .5092 .9029 .7135 .0000 B— -.6604 .050 -409.145 30.9023 -13.240 .001 .4361 .8885 -.6604 .0000 INTERCEPT 144.722 For each t, df = 22 MULTIPLE OVERALL DF R R-squared F num den P<= .9722 .9453 189.975 2 22 .001 RESIDUALS 75 I : ! : I : : : : I s : : I ; : : : I 50 25 + - f + + + + + + + + +++ + + + + + -25 I::s:I::::I::::I::::I::s:I::::I::::I::::I::::I::::I 50 100 150 200 A 250 300 PREDICTED COST (Y) 295 WEIBULL DFR - "BEST CASE" PRO RATA,RENEWING SALES AFTER W WARRANTY =1.25 LIFECYCLE = 20W C = 1.2R COST 400 I : : : I 300 200 100 + 4 . 2 + + + 2 + + + + + 0 .2 .4 .6 .8 1 1.2 WEIBULL DFR A 296 WEIBULL DFR - "BEST CASE" PRO RATA,RENEWING SALES AFTER I I I WARRANTY = 1.25 LIFECYCLE = 20W COST 400 I::::I::::I:sssi:tssi:: : s s i s s i s s : s i 300 200 + + 2 2 + + + + + + + + + 100 : + + + + 0 I::::I::::I::::I::::Is:::I:;::I::::I::::I::::I::::I .2 .4 .6 .8 1 1.2 WEIBULL DFR B C = 1.2R 297 WEIBULL DFR - "WORST CASE" PRO RATA,RENEWING NO SALES AFTER UJ WARRANTY = 0.75 LIFECYCLE = 2UJ C = 1.6R IV BETA/SE RAW B / SE A— -.0871 -3.36414 .166 6.41989 B— -.6206 -23.9807 .166 6.41989 INTERCEPT 27.0776 2 2 t P<= sr pr -.524 .606 .0076 .0123 -3.735 .001 .3851 .3881 r udth . . . Y IW -.0871 .0000 -.8206 .0000 For each t df = 22 MULTIPLE OVERALL R R-squared F .6267 .3927 7.114 DF num den P<= 2 22 .004 RESIDUALS S 51 4 + ++ 4 4 + 4 4 4 0 + + 4 44 4 4 4 44 4 4 4 4 -5 -10 4 8 10 12 14 16 ' * 1 18 PREDICTED COST (V) 298 WEIBULL DFR - "WORST CASE" PRO RATA, RENEWING NO SALES AFTER W WARRANTY =0.75 LIFECYCLE = 2W C = 1.6R COST + + + + + + 15 + + 2 + + + 2 + + 2 2 + + 10 + 5 + .2 .4 .6 • .8 1 1.2 WEIBULL DFR A 299 WEIBULL DFR - "WORST CASE". PRO RATA, RENEWING NO SALES AFTER W WARRANTY = 0.75 LIFECYCLE = 2W C = 1.BR COST 20 Issssls::si:::sis:::I::::I::::I::::I::::I::::I::::I • ; 2 : + + s 2 15 : + + : 2 + : 2 + + : 2 + 3 : 2 10 : + 5 ; j + 0 Its: slsss s i s s i s s; si's s: sis s s sis s s sis ssslsts si .2 .4 .6 .8 ' 1 1.2 WEIBULL DFR B 300 WEIBULL IFR - "BEST CASE" PRO RATA, RENEWING SALES AFTER IU WARRANTY = 1.0 LIFECYCLE = 20UI C = 1.2R IV BETA/SE RAW B / SE A— -.3544 -28.4414 .055 4.39511 B— -.8991 -97.717 .055 5.95192 INTERCEPT 181.858 2 2 t P<= sr pr -6.471 .001 .1256 .6556 -16.418 .001 .8084 .9245 r with . . . Y I \ l -.3544 .0000 -.8991 .0000 For each t df = 22 MULTIPLE OVERALL R R-squared F .9664 .9340 155.709 DF num den P<= 2 22 .001. RESIDUALS 1C T....T....T....T. + , • + + + 5 + ++ + + + + 0 + + + + + -5 + + + + + +4- + + -iu ± : j . : ; : ; _ L ; ; ; : ±: -20 0 20 40 A BO PREDICTED COST (Y) 80 301 WEIBULL IFR - "BEST CASE" PRO RATA, RENEWING SALES AFTER W WARRANTY = 1.0 LIFECYCLE = 20W C = 1.2R COST 150 I : : : I : : : : I : s s i s ; : I s s i s s s s i 100 50 + + + + -50 1 1.2 1.4 1.6 1.8 2 WEIBULL IFR A 302 WEIBULL IFR - "BEST CASE" PRO RATA, RENEWING SALES AFTER W WARRANTY =1.0 LIFECYCLE = 20W COST 100 + + 2 50 s + + + + 2 + + + 32 + 2 2 2 -50 I::;:I:t:iI::t:I::::I::::I::::I:s::I::::I:::!l::s;I .8 1 1.2 1.4 1.6 1.8 WEIBULL IFR B C = 1.2R 303 WEIBULL IFR - "UORST CASE” PRO RATA, RENEWING SALES AFTER W WARRANTY = 0.75 LIFECYCLE = 2W C = 1.2R IV BETA/SE RAW B / SE 2 2 r with • • • t P<= sr pr Y IV A— -.5599 .113 -3.65701 .737118 -4.961 .001 .3135 .5280 -.5599 .0000 B— -.6374 -5.63721 -5.647 .001 .4062 .5918 -.6374 .0000 .113 .99821A INTERCEPT 1 A.0747 For each t, df = 22 MULTIPLE OVERALL DF R R-squared F num den P<= .8484 .7198 28.253 2 22 .001 RESIDUALS 7.5 I:: ::Is:: 5.0 2.5 .0 + + + + ++ ++ + ++ + + ++ ++ + + + ++ + -2.5 I::::I::::I::::I::::I::::I::::I::::I:s::I::::I::::I -2 0 2 4 A 6 B PREDICTED COST (Y) 304 WEIBULL IFR - "WORST CASE" PRO RATA, RENEWING SALES AFTER W WARRANTY =0.75 LIFECYCLE = 2W C = 1.2R COST 10.0 I::jsis:ssi: 7.5 5.0 + + 2.5 s 3 + + 2 + 2 3 + .0 Isst:I:s:sis::sis:::1s::sis::sis::sis::sis 1 1.2 1.4 1.6 1.8 WEIBULL IFR A 305 WEIBULL IFR - "UJORST CASE" PRO RATA, RENEWING SALES AFTER W WARRANTY = 0.75 LIFECYCLE = 2W C = 1.2R COST 10.0 I : : : I : : : : I : s s i s s s s i s s i s s s i 7.5 5.0 2.5 .0 I: .8 + 4* + 1.2 1.4 1.6 lilEIBULL IFR B :::I 1 .8 306

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Asset Metadata

Creator Hill, Vickie Lee (author)

Core Title A quantitative model for the analysis of warranty policies

Contributor Digitized by ProQuest (provenance)

Degree Doctor of Philosophy

Degree Program Business Administration

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag business administration, general,OAI-PMH Harvest

Language English

Advisor Blischke, Wallace R. (committee chair), [illegible] (committee member)

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c17-743615

Unique identifier UC11344985

Identifier DP22591.pdf (filename),usctheses-c17-743615 (legacy record id)

Legacy Identifier DP22591.pdf

Dmrecord 743615

Document Type Dissertation

Rights Hill, Vickie Lee

Type texts

Source University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

Access Conditions The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the au...

Repository Name University of Southern California Digital Library

Repository Location USC Digital Library, University of Southern California, University Park Campus, Los Angeles, California 90089, USA