Cost Modelling Using Automobile Warranty Data
Cost ModellingUsingAutomobile Warranty DatabyRaymond SummitVICTORIA eUNIVERSITYzoooA thesis submitted at Victoria University of Technology infulfilment of the requirements for the degree of Doctor of Philosophy.School of Computer Science and MathematicsVictoria UniversityMelbourne, AustraliaAugust 2004
FTS THESIS338.45629222 SUMtext30001008593826Summit, RaymondCost modelling usingautomobile warranty data(c)2004Copyright by Raymond SummitAll rights reserved11
DisclaimerThis thesis contians no material that has been accepted for the award of any other degree or diploma in any university or tertiary institution. To the best of my knowledge andbelief, it contains no material previously published or written by another person, exceptwhere due reference is made in the text of the thesis.I confirm that this thesis does not exceed 100,000 words (excluding the bibliographyand appendices).Raymond SummitAugust 2004111
AbstractThis thesis sets out to model, fi-om the manufacturer's point of view, the warrantycost of a repairable product. The product can be a complex one made up of numerouscomponents, all of which are replaced upon failure and are non-repairable. The warrantycost model is used to extrapolate the cost when the warranty is extended. Both point andinterval estimates of the current and extended warranty costs are evaluated in this study.The modelling in this diesis is based upon real data obtainedfi-oman Australian carmanufacturer. As such, the thesis starts out with a detailed discussion of the importantissue of data checking and cleaning.Survival methods are used to model the product as a repairable system, with eachrepair consisting of the replacement of one or more failed components. Thus, the repair is taken to be as good as new, and the repair process can be modelled as a renewalprocess. Of interest is the expected number of replacements of each component duringthe warranty period, which can be estimated using renewal theory. To make the modelling manageable, the failure of each component is taken to be independent of the failureof other components.The reliability of components with a small number of claims is assumed to followthe exponential distribution, whilst for components with more claims, the Weibull distribution is used. Although other models are considered, the exponential proves to beadequate when tiie number of claims is small compared to the number of items produced.Because of its versatility, the Weibull distribution is an appropriate choice when modellingcomponents with a larger number of claims. Log likelihood methods are used to estimatethe parameters of the models, from which tiie number of renewals during the warrantyperiod are estimated. Numerical methods are employed to do this for the Weibull model.The expected warranty cost for each component is calculated from the expectednumber of replacements and the expected cost of repair. The cost of repair is taken to bea variable quantity in this study. Using the variance of the expected number of renewalsand tiie variance of tiie cost of repak, tiie variance of the warranty cost is obtamed. TheIV
Abstractvestimated warranty costs and variances of all components are used to obtain an expectedwarranty cost per vehicle produced, and a confidence interval on that cost.The variance of the Weibull model proves to be too big to be of practical use. However, the exponential model's variance is quite usefiil. Simulation has been used to obtaina better confidence interval for the Weibull model. Simulations are also used to verify theresults obtained by the modelling used in this thesis.S-Plus has been used extensively throughout the thesis to perform the analysis. Although the number of fimctions in the S-Plus library is quite comprehensive, many newfiinctions have been written by the author to undergo the analysis needed for this study.
Acknowledgements/ would like to gratefully acknowledge the assistance of my principal supervisor. Associate Professor Peter Cerone, Head of the School of Computer Science andMathematics. Without his encouragement to apply for candidature and a scholarship, this project would not have begun; without his guidance and patience it wouldnot have progressed to completion. I would also like to express my appreciation toAssociate Professor Ned Barnett, who, as Head of the School awarded me a Schoolscholarship in the later part of my studies. I also appreciate the facilities providedat the School to enable me to conduct this project. Without these financial and otherassistance, I would not have been able to undertake this project.I gratefully appreciate the assistance of my co-supervisor, Dr Neil Diamond,whose assistance with statistical and S-Plus matters has been invaluable. Again,without his assistance, this project would not have come to completion.I would also like to thank the Warranty Manager of the car manufacturer thathas made the data available for this study. His generosity is much appreciated.Obviously without that data, this project would not have been possible. He and hiscompany shall have to remain nameless so that the data is not misused against thecompany.I would also like to thank my wife, Isobel, for her assistance in proof-reading,and also for providing moral and physical support throughout this project I wouldlike to dedicate this work to her and my children Katherine and Jessica, each ofwhom have had to go with only a limited amount of my time and attention duringthis project. I would also like to dedicate this work in loving memory of my parents,Mary and Charles, who encouraged me when a child, to learn and inquire.VI
Contents1 Thesis Introduction11.1 Scope and Significance of This Study11.2 Thesis Outline42 Warranty Perspectives72.1 Introduction72.2 A History of Warranties72.3 Definition of a Warranty102.4 Classification of Warranties112.5 Two-Dimensional Warranty Regions122.6 Warranty and Management152.6.1 Decision-Making162.6.2 Marketing172.6.3 Warranty Servicing PoHcies182.6.4 Warranty Reserves192.6.5 Legal Obligation192.6.6 Data Collection and Management202.7 Warranty and Engineering202.7.1 Manufacturing Stages202.7.2 Quality Control212.7.3 Predictive Model and Warranty Data222.7.4 Maintenance Scheduling232.8 Conclusion23VII
viiiContents3 Warranty Modelling-A Review253.1 Modelling the Warranty Process253.1.1 Modelling Failure Mode963.1.2 Modelling Time to First Failure3.1.3 Modelling Customer Claim Behaviour 3.1.4 Modelling Rectification Action 3.1.5 Modelling Subsequent Failures 3.1.6 Modelling the Cost of Rectification313.2 Modelling Two-Dimensional Warranties323.2.1 One-Dimensional Approach333.2.2 Two-Dimensional Approach333.2.3 Comparison of the Two Approaches343.2.4 Cost Models for Free Replacement Warranty353.3 Modelling Product Reliability363.3.1 Lifetime Distribution363.3.2 Censoring Times373.3.3 Reporting Lags383.4 Survey of Approaches to Warranty Modelling383.4.1 Warranty Claims Distribution383.4.2 Bayesian Approach3.4.3 Dynamic Lmear Modelling and Neural Networks40403.4.4 Markovian Approach413.4.5 Renewal Process Approach423.4.6 Simulation Approach433.5 Conclusion44
Contents4 The Manufacturer's Dataix454.1 Introduction454.2 Importing The Data464.3 Exploratory Data Analysis484.4 Data Cleaning534.4.1 Errors In The Repair Date554.4.2 Errors In Sale Date554.4.3 Repair Date Between Production And Sale564.4.4 Check age2 Against age And dist574.4.5 Cleaned Database584.4.6 Missing Records From Sales Database584.5 Component Ages in Subsequent Repairs594.6 Database of Failed and Censored Times594.7 Conclusion605 Survival and Cost Models625.1 Introduction625.2 Assumptions625.2.1 Validity of Claims635.2.2 Timing of Failures635.2.3 Repair Time645.2.4 Nature of Repair645.2.5 Independence of Component Failures645.2.6 Uniformity of Parts655.2.7 Location Factors655.2.8 Dimensionality of Warranty65
Contents5.3 Survival Analysis Methodsfn5.3.1 Censoring Times5.3.2 Hazard Rate 705.4 Non-Parametric Estimation of Component Reliability725.5 Parametric Models of Component Reliability735.5.1 Log Likelihood Function5.5.2 Exponential Model 5.5.3 Weibull Model' 5.6 Modelling Failure Using a Renewal Process5.6.1Formulation' ' 5.6.2 Solution for the Exponential Distribution815.6.3 Numerical Solution for tiie Weibull Distribution825.7 Modelling The Cost of Component Repair845.7.1 Choice of a Representative Cost Statistic875.7.2 Variance of the Cost of Repair895.8 Modelling The Cost of a Warranty895.8.1 Point Estimate895.8.2 Variance905.8.3 Total Warranty Cost905.8.4 Pomt and Interval Estimates Using the Bootstrap Method915.9 Limitations of the Models935.10 Conclusion946 Implementation of the Survival Models966.1 Introduction966.2 Frequency Distribution of Claims97
Contents6.3 Parametric Models of Reliability6.4 Exponential Modelxi991036.4.1 Expected Warranty Cost of Components1046.4.2 Implementation in S-Plus1056.4.3 Results1076.5 Weibull Model1096.5.1 The S-Plus Function survReg1096.5.2 Point Estimate: Implementation in S-Plus1146.5.3 Variance: Implementation in S-Plus1156.5.4 Results1186.6 Comparison of Exponential and Weibull Models1186.7 Model Estimates of Warranty Cost1206.7.1 Pomt Estimate1206.7.2 Variance1226.8 Costing an Extended Warranty1246.9 Sensitivity Analysis1266.9.1 Range of Parameter Values1276.9.2 Results1286.9.3 Interpretation of Results1286.10 Conclusion7 Simulation Studies1321347.1 Introduction1347.2 Bootstiap Estimate of Total Warranty Cost.1357.2.1 Purpose1357.2.2 Method135
xiiContents 7.2.3 Results1 -J/T7.3 Simulated Warranty Costs: 3 and 5 Year Warranties7.3.1 Purpose 7.3.2 Metiiod 7.3.3 Results7.4 Simulated Warranty Costs: Age and Distance Limitations 1407.4.1 Purpose1" 7.4.2 Metiiod1407.4.3 Results1407.5 Reliability Estimation Using Same-Age Data1417.5.1 Purpose and Background1417.5.2 Metiiod1427.5.3 Results1437.5.4 Interpretation of Results1467.6 Reliability Estimation Using Varying-Age Data1487.6.1 Purpose and Background1487.6.2 Method1487.6.3 Results1517.7 Reliability Estimation Usmg Varymg-Age Data From 1 Year's Production. 1547.7.1 Purpose1547.7.2 Metiiod1547.7.3 Results1557.7.4 Interpretation of Results7.8 Conclusion,157157
Contentsxiii8 Concluding Remarks and Future Directions8.1 Introduction'.1591598.2 Motivation For This Sttidy1598.3 Findings of This Stiidy1608.3.1 Comparison of Models1618.3.2 Findings of the Simulations1628.4 Extensions to This Research1638.5 Conclusion164References165A S-Plus Scripts and Outputs for Chapter 4174A.l Exploratory Data Plots and Summaries (PlotSummary.ssc)174A.2 Function to Check the Data (Check.ssc)175A.3 Output offCheck177A.4 Function to Clean the Data (Clean.ssc)178A.5 Function to Add Sales Records (AddSale.ssc)180A.6 Function to Calculate Components' Ages (compAge.ssc)180A.7 Function to Construct a Survival Database (Survive.ssc)182B S-Plus Scripts for Chapter 5B.l Calculation of Usage Rates (Usage.ssc)C S-Plus Scripts for Chapter 6185185187C. 1 Exponential Modelling (CostExp.ssc)187C.2 Exponential Fail Rate (FailRate.ssc)189C.3 Exponential Modelling with Grouping (CostExpGp.ssc)191C.4 Renewal Function Solver (Renew.ssc)194
xivContentsC.5 Weibull Modelling (CostWbl.ssc)C.6 Confidence Region (ConfRgn.ssc) 197C.7 Combined Exponential and Weibull Modelling Function (CostExpWblssc).199C.8 Estimate of Total Warranty Cost (WrntCstTot.ssc) C.9 Extended Warranty Cost (ExtdWrnt.ssc)205CIO Sensitivitiy Analysis (Sensitivity.ssc)2 D Component Reliability Graphs for Chapter 6210D.l Survival Plots of Components With Ill-fitting Weibull Models210D.2 Age Density Plots of Components Witii Ill-Fitting Weibull Models214D.3 Age Density Plots of Components With Good-Fitting Weibull Models218E S-Plus Scripts for Chapter 7E.l Warranty Cost Simulation (WmtCstSimulssc)220220E.2 Simulated Warranty Cost With Distance Limitation (WrntCstSmulDist.ssc) . 222E.3 Sample Generating Function (SmpDisc.ssc)224E.4 Production Density (ProdDens.ssc)226E.5 Sales Delay Density (DelayDens.ssc)226E.6 Random Samples With Continuous Production (SmpCont.ssc)227E.7 Random Samples With 1 Year's Continuous Production (SmpContlyr.ssc).230
Chapter 1Thesis Introduction1.1 Scope and Significance of This StudyExpensive, complex products are almost always sold with a warranty, so the cost of servicing a warranty is significant. Wasserman (1992) suggested that warranty claims couldamount to 10-30% of production costs. Menezes and Quelch (1990) claimed that warranties represent an increasing cost to the manufacturer. They reported that the automotive industry in the United States of America spent over US 5 bilUon on product warrantyin 1988, up from just over 700 million in 1965. Majeske and Herrin (1998) stated thatin 1992, the combined total warranty payment bill for Ford, General Motors and Chryslerwas 9.2 bilHon.The cost of servicing warranty claims is to be paid for by the manufacturer, whomust cover this cost in the sale price of the product. The most important reason fordetermining the cost of a warranty is the need to price the warranty (Hill and Blischke,1987). Analysis of warranty data provides many benefits: it enables an estimate of thewarranty cost to be made; it can predict the new costs if the terms of the warranty arealtered; and it provides feedback to engineers about the reliability of the product, whichmay be valuable in reviewing the design of a product or the manufacturing process.Manufacturing in Austialia has become more competitive in the world market withthe reduction or removal of import tariffs. The building of cost-effective, quality products has become an aim of many Australian (and international) companies. This is clearlyexemplified in the car manufacturing industry, where the decrease of import duties hasresulted in quality, imported vehicles becoming more competitively priced. Local manufacturers have had to build a reliable product at a cost-effective price to compete in thisfierce market place. Over recent times, the standard warranty that comes with Australianbuilt cars hasrisenfromone year to three years, but in confrast, many importers currentiyoffer a more generousfive-yearwarranty. Since the extent of a warranty is often used tojudge the rehability of a product, it can be a powerful marketing tool.
2Chapter 1 Thesis IntroductionThe modelling of warranty costs is a complex stiidy because of the variety otwarranty terms, and because of the stochastic natiire of product or component failures.Singpurwalla and Wilson (1993) affirmed that "the warranty problem is multidisciplinary, involving topics as diverse as economics, game tiieory, law, marketing, operationsresearch, psychology, probability, and statistics."One of tiie most difficult warranties to analyse is the two dimensional warranty,which is limited by both calendar time and usage. Altiiough some literatiire has appearedsince Singpurwalla and Wilson (1993) declared tiiat "not much has been done witii regardto two-dimensional warranties", there still appears to be a need for further work with twodimensional warranty modelling. Although the data used in this tiiesis is two-dimensional,the approach used is a one-dimensional approach. This approach, as discussed in Subsection 5.2.8, is suitable for the terms of the manufacture's warranty. This simplifiedapproach has been used in this thesis because of the large number of components to beanalysed.As much has already been written about the modelling of warranty costs, one mightconsider the subject to be in no need of further attention. However, most papers havetaken the form of a theoretical treatise, with few studies investigating the application ofthe models. Blischke (1990) stated:Perhaps more pressing, however, is the needfor practical applications-oriented research. Before they find widespread use in practice, the elegant models that havebeen developed must be incorporated into approaches that identify all significantcost factors and the associated data requirements, that emphasize the use of information that could realistically be attainable and that realistically model operationalwarranty progress. Methodological papers along these lines have not appeared inthe statistical or management science literature (perhaps not because such studieshave not been done, but because the results are proprietary).Thirteen years later, Jablonowski (2003) is still echoing these sentiments. He talks aboutthe widenmg gap between theoretical developments and practice in risk managementmodelling, and what needs to be done to reduce the gap:The existence of a wide gap, however, may be indicative that theoretical develop-
1.1 Scope and Significance of This Studyment in afield is not fruitful3. The key is moving formulas and equations frompaper to practice. . Rewarding practical success, not just formal rigor, wouldgo a long way to help assure that proper attention is paid to the development of"application friendly " theory. The corresponding threat is that without practicaljustification, or at least its promise, the theory will not be taken seriously.This thesis attempts to reduce the gap between theory and practice. It uses real datafrom an automobile manufacturer's warranty database. Practical issues that concern thehandling of real data are discussed in this study, from the methodology involved in thechecking and cleaning of the data, to the implementation of theoretical models. Thesepractical issues are an important aspect of this thesis, and therefore much effort has beendevoted to them in this study.Murthy and Blischke (2000) acknowledged that "For most companies, warrantycosts are a closely guarded secret, as evidenced by the fact that very little warranty data isavailable in the public domain." In the quote of the previous paragraph, Blischke (1990)pointed out the proprietary nature of warranty data. Some studies have analysed carwarranty data, for example, Kalbfleisch, Lawless and Robinson (1991), Lu and Vance(1997), Majeske and Herrin (1995 and 1998), and Lu (1998). However, because of spacelimitations, these papers have discussed the analysis of one or two parts only. There is aneed for a more extensive study that includes the warranty cost covering all componentsof a complex product, such as the automobile. There is a need to explore the processesinvolved in a larger study, and there is a need for practically-based research. In addition,the literature appears not to contain any warranty data from an Australian manufacturer.This thesis models the warranty cost of an entire vehicle manufactured in Ausfralia, Italso discusses in detail the practical issues that need to be resolved when analysing thereal data.It is common practice in the literature to assume that the cost of repairing a productis a constant. However, a manufacturer can only indicate a range of costs that it is willingto pay to the repair agent for particular repairs, rather than one particular price. The dataobtained for this study reveals that the cost of replacing a particular component resultingfrom a warranty claim is far from constant. By allowing the cost of repair to be a variable,this study extends the current warranty models.
4Chapter 1Thesis IntroductionThe current literature on the use of simulations in warranty-cost modelling is complemented by the use of simulation to verify the models that are established in the tinsstudy, and by the use of simulation to validate the techniques used in developing the models. Some of the simulations produce new results that complement the current literatureon warranty sunulation.An integral part of tiiis tiiesis is the extensive use of S-Plus in developing the manyfimctions needed to complete the analysis. Although the S-Plus library of functions isvast, new fiinctions have been developed to do the processing required in this study.To summarise, this thesis extends the current warranty-cost models in the literatureby presenting a practically-oriented, extens
3.2.4 Cost Models for Free Replacement Warranty 35 3.3 Modelling Product Reliability 36 3.3.1 Lifetime Distribution 36 3.3.2 Censoring Times 37 3.3.3 Reporting Lags 38 3.4 Survey of Approaches to Warranty Modelling 38 3.4.1 Warranty Claims Distribution 38 3.4.2 Bayesia 4n Approach 0 3.4.3 Dynamic Lmear
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Here, warranty cost for – A. Manufacturer :- Sum of production cost of item and servicing cost of item if it fails in original warranty period (W). B. Customer :- Purchase prize and cost related to warranty claims. II. METHODOLOGY FOR WARRANTY COST ANALYSIS . Following File Size: 1MB
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