Modelling and Analysis of Reliability and Costs for Lifetime Warranty and Service Contract Policies

Transcription

1 Modelling and Analysis of Reliability and Costs for Lifetime Warranty and Service Contract Policies by Anisur Rahman Master of Engineering (Research and Thesis), M.Sc. Engineering (Eng. Management), PG Dip in Personnel Management, B.Sc Engineering (Mechanical) A Thesis Submitted to Queensland University of Technology for the degree of DOCTOR OF PHILOSOPHY School of Engineering Systems Queensland University of Technology January,

2 ABSTRACT Reliability of products is becoming increasingly important due to rapid technological development and tough competition in the product market. One effective way to ensure reliability of sold product/asset is to consider after sales services linked to warranty and service contract. One of the major decision variables in designing a warranty is the warranty period. A longer warranty term signals better reliability and provides higher customer/user peace of mind. The warranty period offered by the manufacturer/dealer has been progressively increasing since the beginning of the 20 th Century. Currently, a large number of products are being sold with long term warranties in the form of extended warranty, warranty for used product, long term service contracts, and lifetime warranty. Lifetime warranties and service contracts are becoming more and more popular as these types of warranties provide assurance to consumer for a long reliable service and protecting consumers against poor quality and the potential high cost of failure occurring during the long uncertain life of product. The study of lifetime warranty and service contracts is important to both manufacturers and the consumers. Offering a lifetime warranty and long term service contracts incur costs to the manufacturers/service provider over the useful life of the product/contract period. This cost needs to be factored into the price/premium. Otherwise the manufacturer/ dealer will incur loss instead of profit. On the other hand, buyer/user needs to model the cost of maintaining it over the useful life and needs to decide whether these policies/service contracts are worth purchasing or not. The analysis of warranty policies and costs models associated with short-term or fixed term policies have received a lot of attention. A significant amount of academic research has been conducted in modelling policies and costs for extended warranties and warranty for used products. In contrast, lifetime warranty policies and longer term service contracts have not been studied as extensively. There are complexities in developing failure and cost models for these policies due to the uncertainties of useful life, usage pattern, maintenance actions and cost of rectifications over longer period. 2

3 This thesis defines product s lifetime based on current practices. Since there is no acceptable definition of lifetime or the useful life of product in existing academic literatures, different manufacturer/dealers are using different conditions of life measures of period of coverage and it is often difficult to tell whose life measures are applicable to the period of coverage (The Magnuson-Moss Warranty Act, 1975). Lifetime or the useful life is defined in this thesis provides a transparency for the useful life of products to both manufacturers/service provider and the customers. Followed by the formulation of an acceptable definition of lifetime, a taxonomy of lifetime warranty policies is developed which includes eight different one dimensional and two dimensional lifetime warranty policies and are grouped into three major categories, A. Free rectification lifetime warranty policies (FRLTW), B. Cost Sharing Lifetime Warranty policies (CSLTW), and C. Trade in policies (TLTW). Mathematical models for predicting failures and expected costs for different one dimensional lifetime warranty policies are developed at system level and analysed by capturing the uncertainties of lifetime coverage period and the uncertainties of rectification costs over the lifetime. Failures and costs are modelled using stochastic techniques. These are illustrated by numerical examples for estimating costs to manufacturer and buyers. Various rectification policies were proposed and analysed over the lifetime. Manufacturer s and buyer s risk attitude towards a lifetime warranty price are modelled based on the assumption of time dependent failure intensity, constant repair costs and concave utility function through the use of the manufacturer s utility function for profit and the buyer s utility function for cost. Sensitivity of the optimal warranty prices are analysed with numerical examples with respect to the factors such as the buyer s and the manufacturer/dealer s risk preferences, buyer s anticipated and manufacturer s estimated product failure intensity, the buyer s loyalty to the original manufacturer/dealer in repairing failed product and the buyer s repair costs for unwarranted products. 3

4 Three new service contract policies and cost models for those policies are developed considering both corrective maintenance and planned preventive maintenance as the servicing strategies during the contract period. Finally, a case study is presented for estimating the costs of outsourcing maintenance of rails through service contracts. Rail failure/break data were collected from the Swedish rail and analysed for predicting failures. Models developed in this research can be used for managerial decisions in purchasing life time warranty policies and long term service contracts or outsourcing maintenance. This thesis concludes with a brief summary of the contributions that it makes to this field and suggestions and recommendations for future research for lifetime warranties and service contracts. 4

5 ACKNOWLEDGEMENT The preparation of a substantial work such as this thesis is not possible without the assistance and support from a large number of people. I would like to take this opportunity to acknowledge all those people who have contributed to complete this project. My supervisor, Dr. Gopinath Chattopadhyay for his sincere and constant, tireless support, encouragement, and guidance throughout this project. He spent his valuable time in discussing various solutions related to problems during this project. I am indebted to him for his patience during discussions, detailed examination of this manuscript. His critical insight and valuable suggestions have contributed to a great extent to the final form of this dissertation. My associate supervisor, Professor Erhan Kozan of School of Mathematical Science for his assistance and direction in developing models and preparation of refereed papers. I am very much grateful to Professor Doug Hargreaves, Head of School, School of Engineering Systems, QUT for his support and financial assistance; without which it would have been impossible for me to continue this research. Professor Joseph Mathew, Chief Executive Officer, CIEAM and Associate Professor Lin Ma, faculty of BEE for providing financial support. Dr. Venkatarami Reddy, Mr. Ajay Desai and Dr. Nguen Than, for helping me time to time in preparation of this thesis and developing models and programming. My Brother in law Md. Keramatullah Furuki for his inspiration and support which helped me to finish this research work on time. Peter Nelson, for his great help in English corrections and useful advice. Finally, I express my heart felt appreciation to my wife Roushan and my daughter Afnaan and sun Rakin for their love, support and continuous sacrifice and encouragement throughout this doctoral program. 5

6 STATEMENT OF ORIGINALITY I declare that to the best of my knowledge the work presented in this thesis is original except as acknowledged in the text, and that the material has not been submitted, either in whole or in part, for another degree at this or any other university. Signed:.. Anisur Rahman Date: 6

7 LIST OF RESEARCH PUBLICATIONS PUBLICATIONS RESULTING FROM THIS THESIS Referred International Journal papers (Published): 1. Chattopadhyay, G. and Rahman, (2007) A., Development of Lifetime Warranty Policies and Cost Model for Free Replacement Lifetime Warranty (FRLTW) Policy, Reliability Engineers and System Safety, Published Online March, (Based on the Chapter 3 and 4). 2. Rahman, A. and Chattopadhyay, G.N (2006), Review of Long Term warranty policies, Asia Pacific Journal of Operational Research Vol 22(4), p (Based on Chapter 2). 3. Chattopadhyay, G.N. and Rahman, A. 2004, Optimal Maintenance Decisions for Power Supply Timber Poles, Vol 5(4), p ISSN (Based on Chapter 6). Referred International Journal papers (under review/in process): 4. Rahman, A. and Chattopadhyay, G., Modelling Failures and Costs for Service Contracts, submitted for publication in International Journal of Reliability and Application. (Based on Chapter 6 and 7). 5. Chattopadhyay, G. and Rahman, A., Modelling risks to manufacturers and customers for lifetime warranty policies when failures follow Nonhomogeneous Poisson Process, for IEEE Transactions on Reliability (submitted). (Based on Chapter 5) 6. Chattopadhyay, G., Murthy, D.N.P., and Rahman, A., Warranty cost models for second-hand products A review, prepared for submitting in the International Transactions on Operations Research (Based on Chapter 2) Refereed International Conference Papers 1. Chattopadhyay, G., Rahman, A. (2007), Modelling Total Cost Of Ownership Of Rail Infrastructure For Outsourcing Services, The 2nd World Congress On Engineering Asset Management And 4 th International Conference On Condition Monitoring, Harrogate, UK, pp (Based on Chapter 6 and 7) 2. Rahman, A., Chattopadhyay, G. (2007), Modelling Cost Sharing Policies For Lifetime Warranty, The 20 th International Congress on Condition Monitoring and Diagnostic Engineering Management, COMADEM 2007, University of Algarve, Faro, Portugal, Based on Chapter Rahman, A., Chattopadhyay, G. (2006), Conceptual Model for Outsourcing Rail Network Asset Using Long-Term Service Contracts, Congress on Condition Monitoring and Diagnostic Engineering Management, COMADEM 2006, Lulea University of Technology, Lulea, Sweden, pp Based on Chapter 6 and 7 7

8 4. Chattopadhyay, G., Rahman, A. (2006), Analysis of Rail Failure Data for Developing Predictive Models and Estimation of Model Parameters First World Congress on Engineering Asset Management (1st WCEAM), Gold Coast, Australia, paper 57. (Based on Chapter 7) 5. Rahman, A., Chattopadhyay, G. (2005) Modelling Failures of Repairable Systems and Costs of Service Contracts, Smart Systems Postgraduate research conference, Queensland University of Technology, Brisbane Australia, December pp (Based on Chapter 6). 6. Chattopadhyay, G.N., Rahman, A. (2005) Modelling Costs and Risks to Manufactures and Buyers for Lifetime Warranty Policies, Proceedings of the 18th International Congress on Condition Monitoring and Diagnostic Engineering Management, COMADEM 2004, Cranfield University, U.K. August September, ISBN pp ( Based on Chapter 5) 7. Rahman, A. and Chattopadhyay, G.N. (2004), Lifetime Warranty Policies: Complexities in Modelling and Potential for Industry Application, Asia Pacific Industrial Engineering and management Systems Conference (APIEMS ), Dec 2004, Gold coast, Australia. ISBN (Based on Chapter 3 and 4) 8. Chattopadhyay, G.N., Rahman, A. (2004) Modelling Costs for Lifetime Warranty Policies, Proceedings of the 17 th International Congress on Condition Monitoring and Diagnostic Engineering Management, COMADEM 2004, Cambridge, U.K. August ISBN pp (Based on Chapter 3 and 4) Non-refereed National Conference Papers 1. Rahman, A., Chattopadhyay, G. (2005) Modelling Cost of Service Contracts, Proceedings of the 6 th Operations Research Conference of the Australian Society for Operations Research Queensland Branch. Marriott Hotel, Brisbane, Australia, 12 August, pp 7-8. (Based on Chapter 6 ) OTHER PUBLICATIONS 1. Rahman, A. and Chattopadhyay, G.N., (2007) Soil factors behind inground decay of Timber Poles: testing and interpretation of results, IEEE Transactions on Power Delivery, Vol. 22(3), pp Rahman, A., Chattopadhyay, G., Wah, Simon (2006), Application of Just In Time and Kanban Strategies for Component Stocks At Cox Industries, Congress on Condition Monitoring and Diagnostic Engineering Management, COMADEM 2006, Lulea University of Technology, Lulea, Sweden, pp Rahman, A. and Chattopadhyay, G.N., (2003); Identification and Analysis of Soil factors for Predicting Inground Decay of Timber Poles in Deciding Maintenance Policies ; Proceedings of the 16 th International Congress on Condition Monitoring and Diagnostic Engineering Management, 8

9 COMADEM 2003, Vaxjo, Sweden, August ISBN pp Rahman, A. and Chattopadhyay, G.N., (2003); Estimation of Parameters for Distribution of Timber Pole Failure due to In-ground Decay ; Proceedings of the 5 th Operations Research Conference on Operation Research in the 21 st Century, the Australian Society of Operations Research, Sunshine coast, Australia, 9-10 May, Chattopadhyay, G.N., Rahman, A. and Iyer, R.M (2002); Modelling Environmental and Human Factors in Maintenance of High Volume Infrastructure Components ; 3 rd Asia Pacific Conference on System Integration and Maintenance, Cairns, Sept ISBN pp

10 Nomenclatures Notations Used in Modelling Lifetime Warranties α Increasing rate of cost due to inflation and other factors β Shape parameter δ Discount rate (annuity) η Characteristic life parameter of the product ρ Parameter for the truncated exponential distribution used in the life distribution of products μ Mean of the failure distribution λ Failure intensity λ(t) Intensity function for system failure λ I (t) Intensity function for system failure due to component/s Set I ( included in the warranty) λ E (t) Intensity function for system failure due to component/s Set E ( excluded in the warranty) L Defined lifetime of the product. H(a) Distribution function of the lifetime (useful life) h(a) Density function associated with H(a) l Lower limit of the defined lifetime u Upper limit of the defined lifetime F(.) Failure distribution of product R(.) Product reliability distribution f(.) Density function associated with F(.) r(.) Hazard rate function associated wit F(.) M(L) Number of renewal up to the lifetime L X 1 Age of the item at its firs failure M g (.) Renewal function associated with distribution function G(.) M d (.) Renewal function associated with distribution function F(.) C x c x Total warranty cost over the lifetime associated with a ordinary renewal Process Average cost of each failure replacement associated with an ordinary renewal process 10

11 C y c y G(c) g(c) c Total warranty cost over the lifetime associated with a delayed renewal process Av. cost of repair associated with an delayed renewal process Cost distribution function Density function associated with G(c) expected cost of each rectification over the lifetime (system level) γ Parameter for cost distribution E[N(L)] Expected number of failures over the lifetime. E[C(L)] Total expected cost for model L1 over the lifetime. E[N I (L)] Expected number of failures of the component/s Set I. E[N E (L)] Expected number of failures over the lifetime: component/s Set E. E[C I (L)] Total expected costs for model L2 over the lifetime: component/s Set I. E[C I (L)] Total expected costs for model L2 over the lifetime: component/s Set E. c I Individual failure(claim) cost limit(associated with Model L3 & L5) C j Total rectification costs for jth failure(claim) (associated with Model L3 & L5) M j Manufacture/dealer s cost for jth claim (associated with Model L3) B j Customer/ buyer s cost for jth claim (associated with Model L3) c m Expected cost for each rectification to the manufacturer c b Expected cost for each rectification to the customer c T C t TC j TM j TB j Manufacturer s cost limit for all failures over the lifetime for model L4 and L5 Total cost to the manufacturer by time t (associated with models L3, L4 and L5) Total cost of rectification of the first j failures subsequent to the sale Cost to the manufacturer associated with j number of failure Cost to the customer associated with j number of failure E[C m (L)] Expected cost to the manufacturer for models L4 and L5 E[C b (L)] Expected cost to the buyer/customer for models L4 and L5 C L V(c ) v(c) Cost of rectification of all failures over the lifetime for Model L4 and L5 Distribution function for C L density function associated with V(c) 11

12 G (r) (c) The r-fold convolution of G(c) T b Time at which warranty expires for model L5 V m (c; t) Distribution function for the total costs to the manufacturer v m (c; t) Density function associated with V m (c; t) Q(t; c T ) Distribution function for T b q(t; c T ) Density function associated with Q(t; c T ) Notations Used in Modelling Risk Preference S p Number of total product sold Proportion of buyers who accept the warranty offer (1- p) Proportion of buyers who do not accept the warranty offer L Products lifetime k The proportion of buyers without warranty coming back to manufacturer for repairing of the faults/defects. N b (L) Number of valid claims made by the buyer per item N(L) The total number of possible claims for S items if sold for lifetime warranty. E[N b (L)] Expected number of failure per item experienced by the buyers over the lifetime. U m (Y) Manufacturers continuous utility functions for a monetary asset Y U b (X) Buyers continuous utility functions for a monetary asset X. U m An individual manufacturer s utility function U b The aggregate utility function representing the entire buyer s risk preference as a whole. c a Buyer s risk parameter Manufacturer s risk parameter Notations Used in Modelling Service Contracts Λ m (t) The manufacturer s failure intensity. 12

13 Λ b (t) Per item failure intensity for an individual buyer during the lifetime r b Cost of rectification (repair cost) for a buyer in each occasion of failure if the item is not warranted. When all rectifications are carried by the r m d Manufacturer s repair cost per occasion Difference between the buyer s and manufacturer s repair cost Λ pm (t): Failure intensity at time t, with maintenance. Λ(t) N N i Original failure intensity at t when no maintenance is performed. number of times the planned servicing is performed during the contract period number of times the planned servicing is performed during the ith replacement i = 1, 2, 3,. M L number of replacements corrective actions. Duration (length) of service contract k number of times PM is carried up to t. τ α C re C mr C pm C cl Age restoration after each PM. τ = αx, Quality or effectiveness of preventive maintenance action cost of replacement cost for each minimal repair. cost for each PM expected cost for the last cycle. 13

14 CONTENTS ABSTRACT ACKNOWLEDGEMENT Statement of Source List of Publications LIST OF TABLES...17 LIST OF FIGURE...18 CHAPTER 1 SCOPE AND OUTLINE OF THESIS INTRODUCTION LIFETIME WARRANTY AND SERVICE CONTRACTS OBJECTIVES OF THIS RESEARCH THESIS OUTLINE...24 CHAPTER LONG TERM WARRANTIES AN OVERVIEW INTRODUCTION CONCEPT AND ROLE OF WARRANTY PRODUCT CLASSIFICATION Consumer durable products Industrial or commercial products Government or defence products WARRANTY TAXONOMY (WARRANTY CLASSIFICATION) WARRANTY STUDY WARRANTY COSTS REVIEW OF BASIC WARRANTY MODELS Modelling Warranty Cost Warranty Engineering LONGTERM WARRANTY POLICIES AND COST MODELS AN OVERVIEW Extended Warranty Warranty for used product Service Contracts Lifetime Warranties CONCLUSIONS...65 CHAPTER 3 MODELLING POLICIES AND TAXONOMY FOR LIFETIME WARRANTY INTRODUCTION LIFETIME WARRANTY TAXONOMY OF LIFETIME WARRANTY POLICIES CONCLUSIONS...75 CHAPTER 4 MODELLING COST FOR LIFETIME WARRANTY INTRODUCTION PRELIMINARIES: MODELLING PRODUCT FAILURES Modelling at Component Level Modelling warranty at System Level: MODELLING LIFETIME WARRANTY COSTS AT SYSTEM LEVEL

15 4.3.1 Modelling Uncertainties of Lifetime Modelling Rectification Cost Model L1: Free Rectification Lifetime Warranty (FRLTW) Model L2: Specified Parts Exclusive Lifetime Warranty ( Policy 3): Model L3: Limit on Individual Cost Lifetime Warranty (Policy 4) Model L4: Limit on Total Cost Lifetime Warranty (Policy 5) Model L5: Limit on Individual and Total Cost Lifetime Warranty (Policy 6) ANALYSIS OF THE MODELS Analysis of Model L1 [FRLTW]: Analysis of Model L2 [SPELTW]: Analysis of Limit on Individual Cost (LICLTW) Policy CONCLUSIONS CHAPTER 5 MODELLING RISKS TO MANUFACTURER AND BUYER FOR LIFETIME WARRANTY POLICIES INTRODUCTION OVERVIEW RISK ATTITUDE AND UTILITY FUNCTION Utility Theory, Utility function and Concept of certainty equivalent MODEL FORMULATION Notations: Modelling Risks in Lifetime Warranty SENSITIVITY ANALYSIS OF THE RISK MODELS Sensitivity Analysis of Buyer s Willingness to Pay for Warranty Price Sensitivity Analysis of the Manufacture s Warranty Price CONCLUSIONS CHAPTER 6 MODELLING POLICIES AND COSTS FOR SERVICE CONTRACTS INTRODUCTION SERVICE CONTRACT - BACKGROUND SERVICING STRATEGIES DURING THE CONTRACTS MODELLING POLICIES FOR SERVICE CONTRACT MODELLING COSTS OF SERVICE CONTRACT FOR DIFFERENT POLICIES Assumptions Notations and Reliability Preliminaries Modelling Cost for Service Contract Policy Modelling Cost for Service Contract Policy Modelling Cost for Service Contract Policy Parameters Estimation CONCLUSIONS CHAPTER 7 A CASE STUDY OUTSOURCING RAIL MAINTENANCE THROUGH SERVICE CONTRACTS INTRODUCTION DEGRADATION OR FAILURE OF RAIL TRACK MODELLING RAIL BREAK/FAILURES ESTIMATING COSTS OF OUTSOURCING RAIL MAINTENANCE ANALYSIS OF THE MODELS FOR RAIL Estimation of rail failure parameters Estimating Costs of Different Service Contracts for Rail CONCLUSIONS CHAPTER 8 CONCLUSIONS AND SUGGESTIONS FOR FUTURE RESEARCH INTRODUCTION CONTRIBUTION OF THIS THESIS

16 8.3 SCOPE FOR FUTURE RESEARCH Lifetime warranty policies and cost models Risk Preference Models Service Contract Policies and Cost Models Other Scope REFERENCES APPENDICES

17 List of Tables Table 3.1: Lifetime warranty policies at a glance...76 Table 4.1: Expected lifetime warranty cost ($) to the Manufacturer for FRLTW Table 4.2: Expected lifetime warranty cost as product failure intensity varying Table 4.3: Expected cost with the product lifetime parameter variation Table 4.4: Warranty cost ($) to the Manufacturer for lifetime SPELTW: App Table 4.5: Warranty cost ($) to the Customer for lifetime SPELTW: App Table 4.6: Manufacturer s costs for Specified parts excluded policy Approach Table 4.7: Customer s cost for Specified parts excluded policy Approach Table 4.8: Manufacturer s/dealers costs for policy LICLTW Table 4.9: Customer s costs for policy LICLTW Table 5.1: Basic expressions (Product can be sold with or without warranty) Table 5.2: Warranty price (W) in $ for different Buyer s risk preferences (c ) Table 5.3: Warranty price (W) in $ for different Buyer s repair cost (r b ) Table 5.4: Effect of Buyer s repair cost on the Warranty price Table 5.5: Effect of buyer s anticipated failure parameters on the warranty price Table 5.6: Warranty price for different manufacturer s risk preferences a and n m *.136 Table 5.7: Warranty price (W) in $ for different manufacturer s failure intensity Table 5.8: Warranty price for different failure intensity and Shape parameter Table 5.9: Buyer s repair cost vs Warranty price (W) in $ Table 5.10: Buyer rate of return (k) vs Warranty price (W) Table 6.1: Service Contract Policies Table 7.1 Rail breaks in Million gross tonnes (MGT)

18 List of Figure Figure 2.1 :Stake Holders in Warranty policies...28 Figure 2.2: Taxonomy for new products warranty policies...31 Figure 2.3: A sample for warranty regions for 2-D policies...34 Figure 2.4: Taxonomy for warranty policies for Second-hand products...38 Figure 2.5: System Approach to Problem Solution...42 Figure 2.6: Simplified system approach for warranty cost analysis...43 Figure 2.7: Frame work for warranty rectification...45 Figure 2.8: A framework for long-term warranty...53 Figure 3.1: A sample of lifetime warranty offer...69 Figure 3.2: Taxonomy for lifetime warranty policies...71 Figure 3.3 Combined Trade in with lifetime warranty [CTLTW] policy...74 Figure 4.1: Failure intensity over the basic and lifetime warranty coverage period...87 Figure 4.2: Effects of discounting for warranty costs under longer coverage Figure 4.3: A Schematic diagram of Limit on Total Cost Lifetime Warranty...96 Figure 4.4: Expected lifetime warranty cost ($) to the Manufacturer FRLTW Figure 4.5: Expected lifetime warranty cost ($) for SPELTW-Approach Figure 4.6: Expected lifetime warranty cost ($) for SPELTW-Approach Figure 4.7: Expected lifetime warranty cost ($) for LICLTW Figure 5.1: Marginal utility curves Figure 5.2: Effect of buyer s risk preference parameter on the Warranty price Figure 5.3: Effect of buyer s anticipated product failure on the Warranty price Figure 5.4: Effect of buyer s risk parameters and number of failures on Warranty.131 Figure 5.5: Effect of buyer s repair cost on the Warranty price Figure 5.6: Effect of buyer s anticipated failure intensity on the Warranty price Figure 5.7: Combined effect of failure parameters on the Warranty price Figure 5.8: Effect of manufacturer s risk preference parameter on Warranty price..137 Figure 5.9: Effect of manufacturer/dealer expected product failure on Warranty Figure 5.10: Effect of failure intensity over warranty price Figure 5.11: Effect of failure parameters on the manufacturer s Warranty price Figure 5.12: Effect of buyer s cost of repair over the manufacture Warranty price..141 Figure 5.13: Effect of buyer s cost of repair on the manufacture s Warranty price..143 Figure 6.1: Failure rate with effect of various maintenance actions Figure 6.2: Graphical representation of the Service contract policy Figure 6.3: Failure intensity curve for Service contract policy Figure 6.4: Failure intensity curve for for Service contract policy Figure 7.1: Rail Profile and wear area Figure 7.2: The wear rate (mg m -1 ) vs hardness (HV) of rail steel Figure 7.3: Cumulated Rail break vs. accumulated MGT Figure 7.4: MATLAB generated Weibull graph for rail failure data Figure 7.5: Framework for service contract cost model

19 CHAPTER 1 SCOPE AND OUTLINE OF THESIS 1.1 INTRODUCTION The reliability of products is becoming increasingly important because of the increasing cost of downtime, competition, and public demand. The reliability of a product/system is its characteristic expressed in terms of conditional probability that it has not failed up to a certain point and it will perform its required function under defined environmental and operational condition for a stated time of period. Blischke and Murthy (2000) stated reliability conveys the concepts of dependability, successful operation or performance which means absence of failure. A product or system is designed to sustain certain nominal stresses due to surrounding environmental and operating conditions. But failure or breakdown is evident over the life of a system. Failures or breakdowns may occur due to faulty design, bad workmanship, age, usage or the increase of operational and environmental stresses above the designed level. It is impossible to totally avoid all failures. The manufacturer/service provider can prevent or minimise the effects of such failure by ensuring after sales service through warranty and service contract. Warranty has been defined in different ways as illustrated by the following: Blischke and Murthy (2000) defined A warranty is a manufacturer s assurance to a buyer that a product or service is or shall be as represented. It may be a contractual agreement between a buyer and manufacturer (or seller) that is entered into upon sale of the product or service. 19

20 Thrope and Middendorf (1979) expressed A warranty is the representation of the characteristics of quality of product. The National Association of Consumer Agency Administration, USA (1980) advocates A warranty is an expression of the willingness of business to stand behind its products and services. As such it is a badge of business integrity - In today s context, a warranty is a contractual obligation of a manufacturer/dealer associated with the sale of a product which looks into the after sale service. Terms and conditions for service contracts are similar to warranty contracts, but the scope and coverage can vary and may be negotiated by the buyer and the service provider (Blischke and Murthy, 2000). Warranty limits the liabilities of both manufacturer and consumer in the event of premature failure. It protects the manufacturer by limiting the manufacturer s liability for the product failure and at the same time it acts as a promotional banner for product quality and reliability. To the consumer, it provides signals with information on the reliability and quality of the product and acts as an insurance against the early failure of the product. By offering a warranty, the manufacturer or dealer gives a guarantee or assurance for the satisfactory performance of the product for a certain period of time, called the warranty period or the warranty coverage period. In the case of product failure, the manufacturer/dealer repairs/replaces at no or a fraction of the rectification cost to buyers or refunds full or part of the sale price to the buyer as per the warranty terms. The legal obligation of the manufacturer/dealer to protect the buyers against the unsatisfactory performance has become a major focus in recent years. McGuire (1980) showed that servicing of warranty results in an additional cost to the manufacturer which is about 1 to 15 percent of net sale. However, it has promotional value for better terms or longer coverage period than the competitors, resulting in competitive advantage. It acts as a marketing tool to boost sales. To the consumer, longer terms of warranty mean peace of mind. As a result, the warranty period offered by the manufactures or dealers, in recent years has been progressively increasing with time (Murthy and Jack 2003). In early days of the last Century, the warranty period of a new product was three/six months which became two to three years at the end of the Century. Currently, Daewoo, 20

21 Hyundai ( introduced a five year warranty for the whole car and six or more years warranty for selected parts such as body frame. A large number of products are being sold with long-term warranty policies in the form of lifetime warranty policies and service contracts. Both lifetime warranty and service contracts protect the buyer/user through the redressed actions which includes free or cost sharing repair/replacement of product, or return of full or partial amount of money by the manufacturer in case of a future failure of the sold product during the longer coverage period. The contract specifies both the performance that is to be expected and the redress available to the buyer/user if a failure occurs. On the other hand, it protects the manufacturer against any misuse or intentional abuse of the product. The study of warranty is important to both manufacturers and the consumers. Analysis of warranty policies and costs models associated with short-term or fixed term policies have received a lot of attention. In contrast, lifetime warranty policies and longer term service contracts have not been studied well. There are complexities in developing cost models for these policies due to the uncertainties of coverage period, failures over longer terms, acquisition of quality data over longer term and uncertainty of costs over that period. The motivation for the research reported in this thesis is based on the need for the study of lifetime warranty policies and longer term service contracts, the modelling and analysis of failures, expected servicing costs for such policies, and the risks associated with such policies to the manufacturers and the buyers. 1.2 LIFETIME WARRANTY AND SERVICE CONTRACTS Due to rapid technological advancement and customer demands, industrial/commercial products and consumer durable goods are appearing on the market at an ever increasing pace. In addition to technological advancement continuous innovation and more complexities reduce the ability of the customer to evaluate product performance due to lack of knowledge, expertise or experience. As a result, the manufacturers are under pressure to extend the coverage period of their after sale services. Therefore, a large number of products are now being sold with long-term warranty policies in different formats known as extended warranties, warranties for used products, long term service contracts or lifetime warranty 21

22 policies. Lifetime warranties and long term service contracts are becoming popular as they provide an assurance to consumers for reliable service and greater customer peace of mind for the whole life of the product/asset. Although the lifetime warranty market has grown over the past five years, there is concern about the real value of those warranties. An enquiry commission established by the UK government reported that the lifetime warranty prices are usually much higher than the expected costs (Kumar and Chattopadhyay, 2004). As mentioned earlier, offering a lifetime warranty or a longer term service contract incurs additional costs to the manufacturers for servicing of claims under coverage period. These costs are then needed to be factored into the sales price of the product or after sale service. From the manufacturer or the service provider s point of view, management of lifetime warranty/longer term service contract is important, since it can drastically affect the profitability. In offering such policies, the manufacturer or service provider is faced with three major problems: 1) decisions on coverage period which depends on the type of product and the purpose of warranty, whether it is for protectional or promotional purpose, 2) prediction and estimation of expenses due to warranty/service claims which depend on the coverage period, product reliability and type of rectification actions (such as replacement with new or used, overhauling, major repair or minimal repair) or refund policies (full or part refund), and associated costs and 3) reduction of these costs by using better servicing strategies or by improving the product reliability during the design, manufacturing, and operating stages. Although the concepts of lifetime warranty and long term service contracts have attracted significant attention among the practitioners, research works on modelling policies, prediction of failures and estimation of expected costs for offering lifetime warranty and long term service contracts are very limited. Literature studies show a significant amount of research has so far been conducted on various issues relating to product failures, warranty, and reliability. There is only limited literature currently available on lifetime warranty policies and long term service contracts, the effect of various servicing strategies such as overhauling or major repair on the warranty cost, multi dimensional service contracts and manufacturer and buyer s risk preferences. 22

23 There are complexities in developing lifetime warranty policies and long term service contracts and cost models for these policies due to the uncertainties of measuring the useful life of product (technical, technological and commercial life of products). The complexity is further aggravated due to the uncertainties of usage and maintenance decisions during longer periods of contract, prediction of costs in longer terms, and availability of quality data for modelling and analysis. Therefore, there is a need to define lifetime, formulate new policies for lifetime warranty and long term service contracts and develop models for predicting failures and expected costs associated with these policies. These models should take into account the related issues such as uncertainty of lifetime, uncertainty of costs, servicing strategies involving both corrective maintenance to rectify failures during the lifetime or coverage period and planned or preventive maintenance particularly for service contracts, and risks associated with such policies to both manufacturer/service provider and buyers/user. 1.3 OBJECTIVES OF THIS RESEARCH The objectives of this research are to: Define lifetime or useful life of product based on the current practices; Develop new lifetime warranty policies and develop taxonomy for lifetime warranty policies Develop failure and cost models for different lifetime warranty policies by capturing the uncertainties in decision making over the longer period; Analyse failure data over the longer term to define failure mechanisms and costs of product failure under various operating, maintenance and environmental conditions; Formulate models for manufacturer and buyer s risk preference for lifetime warranty policies; Develop policies and cost models for service contracts considering both corrective maintenance and preventive maintenance. And analyse effects of various warranty servicing strategies and cost-benefit analysis of those alternative strategies; 23

24 Develop decision models for cost-benefit analysis of proposed policies for industrial application such as outsourcing maintenance. 1.4 THESIS OUTLINE The outline of this thesis is as follows: Chapter 1 defines the scope and outline of this research. It clearly defines the complexities of the problem and the need for study and modelling new policies, modelling for product failure, and estimation of expected costs of these lifetime warranty policies and long term service contracts and modelling manufacturer and buyer s risks. Chapter 2, provides an overview of warranties. It covers the role of warranty for stakeholders, existing warranty policies and taxonomy for both new and used products, and service contracts. System approach is used in subsequent chapters to model failures and costs of lifetime warranty policies and long-term service contracts as proposed in this research. Different issues involved in the study of long-term warranties and service contracts are also discussed. In Chapter 3, a definition of lifetime is developed based on the relevant literatures and current practices. In this chapter, taxonomy for lifetime warranty policies is formulated and a variety of one and two dimensional lifetime warranty policies are developed and discussed. In Chapter 4 mathematical models are developed for predicting failures and estimation of costs associated with proposed policies. Since warranty claims occur due to item failure, the starting point for such an analysis is the modelling of the failures over the lifetime. The modelling of failures over the lifetime is complex due to the uncertainties of the measures of lifetime of products/items and the uncertainties of the maintenance decisions taken over the long coverage period. Cost models for different lifetime warranty policies are developed at system level by considering coverage period as the random variable. In Chapter 5, manufacturer and buyer risk preference towards lifetime warranty policies are modelled. These models incorporate risk preferences in finding the optimal warranty price through the use of the manufacturer s utility function for manufacturer s cost and the buyer s utility function for repair cost. The sensitivity of 24

25 the models are also analysed in terms of manufacturer and buyer s repair costs, product failure intensity and buyer and manufacturer risk factors. In Chapter 6, three new service contract policies are developed; models are developed for expected costs for long term service contracts with various servicing strategies. Both the corrective maintenance actions (rectification on failure) and the planned preventive maintenance actions to prolong the reliability of the asset/equipment are considered. In Chapter 7, a case study is used for illustrating failures and estimation of costs for outsourcing rail maintenance using service contracts. Short and long term service contract policies are analysed and cost models are developed. The estimated costs can be considered for managerial decisions for outsourcing the maintenance of major infrastructures in asset intensive industries. Finally, Chapter 8 provides a summary of this thesis, the contribution of the thesis to this field of research, and a discussion of the potential extensions and future topics of research. 25

26 CHAPTER 2 LONG TERM WARRANTIES AN OVERVIEW 2.1. INTRODUCTION The role and concept of warranty, along with warranty related various issues, and a brief overview of studies and research studies on long-term warranty are being discussed in this chapter. The outline of this chapter is as follows: The concept of warranty and its role for various stakeholders are briefly discussed in Section 2.2. Product classification and the nature of warranty are defined in Section 2.3. Section 2.4 briefly describes the taxonomy of different warranty policies for new and used second hand products. This section also discuses different warranty policies available for both new and used products. In Section 2.5, studies on warranty and service contracts involving many diverse concepts from different disciplines are discussed. This section also discusses the system approach for the study of warranty. Section 2.6 describes the two main types of warranty costs. Section 2.7 briefly reviews the academic research-based literature on product warranty in general. A review of long-term warranty policies and related cost models are presented in Section 2.8. Section 2.9 summarises the chapter and it highlights the issues that will be applied in the later chapters. 26

27 2.2. CONCEPT AND ROLE OF WARRANTY The concept of warranty is not new. The history of warranty can be traced back to 1800 B.C. when Hammurabi, the king of ancient Babylon, defined the penalties for craftsmen found guilty of making faulty or defective products in one of his codes of trade law. According to this law, the craftsmanship was warranted with the thumb of the craftsman. Similarly, in the Egyptian civilisation, the builders were asked to stand underneath the newly built roof, when all the supports were taken away. Up to the middle of the nineteenth century caveat emptor ( let the buyer beware ) was the accepted norm. This was acceptable as long as the products were simple and the product mechanism and performance evaluation were easily understood by the users. Today, the situation has changed. Due to the growth of rapidly changing technologies, the global market, nearly identical products, and better educated and more demanding customers, products have become more complex and sold to a larger segment of the community. Products are now warranted in terms of age and usage, implying the assurance of quality and reliability of the products over the coverage period. At the end of the nineteenth century and at beginning of the twentieth century, laws began to make exceptions to the above mentioned brutal rules and refused to enforce unfair terms of any warranty. The United States Uniform Commercial Code included express warranties (UCC, 2-313) and implied warranties of merchantability (UCC, 2-315). A warranty offer involves a number of parties or stakeholders. The manufacturer produces a product and sells it to the customer either directly, or through a dealer with a product warranty (if the product is warranted). The warranty claims are handled either by the manufacturer/dealer himself or through a third party underwriting insurance company. The public policy maker plays statutory role to enact laws to see that the warranty terms are fair and effective to all parties involved. Figure 2.1 describes the interactions between these parties associated with the warranty offer 27

28 Manufacturer WARRANTY Public Policy Maker Dealer Insurance Provider Consumer Figure 2.1 : Stake Holders in Warranty policies Roles to Consumer/customer: When a consumer purchases a product the warranty document acts as a source of information about product characteristics and it also acts as an indicator of quality and reliability of the product in the context of complex or innovative products where the customer is unable to evaluate product performance due to lack of knowledge, expertise or experience (Akerlof, 1970) and Spence, 1977). It plays protective role to the customer by acting as an insurance against early failures of an item due to design, manufacturing or quality assurance problem during the warranty period. If the warranty is optional (such as an extended warranty), the consumer has to decide whether the warranty is worth the additional cost. For commercial products, the cost of repairs not covered by warranty can significantly impact profits due to downtimes. Roles to Manufacture/dealer: Warranties protect the manufacturer against any damage/failure to the product due to misuse (beyond the limit specified in warranty terms) or abuse by the user (mobile phone charging through a high voltage line, drying of wet cat in microwave oven). It plays the promotional role to signal the quality and standard of products to the potential customer. It also acts as a powerful advertising tool for a manufacturer/dealer to compete effectively in the market. If used properly as a marketing tool, warranties increase sales and generate extra revenue. Roles to Public policy maker: Governments and regulatory bodies intervene in the product market with the aim of making the market more competitive and fair to all 28

29 parties through fair warranty terms. These interventions act as instruments to protect consumers against defective items. Laws are enacted to see that warranty terms are not favouring manufacturers at the cost of consumers. Administrative machinery set up by statutory bodies takes appropriate measures to resolve conflicts and settle disputes if any manufacturer/dealer refuses a valid warranty claim. Third party under writing insurance company: Currently, most manufacturers or dealers use insurance companies to underwrite the costs incurred due to warranty service claims. Manufacturers/dealers pay preset premium to insurers. It is the insurer s responsibility to evaluate the validity of all claims and pay according to the terms of the warranty policy. 2.3 PRODUCT CLASSIFICATION The type of warranty policy depends on the type of products. Products can be either new or used (second hand). The new products can be further divided into three major groups. These are: Consumer durable products This type of products include a large number of items ranging from everyday household goods such as TV sets, fridge, electronics appliances to car and building materials and are purchased by a large number of buyers mostly for long-term use. Criteria to judge the performance of these goods can be simple or complex. Normally the warranty length and type is set by the manufacturer/dealers at the time of sale but warranty terms and conditions for such products are regulated by federal or state government bodies. Items of the same kind are sold with identical terms and conditions but may have a different coverage period and type of policy for different brands Industrial or commercial products There are a comparatively small number of customers and manufacturers for this type of products. Examples of these types of products include plants and machinery used for industrial or commercial purposes. Warranty terms are more complex and involve many performance measures such as reliability, maintainability, availability, 29

30 efficiency, etc. Usually the warranty terms are determined through negotiation between the buyer and seller Government or defence products Products such as war and defence goods including tanks, missiles, and ammunitions are common examples of this category of products. Government or defence agencies are normally the customers of these products and the customers have strong bargaining power to dictate the warranty terms and conditions. The number of manufacturers of such products is very small and these types of products often involve new technologies and one custom built. Warranty terms are complex due to multiple performance measures associated with these products. Used products are, in general, sold individually and can be consumer durable, industrial or commercial products. 2.4 WARRANTY TAXONOMY (WARRANTY CLASSIFICATION) Many different types of warranty policies for new and old (used) products have been studied and proposed. Blischke and Murthy (1992) proposed taxonomy for the warranty policies for the new products to integrate these policies (See Figure 2.2). Similarly, Murthy and Chattopadhyay (1999) proposed policies and taxonomy for used product warranty (See Figure 2.4). These warranty taxonomies are discussed as follows: Warranty taxonomy for new products Warranty policies for new products are classified based on whether or not they require development after sale of the product (Blischke and Murthy, 1992). Policies which do not involve product development can be further divided into two groups - Group A, consisting of policies for a single item, and Group B, policies for groups of items (lot, fleet or batch sales). Policies in Group A can be subdivided into two subgroups, based on whether the policy is renewing or non-renewing. A further classification may be "simple" or "combination" policies. The free replacement (FRW) and pro-rata (PRW) policies are simple policies. A combination policy is a simple policy combined with some additional features, or a policy which combines the terms of two or more simple policies. 30

31 Warranty policies Not involving product development C Involving product development A Single item B Group of items Simple Combination B1 B1 Non Renewing Renewing Simple Combination Simple Combination A1 A2 A3 A4 Figure 2.2: Taxonomy for new products warranty policies Each of these four groupings A1 - A4 can be further subdivided into two sub-groups based on whether the policy is one-dimensional or two- (or more) dimensional. Policies in Group B can be subdivided into B1 and B2 categories based on whether the policy is "simple" or "combination." B1 and B2 can be further subdivided based on whether the policy is one-dimensional or two-dimensional. Finally, policies involving product development are discussed in Group C. Warranties of this type are typically part of a maintenance contract and are used principally in commercial applications and government acquisition of large, complex 31

32 items. A number of important characteristics in addition to failure coverage within certain age and usage, for example, fuel efficiency are used in this category. A complete list of these warranty policies can be found in Blischke and Murthy (1992 and 1994). The following notation is used in defining some of these policies: W = Length of warranty period C = Sale price of the item (cost to buyer) X = Time to failure (lifetime) of an item Type A Policies: These policies are applicable to single items only and these policies can be one or two dimensional. Type A1 warranty policies. The two most common Type A1 policies are the following: Policy 1: Free Replacement Policy (FRW) The manufacturer agrees to repair or provide replacements for failed items free of charge up to time W from the time of the initial purchase. The warranty expires at time W after purchase. Policy 2: FRW-Rebate under this policy, if an item fails during the warranty period, W, a full refund of the original price is given to the customer. Policy 3: Pro-Rata Rebate Policy (PRW) The manufacturer agrees to refund a fraction of the purchase price should the item fail before time W from the time of the initial purchase. The buyer is not constrained to buy a replacement item. The function characterizing the fraction refunded can be either a linear or non-linear function of the age of the item at failure. Type A2 policies typically combine the features of two or more Type A1 policies as illustrated by the following. Policy 4: Combination FRW/PRW- The manufacturer agrees to provide a replacement or repair free of charge up to time W 1 from the initial purchase; any failure in the interval W 1 to W (where W 1 < W) results in a pro-rated refund. The warranty does not renew. 32

33 Type A3 policies are similar to Type A1 policies as illustrated by the following: Policy 5: Renewing FRW -Under this policy the manufacturer agrees to either repair or provide a replacement free of charge up to time W from the initial purchase. Whenever there is a replacement, the failed item is replaced by a new one with a new warranty whose terms are identical to those of the original warranty. [Note: Under this policy, the buyer is assured of one item that operates for a period W without a failure.] Policy 6: Pro-rated renewing- Under this policy, should a failure occur during the warranty period, the customer is supplied with a replacement at a reduced price with a new warranty identical to the original one. It is a conditional rebate warranty tied to replacement purchase. This policy is offered for non-repairable items such as batteries. Policy 7: Renewing combination policies (Combined FRW and PRW)- Under this policy if an item fails in the period [0,W 1 ), the manufacturer agrees to replace it free of cost to the customer. If it fails in the period [W 1, W), the item is replaced at a pro-rated cost and the replacement comes with a new warranty identical to the original one. Simple Two Dimensional Policies The two dimensional warranties policy involves two variables- time/age and usage, and the warranty is described in terms of a region in the two dimensional plane, either representing time or age and representing usage. Item failures are points on a plane with the horizontal axis representing warranty age (W) and the vertical axis representing warranty usage (U). Different shapes for the region characterise different policies and many different shapes have been proposed (see Blischke and Murthy 1994). Figure 2.3 shows the typical regions for different 2-D warranty policies. 33

34 Usage U U 0 W Age 0 W Age (a) (b) U 1 Usage U Usage U 1 0 W 1 W 2 Age 0 W Age ( c) (d) Usage U 1 PRW U 1 0 FRW W 1 W 2 Age (e) Figure 2.3: A sample for warranty regions for 2-D policies Policy 8: Free non renewing: under this policy if an item fails within the region shown in the Figure 2.3(a), the manufacturer rectifies the fault free of cost to the customer. for example, all Daewoo cars are now selling with six years or 100,000 km bumper to bumper warranty which comes first. 34

35 Policy 9: under this policy the manufacturer agrees to repair or provide a replacement for failed items free of charge up to minimum time W from the time of initial purchase and up to a minimum total usage U. The warranty region is given by two strips (please see Figure 2.3(b)). Policy 10: under this policy the manufacturer agrees to rectify all defects free of charge up to time W 1 from the time of the initial purchase, provided the total usage at failure is below U 2 and up to a time W 2 if the total usage does not exceed U 1. The warranty region is shown in the Figure 2.3(c). The low rate user is covered up to W 2 or U 1 depending on which occurs first and the high rate user is covered up to W 1 or U 2 whichever occurs first. Policy 11: under this policy the manufacturer agrees to rectify all defects free of charge up to a maximum time W from the time of the initial purchase and for a maximum usage U. The warranty region of this policy is characterised by a triangle as shown in Figure 2.3(d). Any failure with the age and the usage outside the triangle is not covered by warranty. If X is the time since purchase and Y is the total usage at failure, the item is covered under this warranty if [ Y ( U ) X ] < U +. W Policy 12: Pro-rata replacement policy- under this policy the manufacturer agrees to refund to the customer a fraction of the original purchase price should the item fail before time W from the time of the initial purchase and total usage at failure is bellow U. The warranty region of this policy is characterised by a rectangle [0,W] x [[0,U] ( please see Figure 2.3(a)). The functions for the refunds are as follows: i) R( t x) = max{ 0, 1 ( tx )} C s, and WU ii) R( t x) = max{ 0,1 ( t ), 1 ( x )} C s,, where C W U s is the sales price and t and x represent the age and usage respectively since initial purchase. Policy 13: Combined replacement policy- under this policy the manufacturer agrees to provide replacements for failed items free of charge up to a time W 1 from the time of initial purchase provided the total usage at failure is bellow U 1. Any failure with time greater than W 1 but less than W 2 is replaced at a pro-rated cost. The warranty region is shown in Figure 2.3(e) 35

36 Type B policies. Type B policies are called cumulative warranties (or fleet warranties) and are applicable only when items are sold as a single lot of n items and the warranty refers to the lot as a whole. Under a cumulative warranty, the lot of n items is warranted for a total time of nw, with no specific service time guarantee for any individual item. Cumulative warranties would quite clearly be appropriate only for commercial and governmental transactions since individual consumers rarely purchase items by lot. In fact, warranties of this type have been proposed in the U.S. for use in acquisition of military equipment. The rationale for such a policy is as follows. The advantage to the buyer is that multiple-item purchases can be dealt with as a unit rather than having to deal with each item individually under a separate warranty contract. The advantage to the seller is that fewer warranty claims may be expected because longer-lived items can offset early failures. Policy 14: Cumulative FRW A lot of n items is warranted for a total (aggregate) period of nw. The n items in the lot are used one at a time. If S n < nw, free replacement items are supplied, also one at a time, until the first instant when the total lifetimes of all failed items plus the service time of the item then in use is at least nw. Policy 15: Cumulative PRW: A lot of n items is purchased at cost nc and warranted for a total period nw. The n items may be used either individually or in batches. S n, the total service time, is calculated after failure of the last item in the lot. If S n < nw, the buyer is given a refund in the amount of C(n - S n /W), where C is the unit purchase price of the item. Policy 16: Cumulative combination policy- under this policy free replacements are provided if S n < nw 1 (where, W1< W). if nw 1 < S n < nw, the customer is given a rebate of (n-s n) C s. Type C policies. The basic idea of a Reliability Improvement Warranty (RIW) is to extend the notion of a basic consumer warranty (usually the FRW) to include guarantees on the reliability of the item and not just on its immediate or short-term performance. This is particularly appropriate in the purchase of complex, repairable equipment that is intended for relatively long use. The intention of reliability improvement warranties 36

37 is to negotiate warranty terms that will motivate a manufacturer to continue improvements in reliability after a product is delivered. Under RIW, the contractor's fee is based on his or her ability to meet the warranty reliability requirements. These often include a guaranteed MTBF (mean time between failures) as a part of the warranty contract. The following illustrates the concept: Policy 17: Reliability Improvement Warranty (RIW) Under this policy, the manufacturer agrees to repair or provide replacements free of charge for any failed parts or units until time W after purchase. In addition, the manufacturer guarantees the MTBF of the purchased equipment to be at least M. If the computed MTBF is less than M, the manufacturer will provide, at no cost to the buyer, (1) engineering analysis to determine the cause of failure to meet the guaranteed MTBF requirement, (2) Engineering Change Proposals, (3) modification of all existing units in accordance with approved engineering changes, and (4) consignment spares for buyer use until such time as it is shown that the MTBF is at least M Taxonomy for Second Hand Warranties Policies and Taxonomy (Murthy and Chattopadhyay, 1999, Chattopadhyay, 1999), for different types of one-dimensional warranty policies for second hand products are shown in Figure 2.4. The policies can be divided into two groups based on whether the policies offer a buy-back option or not. Policies without buy-back options imply that the seller has no obligation to take back an item sold. As a result, the warranty expires after the duration indicated in the warranty policy. Any failures within the warranty period are rectified according to the terms of the warranty policy. These policies can be further subdivided into two subgroups - Group A: Non-renewing policies and Group B: Renewing policies. Under a non-renewing warranty, the terms of the warranty do not change during the warranty period. As a result, if an item fails during the warranty period, it is rectified by the dealer and returned to the buyer without any changes to the original warranty terms. Under a renewing warranty, the warranty terms can change, for example, after failure, the item is returned with a new warranty either identical to, or different from, the original warranty terms. Each of these can be further subdivided into two 37

38 subgroups - Group A1 (B1): Simple policies and Group A2 (B2): Combination policies Warranty policies for second-hand products Non-buyback warranties Buy-back warranties C Non-renewing Warranties A Renewing Warranties B Simple C1 Combination C2 Simple A1 Combination A2 Simple B1 Combination B2 Figure 2.4: Taxonomy for warranty policies for Second-hand products Under the buy-back option, the buyer can receive a monetary refund (either full or a fraction of the sale price) by returning the purchased item any time within the warranty period and the warranty terminates when this occurs. Such policies can be categorised under Group C. All failures before the termination of the warranty are rectified according to the terms of the warranty. Type A [Non-renewing] Policies: Free Repair/replacement Warranty [FRW] policies. These are as follows: No Cost to Buyer: The policy is identical to Policy 1 for new products and the cost of each rectification under warranty is borne by the dealer. Cost Sharing Warranty [CSW] Policies: under the cost sharing warranty the buyer and the dealer share the repair cost. The basis for the sharing can vary as indicated below. Policy 9 - Specified Parts Excluded [SPE]: under this policy, the dealer rectifies failures of components belonging to the Set I at no cost to the buyer over the warranty period. The costs of rectifying failures of components belonging to the set 38

39 E are borne by the buyer. [Note: Rectification of failures belonging to the Set E can be carried out either by the dealer or by a third party]. Cost Limit Warranty [CLW] Policies: under the cost limit warranty policy, the dealer's obligations are determined by cost limits on either individual claims or total claims over the warranty period. Policy 10 - Limit on Individual Cost [LIC]: under this policy, all claims under warranty are rectified by the dealer. If the cost of a rectification is below the limit c I, then it is borne completely by the dealer and the buyer pays nothing. If the cost of a rectification exceeds c I, then the buyer pays the excess (cost of rectification - c I ). Policy 11 - Limit on Total Cost [LTC]: under this policy the dealer's obligation ceases when the total repair cost over the warranty period exceeds c T. As a result the warranty ceases at W or earlier if the total repair cost, at any time during the warranty period, exceeds c T. Pro-rated Refund Warranty [PRW] policies. The policy is identical to Policy 2 for new products. FRW-PRW Combination Policies. These are as follows: No Cost to Buyer: The policy is identical to Policy 3 for new products. Combination Policies with Limits [CLW] Two policies in this group are as follows: Policy 12 - Limits on individual and total cost [LITC]: Under this policy, the cost to the dealer has an upper limit (c I ) for each rectification and the warranty ceases when the total cost to the dealer exceeds c T or at time W, whichever occurs first. The difference in the actual cost of rectification and the cost borne by the dealer is paid by the buyer. [This combines the features of Policies 9 and 10] Policy 13 - [FRW-LIC]: Under this policy the dealer repairs all failures in the interval [0, W 1 ) at no cost to the buyer. The rectification of a failure in the interval [W 1, W) results in no cost to the buyer if it is below c I and an amount which is the excess if it exceeds c I. Type B [Renewing] Policies Policy 14 - Partial renewing free replacement warranty [PR-FRW]: Under this policy the dealer rectifies all failures in the interval [0, W 1 ) at no cost to the buyer. The warranty period after rectification is the remaining period of the original 39

40 warranty. If a failure occurs in the interval [W 1, W) then the item is rectified by the dealer at no cost to the buyer and returned with a new warranty of duration (W - W 1 ). [This implies that the warranty ceases only when there is no failure in the period [W 1, W) under the original warranty or there is no failure under the new warranty.] Type C [Buy-Back] Policies Simple Buy-Back Warranty [BBW] Policies. Under a buy-back warranty policy, the buyer has the option of returning an item within the warranty period with the dealer refunding either a fraction or the full sale price. The refund can also include payment for associated costs incurred by the buyer. This option can be either unconditional (money back guarantee) or conditional on certain events. For example, if the number of failures over the warranty period exceed some specified limit. Policy 15 - Money-back guarantee [MBG]: Under the money-back guarantee policy all failures over the warranty period [0,W) are rectified at no cost to the buyer. If the number of failures over [0,W) exceed a specified value k (k 1), then at the (k+1) failure, the buyer has the option of returning the item for a 100% refund and the warranty ceases when the buyer exercises this option. If the number of failures over [0,W) is either k or the buyer does not exercise the buy-back option when the (k+1) failure occurs, then the item is covered for all failures till W. [Note that this policy does not include any associated cost.] Combination Buy-Back Warranty [BBW] Policies. A combination buy-back policy is one where the buy-back option is combined with one or more of the non-buyback policies as indicated below: Policy 16 [MBG-FRW]: Under this policy all failures over warranty period [0,W) are rectified at no cost to the buyer. If the number of failures over [0, W 1 ), for W 1 <W, exceeds a specified value k (k 1), then at the (k+1) failure the buyer has the option of returning the item for a 100% refund and the warranty ceases when the buyer exercises this option. If the number of failures over [0, W 1 ) is either k or the buyer does not exercise the buy-back option when the (k+1) failure occurs, the item is covered for all failures until W. [Note: This policy can be viewed as a combination policies. When W 1 = W, the policy reduces to Policy 15.] 40

41 2.5 WARRANTY STUDY The study of warranty and service contract involves many diverse concepts from different disciplines (e.g. engineering, accounting, marketing, law etc). Because of this diversity, product warranty has received the attention of the researchers from many diverse disciplines. Warranties have been studied and analysed from many different perspectives and they deal with different issues as illustrated by the following list (Murthy and Djamaludin, 2002): 1. Historical: origin and use of the notion. 2. Legal: court action; dispute resolution, product liability. 3. Legislative: Magnusson-Moss act; federal trade commission, warranty requirements in government acquisition (particularly military). 4. Economic: market equilibrium, social welfare 5. Behavioural: buyer reaction, influence on purchase decision, perceived role of warranty, claims behaviour. 6. Consumerist: product information, consumer protection. 7. Engineering: design, manufacturing, quality control, testing. 8. Statistics: data acquisition and analysis, data-based reliability analysis. 9. Operations Research: cost modelling, optimization, scheduling. 10. Accounting: tracking of costs, time of accrual. 11. Marketing: assessment of consumer attitudes; assessment of the marketplace; use of warranty as a marketing tool, warranty and sales 12. Management: integration of many of the previous items, determination of warranty policy, warranty servicing decisions. 13. Society: public policy issues. Djamaludin et al (1996) provide a list of over 1500 papers on warranties classified into different categories. For detailed discussion of some of these issues, see Blischke and Murthy (1996). The study of warranty/service contract involves integrating these diverse concepts. The system approach provides a framework to study warranty related problems in an integrated and unified manner. Murthy et al (1990), Murthy and Blischke (1992a, (1992 b ) use this approach to study a variety of warranty related problems. The basic steps in the system approach are shown in Figure

42 Real world problem: Long-term warranty policies Problem definition (Step 1): warranty policies and cost model System characterisation (Step 2) Make change Model formulation (Step 3) Parameter estimation (Step 4) Analysis (Step 5) Not adequate Validation (Step 6) Adequate model Solution to problem Figure 2.5: System approach to problem solution Problem Definition: The problem definition is the statement which helps to understand the problem. It defines the objective/s for solving the problems and serves as a starting point to gather further information to understand the issues involved. In this research context, the problem is to predict failures by developing multi dimensional models. It will develop policies, estimate the expected cost for offering lifetime warranty by integrating different warranty service strategies such as perfect repair or overhauling in service contracts System Characterisation System characterisation is the identification of the significant parameters and variables, and their interrelationships. A system characterisation for warranty is 42

43 needed to be developed to enable to build models for carrying out a cost analysis from the perspectives of both manufacturer and consumer. This can be extended to the deregulated environments of service providers of complex infrastructure and users of those services. Manufacturer Consumer Warranty Product Characteristics Product Performance Usage Satisfactory No Warranty Costs Warranty costs Figure 2.6: Simplified system approach for warranty cost analysis A simplified system characterisation for warranty cost analysis is presented in Figure 2.6. Characterisation of each of the elements depends on a number of variables. Manufacturer produces product/good with specified product characteristics determined by design and manufacturing decisions during the manufacturing process. The product is then sold to the customer with warranty and the warranty terms depend on the product characteristics and marketing factors such as competition. Customers/buyers purchase the product and the decision to purchase is influenced by several factors such as quality brands and provisions for after sales service such as warranty. Product performance is determined by the interaction between product characteristics and the usage of product (determined by the consumer). If the customer is not satisfied with the product performance during the coverage period a claim under warranty results and the magnitude of the warranty cost to the manufacturer for servicing the claim depends on the warranty terms and condition. 43

44 2.5.3 Model Formulation: Model formulation is the linking of a system characterisation to an appropriate mathematical formula. Modelling of product failure is the first step to model the warranty costs since the warranty cost of a product is dependent on failure distribution of that product. These costs, over the warranty period are then needed to be modelled. Modelling Product Failure and Warranty Costs The warranty costs are different for buyer and manufacturer. These costs are random variables, since claims under warranty and the cost to rectify each claim are uncertain. Warranty claims occur due to item failures (real or perceived). An item is said to have failed when it is unable to perform its function in a satisfactory manner. In modelling failure, two approaches can be used. These are: Black box (empirical) and White box (physically based). In the Black box approach, an item is characterised in states of working or failed and it models time to failure directly as a random variable with a distribution function based on the modeller s intuitive judgement or on historical data. In the White box approach, one can model the number of failures over time by a suitable counting process and this physically based model involves a more detailed system characterisation of the physics of the failure (Blischke and Murthy (1994)). In this project, we confine our attention to the Black box modelling. Modelling rectification action The evaluation of the warranty cost or any other parameter of interest in modelling warranties depends on the failure mode, rectification strategies, and the assigned preventative warranty maintenance actions for the items. The rectification can be classified according to the depth of restoration that is the degree to which they restore the ability of the item to function. The post-failure rectification affects products in one of the following ways (Chattopadhyay, 2002): (i) Replacement: this type of action is taken either in the case of complete failure of an item/system or in the case of a non-repairable item. 44

45 (ii) Overhauling or perfect repair is a restorative maintenance action that is taken before an item has reached to a defined failed state that enables the system to be as good as new condition (Jardine 1973). This means the failure rate of the system is restored to zero. (iii) Imperfect repair restores a substantial portion and the hazard rate falls in between as good as new and as bad as old (Coetzee 1997). (iv) A minimal repair makes insignificant improvements and the condition after maintenance is as bad as old. Most real life systems/items are complex in the sense that these items are viewed as systems comprising of several components and failure of one or more components may result in the complete failure of a system. Therefore rectification actions over the warranty period can be modelled either at the system (item) level considering failure of all the components or at the component level considering failure distribution of only one component at one time (see Figure 2.7). Rectification Action System level Component level Minimal repair Non-repairable Repairable Major repair Replacement Replacement with New one Replacement with used second hand one Perfect repair /Major repair Imperfect repair Replacement Figure 2.7: Framework for warranty rectification At system level, the rectifications can be viewed as minimal repairs because repair or replacement of one or more components do not change the failure rate of the system if the other parts of the system remains unchanged (Barlow, 1960). Components can be repairable and non-repairable. In case of component level rectification, the 45

46 manufacturer has the option of replacing the failed component with a new or used one or repairing the components by means of overhauling, imperfect or minimal repair. While for non-repairable component, the only option for the manufacturer is to replace the failed component with a new or used one component Parameter Estimation Some of the methods of parameter estimation are: method of Least square, method of Moments, and method of Maximum likelihood. However, non-parametric analysis may needed to be applied if the data requires such an approach based on the validation using industry data. Suzuki (1985) proposed parametric and non parametric methods of estimating lifetime distribution from field failure data with supplementary information about censoring times obtained from following up a portion of the products that survived warranty time. The method of Maximum Likelihood (MLE) is one of the most popular methods of parameter estimation. The MLE are the values, which maximise the log likelihood function. These are given by the solution to the equations obtained from the first order necessary condition. The non-linear equations need to be solved using numerical method or using some mathematical software Mathematical Analysis and Optimisation The next step after the development of a mathematical model is to obtain the behaviour of the model with the exposure of the real life situation. This can be carried out by the analysis of the formulated model using appropriate mathematical techniques. The analysis of mathematical formulation can either be qualitative or quantitative. The qualitative approach deals with the study of qualitative aspects of the models without solving it. On the other hand, quantitative analysis is concerned with finding the explicit solution which satisfies the given mathematical solution. This approach can be divided into two methods: Both analytical and computational method can be used to analyse developed models. a) Analytical method: - Here, the solution of a mathematical model is obtained in terms of an analytical expression involving the parameters and the variables related to this model. These methods of analysis yield the parameterised solutions which are 46

47 the analytical expressions of parameters. These types of closed form solution are possible only for simple models. b) Computational method: - For complex models, computational techniques (numerical method or simulation) are needed. Murthy et al (1990) discussed the modelling and analysis of new products. In this method the model solutions are obtained by the application of computer software such as MATLAB, FOTRAN and the solutions are approximate and depend on the type of software used. These methods yield solutions for specific numerical values of the parameters. The solutions can be recomputed every time the parameter values are altered Model Validation Model validation is used to establish whether or not the model is a good representation of the real world and can give adequate and meaningful solution to the problem under consideration. It involves hypothesis testing. If the model is not adequate, changes must be made to the system characterisation or mathematical formulation. In carrying out a model validation, it is essential to assign specific values to model parameters. If the models are not adequate, changes will be made in the model based on the correlation with industry data. The process continues until an adequate model is found. Assessment of predictive power of a model is basically a statistical problem. The idea is to use an estimated model to predict outcomes of other observations and to evaluate the closeness of the prediction to observed value. A key requirement for credibility is that the data used for validation not be part of the data base used in parameter estimation. There are many approaches to solve this problem such as Coefficient of determination R-square, Goodness of fit approach, Mean square error (MSE), Chi-Square test of various form and many more. The Coefficient of determination R-square: the correlation coefficient r is a measure of strength of the relationship between two variables; its square is the measure of the reduction in variation in a regression relationship. With more than two variables, usually a set of predictors and a response, R 2 measures the strength of the relationship between the response and group of predictors. Two advantages of the correlation coefficient are (1) it is an index number and (2) there are some relatively easy tests for these interference problems. To validate a model we will look for evidence of a strong 47

48 relationship. A more appropriate test is one of the null hypotheses H 0 : ρ ρ 0, where ρ 0 is some minimal value that will be accepted as evidence of model validation. 2.6 WARRANTY COSTS Offering a warranty incurs additional costs to the manufacturer or dealer for servicing the warranty claims, even if, any claim is found to be invalid. Warranty costs are made up of one or more of the elements such as administration and handling costs, the cost of replacement of an item, the cost of repair, labour and parts, compensatory costs (for example, a replacement for the duration of the repair). The additional cost due to warranty servicing is important to the manufacturer as it directly influences the selling price and profit. From the manufacturer s point of view, a warranty offer will be worthwhile if the profit considering the total cost, including the additional cost due to warranty is improved through an increase in sales and reduction in warranty servicing costs. The following costs are important for the context of warranty: The following costs are generally modelled and analysed for warranty policy decisions. Warranty costs per unit sale: whenever a warranty claim occurs, it incurs additional costs to the manufacturer/dealer. Warranty servicing cost is a random variable since claims under warranty and rectification of each claim are uncertain. Warranty cost per unit sale can be estimated from the total cost of warranty and the number of units sold. The total cost includes repair or replacement costs, and/ or downtime cost, and/or the product improvement cost along with administrative costs. Life cycle costs (LCC): this cost is important to both buyer and manufacturer for complex and expensive products and is dependent on the life cycle of the product. Life cycle starts with the launch of a product onto the market and ends when the manufacturer stops producing the product or when it is withdrawn from the market due to the launch of a new product. This cost over the product life cycle is a random variable. (For details see Blischke and Murthy, 1994) 48

49 2.7 REVIEW OF BASIC WARRANTY MODELS Modelling Warranty Cost Warranty cost modelling and analysis has received significant attention in the literature. Early models dealt with the problem of estimating the expected warranty costs for products sold under different policies. Other than modelling costs, many different aspects of warranties have been studied by researchers from diverse disciplines such as Marketing, Operation research, Law, Economics, accounting and others. A literature review on the warranties for new products follows: Warranty costs per unit sale: Whenever a valid warranty claim is made, the manufacturer/dealer incurs additional costs that are randomly variable since claims under warranty and to rectify each claim incur uncertain costs. Per unit warranty cost may be calculated as the total cost of warranty divided by the number of units sold. The total cost includes repair or replacement costs, and/or downtime cost, and the product improvement cost. This per unit sale warranty cost is important in the context of pricing the product. Predicting failure, warranty cost modelling and reliability analysis for one dimensional (considering only age or usage) warranty has received a lot of attention in the literature. The modelling for warranty cost analysis from an engineering point of view has been extensively covered in Blischke and Murthy (2000). Lowerre (1968) and Menke (1962) developed the earliest probabilistic warranty cost models for rebate policy. They considered free replacement policy and an exponential failure distribution. Amato and Anderson (1976) extend Menke s model to allow for discounting and price adjustment. They extended Menke s model to estimate the warranty cost for products with general failure time distribution. Karmarkar (1978) and Balachandran et al (1981) proposed models for warranty cost of a failure free warranty within a fixed period. Warranty cost models for some other policies had been developed by Heschel (1971) and Thomas (1981). They studied the expected costs and profit per unit time for an infinite life cycle for various warranty policies. The weakness of these models is that they ignored the effect of replacement over the product life cycle. To overcome this problem, Blischke and Scheuer (1975) had approached the problem of calculating the costs over the product life cycle. They estimated the failure distribution function from incomplete data using Kaplan and Meier s (1958) technique to analyse FRW and PRW policies. They looked into the 49

50 problem both from the manufacturer and consumers perspective by comparing the product life cycle costs and the profits when items are sold with or without warranty. Blischke and Murthy (1992) analysed warranty costs from both manufacturer and consumer s point of view for Exponential, Weibull, gamma, lognormal and mixed exponential distribution function. Wu et al (2006a) proposed a cost model to determine the optimal burn-in time and warranty length for non-repairable products under the fully renewing combination free replacement and pro-rata warranty (FRW/PRW) policy. They provided numerical examples considering failure time of the product follows either the mixed exponential distribution or the mixed Weibull distribution. They concluded with the comment that the fully renewing combination FRW/PRW is always better than the fully renewing policy in terms of cost. For products sold with the free renewal warranty policy, Wu et al (2006b) proposed a decision model for manufacturers to determine the optimal price, warranty length and production rate to maximise profit based on the pre-determined life cycle. A number of researchers paid attention to two dimensional (considering both age and usage of the product) models for predicting failures and estimating warranty cost. Moskowitz et al (1988) analysed the models for two dimensional warranty cost by using a Baysean technique for non-renewing 2D FRW policy. Wilson and Murthy (1991) discussed the concept of two-dimensional warranties involving age and usage of the product. In a two dimensional warranty, failures are described in terms of a region in the two dimensional points on a plane with horizontal axis representing time and vertical axis representing usage. Manna et al (2006) proposed a methodology to determine the optimal 2D warranty region where customers utility is measured by the length of warranty coverage time. Singpurwalla and Wilson (1993) proposed models for two dimensional failures. They modelled the time to failure by univariate distribution while total usage was kept on condition. The total usage as a function of age was modelled by another univariate distribution. Iskander (1993) used Moskovitz and Chun s two-dimensional approach to analyse warranty costs for two dimensional policies. Murthy et al (1995) modelled failure as a two dimensional point process. 50

51 2.7.2 Warranty Engineering The additional cost due to warranty is important to the manufacturer as it directly affects the selling price. The expected number of failures and warranty cost can be reduced by improving the product reliability through decisions made in the design and development stage and also through improved quality control during the manufacturing stage. The warranty servicing cost can also be minimised by taking cost effective maintenance strategies for the claims during the warranty periods. Murthy and Nguyen (1985a) modelled a case of two component products with failure interaction whereby a component induces the failure of the other component. Murthy and Nguyen (1985b) extended this model to n-components. Reliability improvement through overhaul or imperfect maintenance has received some attention in the reliability literature. Jack and Dagpunar (1994) studied the optimal imperfect maintenance over the warranty period. Dagpunar and Jack (1994) also developed a preventive maintenance strategy over the warranty period. Murthy and Jack (2003) studied and analysed the warranty servicing cost. They showed that this can be minimised through optimal corrective maintenance decisions. Perfect or imperfect preventive maintenance can be useful to reduce the likelihood of a failure or to prolong the life of the item. This type of maintenance action is worthwhile for the manufacturer if the warranty costs exceed the additional cost of PM. Chukova et al (2004) presented an approach to analyse warranty cost when imperfect repair was taken as warranty servicing for post failure claim. They showed that the evaluation of warranty cost depends on the failure distribution of the item, type of repair process in servicing warranty claims. However, they could not model the effect of imperfect repair on the warranty cost. Iskander et al (2005) proposed a new repair replacement strategy for product sold with two dimensional warranties where they suggested the replacement of product when it fails for the first time in a specified region of the warranty or otherwise a minimal repair policy to minimise the total expected warranty servicing cost. Chukova and Johnstone (2006) proposed a warranty repair strategy, related to the degree of the warranty repair, for non-renewing, twodimensional, free of charge to the consumer warranty policy. They subdivided a rectangular 2-D warranty region into three disjoint subregions, so that each of these subregions has a pre-assigned degree of repair for a faulty item. They also provided a comparison between their work and the other researcher s strategies. 51

52 Adopting appropriate warranty servicing strategies during the warranty period For non repairable items/components replacements can either be new, cloned or used. For repairable items/components warranty servicing can be by replacement or repair. In the case where corrective maintenance actions are used, two possible strategies are: a. Repair versus replacement: In case of a claim, the manufacturer has the option of either replacing the failed item with a new or used one or repairing the item by means of perfect (overhauling), imperfect or minimal repair. A detailed discussion of this strategy is made in section b. Repair cost limit strategy: Here, the decision on repair or replacement strategy is based on the limit of the claim. Most of the researchers while modelling optimal servicing strategies, divide warranty period into two intervals (Murthy and Jack,.2003). A failed item is always replaced by a new item that occurs any time within the first interval while the item is repaired during the next interval of the warranty period. Nguyen and Murthy (1989) extended this by adding a third interval where a new warranty is given to a repaired or replaced item in case of a failure. Murthy and Nguyen (1988) proposed that the failed item is first inspected and assessed for rectification cost. If this cost exceeds a certain limit it is replaced, otherwise repair is carried out at no cost to the buyer. Preventive Maintenance (PM) action throughout the warranty period is another strategy used by the manufacturer to reduce the probability of failure. But the application of PM strategy during the warranty period is worthwhile only if the reduction in the warranty cost exceeds the additional cost of PM. Pascuala and Ortegab (2006) proposed a cost optimization model for a product sold with warranty to determine the optimal levels of preventive maintenance which takes into account the overhauling, repairs and replacement. The above studies related to warranty and service contract, and an extensive literature review shows that huge numbers of research have so far been conducted on various issues related to product failures, reliability, warranty policies and warranty cost analysis for both new and used products. Although the coverage period and after sale service contract offered by the manufacturer/dealer has been progressively increasing with time, there is limited literature so far available on long-term warranty policies and service contracts, complexities in modelling cost and decision on service strategies for such policies and the effect of various servicing strategies such as 52

53 overhauling or major repair on the cost of one and multi dimensional policies and decisions linked to manufacturer and consumer risk preferences. The following section reviews the available literature on the long-term warranty policies and the cost models associated with those policies. 2.8 LONGTERM WARRANTY POLICIES AND COST MODELS AN OVERVIEW In the previous sections of this Chapter, we discussed various warranty aspects in general and some of the important generic warranty cost models. This section reviews different long-term warranty policies and mathematical models for estimation of warranty servicing costs. Due to the fierce competition in the market and customer demands, manufacturers in recent years have started to extend the coverage period of warranty policies. As a result, the manufacturers have started offering long-term warranty policies for their products in the form of extended warranty, warranty for used products, lifetime warranties and service contracts. Rahman and Chattopadhyay (2004) developed a framework for long-term warranty policies (Figure 2.8). All these policies and the associated cost models are overviewed and discussed in the following section. Figure 2.8: A framework for long-term warranty 53

54 Extended Warranty An extended warranty is the extension of the base warranty which is an obligation or responsibility assumed by the manufacturer or dealer for further service to buyers beyond the base warranty period for a certain premium. The base warranty is an integral part of a product sale and its cost is factored into the sales price (Murthy and Jack 2003). Consumers who prefer extra protection purchase additional coverage in the form of extended warranty or the extended service contract. It has been observed that on average 27% of new car buyers purchase extended warranty (Kumar and Chattopdhyay, 2004). Extended warranty generates a significant amount of revenue for manufacturers. Sears Roebuck reported a revenue generation of nearly US$ 1 billion from the sale of extended warranties. The annual sale of extended warranties in the UK alone is more than US$1 billion. These are sold by the manufacturer, dealers or by third parties. An independent insurer/ under-writer can change the manufacturer s warranty and pricing policies which can have an impact on manufacturers profit and consumer s purchasing intention (Padmanabhan and Rao, 1993). Extended warranty has attracted significant attention among practitioners. However, the academic research on extended warranty is limited. Padmanabhan (1995) analysed the impact of usage heterogeneity and extended service contracts. He modelled the buyers and manufacturers preference attitudes to risk using utility function. He argues that variations in consumer demands depend on their usage habit. The heavy users of the product prefer extended warranty more than the light users. So it would be less risky for the service providers to exclude the heavy users if the number of heavy users was small compared to the number of light users. Lutz and Padmanabhan (1998) analysed the effect of extended warranties on a manufacturer s warranty policy under conditions of manufacturer s moral hazard. They proposed that the consumer s income influences the demand for the extended warranty. The high income earners prefer to buy this type of warranty as they have lower marginal utility for wealth. They stated the effect of competition from extended warranties in a market made up of consumers who differed in the value they receive from a working unit. They advocated that, because of two types of consumers (such as low and high valuation in the market), the manufacturer can attempt to sell a different version of the products. It results in offering a product at price p L with warranty w L, and quality 54

55 q L to low valuation consumers, and price p H, with warranty w H and quality q H to high valuation consumers. Since the manufacturer cannot identify any individual consumer s type, he will have to choose a menu of contracts ( p L, w L ) and ( p H, w H ) to satisfy constraints imposed by the consumers behaviour that maximizes utility. The manufacturer s maximised profit can then be given by P max L L L L H H H H ( p ( 1 q ) w c( q ) + ( 1 λ) ( p ( q ) w c( q ) = λ 1 (2.1) Subject to the constraints L L L H H H ( L, q, p, w ) max{ 0, E( L, p, q w )} E, and H H H L L L ( H, q, p, w ) max{ 0, E( H, p, q w )} E, Hollis (1999) carried out an economic study of the extended warranty market with a focus on duration of extended warranty. Mitra and Patanakar (1997) proposed a cost model where the product is sold with a rebate policy and the buyer has the option to extend the warranty till the product failure has not occurred during the initial warranty period. They proposed following two warranty policies: 1) An initial warranty period is w 1, and an additional warranty is offered up to time w (from the date of sale) on items that did not fail during the initial warranty period. The rebate function is given by: r () t c c = 0 t w w 1 1 t w (2.2) Where, c and c * are the price of the product and amount of lump-sum rebate offered during the second period. and 2) Lump-sum rebate for the initial warranty is c, the price of the product and for the renewable part, the rebate is pro-rated starting from a value c*, which is less than c and reduces to zero at time w. The rebate function is given by r () t c, t w1 c 1 w w 2 =, 1 0 t w w 1 1 t w (2.3) Rinsaka and Sandoh (2001) studied the extension of the contract period. Yeh and Peggo (2001) developed extended warranty policies and analysed from consumers and manufacturers perspective. They proposed optimal cost models for extended 55

56 warranty by considering two cost criteria namely; total expected discounted cost and long-run average cost per unit time. These costs are given respectively by Total expected discounted cost to a consumer ( k) ( 1 e ) ( δw ) ( kl) cn e 1 = cr + ( δl ) ( δ ( W kl) ) T cn + 1 e 1 e (2.4) And total expected discounted manufacturer s cost ( c c k) ( W + kl) ( ) δkl ( ) ( )( 1 e ) ( W kl) ( 1 e )( 1 e ) ( δl ) δw cr + cm cr 1 e + cn e = ( W + kl) δ ( δl ) δ T mn p N + 1 e ( (2.5) Where, c R, c N, c p c m, and c r are the cost of initial purchase price, renewal cost to the consumer, repair cost, manufacturing cost and manufacturer s repair cost of each rectification respectively. W and L are the normal and extended warranty period, k( =1, 2,.. ) is the number of renewal or repair up to L. δ represents the discount rate over time. Average long run cost for consumer A cn (k), and manufacturer A mn (c p,c N,k), for k renewals are given by and A A cn mn ( k) cr + kcn = (2.6) W + kl ( c c, k) p ( M ( W + kl) ) ( c c ) c p R r kcn, N = (2.7) W + kl Where, M(W+kL) is the renewal function over the period W+kL and c p is the repair cost when warranty is not renewed. Yeh and Peggo (2001) found a critical point at which consumer is indifferent to the policies and derived optimal extended warranty policy for the manufacturer around that point. Kumar and Chattopadhyay (2004) developed mathematical models to estimate the minimum duration of extended warranty that would benefit the customers opting for extended warranty. They modelled the associated costs for manufacturer for providing extended warranty. Under minimal repair strategy, the cumulative hazard function, H (t), is given by. 56

57 H t () t = h( x)dx 0 For Weibull distribution, it is given by () η H t = T Minimum duration of extended warranty, t ew, a customer should purchase under the minimal repair policy was given by β t ew = β β β ( T + ) T η 1 (2.8) where η is the scale parameter and β is the shape parameter of the Weibull distribution. A customer needs to make a decision whether to buy an extended warranty and a manufacturer needs to decide the price of offering it to their customer. The expected total cost to manufacturer for extended warranty is given by: C FRW ( T + tew) = Cs[1 + M ( T + tew)] (2.9) Where C s is the cost of the product and M(t) is the renewal function for the expected number of failures and is given by: M ( T + t ew ) = F( T + t ew ) + T + t ew M ( T + t 0 ew x) df( x) (2.10) F(t) is the cumulative failure distribution function. The cost of providing extended warranty, C ew (t ew ), for t ew is given by: C ew ( t ) = C [ M ( T + t ) M ( T )] ew s ew (2.11) For third party (underwriting insurer) under minimal repair, the cost of providing extended warranty for t ew is given by: C ew ( t ) = C [ H ( T + t ) H ( T )] ew r ew (2.12) where C r is the mean repair cost. For Weibull distribution the cost can be given by: 57

58 β β ( T + t T ) C C t r ew( ew) = β ew) η (2.13) Warranty for used product Warranty for used product is offered with the sale of second hand products when sold by the dealer (for details see Chattopadhyay and Murthy (2000, 2001, 2004). The market for second hand products as a fraction of the total market (new + secondhand) has been continuously increasing as new products are appearing at a faster rate in the market and the expected life of products is increasing due to rapid advances in technology. In the US car market, the trade for second hand cars was 40% of that for new cars in numbers and 22% in terms of dollar sales (Genesove( 1993)). Buyers are demanding warranties for items purchased from the dealers of second-hand or used products. In response to consumer demand, public policy makers have begun enacting laws requiring dealers to offer warranties for second-hand products and service warranty claims. Dealers of used products have now started using warranties to promote sales. Decisions related to second-hand products are more complex compared to new products due to the fact that each second-hand product is statistically different from other similar products due to variation of age, usage and previous maintenance history. Therefore, dealers of second-hand products need to formulate policies and conduct a proper cost analysis of warranty policies applicable to second-hand products. This is necessary to avoid making a loss instead of profit. Chattopadhyay and Murthy (1996) is the earliest paper to develop a cost model of a warranty policy for second-hand products. Davis (1952) provides an opportunity to improve the reliability of used items through reconditioning/ overhaul. He modelled reliability of reconditioned bus engines actions to analyse the effect on failure rate. Reconditioning/ overhaul allows dealers to offer better warranty terms and sell an item at a higher price. However, reconditioning/ overhaul is worthwhile if the expected saving in the warranty servicing cost or the increased profit due to increased sale price exceeds the costs associated with the improvement. Bhat (1969) introduced the application of useditem in rectification. Nakagawa (1979) developed optimum replacement policies for used units. Malik (1979) introduced a "degree of improvement" in the failure rate and called it the "improvement factor. Kijima et al (1988) and Kijima et al (1989) 58

59 modelled virtual age where the improvement results in a reduction in the age of the system from. Wogalter et al. (1998) studied the availability of operator manuals for used (secondhand or resold) consumer products (e.g. car, computer, lawnmower, bicycle, etc.). Chattopadhyay and Murthy (2000) proposed stochastic models to estimate the expected warranty cost for second-hand products. They discussed two approaches: Approach l: the system is viewed as a black-box and the claims are modelled as point process with an intensity function Λ(t) where t represents the age of the system. The function Λ(t) is an in-creasing function of t indicating that the number of claims (in a statistical sense) increases with age. This type of characterisation is appropriate when system failures occur due to a component failures and the system can be made operational through repair or replacement of the failed component using minimal repair strategy. For this approach the expected warranty cost (with free replacement warranty) is given by C(W; A) when the item is of age, A, at sale. A is a random variable assuming values in the range of age [L, U] and characterised by a distribution function H(a), with H(L) = 0 and H(U) = 1. h(a) denotes the density function associated with H(a). C [ r. (2.14) L A U A+ w = ( w) E C( w; A) ] = C Λ ( t) dt h( a)da C r denotes the expected cost of each rectification action. Using C(w) instead of C(w; A) in the pricing decision implies that the dealer is charging the same amount for warranty for items within defined age range. They provided a linear refund function for the pro-rata warranty (PRW) policy. The item fails after a period x subsequent to the sale. Then, the refund function, R(x), is given by R ( x) C = s x 1 w 0 for 0 x w for x > w (2.15) And the expected warranty cost for PRW policy is given by 59

60 C W W 1 = s 0 w 0 ( w) R( x) df ( x) = C F ( w) xdf ( x) (2.16) where, C s denotes the sale price. Approach 2: the claims were characterised through failures at component level. The total expected warranty cost for the system is given by the sum of the costs for the different components where the component failures are independent. First failure: If the age A i of the component is known and is given by A i = a i, then the time to first failure, X i1, is given by the distribution function F i1 (x). It is given by F ( x) F ( ai + x) F( ai ) 1 F ( a ) i i1 = (2.17) i i The expected number of failures over the warranty period is modelled as a modified renewal process with first failure given by F i1 (x) and subsequent failures by F i (x). Then the expected warranty cost, C i (w; a i ), is given by: C W ( w a ) = c F ( w) + M ( w x) df ( x) ; i i i1 i i (2.18) 0 i i 1 where M i (.) is the renewal function associated with F i (.), c i is the cost of each failure rectification. Chattopadhyay and Murthy (2001) developed and analysed stochastic models for three new cost sharing warranty policies for second-hand products. Specific parts exclusion [SPE]:- The components of the product are grouped into two disjoint sets, Set I (for inclusion) and Set E (for exclusion). Under this policy, the dealer rectifies failed components belonging to set I at no cost to the buyer over the warranty period. The costs of rectifying failed components belonging to Set E are borne by the buyer. They modelled the failures during warranty using two different approaches. Approach 1: Here the failures were modelled by a point process with common intensity function Λ(t). Here, percentage of warranty included parts and excluded parts are given by p and q =1-p respectively The expected number of failures for an item of age A at sale covered under warranty for a period W was given by 60

61 A+w E [ N ( w; A)] = (1 p) Λ( t) dt (2.19) I A Similarly, the expected number of failures not covered under warranty was given by A+w E [ N ( w; A)] = p Λ( t) dt (2.20) E A Approach 2: Here component failures belonging to set I were modelled by an intensity function Λ I (t) and for those belonging to set E were modelled by another intensity function Λ E (t). Both of these were increasing functions of time t. Then the expected number of failures N I (W;A) for the included parts and N E (W;A) for the excluded parts over the warranty period were distributed according to non-stationary Poisson process with intensity functions Λ I (t) and Λ E (t) respectively and therefore, the expected costs of warranty are given by the Equations (2.21) and (2.22) respectively. and A+w E [ C ( w; A)] = c Λ ( t) dt (2.21) I A I A+w E [ C ( w; A)] = c Λ ( t) dt (2.22) E A E The cost of each rectification resulting from failures due to components belonging to set E and set I were modelled by distribution functions G E (c) and G I (c) with mean c E and c I respectively. Expressions for the mean values are given by = 0 c cg( c) dc using the appropriate distribution function. g(c), the density function derived from distribution function G(c). Limit on individual cost [LIC]:- if the cost of a rectification is below a limit c i, it is borne completely by the dealer whereas if the cost of a rectification exceeds c i, then the buyer pays the rest. Here, the total expected warranty cost to the dealer, E[C d (w;a)], is given by 61

62 A+w E [ C ( w; A)] = c Λ( t) dt (2.23) d d A and the total expected cost to the buyer, E[C b (w;a)], is given by A+w E [ C ( w; A)] = c Λ( t) dt (2.24) b b A Limit on individual and total cost [LITC]:- the total cost to the dealer over the lifetime has an upper limit (c T ) for each rectification and the warranty ceases when the total cost to the dealer (subsequent to the sale) exceeds c T or at time W, whichever occurs first. The buyer pays the difference between rectification cost and the dealer s cost. Then expected total cost to the dealer is given by ct E[ C ( w; A)] = zv ( z) dz + c V ( c ) (2.25) d 0 d T d T where, v d (z; t) is the density function associated with V d (z; t) (the distribution function for the total cost to the dealer by time t and is given by Vd ( z; t) Gd ( z)[pr ob{ N( t) = = r= 0 r r}] (2.26) where, G r d(z) is the r fold convolution of distribution function of dealers cost of rectifications using G d (z). The expected cost to the buyer until warranty expires when Z t (total cost to the dealer by time t) first time crosses c T and is given by E [ Cb ( w; A)] = E[ N( tb )] ( c ci ) g( c) dc (2.27) c I and expected number of failures over [0,T b ) is given by w A+ t E[ N( t )] = [ Λ( x) dx] q( t; c ) dt (2.28) b 0 A T where q(t; c T ) is the density function of T b 62

63 The authors suggested simulation approach for solution because the model is analytically complex. Chattopadhyay and Murthy (2004) proposed optimal decision models on reliability improvement through upgrading actions by dealers before sale of second-hand products sold with warranty. Two problems are studied: Problem 1: The sale price and warranty terms are specified and do not depend on the reliability improvement. The upgrade (improvement) is used to reduce the expected warranty servicing cost. For this instance, the expected profit is given by ψ ξ β β β J ( w, τ ; A) = S( w; A) cτ A cλ {( A + w τ ) ( A τ ) } P( A) (2.29) Where, S(w; A) and P(A) are the sales price and the dealer s purchase price repetitively. Where, τ is the age restoration, c and c are the cost of improvement and expected cost of repair respectively and ψ, ξ are the parameters of cost functions for upgrade action. λ and β are the Weibull parameters. The optimal τ * ( 0 < τ * < τ ) can be obtained by the usual first order condition (if it * is an interior point of the admissible region) and if not, τ is either zero (implying no reliability improvement) or = τ (implying maximum reliability improvement). Problem 2: The sale price depends on the warranty terms. Upgrade (improvement) provides an opportunity to offer better warranty terms and this impacts on the profit. As a result, both upgrade (improvement) and warranty period are decision variables to be selected optimally to maximise the expected profit. Here, the expected profit is given by J ψ ξ β ( w θ; A) = S( w; A) cθ A cλ ( 1 θ ) ( A + w) β β β [ { A } + θw ] P( A) ; (2.30) Where, θ is degree or level of improvement with θ = 0 means no upgrade and θ = 1 means repair quality as good as new and c is the expected cost of repair. Chattopadhyay (2002) discussed parameter estimation for warranty cost associated with the sale of second-hand products. The claims (at the system level) for 1335 second-hand cars were obtained from an Australian insurance company, which acts as an underwriter for second hand car dealers. 558 warranty claim data were used for 63

64 the analysis of expected warranty costs. The maximum likelihood estimates (for more details see Crowder et al (1991) were used for parameter estimation Service Contracts For expensive and complex equipment (e.g., a power generation turbine), the maintenance service provider of the equipment needs to have expertise and specialised maintenance facilities to carry out appropriate maintenance. In many cases, owners find it economical to contract out the maintenance to an external agent (Ashgarizadeh and Murthy 2000). The external agent could be the manufacturer or an independent third party. Karmarkar (1978) proposed a future cost models for a service contract for consumer durable goods. Blischke and Murthy (2000) proposed a service contract with a scope for negotiation on coverage. In recent years the concept of the service contract has received significant attention. Murthy and Yeung (1995) proposed two models for a service contract. They considered it in a demand-supply framework, where the user generates demand for maintenance service and the service provider acts as the supplier of the service. In Model one, the service provider provides an immediate replacement on demand if the system fails before it reaches an age T i. This could result in high inventory holding as the agent needs to hold spares as inventory. One way of reducing this cost is to order a spare at time T 0 (0 < T 0 < T i ) after a maintenance action is executed. The inventory cost decreases as T 0 increases. In Model two, the service provider is not in a position to provide an immediate replacement on demand. The provider compensates the user for the loss of revenue till the failed system is replaced. Murthy and Ashgarizadeh (1995) developed a game theoretic model for a service contract to characterise the optimal strategies for a single customer and a single agent. They considered exponential distribution for failure and repair times. Here, the service agent has two options: (i) to rectify all failures over the life of the equipment for a fixed price (P) along with a penalty if the repair is not completed within the specified time, and (ii) to rectify each failure at a fixed price (C s ) without penalty. The customer chooses the optimal decision to maximise its utility. Ashgarizadeh and Murthy (2000) extended this to develop a stochastic model for the service contract to examine the optimal strategies for customers as well as the agent in a game theoretic setting. Many of the models discussed in this section considered a 64

65 number of assumptions such as constant failure rate, replacement only with new items (if replacement is the only option), a constant repair cost and identical customers with risk neutral attitude. Therefore, these conditions could be relaxed to develop strategies for situations close to real life Lifetime Warranties Lifetime warranty is important due to its application to longer life assets and enhanced customer s demand for service for a product instead of procurement of products. As a result, in recent times, manufacturers of a large number of consumer durable products have started offering lifetime warranties. By a lifetime warranty we mean that the manufacturer/dealer has committed to repair or replace the sold failed item during the time that the customer owns the item. It will be assumed that the useful life of the item will terminate in some finite but random amount of time. For example, some manufacturers/dealers of automobile mufflers are offering nontransferable lifetime warranty upon installation of a muffler. This warranty may terminate due to the change of ownership of the car, the buyer s failure to enforce the warranty, or the scraping of the car after a major accident that seriously damaged one or more critical sub-systems such as the chassis or engine or it could be due to the obsolescence of the exhaust system technology as a result of new technology innovation. Although the concept of lifetime warranty has attracted significant attention among practitioners, especially in recent years, the academic research on lifetime warranty is very limited. A literature survey shows that only a few research studies have so far been conducted in modelling policies and costs for lifetime warranties CONCLUSIONS A literature review of product warranty, warranty policies, service contracts and generic cost models for various warranty policies was conducted and presented in this chapter. This chapter also studied and analysed the currently available literatures on policies and cost models for various long-term warranties. The literatures review shows that although a huge amount of research work has been conducted on some types of long-term warranty policies such as extended warranty, warranty for used or second hand product, only a limited number of academic research has so far been 65

66 carried out on formulation of new policies and modelling costs for lifetime warranty policies and service contracts. It also shows that little or no research has been carried out on developing models that consider buyer s and manufacturer s risk preferences, the effect of various servicing strategies on the lifetime warranty and service contracts. Therefore, this research aims to study, analyse and develop various policies, servicing strategies and cost models for lifetime warranty and service contracts that cover the burning issues such as the effect of various servicing strategies, and to analyse the risks to manufacturers and customers. These are discussed in the subsequent chapters of this thesis. Lifetime warranty policies and models for estimating costs for these policies are developed and discussed in Chapter 3 and Chapter 4 respectively. 66

67 CHAPTER 3 MODELLING POLICIES AND TAXONOMY FOR LIFETIME WARRANTY 3.1 INTRODUCTION Market survey shows that a large number of products are now being sold with lifetime warranties which indicate the high popularity of lifetime warranty among customers. This is because it signals higher product reliability and greater customer peace of mind. Lifetime warranties are a relatively new concept. Modelling of failures during the warranty period and the costs for such policies is complex since the lifespan in these policies is not well defined and it is often difficult to tell about life measures for the longer period of coverage due to usage pattern/maintenance activities undertaken and uncertainties of costs over the period. In Chapter 2, an overview of warranties in general and a literature review of longterm warranty were carried out which revealed the need for research study on lifetime warranties and service contracts. In this Chapter, lifetime warranty is defined based on the lifetime warranty offer literatures provided by the manufacturers or dealers. Based on this, eight new one dimensional lifetime warranty policies are developed based on the current practice. Taxonomy for those policies is also proposed in this Chapter. 67

68 Definition of lifetime warranty and complexities in defining the useful lifespan of a product under lifetime warranty policy are discussed in the Section 3.2. Taxonomy of lifetime warranty policies is proposed in the Section 3.3. In this section, potential one-dimensional lifetime policies are discussed briefly. Finally, a summary of this chapter is made. 3.2 LIFETIME WARRANTY Lifetime warranties are covered for the life defined in the warranty policy and in general it can be defined as the manufacturer/dealer s commitment to provide free or cost sharing repair or replacement of the sold product in case of failure due to design, manufacturing defects or quality problems throughout the useful life of the product or the buyer s ownership period. The concept of lifetime/useful life is not clear in the warranty literatures and can be a source of confusion to the customers. Because it is often difficult to tell just whose life measures the period of coverage and this makes difficulty to tell about measures of the period of coverage (Magnuson-Moss warranty Act, 1975). For example: Sealevel Systems Inc. ( warrants their I/O products to conform and perform in accordance with published technical specifications to be free from defects due to materials and workmanship for the lifetime. "Lifetime" is defined as seven years after Sealevel discontinues manufacturing the product. The warranty period would, therefore, be technically around ten years from the date of purchase. Celestron warrants their binoculars to be free from defects of materials and workmanship for its usable lifetime ( SOG Corporation Limited (please see Figure 3.1), a manufacturer of tools and blades provides warranty for their product as long as the customer uses the products for his/her own purpose and the coverage 68

69 terminate with resell or transfer of the product ( These examples show that the measurement of a product s useful life (lifetime) varies from product to product and depends on the manufacturer s intention. Figure 3.1: A sample of lifetime warranty offer "Lifetime" is defined in different ways by different providers of lifetime warranty. A manufacturer of an automuffler may intend "lifetime" to be for the life of the car. In this case, the muffler warranty would be transferable to subsequent owners of the car and would remain in effect throughout the car's useful life. However, a "lifetime" warranty can cover as long as the original purchaser of the muffler owns the car. Or, "lifetime" can be as long as the original product survives. This is probably the least common usage of the term. Termination of such warranty may also arise due to technological obsolescence, design modifications, or change of component. For example the lifetime warranty of a picture tube will cease if the television unit is out of service due to the failure of other critical components because of wear out or accident (Wells (1985) or the whole system is declared obsolete due to technological transformation. This was the case 69

70 when black and white televisions replaced by the colour televisions in 1980s or by the LCD and plasma televisions in 2000s. Therefore, it is essential to develop a reasonable definition for lifetime warranty to resolve the confusion about the lifetime /useful life among the stakeholders. This motivates us to develop a definition for lifetime warranty in which useful lifetime is defined as the lifetime of the product in the market and assumed to be terminated in some finite, random time horizon. Outdated technology should not be covered by lifetime warranty. Termination of this type of warranty can also results from the ownership change, technical or commercial reasons. The estimated life of an asset can therefore be defined as: Technical life/ Physical life the period over which the product might be expected to last physically (up to the period when replacement or major rehabilitation is physically required). Technological life the period until technological obsolescence dictating replacement due to the development of technologically superior alternatives. Commercial life/ Economic life the period, over which the need for the product exists, the period until economic obsolescence dictates replacement with an economic alternative. Ownership life/ Social and legal life the period until customer desire or legal ownership is retained or replacement change of ownership occurs. The terms and conditions of lifetime warranty policies are almost similar to those of basic warranty. The difference between a lifetime warranty policy and a basic warranty policy is that the warranty coverage period of normal warranty is fixed for a particular product whereas the coverage period of a lifetime policy is uncertain and randomly variable depending on the useful life of the product. Lifetime warranty policies can involve features such free rectification to customer due to defined failures or faults, sharing rectification costs between the manufacturer/dealer and the customer/owner, cost limits, exclusion, trade-in and so on. It depends on the product and manufacturer/dealer s offer. The terms (eg, coverage) can vary from product to product and can depend on the customer s and 70

71 manufacturer s risk preferences. Taxonomy of lifetime warranty policies is developed and presented in the following section. 3.3 TAXONOMY OF LIFETIME WARRANTY POLICIES Taxonomy for the different types of lifetime warranty policies is shown in Figure 3.1 Life time Warranty Policies A. Free Rectification Lifetime Warranty B. Cost Sharing Lifetime Warranty C. Trade In Lifetime Warranty (TLTW) One- Dimensional FRLTW Two- Dimensional FRLTW-2D One- Dimensional CSLTW Two- Dimensional CSLTW-2D Simple Trade in STLTW Combined Trade in CTLTW Specified Parts Excluded SPELTW Limit on Individual cost LICLTW Limit on Total Cost LTCLTW Limit on Total Cost and Individual cost LTILTW Figure 3.2: Taxonomy for lifetime warranty policies Lifetime warranty policies can be divided into three main groups. These are: A. Free rectification lifetime warranty policies which are again classified into two sub groups based on the dimensions of the coverage such as age and usage - i) One-Dimensional lifetime warranty policies (FRLTW) when it covers either life decided on age (most common case) or life decided on usage (rare case), and ii) Two-Dimensional lifetime warranty policies (FRLTW-2D) where lifetime is decided on both age and usage. B. Cost sharing lifetime warranty policies. These policies are sub grouped into One dimensional (CSLTW) and Two dimensional policies. In line with Chattopadhyay and Murthy (2001) each of these sub groups are again 71

72 classified into four types of policies: i) Specified parts excluded policies (SPELTW), ii) Limit on individual cost policies(licltw), iii) Limit on total cost policies (LTCLTW), and iv) Limit on total and individual cost policies (LTILTW), C. Lifetime warranty with Trade-in policies (TLTW)- these policies are again sub-grouped into two major classes, such as i) Simple trade-in within the lifetime (STLTW) and Combined trade-in within lifetime (CTLTW) policies. Different Lifetime Warranty Policies: The main complexity in this area is the uncertainty with useful life (lifetime) and subsequently the coverage periods. Another complexity is the uncertainty of servicing costs over longer uncertain periods of time. A brief description of all these policies is as follows: A. Free Rectification Lifetime Warranty [FRLTW] Policies. Under the free rectification policy the manufacturer or dealer is obligated to rectify (repair or replace) all faults or failures of the sold product free of cost to the customer (buyer) over the defined lifetime of the product. This category of policies can be divided into two major policies based on their dimensional attributes. These are: Policy 1: Lifetime Warranty on age (One-dimensional case) with no cost to Buyer/Customer [FRLTW]: Under this policy, the manufacturer/dealer is responsible for rectifying all defects and failures of the sold product due to design or manufacturing problems over the useful life or the defined lifetime of the product. Rectification can be a replacement, repair or in some cases refund. Unlike normal warranty, the coverage period for lifetime warranty is uncertain and randomly variable. Policy 2: Lifetime warranty on both age and usage (Two-Dimensional case) with no cost to Buyer/Customer [FRLTW-2D]: Under this policy the manufacturer/dealer rectifies all defects and failures of the sold product due to design or manufacturing problems over the lifelong age and lifelong usage of the product, whichever comes first. Rectification can be a replacement or repair or in some cases 72

73 a refund can be made. Here the coverage terminates at any age or usage due to the ownership change, technological obsolescence, technical or commercial reason. B. Cost Sharing Lifetime Warranty [CSLTW] Policies: Under this category of policies, the customer and the manufacturer/dealer share the repair cost over the uncertain coverage period. The basis for cost sharing can vary from product to product depending on the manufacturer s sharing policies. In line with Chattopadhyay and Murthy (2001), we propose four One-dimensional Cost Sharing Lifetime Warranty (CSLTW) policies. These are: Specific Parts Exclusion (SPELTW), Limit on Individual Cost (LICLTW), Limits on Total Cost (LTCLTW), and Limit on Individual and Total Cost Lifetime Warranty [LITLTW]. These policies are described briefly as follows: Policy 3 - Specified Parts Excluded Lifetime Warranty [SPELTW]: Under this policy, the components of the product are grouped into two disjoint sets, Set-I (for inclusion) and Set-E (for exclusion). Here, the manufacturer/dealer rectifies failed components belonging to Set-I at no cost to the buyer over the defined lifetime of the product. The costs of rectifying the failed components belonging to Set-E are borne by the customer. (Note: The rectification of failed components belonging to set E can be carried out either by the dealer or a third party). Policy 4 - Limit on Individual Cost Lifetime Warranty [LICLTW]: Under this policy, if the cost of a rectification on each occasion is below the limit c I, then it is borne completely by the manufacturer/dealer and the customer pays nothing. If the cost of a rectification exceeds c I, the buyer pays all the costs in excess of c I (i.e cost of rectification - c I ). This continues until the termination of lifetime. Policy 5 - Limit on Total Cost Lifetime Warranty [LTCLTW]: Under this policy the manufacturer/dealer's obligation ceases when the total repair cost over the lifetime exceeds c T. As a result the warranty ceases at an uncertain lifetime L or earlier if the total repair cost, at any time during the lifetime, exceeds a prefixed cut off cost c T. Here, the warranty coverage is uncertain not only for uncertainty in exceeding the total cost limit, but also for the uncertainty of lifetime. Policy 6 - Limit on Individual and Total Cost Lifetime Warranty [LITLTW]: Under this policy, the cost to the manufacturer/dealer has an upper limit (c I ) for each 73

74 rectification and the warranty ceases when the total cost to the dealer (subsequent to the sale) exceeds a cut off cost c T or the termination of the product life due to the defined reasons, whichever occurs first. The customer pays the difference between rectification cost and the dealer s cost if the individual rectification cost exceeds cost limit c I. C. Trade-in with Lifetime [TILTW] Policies: The two main types of trade in with lifetime warranty policies are as follows Policy 7 Simple Trade-in [STLTW]: Under this policy, customer has the option to to have the product replaced at reduced cost of trade-in for the used one. In this type of warranty the old used product is repurchased by the manufacturer/dealers. The repurchased price would be a proportion of the original purchased price depending on the age of the old/ used product i.e the repurchased price (trade in a price) Pt = Po, where P o and a are the original purchased price and age and E(L) condition of the product at the time of trade in. E(L) is the expected lifetime of the product. In real life this can be negotiable. Policy 8 Combined Trade in with warranty [CTLTW] policy: Under this policy the failed or defective product is rectified free of cost to the customer/buyer up to a certain time w and if the product is failed any time beyond w over the rest of the lifetime (L), the failed product is repurchased by the manufacturer/dealer at a reduced price (see Figure 4.3). Free rectification Trade in time 0 w L Figure 3.3 Combined Trade in with lifetime warranty [CTLTW] policy 74

75 In Figure 3.3, w represents the normal free rectification warranty period and L represents the lifetime of the product. If the product failed within period w it is rectified free of cost by the manufacturer or dealer and if the product failed after w but before the defined lifetime L, manufacturer or dealer will purchase the failed product from the buyer at a discount value (Trade-in). The coverage period of this type of policy is clearly divided into two parts. The first part follows the normal free rectification warranty (from 0 to w) and the second part (from w to L) follows an uncertain coverage with one failure CONCLUSIONS Life measures applicable to lifetime warranty policies have defined in this chapter, and a framework for long term warranty policies have developed. Taxonomy for lifetime warranty policies have also developed. The taxonomy dealt with eight potential new lifetime warranty policies and the proposed policies were then discussed briefly. Most of these policies are currently offered by the manufacturers/dealers of products and the others would be of potential interest to the manufacturers/dealers. A summary of these policies is presented in the Table 3.1. In order to judge the cost effectiveness of these policies to both the manufacturer and the customer, it is necessary to develop mathematical models in estimating the cost of offering these types of policies. Mathematical models for estimating costs for different lifetime warranty policies from both the manufacturer and the customer perspectives and analysis of these proposed cost models are aimed to carry out in the next chapter (Chapter 4). 75

76 Table 3.1: Lifetime warranty policies at a glance Group Policy Policy name Abbreviation A Policy 1 One-dimensional Free rectification FRLTW Free lifetime warranty policies rectification Policy 2 Two-dimensional Free rectification FRLTW 2D policies lifetime warranty policies B Policy 3 Specific parts excluded lifetime SPELTW Cost warranty policies sharing Policy 4 Limit on individual cost lifetime LICLTW policies warranty policies Policy 5 Limit on total cost lifetime warranty LTCLTW policies Policy 6 Limit on individual and total cost LITLTW lifetime warranty policies C Trade-in Policy 7 Simple Trade-in within lifetime policies STLTW policies Policy 8 Combined Trade-in with lifetime CTLTW warranty policy 76

77 CHAPTER 4 MODELLING COST FOR LIFETIME WARRANTY 4.1. INTRODUCTION Complexities with lifetime warranty, various lifetime warranty policies and taxonomy for these policies were developed and discussed in Chapter 3. The importance of and need to study, develop and analyse the cost models of the proposed lifetime warranty policies were determined. In this chapter, mathematical models for estimating costs of different lifetime warranty policies are developed from the manufacturer and customer s point of view. Analysis of the developed lifetime warranty cost models are carried out for one dimensional policies. Since warranty claims occur due to failures of product, the starting point for such an analysis is the modelling of the failures over the lifetime (useful life) of the sold product. The next step is the modelling the estimation of rectification costs over the lifetime. Modelling of failures during warranty period and estimation of costs for such policies are complex. The reasons are: 1) the lifetime or the useful life of products are uncertain, randomly variable and assumed to be terminated in some finite time horizon, 2) failures over this uncertain coverage period are also randomly variable, and 3) the uncertainties of costs due to uncertainties of the maintenance decisions taken over the coverage period and the changes of cost parameters due to the inflation, devaluation or discounting of money over such a long period of coverage. 77

78 The outline of this chapter is as follows: Section 4.1 briefly introduces this chapter. Section 4.2 deals with the preliminaries of product failures at component and system level. In Section 4.3, the modelling of item failures during the lifetime of the product and the modelling of costs of rectification are discussed. Based on these, cost models for different types of lifetime warranty policies as described in the Chapter 3 are developed. Section 4.4 analyses and discusses the sensitivities of the developed models with numerical illustrations. In the final Section, the whole chapter is summarised with concluding remarks. 4.2 PRELIMINARIES: MODELLING PRODUCT FAILURES Modelling failures Most products are complex, made up of a number of components (sometimes, more than hundreds). Therefore, a product can be viewed as a system comprising of multiple components. The failure of such product occurs due the failure of one or more component/s. Failures over the lifetime can be modelled either at the component level or at the system level. The component level models are sometime appropriate for certain types of policies but the difficulty with the component level models is that they require data at component level. Often, the manufacturers/dealers do not keep records of those data. In contrast, system level models require aggregated data which are usually available from the manufacture s /dealer s data base Modelling at Component Level Failure of each component can be modelled separately. The modelling of the first failure needs to be treated differently from that of subsequent failures. It depends on (i) whether the component is repairable or not, (ii) the type of rectification and (iii) the type of component (new or used) used as replacement. MTTF of the first failure can be modelled by a probability distribution function. Subsequent failure can be modelled either by ordinary renewal process, when every failure results in a replacement by a new product and the replacement times are negligible or a delayed renewal process or point process when all failures are repaired with negligible repair time and with a specified intensity function (Blischke and Murthy (1994)). When the rectification involves either repair or replacement by a used or cloned part, 78

79 then the modelling is more complex and can be formulated by the modified renewal process (Kijima 1989). Modelling first failure for one dimensional formulation Black box approach Let X 1 denotes the age of an item at its first failure. This is also called time to first failure. Let F(x) and R(x) denote the cumulative distribution function and reliability function (the probability that the first failure does not occur prior to x) for the first time to failure respectively and f(t) is the density function for this case and is given by Here we have, f(x) = df(x)/dx. (4.1) F ( x) P{ X x} = 1 and ( x) F( x) = P{ X x} R 1 > (4.2) = 1 The conditional probability of item failure in the interval [x, x + t], given that it has not failed before x is given by F ( t x) F( t + x) F( x) [ ] R ( x ) = (4.3) The failure rate associated with a distribution function F(t) is defined as r () t ( t x) f ( x) F lim = (4.4) t t R = 0 ( x) For Exponential distribution, the density function, f(x) and failure rate, r(x) are given by f ( x) ( x) λe λ ( x) = λ =, for 0 x <, and λ > 0 (4.5) r (4.6) where, λ is the failure intensity. For Gamma distribution, the density function and failure rate are given by f ( x) β λ x = Γ β 1 e ( β ) λx, for 0 x <, λ > 0 and β > 0 (4.7) r t = x x β 1 λ ( t x) ( x) e dt 1 (4.8) 79

80 For Weibull distribution the distribution function and failure rate are given by F β = 1 [ ], for 0 x <, λ > 0, and β > 0 (4.9) ( λx) ( x) e β 1 β β 1 β β 1 r ( x) = βλ( λx) = x = x (4.10) β 1 β η η η Modelling subsequent failures for non-repairable component and replacement by new one: Let us consider the case where non-repairable items are sold with a free replacement lifetime warranty with useful life (lifetime) L. All failed items will be replaced (renewed) by the manufacturer/dealer at no cost to the buyers. If we assume the time required for replacement is negligible compared to warranty time and is zero, in line with Ross (1970), we can apply an ordinary renewal process (Cox, 1962) as follows. Let the number of renewals over the lifetime [0, L), be M(L), and this is given by ( L) = E[ N( L) ] np{ N( L) } = M (4.11) n= 0 where E[N(L)] is the expected number of renewals during the lifetime L of the product/item, and n is the number of failures, and n =0, 1, 2,. Conditioned on X 1 (the time to first failure), M(L) can be expressed by ( L) E N( L) [ X x] df( x) M = 1 = (4.12) 0 According to the renewal property the following expression is valid E [ N( L) X = x] 0, if x L 1 = 1 + M ( L x), if x L (4.13) If the first failure occurs at x L, then the number of renewals over L x occur according to an identical renewal process, and hence the expected number of renewals over the period is M(L - x). Therefore we have M L ( L) = F( L) + M ( L x) f ( x)dx 0 (4.14) 80

81 where, F(L) and f(x) represent Cumulative distribution function during warranty period and probability density function respectively. Therefore, the total warranty cost over the lifetime of the product/item is given by C i ( L) = c F( L) + L M ( L x) f ( x) i 0 dx (4.15) where, c i is the cost of each failure replacement Modelling costs for repairable component sold with non-renewing FRW In this case, all failed components are repaired and the failure distribution of the repaired component will not be the same as the new product/item. Let the new failure distribution be given by G(x), which is different from the failure distribution of the new item F(x). This situation represents the delayed Renewal process (Cox, 1962). A counting process {N(t), t 0} is a delayed renewal process if 1. N(0) = 0 2. X 1, the time to first renewal is a non negative random variable with distribution function F(x). 3. X j, j 2, the time interval between jth and (j-1)th repair are independent and identically distributed random variables with distribution function G(x), which is different from F(x). 4. N(t) = sup{n; S n t}, where S 0 = 0 and, for n 1. n S n = X j j= 1 Let M d (L) denote the expected number of renewals over the lifetime [0, L) for the delayed renewal process. Then in line with Ross (1970), we can rewrite the following expressions for products sold with lifetime warranties. L ( L ) = E [ N ( L )] = E N ( L ) [ X x] df ( x ) M d 1 = (4.16) 0 81

82 An expression for this can easily be obtained using the conditional expectation approach used for obtaining M(L) for the ordinary renewal process. Conditioning on T 1, the time to first renewal, we have E [ N( L) X = x] 0, if x > L 1 = 1 + M ( L x) if x L (4.17) g Where, M g (L) is the renewal function associated with the distribution function G(L). This follows from the fact that if the first repair occurs at x L, then over the interval (x, L), the repairs occur according to a renewal process with distribution G(.). Hence M d (L) is given by M d L ( L ) F ( L ) + M ( L x ) f ( x )dx = 0 (4.18) g and The warranty cost is given by C j L ( L) + M ( L x) f ( x) = c j F g dx 0 (4.19) where c j is the average cost of all repairs Modelling warranty at System Level: Failures of a system (the product itself is a system and comprised of multiple components) can be viewed as a point process and the failures can be modelled either by stationary or homogeneous Poisson process with failure rate λ when failures are independent of time (constant failure rate) or by non-stationary Poisson process (also called Non-homogeneous Poisson process) with an intensity function Λ(t) which is an increasing/decreasing function of time t. Modelling failures with independent increments - Homogeneous Poisson Process (HPP) In this case, the number of product failures during the warranty period is distributed as Poisson with intensity λ. This also implicitly implies that the duration of time needed to repair or replace a failed component is negligible when compared to the whole warranty period, so that the failure process is not interrupted during the 82

83 warranty period (Balachandran et al., 1981). The constant failure rate is reasonable particularly for sophisticated products that are composed of a large number of components; a repair or replacement of a component has little effect on the overall failure rate of the whole product. This is also validated when the early failure period in the well-known bath-tub failure patterns is effectively eliminated by a refund policy and only the constant failure period is associated with the warranty period. Here, the number of failures in any interval t is given by P { N( x t) ) N( x) = n} ( λt) n λt e + = (4.20) n! For n= 1, 2, 3,., and all x 0 and t 0 And the expected number of failures over the lifetime, L can be given by E [ N( L) ] λl = (4.21) Modelling time dependent failures - Non-homogeneous Poisson Process (NHPP) F(t) denotes the cumulative failure distribution and if modelled as non homogeneous Poisson process is given by: β F( t) = 1 exp( ( λt) ) (4.22) In this case the failure intensity Λ(t) is a function of time and is given by β 1 β f ( t) λβ ( λt) exp( ( λt) ) β 1 Λ(t) is given by: λ( t ) = = = λβ ( λt) (4.23) β 1 F( t) 1 (1 exp( ( λt) ) with β > 1 and λ > 0. where β and λ are Weibull parameters. The expected number of failures E[N(w)] over the warranty period w, is given by E[ N( w)] Λ ( t) dt (4.24) = w 0 The expected warranty cost to the manufacturer for unit sale, E[C m (.)], as modelled by Blischke and Murthy (2000), is given by 83

84 E [ C ( w) ] C [ E[ N( w) ] = 1 (4.25) m s + where, C s is the total manufacturing cost per unit MODELLING LIFETIME WARRANTY COSTS AT SYSTEM LEVEL In this section, modelling of product failures during the lifetime of the product and the cost of rectification are developed and analysed. A product is treated as a system comprising of several components and failures are modelled at the system or subsystem level. We assume product cumulative failure distribution F(t) with density function f(t) = df(t)/dt and the failure intensity function can be estimated from the Equation Λ () t = f () t ( 1 F() t ). Assumptions: For the purpose of model simplification, the following assumptions are made. One can relax one or more assumptions, but this results in complexity in analysis. The analysis of such models may in some instances be analytically intractable. The assumptions are: Item failures are statistically independent; Item failure, in a probabilistic sense, is only a function of its age; The time to carry out a rectification action by repair or replacement is negligible compared to the mean time between failures and this time can be ignored; An item failure results in an immediate claim and all claims are valid; All replacement is made with new and identical one; and Failures over the warranty period are modelled at the system (or item) level. Notations: We use the following notations for the purpose of this section α β δ η Increasing rate of cost due to inflation and other factors Shape parameter Discount rate (annuity) Characteristic life parameter of the product 84

85 ρ Parameter for the truncated exponential distribution used in the life distribution of products μ Mean of the failure distribution λ Failure intensity Λ(t) Intensity function for system failure Λ I (t) Intensity function for system failure due to component/s Set I ( included in the warranty) Λ E (t) Intensity function for system failure due to component/s Set E ( excluded in the warranty) L Defined lifetime of the product. a Useful life of product conditioned on L = a H(a) Distribution function of the lifetime (useful life) h(a) Density function associated with H(a) l Lower limit of the defined a u Upper limit of the defined a F(.) Failure distribution of product R(.) Product reliability distribution f(.) Density function associated with F(.) r(.) Hazard rate function associated wit F(.) M(L) Number of renewals during the lifetime L X 1 Age of the item at its firs failure M g (.) Renewal function associated with distribution function G(.) M d (.) Renewal function associated with distribution function F(.) C x c x C y c y G(c) g(c) c γ Total warranty cost over the lifetime associated with an ordinary renewal process Average cost of each failure replacement associated with an ordinary renewal process Total warranty cost over the lifetime associated with a delayed renewal process Av. cost of repair associated with a delayed renewal process Cost distribution function Density function associated with G(c) Expected cost of each rectification over the lifetime (system level) Parameter for cost distribution 85

86 E[N(l,u)] Expected number of failures over the lifetime. E[C(l,u)] Total expected cost for Model L1 over the lifetime E[N I (l,u)] Expected number of failures of the component/s Set I E[N E (l,u)] Expected number of failures over the lifetime: component/s Set E E[C I (l,u)] Total expected costs for Model L2 over the lifetime: component/s SetI. E[C I (l,u)] Total expected costs for Model L2 over the lifetime: component/s SetE. c I C j Individual failure (claim) cost limit(associated with Model CSLTW) Total rectification costs for jth failure (claim) (associated with Model L3 & L5) M j Manufacturer/dealer s cost for jth claim (associated with Model L3) B j Customer/ buyer s cost for jth claim (associated with Model L3) c m Expected cost for each rectification to the manufacturer c b Expected cost for each rectification to the customer c T C t TC j TM j TB j Manufacturer s cost limit for all failures over the lifetime for model L4 and L5 Total cost to the manufacturer by time t (associated with models L3, L4 and L5) Total cost of rectification of the first j failures subsequent to the sale Cost to the manufacturer associated with j number of failure Cost to the customer associated with j number of failure E[C m (l,u)] Expected cost to the manufacturer for Models L4 and L5 E[C b (l,u)] Expected cost to the buyer/customer for Models L4 and L5 C L V(c ) v(c) Cost of rectification of all failures over the lifetime for Model L4 and L5 Distribution function for C L Density function associated with V(c) G (r) (c) The r-fold convolution of G(c) T b Time at which warranty expires for Model L5 V m (c; t) Distribution function for the total costs to the manufacturer v m (c; t) Density function associated with V m (c; t) Q(t; c T ) Distribution function for T b q(t; c T ) Density function associated with Q(t; c T ) 86

87 4.3.1 Modelling Uncertainties of Lifetime In any basic warranty case, the warranty coverage period is certain and fixed (see Figure 4.1a) and the failures occur randomly over this constant period of coverage. But in case of lifetime warranty, the upper limit of the coverage is uncertain since the termination of life is uncertain and randomly variable. Product failure also occurs randomly over this uncertain coverage period (see Figure 4.1b). Conditioned on the upper limit of coverage L = a, one can capture this uncertain coverage period by binding a with a lower limit l and upper limit u at statutory base since the termination of lifetime is assumed to be terminated in some finite, random time horizon. One can model this as a random variable with a distribution function H(a) with H(l) = P(a l) = 0 and H(u) = P(a u) = 1 Failure intensity Λ(t) l u 0 w 0 L= a Coverag Coverag a) Basic warranty case b) Lifetime warranty case Figure 4.1: Failure intensity over the non lifetime and lifetime warranty coverage period h(a) is the probability density function of coverage period a associated with H(a) and ( a) dh h ( a) = (4.26) da One form of H(a), which is analytically tractable, is the following truncated exponential distribution ( see Chattopadhyay and Murthy, 2000): 87

88 H(a) is e e ρl ρl e e ρa ρu which gives a h( a) ρa ρe = (4.27) ρl ρu e e The mean value of useful life of the sold product can be expressed by μ L = E ρl ρu ρl ρu ( ) ( le ue ) + ( e e ) a = e ρl e ρu / ρ (4.28) In real life distribution of lifetime coverage might not be possible to model using a particular distribution and can be modelled using a probability mass function Modelling Rectification Cost The cost of each rectification C is, in general, a random variable because an item failure is due to the failure of one or more of its components and cost of repair or replacement varies with components. We assume G(c) and g(c) as the cumulative distribution function and the probability density function of rectification costs of the product over the lifetime i.e. G( c) = P{ C c} (4.29) Then the expected cost of each rectification action, E ( C) = c, is given by ( C) = c = E cg( c) dc (4.30) 0 For the purpose of simplification of the model analysis, we consider a case where the cost of rectification is distributed exponentially with exponential parameter γ. This consideration is true for most of the simple products. For exponential distribution case with cost parameter γ, G(c) can be expressed by γc G( c) = 1 e (4.31) Then the expected cost of each rectification with cost parameter γ is given by c = [ 1/ γ ] (4.32) 88

89 Moreover the cost can vary subsequently in the longer uncertain period of a contract and a negotiation clause can possibly be included. For a very long product life, a cost trend and discounting factor can be included in the cost model Model L1: Free Rectification Lifetime Warranty (FRLTW) Under this policy, the manufacturer/dealer is obliged to rectify all defects and failures of the sold product due to design or manufacturing problems over the defined lifetime. For products with longer lifetimes such as cars, the total warranty cost is uncertain over a longer period of time due to the uncertainties of lifetime and costs of servicing claims. If the lifetime is very long (more than one year), the future cost will be affected by increased labour cost, inflation and devaluation of money over time. This can be captured by following way (see Figure 4.2): Warranty cost Discounted Discounted Discounted 0 L N1 =Year1 N1 =Year2 N1 =Year3 Lifetime in years Figure 4.2: Effects of discounting for warranty costs under longer coverage period. Let the life of the product be L years. This L is itself uncertain due to ownership change, technological obsolescence, technical and commercial reasons. 89

90 Let the expected number of failures in year i and the expected cost of rectification for each occasion within the year i be E(N i ) and E(C i ) respectively. Costs of premiums are assumed to occur at the beginning of every year. Then the present value of the total expected cost for lifetime warranty can be modelled as E a ( C) E( N ) E( C ) P( N ) = i i i i= α + δ i 1 (4.33) where, P(N i ) = Probability that lifetime terminates at year i. P(N i ) is a function of ownership change, technological, obsolescence, technical and commercial life of products. α denotes increasing rate of cost due to inflation and other factors and δ represents the discount rate of money since the expected cost of warranty would be affected by the discounting of money for a product with a long lifetime. This model is a complex one. Easier ways to calculate the discount are the use of real discount rate or use of Fisher s formula. A simulation approach can also be used to solve this model. For the purpose of model simplification, we assume values of both α and δ equal to 1. This assumption is realistic for products with shorter lifetimes i.e. product with lifetimes close to one year. The total cost of warranty during the lifetime can therefore be expressed as ( C ) E( C) E[ N( L) ] E t = (4.34) where E(C) is the expected cost of warranty for each occasion of failure and E[N(L)] is the expected number of failures over the lifetime coverage period a. Here, we consider the product as a system composed of a number of components. Rectification can be a repair or replacement of one or more components. Product failures are modelled as occurring according to a point process with an intensity function Λ(t) where t represents the age of the product. Λ(t) is an increasing function of t indicating that the number of failures, in a statistical sense, increases with age. As a result, N(t), the number of failures over the warranty period is a random variable with 90

91 a n ( t) dt a Λ Λ( t) dt e 0 0 P{ N( t) = n} = (4.35) n! Since, the useful life of the product can be terminated at any time due to ownership change, technological obsolescence, technical or commercial reasons, the termination point of lifetime coverage can be bounded by a lower limit (l) and upper limit (u) at a statutory base. The expected number of failures during the lifetime can be modelled using conditional probability of the upper limit of coverage = a and then lifting the condition by integrating the probability density function and limiting it between l and u as in the Equation (4.36) E[ N ( l, u)] { Λ( t) dt} h( a) da (4.36) = u l a 0 Therefore, the total cost for FRLTW due to warranty claims over the lifetime of the product can be given by [ ( l, u) ] = E( C) E[ N( l, u) ] = c { Λ( t) dt} h( a) da E C u l a 0 This can be expressed as Equation (4.38) by substituting value of expected per occasion rectification cost from Equation (4.32) E Λ γ l 0 1 u a ( C( l, u) ) = { ( t) dt} h( a) da (4.38) Model L2: Specified Parts Exclusive Lifetime Warranty ( Policy 3): Here the components of the item are grouped into two disjointed sets; I and E. Failures of components belonging to set I are covered by lifetime free replacement warranty and those belonging to set E are not covered under warranty. Let N I (l, u) and N E (l, u) denote the number of failures over the lifetime warranty period. One can model the failures during warranty using two different approaches. We assume that the total costs are not affected by the inflation or the discounting of money. Approach 1 91

92 Here the failures are modelled as in Model L1( for FRLTW) by a point process with intensity function λ(t). However, with each point (corresponding to a failure) there is a mark which indicates whether the item failure is covered under warranty or not. This mark is modelled as a binary random variable Y with Y = 1 indicating that the failure is covered and Y = 0 indicating that the failure is not covered under warranty. Let P{Y = 0} = p and P{Y = 1} = 1 p = q In this case, the expected number of failures of the components of the sold product that are covered with free replacement warranty over the lifetime is given by = u a E[ N I ( l, u)] q Λ() t dt h( a) da (4.39) l 0 where, l and u are the lower and upper limits of the useful life distribution and a is the lifetime coverage of the product. Similarly, the expected number of failures of components that are not covered with free replacement under this warranty policy is given by u a E[ N E ( l, u)] = p Λ() t dt h( a) da (4.40) l 0 Approach 2 Here, the component failures belonging to Set I are modelled by an intensity function Λ I (t) and those belonging to Set E are modelled by another intensity function Λ E (t). Both of these are increasing functions of time t. Then N I (L) and N E (L) are distributed according to non-stationary Poisson processes with intensity functions Λ I (t) and Λ E (t) respectively. In this case, the expected number of failures over the useful life of the products can be expressed as u a E[ N I ( l, u)] = Λ I () t dt h( a) da (4.41) l 0 and 92

93 u a E[ N E ( l, u)] = Λ E () t dt h( a) da (4.42) l 0 Note: If Λ ( t) = pλ( t) and Λ ( t) = qλ( t), then both approaches yield the same E I expected values Model L3: Limit on Individual Cost Lifetime Warranty (Policy 4) Here the cost of individual claims to the dealer is limited to fixed cost c I whereby the manufacturer/dealer carries out all rectification action free of cost to the customer if the cost of rectification is below a limit c I. If the cost of rectification exceeds c I, then the customer pays the difference between the cost of rectification and c I. That is the manufacturer/dealer pays all costs up to c I and the customer pays an amount (C-c I ), where C is the total rectification costs of an individual claim. We assume that the total costs are not affected by inflation or the discounting of money. Then for the j th failure, M = min{ C c } (4.43) j j, I B j = min{ 0,( C c )} (4.44) j I Where M j and B j represent the manufacturer s cost and customer s costs for jth failure respectively. Let the individual cost of rectification, C j be given by a distribution function G(c ) with density function g(c ). Therefore, the expected cost of each rectification to the manufacturer/dealer is given by c m c = I cg( c) dc + ci G ( ci ) (4.45) 0 c I where, G ( c ) = g( c) I dc and that to the buyer is given by 93

94 c b = ( C ci ) g( c) dc (4.46) c I The expected number of failures is given by E[ N ( l, u)] { Λ( t) dt} h( a) da (4.47) = u l a 0 The total expected warranty cost to the manufacturer/dealer, E[C m (l,u)], is given by u E [ C ( l, u)] = c [ { Λ( t) dt} h( a) da] m m (4.48) l a 0 and the total expected cost to the customer over the lifetime, E[C b (L)], is given by u E [ C ( l, u)] = c [ { Λ( t) dt} h( a) da] b b (4.49) l a Model L4: Limit on Total Cost Lifetime Warranty (Policy 5) Under the conditions of this policy, the warranty ceases either at the point of termination of the useful life or the total warranty cost reached at a prefixed cost limit c T, whichever occurs first. Let TC j denote the cost of rectifying the first j failures subsequent to the sale. It is given by TC j = C i j i= 1 ( j =1,2, ) (4.50) where C i is the cost of rectification of the i th failure. Since the total cost (of claims) to the manufacturer/dealer over the life period is limited to c T, we have for the i th failure, the cost to the manufacturer/dealer is M i = min{ Ci,( ct TC( i 1) )} (4.51) and the cost to the customer is B = max{ 0,( TC c )} (4.52) i i T 94

95 TM j denote the cost to manufacturer/dealer associated with the first j numbers of failures and this can be expressed by TM j = M i j i= 1 (j=1,2..) (4.53) TB j denote the cost to buyer associated with the first j failures. It is given by TB j = B i j i= 1 (j=1,2..) (4.54) with M i and B i given by (4.51) and (4.52) respectively. TM 0 = TB 0 = 0. The warranty can cease either at lifetime a with number of failures N(t) or earlier at the j th failure if TM (j-1) < c T TM j and j < N(t). Figure 4.3 shows this in a diagrammatic manner. The total cost to the manufacturer/dealer is a random variable as follows: Let C L be the cost of rectifications of all the failures over the period [0, a) and V(c) be the distribution function for C L. Then using conditional argument r= 0 V ( c) = Pr ob.{ C c N( t) = r}pr ob.{ N( t) = r} (4.55) L Note that C L, conditional on N(t) = r, is the sum of r independent and identically distributed random variables with distribution G(c). Since N(t) is time dependent Poisson distributed (Non-homogeneous Poison process) u a u a r { [ Λ( t) dt] h( a) da} exp( [ Λ( t) dt)] h( a) da ( r) l 0 l 0 V ( c) = G () c (4.56) r= 0 r! where, G (r) (c) is the r-fold convolution of G(c) with itself. The cost to the manufacturer/dealer equals C L if C L c T, and equals c T if C L > c T. In the former case, the warranty ceases at a which is a random variable and in the later case it ceases before a. E[C m (l,u)] denote this expected cost to the manufacturer and it is given by 95

96 c T E[ Cm ( l, u)] = cv( c) dc + ctv ( ct ) (4.57) 0 where v(c) is the density function (= dv(c)/dc) associated with the distribution function V(c). Level crossing C 13 Z L Cumulative cost for item1 C 24 Z L for item2 C T Total warranty cost C 11 C 12 C 22 C 23 C 32 Z L for item3 Z L C 21 C 31 Time in warranty period Figure 4.3: A Schematic diagram of Limit on Total Cost Lifetime Warranty The cost to the buyer over the period [0, a) is max {0, C L c T }. Therefore the expected cost to the buyer can be expressed by E [ C ( l, u)] = ( c c ) v( c) dc (4.58) b c T T When G(c) is exponentially distributed and Λ(t) = λ (ie, occurring according to stationary Poisson process) then v(c) can be obtained analytically (Cox, 1962). For a general Λ(t), it is not possible to obtain v(c) analytically and simulation approach can be used. 96

97 4.3.7 Model L5: Limit on Individual and Total Cost Lifetime Warranty (Policy 6) Under this policy, the cost to the manufacturer/dealer has an upper limit (c I ) for each rectification and the warranty ceases when the total cost to the manufacturer/dealer (subsequent to the sale) exceeds c T or at the termination of useful life a, whichever occurs first. The customer pays the difference between rectification cost and the manufacturer/dealer s limit cost. That is the total cost of claims to the manufacturer/dealer is limited to c T and the cost of an individual claim is limited by c I, then for the j th failure M j min{ C = j, ci,( c T 0 TM ( j 1) )} for for TM TM ( j 1) ( j 1) c > c T T (4.59) and B j max{0,( C = j c ),( TM I C j j c T )} for for TM TM ( j 1) ( j 1) c > c T T (4.60) where TM j denotes the sum of the costs incurred by the manufacturer/dealer with the servicing of the first j failures under warranty. Note that the warranty can expire before a if the total cost to the manufacturer/dealer exceeds c T. Let T b denote the time at which the warranty expires. It is given by the first passage of a process C t which represents the total cost to the manufacturer/dealer by time t. It is seen that T b > t if C < c (4.61) t T Let V m (c; t) denote the distribution function for the total cost to the manufacturer/dealer by time t. Then V m (c; t) is given by Vm ( c; t) Gm ( c)[pr ob{ N( t) = = r= 0 r r}] (4.62) where G m (c) is given by G m G( c) for c < ci ( c) = (4.63) 1 for c ci 97

98 Since, the cost of each single rectification action to the manufacture/dealer is constrained to be less than c I. Note that u a ( t) dth( a) da u Λ a r 0 Pr ob{ N( t) = r} = [ { Λ( t) dt} h( a) da] [ e l ]/ r! (4.64) l 0 Let Q(t; c T ) denotes the distribution and q(t; c T ) is the density function for T b. Then, we have ct q( t; ct ) dx = v m ( t; c) dc (4.65) t 0 The expected warranty cost to the manufacture/dealer can be obtained by using a level crossing of the above cumulative process. Let v m (c) be the density function associated with distribution function V m (c). Then the expected total cost to the manufacturer/dealer is given by ct E[ C ( l, u)] = cv ( c) dc + c V ( c ) (4.66) m 0 m T m T The expected cost to the customer/buyer until warranty expires, when C t first time crosses c T at T b, is given by E [ Cb ( l, u)] = E[ N( Tb )] ( c ci ) g( c) dc (4.67) c where expected number of failures over [0,T b ) is given by I E[ N( t )] = a Λ [ t ( x) dx] q( t; c ) dt (4.68) b 0 0 T It is not possible to evaluate the costs analytically and one needs to use a simulation approach to obtain these costs. 98

99 4.4 ANALYSIS OF THE MODELS Analysis of Model L1 [FRLTW]: Here the rate of product failure is a function of age (as assumed). Since the number of failed components at each failure is very small relative to the number of components in the item, the rectification action can be viewed as having a negligible impact on the failure rate of the product as a whole. In other words, the failure rate after a repair is nearly the same as that just before the failure. Such a repair action is called minimal repair (Barlow and Hunter, 1960). This type of failure can be modelled as the Non-Homogeneous Poisson s Process (NHPP). In this case, Λ(t) is the failure rate associated with the failure distribution for the product. A simple form for Λ(t) is as follows: ( β 1) Λ( t ) = λβ ( λt) (4.69) with the parameters β > 1 and λ > 0. Λ(t) is an increasing function of t. Note that this corresponds to the failure rate of a two-parameter Weibull distribution. By substituting Equation 4.69 in Equation 4.38, the total expected cost can be given by 1 u a E[ C( l, u)] = [ { λβ γ l 0 which can be expressed as Equation (4.70) β 1 ( λt) dt} h( a) da] 1 u E[ C( l, u)] = [ β da γ l ( λa) h( a) ] (4.70) Now substituting Equation 4.28 in Equation 4.70, it can be expressed as Equation u ρa e E[ C( l, u)] = [ β ρ da ρl ρu γ l e e ( λa) ] (4.71) Now the Equation (4.71) can be solved by using mathematical software such as MATLAB, MAPLE etc. For model analysis purposes, car failure data for

100 second hand cars sold in Perth, Australia were collected from an Australian insurance company (identified by the code WNS) which acts as an underwriter for car dealers. This data was obtained from the source under a strict confidential agreement. A preliminary evaluation indicated several shortcomings such as the data indicating a claim but not indicating which component part failed and the exact cost of rectification; the population of cars involved is comprised of various makes and many different models. However, we used these data for modelling failure distribution of cars by making some assumptions and one of the assumptions is that the claims occur according to a non-homogeneous Poisson process. The estimated failure parameters are, the shape parameter β = 2 and inverse characteristic life λ = per year. This implies that the mean time to first failure, μ = 2 years. Let the expected cost of each repair be $100 and let the lifetime coverage distribution parameter ρ = 0.4. The expected lifetime warranty costs, E[C(L) for different combinations of l and u are shown in Tables 4.1. From the Table 4.1 we can generate the Figure 4.4. Table 4.1: Expected lifetime warranty cost ($) to the Manufacturer for FRLTW l u The Figure 4.4 shows that with the given failure distribution the warranty cost increases as the range of useful life increases this implies the longer the span of useful life the more the number of failures and higher the cost to the 100

101 manufacturer/dealer. Therefore the model is sensitive to the distribution of the span of lifetime. Warranty costs are also sensitive to the failure distribution and the lifetime parameter ρ. The higher the failure rate the higher the warranty costs. Figure 4.4: Expected lifetime warranty cost ($) to the Manufacturer FRLTW This can be verified with the Table 4.2 where we use different failure parameter for a fixed life span. Assuming lower and upper limit of lifetime as 0.5 years and 3 years respectively and keeping all the variables same, the Table 4.2 shows that the expected lifetime warranty cost increases as the product failure intensity increases. Similarly, as seen in the Table 4.3, the expected warranty cost decreases as the lifetime parameter ρ increases. Increase of ρ implies a shorter lifetime or useful life. Table 4.2: Expected lifetime warranty cost as product failure intensity varying λ E[C(l,u)] Table 4.3: Expected cost with the product lifetime parameter variation ρ E[C(l,u)]

102 4.4.2 Analysis of Model L2 [SPELTW]: Model L2 (SPELTW) is used to estimate the expected warranty cost to the manufacturer/dealer and the customer/buyer based on both approaches. Approach 1: For the jth failure of ith component of an item/product, we can define Y ji as follows: Y ji 1 = 0 for, for, i I i E (4.72) Let M j and B j denote the cost to the manufacturer/dealer and the customer associated with the j th failure under warranty. Since the costs are shared, we have for the j th failure, M = Y C (4.73) j ji j B = ( 1 Y ) C (4.74) j ji j As a result, the total cost to the manufacturer/dealer over the warranty period, C M (a), is given by N ( a) C ( a) = (4.75) m M j j= 1 where M j is given by (4.73). Similarly, the total cost to the customer over the warranty period (because of non covered components), C b (a), is given by N ( L) C ( a) = (4.76) b B j j= 1 where B j is given by (4.74) Let p denote the probability that a product failure is due to the failure of components from set E. Then E[Y ji ] = q. The expected number of failures covered under warranty is given by (4.39) and not covered under warranty is given by (4.40). Then, the expected warranty cost to the manufacturer/dealer, E[C m (l,u)] associated with the covered components, is given by 102

103 u a E[ Cm ( l, u)] = E[ Y jic j ] E[ N( L)] = cq Λ() t dt h( a) da (4.77) l 0 When it follows the Weibull distribution, the expected cost can be expressed as E[ C ( l, u)] E[ Y C ] E[ N( L)] = c q I m = ji j u ρa ρe ( λ a) β da and finally, ρl ρu l e e u a E[ C = ( ) m ( l, u)] c q λβ λt β 1 dt h( a) da (4.78) I l 0 And the expected cost to the customer over the warranty period, E[C b (L)], is given by u ρe ρa E[ C b ( l, u)] = c p ( λa) β da (4.79) E l u l e ρ e ρ Where, the cost of each rectification resulting from failures due to components belonging to set E and set I are C E and C I respectively. This is modelled by distribution functions G E (c) and G I (c) with mean c E and c I respectively. Expressions for the mean values can be followed by the Equation (4.32) using the appropriate distribution function. For both the equations (4.78) and (4.79) the value of h(a) is substituted from Equation (4.27). Using the same car failure data, the failure parameters values are, β = 2 and λ = per year. Let the lifetime distribution parameter ρ = 0.4. This implies that the mean time to first failure, μ = 2 years. Let expected cost of rectifications for warranty included and warranty excluded components be c I = $100 and c E = $70 respectively and let the probability of warranted excluded component p = 0.4 (40%), and probability of warranty covered component q =1- p =

104 Table 4.4: Warranty cost ($) to the Manufacturer for lifetime SPELTW: Approach 1 l u Table 4.5: Warranty cost ($) to the Customer for lifetime SPELTW: App. 1 l u

105 The expected warranty costs to the manufacturer/dealer and the customer are shown in Table 4.4 and Table 4.5 respectively. From Tables 4.4 and 4.5 we can develop the Figure 4.5. Figure 4.5: Expected lifetime warranty cost ($) to the Manufacturer and the Buyer for SPELTW-Approach 1 This Figure 4.5 shows that both the manufacturer s and buyer s cost increase with the increase of lifetime. In this example, with the same failure intensity, the buyer s cost curve (Blue) is flatter than that of the manufacturer (Pink) cost curve. This is dependent on the proportion of warranty covered components and the expected cost of rectification of the warranty covered components. The higher the proportion of warranty covered components and the expected rectification cost, the higher will be the manufacturer s expected warranty cost and the more steep will be the manufacturer s cost curve and vice versa. Note that under this policy, the manufacturer has to bear the costs of failures of all warranted components whereas the buyer has to bear the rectification costs of all non-warranted components. Approach 2: 105

106 Let the mean costs of rectification of each part belonging to the set E and set I be c E and c I respectively. Expressions for the mean values are given by the Equation 4.32 using the appropriate distribution function. Here, intensity functions of Set E component and Set I components are different. The expected number of failures covered under warranty is given by the Equation (4.41) and that of not covered components is given by the Equation (4.42). As a result, the expected warranty cost to manufacturer/dealer, E[C m (l, u)], is given by u a E[ Cm ( l, u)] = ci λ () t dt h( a) da (4.80) I l 0 If the failure of components belonged to Set I follow the non-homogeneous Poisson process with failure parameters λ I and β I, then the Equation (4.41) can expressed as ( λ a) u ρa ρe E[ Cm ( l, u)] = c β I I da (4.81) I ρl ρu l e e Similarly, the expected cost to the customer/buyer over the warranty period, E[C b (L)] for Set E, can be given by ( λ a) u ρa ρe E[ C = b ( l, u)] c β E E da (4.82) E ρl ρu l e e Where, the failure of components follows the Weibull distribution with Weibull parameters λ E and β E, Let the parameter values for failures of components belonging to Set E be λ E = per year and β E = 2.31 and let the expected cost of each rectification c E = $70. The corresponding values for Set I are λ I =0.443 per year, β I = 2 and c I = $100. The expected warranty cost to the manufacturer/dealer E[C M (l,u)] and to the buyer E[C b (l,u)] are shown in Tables 4.6 and Table 4.7 respectively. 106

107 Table 4.6: Manufacturer s costs for Specified parts excluded policy Approach 2 l u Table 4.7: Customer s cost for Specified parts excluded policy Approach 2 l u Figures 4.6 can be developed from the Tables 4.6 and 4.7 which shows that the total expected warranty costs for both the manufacturer and the buyers increase with the increase of the statutory base of the warranty and also with the increase of the useful life of the product. 107

108 Figure 4.6: Expected lifetime warranty cost ($) to the Manufacturer and buyer for SPELTW-Approach 2 The Figure 4.6 also shows that under the assumed failure rates and proportion of warranted component the buyer s cost curve is flatter than the manufacturer s cost. This is because here the failure intensity of warranted components is assumed to be higher than that of the unwarranted components. This is also affected by the proportion of warranted and unwarranted components. The higher the proportion of the warranty covered components, the steeper will be the warranty manufacturer cost curve and lower will be the customer cost curve. This implies that both the manufacturer s and buyer s costs are influenced by the proportion warranted components and the failure intensities of the warranted and unwarranted components if the cost of rectification is same for both the manufacturer and the customer Analysis of Limit on Individual Cost (LICLTW) Policy Under this policy the total expected warranty cost to the manufacturer/dealer, E[C m (l,u)] over the lifetime, is given by the Equation (4.48). We assume failure of 108

109 the product follow the non-homogeneous Poisson process with inverse characteristic life parameters λ and shape parameter β. Therefore, the total expected warranty cost over the lifetime is given by u E[ C ( l, u)] = c [ { λβ ( λt) dt} h( a) da] which is, m m l a 0 ( β 1) c ρa ρe, β da (4.83) I I ρl ρu 0 l e e I u [ ( l u) )] = cg( c) dc + c G ( c ) ( λa) E C m and the total expected cost to the customer over the lifetime, E[C b (l,u)], can be expressed by E[ C ( L)] = b c b which can be expressed as u [ { λβ ( λt) l a 0 ( β 1) dt} h( a) da] ρa ρe, β da (4.84) I ρl ρu c l e e I u [ ( l u) ] = ( C c ) g( c) dc ( λa) E C b In solving equation (4.83) and (4.84) numerically, we assume the rectification costs for the product is exponentially distributed over the lifetime with parameter γ. And γ = Using the car failure data with shape parameter β = 2 and inverse characteric life parameter λ = per year. Let the manufacturer set individual limit cost c I = $125 and we assume the cost of each rectification is not more than $200 which implies that the customer has to pay all the costs rectification from 125 to $200. The expected warranty costs, E[C m (l,u) E[C b (l,u) for different combinations of l and u are shown in Tables 4.8 and Table

110 Table 4.8: Manufacturer s/dealers costs for policy LICLTW u l (lower limit of useful life in years) (upper limit) Table 4.9: Customer s costs for policy LICLTW l u The Figure 4.7 can be generated from the Table 4.8 and Table 4.9. The plots in the Figure 4.7 show that both the manufacturer s and buyer s/customer s expected warranty cost are increasing function of useful life of the products and the cost also increases with the increase of span limits. 110

111 Figure 4.7: Expected lifetime warranty cost ($) to the Manufacturer and Buyer for LICLTW These plots shows that under the conditions and assumptions of this example the manufacturer s expected warranty costs are higher than that of the buyer s. This is because the mean cost of rectification of this particular product is considered $100 1 (as expected cost is estimated, c = ) while manufacturer s cost limit was set γ $125. Most of the costs are less than the manufacturer s limit cost ($125). This implies that the both the manufacturer s and the customer s costs are sensitive to mean cost of rectification and the manufacturer s limit on cost per rectification. Therefore the manufacturer should fix the limit on each cost by carefully estimating the expected cost of rectification. 4.5 CONCLUSIONS Since warranty claims occur due to failures of the sold item/product, this chapter first studied and discussed the mathematical modelling of failures of products, both at the component level and at the system level. These failure models are then used to estimate and analyse the expected lifetime warranty costs to the manufacturer and 111

112 customers/buyers at system level. The reason for considering modelling at system level is that most real life products are complex systems, comprised of a number of components and system level models require aggregated data which are usually available from the manufacture s /dealer s data base. The sensitivity of the developed models were then analysed and illustrated using numerical examples. Comparison of costs to the manufacturer and customers/buyer were made for most of the cases. The application of the developed models in industry requires building and validating models based on real life industry data. For the purpose of validating the lifetime warranty models, this study needed to collect and use data relating to product failures/warranty claims, costs of rectifications over the lifetime coverage and the lifetime termination (product useful life data). The original plan of this research was to validate the models by collecting data from one or more manufacturing firms within Australia. This plan had to be adjusted due to the difficulties in collecting real lifetime warranty related data from the manufacturers/dealers who were offering lifetime warranty policies for their products. It was not possible since 1. the practice of lifetime warranty is very new and the quality lifetime termination data and the costs of rectification data over the coverage period are rare; 2. most of the manufacturers/dealers do not have organised quality data and most importantly This could be a scope for future study. While offering a lifetime warranty for a product/item, both the manufacturer/dealer and the customer/buyer are exposed to uncertainties and risks of warranty pricing and product performance during the life of the product. How much a manufacturer/dealer should charge and how much a buyer is willing to pay for this service depends naturally on the attitudes of the stakeholders (Manufacturer and/or buyers) towards risks. The next chapter is focussed on these two perspectives to develop models for warranty price and risks to manufacturers and buyers for lifetime warranty policies. 112

113 CHAPTER 5 MODELLING RISKS TO MANUFACTURER AND BUYER FOR LIFETIME WARRANTY POLICIES 5.1 INTRODUCTION In the Chapters 3, and 4, different policies and cost models for lifetime warranty were proposed. In this chapter, the risks and uncertainties associated with lifetime warranty policies will be examined. Under the typical situation of a lifetime warranty transaction, a buyer of a product pays for the warranty at the time of product purchase which on some occasions is factored into the product price. The manufacturer/dealer provides rectification service in case of product failures due to design, manufacturing and quality assurance problems during the defined life of the product. Under such situations, both the manufacturer and the buyer are exposed to the uncertainties and risks of warranty pricing and product performance during the lifetime of the product. By offering a lifetime warranty for a product, the manufacturer is risking in warranty pricing whether its offer for such a warranty will be accepted by the buyers. At the same time, customers/buyers are unsure about the benefits of buying products sold with lifetime warranty policies. Anticipation of higher product failure encourages a buyer to pay for the higher warranty price which in turn encourages the manufacturer to charge a higher warranty price. In the case of product covered with warranty, the buyer returns to the original manufacturer for rectification of product failures. But when the product is not covered by warranty, the buyer may take it to other service providers for rectifications. If the original manufacturer is the only one that can 113

114 repair the product because of some technological monopoly, the manufacturer could charge a higher price for services and more buyers would be interested to have warranty cover. If buyers pay a higher repair price for any product failures, they might be interested in warranty cover and even paying a higher warranty price for free repair or replacement policies. There is a need to model costs to buyers and manufacturers which includes their risk preferences. Although there is a long and rich history of research efforts devoted to determining the optimal warranty price, only a few of them consider producer's and customers' risk preferences. Limited work takes into account the manufacturer s and buyer s risk preferences toward how much a manufacturer should charge and how much a buyer is willing to pay for this service. Ritchken and Tapiero (1986) proposed a framework in which warranty policies for non-repairable items can be evaluated according to risk preferences of both the manufacturer/dealer and the buyers where they emphasised the design and pricing of warranties to which the manufacturers are indifferent in an expected utility sense. Given the price warranty schedule, a buyer s response is expressed by selecting the price-warranty which minimises disutility. As a result, a manufacturer can increase profit by tailoring price-warranty schedules to specify buyer s need. Menezes (1989) developed a conceptual framework to examine the impact of warranties on buyers preferences. Based on the framework, predictions of consumer preferences between product with and without a warranty were derived for various consumer segments. Chun and Tang (1995) proposed a warranty model for the free-replacement, fixedperiod warranty policy that determines the optimal warranty price for a given warranty period. They assumed a constant failure rate for the product, constant repair costs throughout the warranty period, and a producer's and customers' risk aversion for future repair costs. Using the exponential utility function and the gamma failure rate distribution, they derived the decision model that maximizes the producer's certainty profit equivalent. The outline of the Chapter is as follows. Section 5.1 briefly introduces manufacturers and buyers risk preferences towards a lifetime warranty policy. Section 5.2 provides a brief overview of risk attitudes and utility function. In section 5.3, risk preference models for both the manufacturer and buyer towards a lifetime warranty policy are developed using the exponential utility function. Sensitivity 114

115 analyses of these models are carried out in section 5.4 to see the effect of various factors on the warranty price from both the buyer s and manufacturer s point of view. Finally, the summary of this Chapter is drawn in Section OVERVIEW RISK ATTITUDE AND UTILITY FUNCTION A warranty reduces the risk associated with product failure and thus could affect the buyer s preferences. The buyer s preference for a product with warranty depends on several factors such as the individual buyer s attitude towards risk, the perceived relationship between warranty and product reliability, the anticipated likelihood of product failure, cost of rectification if not warranted for all failures, the price differential between the product and the perceived inconvenience. There are three attitudes toward risk. A manufacturer or buyer may fall into any of these three risk attitude categories: a) Risk averter person with an aversion to risk: A person is risk averter if he or she prefers the expected consequence of any non-degenerate lottery to that lottery (Keeney and Raiffa (1976)). A non-degenerate lottery is one in which no single consequence has a probability of one of occurring. Given a choice between more or less risky investments with identical expected monetary returns, the risk averter would select the less risky investment. b) Risk prone or risk seeker a person is risk seeker or risk prone if he or she prefers any non-degenerate lottery to the expected consequence of that lottery ( Keeney and Raiffa (1976)) i.e. a risk seeker prefers risk. Faced with the same choice as in a, the person would prefer the riskier investment. c) Risk neutral or an indifference to risk- if he or she is indifferent to between the expected consequence of any non-degenerate lottery and that lottery (Keeney and Raiffa (1976)). Simply, a person who is indifferent of risk would not care which investment he or she received. There is considerable debate about the nature of a person s risk attitude in different domains of outcomes. In the domain of gains, there is reasonable agreement that people are risk averter. However, in the domain of losses, there is much less agreement. Both logic and observation suggests that most manufacturers and buyers 115

116 are predominantly risk averter. Because product failure will, in general, be viewed as a loss, it is concerned here about an individual s risk attitude in the loss domain. The most logically satisfying theories in answering the question why most investors would prefer the less risky one amongst two different risky investments involve utility theory Utility Theory, Utility function and Concept of certainty equivalent A utility is a numerical rating assigned to every possible outcome a decision maker may be faced with. (In a choice between several alternative prospects, the one with the highest utility is always preferred.). In economics, utility is a measure of the happiness or satisfaction gained consuming good and services. Given this measure, one may speak meaningfully of increasing or decreasing utility, and thereby explain economic behavior in terms of rational attempts to increase one's utility. To qualify as a true utility scale however, the rating must be such that the utility of any uncertain prospect is equal to the expected value (the mathematical expectation) of the utilities of all its possible outcomes (which could be either "final" outcomes or uncertain prospects themselves). Utility theory The decision maker (DM) chooses between risky or uncertain prospects by comparing their expected utility values, i.e., the weighted sums obtained by adding the utility values of outcomes multiplied by their respective probabilities. At the heart of the utility theory is the notion of diminishing marginal utility for wealth. For example, if $ 2,500 is given to a wealthless person, he could satisfy his most immediate needs with this. If he is given another $2,500, he could utilise it, but the second $2500 would not be quite as necessary to him as the first $2,500. Thus the utility of the second $2,500 is less than that of the first one and so on additional increments of wealth. Therefore it can be observed that the marginal utility of wealth is decreasing or diminishing. On the other hand, if the money receiver is a professional gambler, the second $ 2,500 can increase his thrust for money so that he can involve himself in more gambling and then third and so on will increase his thrust more and more. Therefore the marginal utility of the gambler for wealth is said to be increasing. 116

117 The Figure 5.1 graphs the relationship between wealth and its utility, where, utility is measured in arbitrary units called utils. Curve A, the one of primary interest, is for someone with positive marginal utility for wealth, but this individual also has marginal utility which increases at a decreasing rate. The individual with $5,000 would have ten utils of happiness or satisfaction. With an additional $ 2,500, the individual s satisfaction would rise to twelve utils, an increase of two utils. But with a loss of $2,500, the individual s satisfaction would fall to six utils, a loss of four utils. Nature of the utility function curve becomes concave. In other wards, a decision maker is risk averter if and only if his/her utility function is concave. Total utility (utils) (C) Increasing marginal utility: Risk seeker (Concave) (B) Constant marginal utility: indifferent to Risk t (A) Diminishing marginal utility: Risk averter (Concave) $2,500 $5,000 $7,500 Figure 5.1: Marginal utility curves This individual with diminishing marginal utility will get more pain from a Dollar lost than pleasure from a Dollar gained. Some one who has a constant marginal utility will get same pleasure or pain for each Dollar gain or loss (Curve B, Figure 5.1). However, the risk lover would be eager to continue to earn more utils by risking his wealth and for him/her the utility function curve is convex ( see curve C, Figure 5.1). 117

118 The certainty equivalent The concept of certainty equivalent is basic to utility theory and this concept is used frequently in the presentation of various risk characteristics of utility function. In the next subsection the concept of certainty equivalent will be discussed and using this concept, different attitudes towards risk taking will be specified. The certainty equivalent for an alternative is the certain amount that is equally preferred to the alternative i.e. an amount that would be accepted in lieu of a chance at a possible higher, but uncertain amount. The concept of certainty equivalent is useful in the situation where the value of a risky alternative to the decision maker is different than the expected value of the alternative due to the risk that the alternative poses of loses. Let ξ be the event yielding monitory consequences x 1, x 2,... x n with probabilities p 1, p 2,... p n respectively. Let x be the uncertain consequence (a random variable) of the event and x be the expected consequences. Therefore, the expected consequence: x E n ( ~ x ) = p x i i i= 1 The expected utility of this event is: E n [ u( ~ x )] = p u( x ) i= 1 i i But the certainty equivalent of that event is an amount xˆ such that the decision maker is indifferent between ξ and the amount xˆ for certain. This gives the following equation u xˆ = E u ~ x ( ) [ ( )] Suppose a buyer is optioned to purchase a product with or without warranty. The product price is $10. For purchasing warranty he has to pay an additional warranty price w. Let the buyer s utility function for warranty be u(w) = w 1/2. There is a 60% probability that the product will fail within the warranted period and the buyer s repair cost (if not warranted) be $6. The buyer must choose between the following options O 1 : Buy warranty 1 w 118

119 O 2 : Don t buy warranty $6 0 ( w ) 1/ 2 = 0.6 * ( 6) 1/ * ( 0 ) 1/ 2 Or, w = 3.6 Then the buyer s certainty equivalent for these alternatives is $3.6. That means under such circumstances the buyer would not mind to pay $3.6 for warranty. By using this concept of the certainty equivalent, it is possible to specify different attitude toward risk taking. Such as: When a person s certainty equivalent for alternative specified in terms of profit is less than the expected profit, the person is said to be risk averse. Similarly, when the certainty equivalent for alternative is greater than expected profit the person is said to be risk seeker and the person is said to be risk neutral if the Certainty equivalent for alternative is equal to the expected profit the person is said to be risk neutral. These definitions are reversed for an uncertain alternative specified in terms of losses. Utility function If certainty equivalents are determined for the alternatives in a decision problem, then it is straightforward to determine the preferred alternative by selecting the alternative with the best certainty equivalent. This can be determined by introducing some functions, called the utility functions. Economists discuss goal-seeking (people have goals and try to obtain those goals) in a mathematical terminology. When they talk about maximising utility functions, they use an abstract, numerical way of saying that people are trying to attain goals. A utility function translates outcomes into numerical ratings such that the expected value of the utility numbers can be used to calculate certainty equivalents for alternatives in a manner that is consistent with decision maker s attitude towards risk taking. The utility function expresses utility as a function of consumption of real goods (pounds, gallons, kilograms, litters) as opposed to nominal goods (dollars, euros). A utility function can be written as ( w w ) U = f 1, 2,... w n 119

120 means that items w 1, w 2, etc. to some nth w all contribute to a person's utility. Utility, as the word is used here, is an abstract variable, indicating goal-attainment or wantsatisfaction. If a person has simple goals or objectives, such as accumulating wealth possessions, then w 1 may represent land and house, w 2 cars, w 3 furniture, w 4 antique, and so on. To implement the expected utility approach, it is necessary to first determine the type of utility function. Both theory and practical experience have shown that Exponential utility functions are the most widely used risk-sensitive utility functions because they can model a spectrum of risk attitudes. The Exponential Utility Function is a particular functional form for the utility function. One simplest and commonly used version of exponential utility function can be expressed as U aw ( w) = e Where, U( ) as the utility function and w as wealth. a is a positive scalar parameter. 5.3 MODEL FORMULATION Notations: Let the number of items sold by the manufacturer, be S and p be proportion of items sold with warranty and (1- p) portion of S items were sold without warranty. Here, we consider two conditions: items are sold with warranty or without warranty. Subscripts m and b stands for manufacturer and buyer respectively. Let, k be the proportion of buyers without warranty coming back to manufacturer for repairing of the faults/defects. N b (L) represents the number of valid claims made by the buyer per item and N(L) is the total number of failures for S items if sold with lifetime warranty over the lifetime L. E[N b (L)] is the expected number of failure per item experienced by the buyers over the lifetime. U m (Y) and U b (X) denote the manufacturers and buyers continuous utility functions for profit Y and repair cost X respectively. 120

121 U m is an individual manufacturer s utility function U b is the aggregate utility function representing the entire buyer s risk preference as a whole. Λ m (t) is the manufacturer estimated failure intensity. Λ b (t) is the per item failure intensity for an individual buyer during the lifetime Let r b represents the cost of rectification (repair cost) for a buyer in each occasion of failure if the item is not warranted. When all rectifications are carried by the manufacturer for some technical reasons, the policy is called manufacturer s technical monopoly. It is assumed that there is no manufacturer s technical monopoly in our case. r m represents the manufacture s per occasion cost of rectification (repair cost) which is the actual rectification cost when the item is covered under warranty. W is the warranty price offered by the manufacturer during the time of purchase. Let the product cumulative failure distribution be F(t) with density function f(t) = df(t)/dt and The failure intensity function can be expressed by Λ() t = f ( t) ( 1 F() t ) Therefore the Table 5.1 represents the basic expressions for manufacturer or buyers when the product is sold with or without warranty. Table 5.1: Basic expressions (Product can be sold with or without warranty) Product Total Number of Items Sold Manufacturer Buyer Revenue Cost Cost/Item Warranted ps psw pn (L)r m W Not warranted (1-p)S (1-p)kN(L)r b (1-p)kN(L)r m N b (L)r b Modelling Risks in Lifetime Warranty Assumptions Item failures are statistically independent. 121

122 Item failure, in a probabilistic sense, is only a function of its age. The time to carry out a rectification action by repair or replacement is negligible compared to the mean time between failures and this time can be ignored. An item failure results in an immediate claim and all claims are valid. Failures over the warranty period modelled at the system (or item) level. Manufacturer s cost of rectification r m and the buyer s cost of each rectification r b are constant over the warranty period. Risk averse manufacturer: The Manufacturer s utility function U m (Y) for profit Y is concave (risk averse) and strictly increasing as most manufacturers prefer a higher profit to a lower profit. Thus, it follows from Jensen' s inequality (Zlobec (2004); Li )2000)), that if Y has finite mean then, E[U m (Y)] < U m [ E(Y)] and [y 1 <y 2 ] [U m (Y 1 ) < U m (Y 2 )] Risk averse buyers: The buyer s utility function U b (X) for repair cost X is concave (also risk averse) and strictly decreasing as buyer prefer a lower cost to a higher cost. Thus, it follows from Jensen s inequality that if Y has finite mean then, E[U b (X)] < U b [ E(X)] and [x 1 < x 2 ] [U b (X 1 ) > U b (X 2 )] Buyer s Acceptances of Warranty In determining the worthy of buying a warranty, a buyer may first estimate the total repair cost r b N b (L) of his or her product during the defined lifetime and then compare it with the given warranty price W in terms of the expected utility. Since r b is constant, a buyer s total repair cost is given by r b N b (L) which is estimated by his or her perceived product failure intensity Λ b (t). The higher the buyer estimate of the product failure rate, the more likely he or she would be willing to buy the warranty. Let n * b be the number of product failure when buyers are indifferent between the warranty price W and the total repair cost r b N b (L) in terms of the expected utility. Certainty equivalent of similar concept can be found in Keeney and Raiffa (1976). By definition, a certainty equivalent of W is an amount of r b N b (L) such that the decision maker is indifferent between W and r b N b (L), therefore, it is given by ( W ) E[ U ( r N (L))] U b = b b b (5.1) 122

123 ( = n ) [ ( r N L) )] U ( r n ) P N ( L) E U b b b ( = b b b b b (5.2) n b = 0 Therefore, it can be expressed as U b [ ] = U ( r n ) P( N ( L) = n ) ( W ) E U ( r N ( L) ) = b b b b n b = 0 b b b b (5.3) But when failures of product follows non-homogeneous Poisson process that is failures are time dependent then the one has the following equation Pr ob{ N b ( L) = n b ) = { L 0 Λ b ( t) dt} n b n b e! L Λb ( t) b dt 0 (5.4) Λ b (t) is the buyer estimated failure intensity function. Failures are modelled as nonhomogeneous Poisson process since failure occurrences of most products are time or age dependent. This means more failures as the product ages. However, there are also some other products which observe constant or bathtub type failure mode. These models can be extended by considering the general failure rate function. One form for Λ b (t) using non-homogeneous Poisson process with β ( b b F t) = 1 exp( ( λ t) ) ] can be expressed by: ( β 1) b Λ ( t) = λ β ( t) (5.5) b b b with the shape parameters β b > 1 and inverse characteristic life λ b > 0. This is an increasing function of time or age t. Note that this corresponds to the failure rate of a two-parameter Weibull distribution. Let the buyers risk aversions be represented by exponential utility functions, which is U b (Y)= e cy, c>0, (5.6) 123

124 Where c is the risk parameters representing the buyers risk preferences. A buyer attitude toward risk can be determined by the nature of c. A buyer is more risk-averse if the risk parameter increases. A risk parameter c = 0, indicates a risk neutral buyer. In this study, the buyer s risk parameter c represents a wide spectrum of buyers' risk preferences. In the exponential utility functions, the absolute risk aversion measure is constant for all X, implying that the exponential function represents only the constant risk-averse case. The reasons for considering the constant risk-averse is the use of functions characterising the behaviour of the increasingly risk-averse manufacturer or decreasingly risk-averse buyer or conversely, the decreasingly risk-averse manufacturer or increasingly risk-averse buyer is more computationally complex, but the response surface has been shown to be quite similar to that obtained using constant risk aversion (Moskowitz and Plante (1984) ). Therefore, from Equations (5.3) and (5.6), it can be expressed as the Equation 5.7 cw crb n e b nb = 0 ( N ( L) = n ) e = P (5.7) b By combining Equations (5.4), (5.5) and (5.7), a new can be expressed as b e cw = L 1 n β { b bt b dt} b λ β cr n 0 e b b * n * 0 nb! b = e L β 1 t b λbβb dt 0 (5.8) Which can be written as Equation (5.9) exp β n ( ) ( ) ( ) [ b ] b β λ exp[ ( ) b bl λbl ] cw exp cr n = n b = 0 b b n b! (5.9) By taking logarithm on both sides of the Equation (5.9) the buyer s expected warranty price can be expressed as in the Equation (5.10) W 1 = ln exp c nb = 0 β ( ) [( ) b nb β ] exp[ ( ) b λbl λbl ] cr n b b n b! (5.10) 124

125 From the above equation it will be worthy for the buyer to accept the warranty offer if the buyer s estimated number of failure is more than and equal to indifferent failure n b * Accordingly, a buyer whose expected failure is higher than n b * would buy the warranty if the total estimated repair cost is higher than the warranty price W. Notice that probability to buy this warranty is determined by n b *, which, in turn, is determined by W. Based on the information about the buyer s willingness to pay for the warranty price W, a manufacturer may determine the warranty price such that the expected total profits during the warranty period is maximised Manufacturer s profit The manufacturer s expected total profit E[π(L)] for warranty and rectification related services during the lifetime under warranty policy can be expressed as E π ( L )] = p ( SW E [ N ( L )] r ) + (1-p) k[e[n(l)]r b E[N(L)]r m ] [ m = ( SW E [ N ( L )] r m ) p + (1-p)kE[N(L)]( r b -r m) (5.11) The manufacturer estimates expected number of failures N(L) over the lifetime from S number of products based on the failure intensity Λ m (t). The number of failures over the warranty period, is a function of L, and is a random variable. Let n * m be the expected number of failure of a product when manufacturer is indifferent between the profit from offering warranty product and profit from the non warranted product in terms of the expected utility. Manufacturer s total expected profit when it is selling with such warranty offer is p SW E [ N ( L )] r ) + (1-p)kE[N(L)](r b -r m ) from ( m S items and the manufacturer s profit when it is not warranted comes from the rectification charges of the defective and failed product (only from those buyer s returning to the manufacturer for rectification) is given by k ( r r ) n. For a riskaverse manufacturer with an increasing utility function U m (Y), the certainty equivalent for profit can be expressed from the following relationship. m [ pws pkn( L) r (1 p) kn( L)( r r )] = E U ( k r r ) U ( m b m b [ Sn ] P[ N ( L) = n ] m b m m m m m m (5.12) 125

126 As the manufacturer estimated failure intensity is time dependent, it follows the Nonhomogeneous Poisson process (NHPP) and here intensity function can be expressed as ( β 1) m Λ m ( t) = λ m β m ( t) (5.13) with the parameters β m > 1 and λ m > 0. This is an increasing function of t. Note that this corresponds to the failure rate of a two-parameter Weibull distribution. Since the items are statistically similar then with more information on product failure manufacture can have probability of n m failures over life L for any product as given by: Pr ob{ N m { L 0 Λ m ( t) dt} n b L Λm ( t) dt 0 ( L) = nm ) = (5.14) n! m e E β β [ N m m m ( L)] = λ m ( L ) (5.15) β β [ m m E N( L)] = Sλ m ( L ) (5.16) Similar to the buyers risk aversions, the manufacturer s risk aversions is assumed to be exponential utility functions, which is given by U m ay ( Y ) e =, a> 0, (5.17) where a is the risk parameters representing to the manufacturer s risk preferences. a > 0 indicates risk averse manufacturer, whereas a < 0 indicates risk seeker and a = 0 means a risk neutral manufacturer. Manufacturers profits for non warranted product, is given by k( r r N( L) b m) Utility of expected profit for warranty and no-warranty strategy for indifferent manufacturer s decision point is as following Equation (5.18): e a [ ps ( W ( λ ) m ) + ( ) ( ) m ml β rm 1 p Sk λml β ( rb rm )] = 126

127 n m L βm 1 m mt a( r b r m ) Skn m λ β 0 e. n m = 0 n m! e L m 1 m m t λ β β 0 (5.18) Let the difference between the buyer s repair cost at each occasion of failure and the manufacturer s actual cost of repair or rectification (r b -r m ) be d and the proportion of non-warranted buyers (1- p) by q. The Equation (5.17) can be rewritten as Equation (5.18) when failures follow the Non-homogeneous Poisson process and by taking inverse logarithm on both the sides of the Equation (5.17), the new Equation (5.18) can be obtained a β ( ) [ ( ( ) m β m ) ( ) m λ L ps W λ L r + qsk λ L d] = ln e adsknm. m m m n β m ( ) [ m λ ] e ml n nm = 0 m! βm From which the Equation (5.19) can be developed (5.19) W = 1 ln aps n β m ( ) [( ) m λ λ L ] e m L adskn m m e. n nm = 0 m! βm + q p k β ( ) m β λ ( ) ( ) m ml rb rm λml rm (5.20) 5.4 SENSITIVITY ANALYSIS OF THE RISK MODELS Models developed in this chapter appear analytically intractable. Search algorithm and simulation can be used to explore the various parameters and warranty price combinations attractive to buyer and the manufacture s point of view. This is influenced by the perception of the buyers on failure intensity during warranty period, buyers and manufacturer s risk preference and costs associated with repair/rectifications. In this section, a sensitivity analysis of the buyer s intension to pay for warranty price with respect to the following factors is presented: (1) buyer s risk preference, (2) buyer s anticipated product failure intensity, and (3) buyer s repair costs, if not 127

128 warranted. Secondly, analysis of the effect of manufacturer offered warranty price on (1) manufacturer s risk preference, (2) manufacturer anticipated product failure intensity, and (3) buyer s repair costs in each occasion if not warranted, and 4) buyers return rate to the original manufacturer for repair if not warranted is presented Sensitivity Analysis of Buyer s Willingness to Pay for Warranty Price For all cases it is assumed that 60% of the sold products are sold with warranty and 40% products are sold without this type of warranty, which implies that p = 0.6 and q = 0.4 for all occasions of our analysis. (1) Effect of buyer s risk preferences on the warranty price In this numerical analysis part, the buyer s risk parameters c is varied systematically from 0.1 to 1.0 representing a wide spectrum of buyer s risk preferences. A buyer become more risk averse as the risk parameter c increases and he/she becomes risk neutral if the parameter is zero. For the purpose of simplicity, in this sensitivity analysis it is assumed that the buyer s anticipated lifetime of the product L = 3 years, the buyer s anticipated nonhomogeneous Poisson process parameters: inverse characteristic life λ = 0.325/year, and shape parameter β = 2, buyer s repair cost of each failure r b = $30, if not warranted and the buyer s repair cost is same for all occasions whether it is repaired by the manufacturer/dealer or by an outside repairer. A computer simulation program generates the Table 5.2 for the buyer s acceptance of warranty price with variation of the buyer s risk preference parameter and the buyer s number of failure over the lifetime. From the Table 5.2, Figure 5.2 can be developed which represents the effect of buyer s risk parameter on the warranty price (the buyer is willingness to pay) 128

129 Table 5.2: Warranty price (W) in $ for different Buyer s risk preferences (c ) Buyer s risk parameter c n * The Figure 5.2 clearly states that the warranty price increases with the increase of the of the buyer s risk preference which indicates that, the warranty price increases as the buyer becomes more risk averse and the warranty price decreases as the buyer becomes less averse. This means that the buyer with higher risk averseness is willing to pay higher warranty price to avoid any risk. Manufacturer or dealers can use this buyer s psychology while being pricing the warranty. Figure 5.2: Effect of buyer s risk preference parameter on the warranty price 129

130 From the Table 5.2 Figure 5.3 can developed which represents the effect of buyer s anticipated number of product failure on the warranty price (the buyer is willing to pay) for buyer s with particular risk parameter. The Figure 5.3 shows that the buyer s willingness to pay warranty price increases as the number of failure increases and it is straight forward. Figure 5.3: Effect of buyer s anticipated product failure over the lifetime on the warranty price. This Figure 5.3 shows that the buyers are ready to pay higher warranty price as the anticipated number of product failure increase. The buyer s such intension increases almost linearly with the anticipated number of failures. The Figure 5.4 shows the combined effect of risk parameters and number of failures on the warranty price 130

131 Figure 5.4: Combined Effect of buyer s risk parameters and number of failures on the warranty price (3) Effect of buyer s repair cost on the warranty price Although the buyer s repair cost is applicable only for the buyers with non-warranted item, it has a significant effect on the warranty price. In this analysis, the buyer s rectification or repair cost for each failure is varied systematically from $30 to $70 representing a wide range of buyer s repair cost. It is noted that it is assumed that this repair costs are constant for a particular product all along its lifetime and the variation is made for policy purpose. Let the lifetime of the product be L = 3 years, the buyer s anticipated nonhomogeneous Poisson process parameters be: inverse characteristic life λ = 0.325/year, and shape parameter β = 2, buyer s risk parameter c = 0.5. The computer program generates the Table 5.3 for the buyer s acceptance of warranty price with variation of the buyer s risk preference parameter and the buyer s anticipated number of failure over the lifetime of 3 years. Figure 5.5 can be developed from Table 5.3 which shows the effect of buyer s repair cost on the warranty price (the price the buyer is willing to pay). 131

132 Table 5.3: Warranty price (W) in $ for different Buyer s repair cost (r b ) n r b Figure 5.5: Effect of buyer s repair cost on the warranty price when buyer anticipated product failure is n = 3. Figure 5.4 shows that the buyer s willingness for warranty price increases linearly as the buyer s repair cost increases which imply that the buyers are ready to pay higher warranty price if repair cost is higher. That is a greater proportion of buyers will be 132

133 interested in buying a warranty as the repair price goes up. In contrast, the buyer s intent is to pay less for the warranty if the repair costs are less. (2) Effect of buyer s anticipated product failure intensity on the warranty price An analysis of the sensitivity of buyer s anticipated product failure rate or intensity of failure on the buyer s acceptance of warranty price is made here. To see the influence of product failure rate on the warranty price the λ b is varied from to.525 per year as in the Table 5.4. It is assumed that the lifetime of the product be L = 3 years, the buyer s anticipated non-homogeneous Poisson process parameters shape parameter be β b = 2, buyer s risk parameter c = 0.5. The computer simulation program generates the Table 5.4 for the buyer s accepted warranty prices with variation of the failure intensity parameter λ b. Table 5.4: Effect of Buyer s repair cost on the Warranty price n λ The Figure 5.5 can be developed from the Table 5.4 which represents the effect of buyer s anticipated product failure intensity on the warranty price (the buyer is willing to pay). The Figure 5.6 shows that the buyer s willingness for warranty price increases as the buyer s anticipated failure intensity increases. This implies that the buyers are ready to pay higher warranty price for higher product intensity of failure. 133

134 Figure 5.6: Effect of buyer s anticipated failure intensity (λ b ) on the warranty price Conversely, when the buyers anticipate lower product failure intensity, they will be willing to pay less warranty price. (3) Effect of failure parameters on the warranty price The combined effects on warranty price (Figure 5.7) can be developed from the data shown in the table 5.5. Table 5.5: Effect of buyer s anticipated failure parameters on the warranty price β λ

135 Figure 5.7: Combined effects of failure parameters on the warranty price The Figure 5.7 shows an increment of warranty price with the increment of both the failure parameters λ and β. This implies that buyers are willing to pay more warranty price when the occurrence of product failure is higher Sensitivity Analysis of the Manufacture s Warranty Price In this subsection, the sensitivity of manufacturer s optimal warranty pricing with the variation of manufacturer s risk preference, manufacturer estimated product failure intensity, buyer s repair cost and buyer return rate to the manufacturer/dealer for repair of the failed product will be analysed respectively. (1) Effect of manufacturer/dealer s risk preference on the warranty price Similar to the Sub-section 5.4.1, claws (1), the manufacturer/dealer risk parameters a is varied systematically from 0.05 to 1.0 representing a wide spectrum of manufacturer/dealer risk preferences. A manufacturer/dealer become more risk averse as the risk parameter a increases and the manufacturer/dealer is said to be risk neutral if its risk parameter a is zero. 135

136 Table 5.6: Warranty price (W) in $ for different manufacturer s risk preferences (a ) and n m * n m a Let the expected lifetime of the product L be 3 years, the manufacturer/dealer s estimated non-homogeneous Poisson process parameters be: inverse characteristic life λ m = 0.325/year, and shape parameter β m = 2 (this indicates an increasing rate of failure over time), buyer s repair cost for each failure r b = $30 and manufacturer s actual cost of each repair r m = $10, so that the difference between these two repair costs d is $20. It is assumed that the rate of returning of buyers to the manufacturer/dealer for repair of failed product is 20% that is k = 0.2. A computer program generates the Table 5.6 for the manufacturer/dealer s optimal warranty price with variation of the buyer s risk preference parameter and the buyer s number of failure over the expected lifetime of the product. The Figure 5.8 can be drawn from the Table 5.6 which shows the effect of manufacturer/dealer s risk parameter on the warranty price. The Figure 5.8 clearly shows that the warranty price increases with the increase of the of the manufacture/dealers risk preference parameter a. The warranty price increases as the manufacturer/dealer becomes more risk averse and the warranty price decreases as the manufacturer/dealer becomes less averse. This implies that, as the manufacturer/dealer becomes more risk-averse, manufacturer/dealer would charge a higher warranty price to the buyer to avoid risking loss. 136

137 Figure 5.8: Effect of manufacturer s risk preference parameter on the warranty price when buyer anticipated product failure is n m = 3. The Figure 5.9 can also be developed from the Table 5.6 to visualise the effect of the number of failures over the lifetime on the manufacturer s charge for warranty price for a particular risk parameter. Figure 5.9: Effect of manufacturer/dealer expected product failure over the lifetime on the warranty price. The Figure 5.9 interprets that the manufacturer s charge for warranty price increases as the number of failure increases. This means that the manufacturer/dealer must charge higher warranty price for higher number of expected product failures. 137

138 (2) Effect of manufacturer/dealer s estimated product failure intensity on the warranty price In this analysis, it is assumed a range of manufacturer/dealer s estimated product failure intensity (rate) to observe the effect on the manufacturer s optimal charge for warranty. To do so, λ m is varied from a range of 0.125/year to Here, the manufacturer/dealer risk parameters a is kept constant as 0.4. Let the expected lifetime of the product L be 3 years, the manufacturer/dealer s anticipated non-homogeneous Poisson process shape parameter be β m = 2 (this indicates an increasing rate of failure over time), buyer s repair cost for each failure be r b = $30 and manufacturer s actual cost of repair be r m = $10, so that the difference between these two repair costs d is $20. Let k =0.2 (this means that 20% of the non-warranted buyers will be return to the manufacturer/dealer for repair of their failed item. Let n* m =3. The computer program generates the Table 5.7 for the manufacturer/dealer s optimal warranty price with variation of the buyer s risk preference parameter and the buyer s number of failure over the expected lifetime of the product. Table 5.7: Warranty price (W) in $ for different manufacturer s failure intensity λ m W The Figure 5.10 can be generated from the Table 5.7 which exhibits the effect of failure intensity over the manufacture/dealer s charge for warranty price. 138

139 Figure 5.10: Effect of failure intensity over the manufacture/dealer s charge for warranty price. Analysis of Figure 5.10 shows that the manufacturer s charge for warranty price increases with the increase of the of the failure intensity λ m. This implies that, the higher manufacturer estimated failure intensity results in higher cost of servicing of warranty claims and this leads the manufacturer to charge higher warranty price and vice versa. (3) Effect of failure parameters on the warranty price Here, it is assumed that a range of manufacturer s estimated product failure intensity (rate) and different shape parameters to observe the effect on the manufacturer s optimal charge for warranty. To do so, λ m is varied from a range of 0.350/year to 0.65 and shape parameter is also varied from a range of 1 to 3.5. Here, the manufacturer/dealer risk parameters a is kept constant as 0.4. By keeping all other parameters and variable as before Table 5.8 can be generated. Now Figure 5.11 can be developed from Table 5.8. This Figure shows that manufacturer s charge for warranty price increases with the increase of both the failure intensity and the failure shape parameters. At β =1 which is a constant failure distribution, the warranty charge increases very slowly with the increase of failure intensity whereas at β = 3.5, the charge for warranty increases sharply. 139

140 Table 5.8: Warranty price for different failure intensity and Shape parameter β m λ m Figure 5.11: Effect of failure parameters on the manufacturer s charge for Warranty price This is because at constant failure rate situation, aging is not in effect but for higher β aging is also affecting the charge for warranty from the manufacturer s point of view. (4) Effect buyer s repair costs on warranty price Now, it is time to look into the effect of buyer s repair costs on the manufacturer/dealer s charge for warranty price. Let the buyers repair cost r b be varied from a range of $ 40 to $80 per occasion. It is assumed that buyer s repair 140

141 cost remain constant over for each case. Let the manufacturer/dealer risk parameters a is kept constant as 0.4. And the expected lifetime of the product L be 3 years. Manufacturer/dealer s anticipated non-homogeneous Poisson process parameters are: inverse characteristic life parameter λ m =.443 and shape parameter β m = 2 (this indicates an increasing rate of failure over time). Let it also be assumed that the manufacturer s actual cost of repair r m = $10 for each occasion and constant over the warranty period. Let buyer s return rate to the original manufacturer for repair k =0.2 (this means that 20% of the non-warranted buyers will be return to the manufacturer/dealer for repair of their failed item). Let n * m=3. The computer program generates the Table 5.9 for the manufacturer/dealer s optimal warranty price with variation of the buyer s risk preference parameter and the buyer s number of failure over the expected lifetime of the product. Table 5.9: Buyer s repair cost vs manufacturer set Warranty price (W) in $ r b W Figure 5.12 generates from Table 5.9 which shows the effect of buyer s cost of repair over the manufacture/dealer s charge for warranty price. Figure 5.12: Effect of buyer s cost of repair over the manufacture/dealer s charge for warranty price. 141

142 Analysis of Figure 5.12 shows that the manufacturer s charge for warranty price decreases with the increase of the of the buyer s cost of repair. This occurs because when the buyer s repair cost is higher, the more buyers will prefer warranty rather than repairing the failures or defects and this in turns provides an opportunity for the manufacturer to lower the warranty price to compete effectively in the market. As a result the manufacturer/dealer can make more money through the warranty sale since in the repair market manufacturer/dealer is not the only competitor in most of the cases. (5) Effect of buyer rate of return (k) to the manufacture/dealer for repair on warranty price A portion of non-warranted buyer may come back to the original manufacturer/dealer for repair of their failed products. Here, the effect of buyer rate of return (k) to the manufacture/dealer for repair on the warranty price will be analysed. For this purpose proportion of returning buyers be varied from 0% to 100%. This implies different values of k ranging from 0 to 1. Where, k = 0 means buyers are not coming back to the original manufacture/dealer for rectification of their failed product and k = 1 means 100% of the buyers are coming back to the original manufacturer or/dealers for repairing the failed product. This is possible if the manufacturer holds the technical monopoly. Let the expected lifetime of the product L be 3 years, the manufacturer/dealer s anticipated non-homogeneous Poisson process parameters are inverse characteristic life parameters are λ m =.443 and β m = 2. Let buyer s repair cost of each failure r b = $30 and manufacturer s actual cost of repair r m = $10, so that the difference between these two repair costs d is $20. Let n m =3. The computer program generates the Table 5.9 for the manufacturer/dealer s optimal warranty price with variation of the buyer s risk preference parameter and the buyer s number of failure over the expected lifetime of the product. Table 5.10: Buyer rate of return (k) vs Warranty price (W) k W

143 The Figure 5.13 can be generated from the Table 5.10 which shows the effect of buyer s rate of return to the original manufacturer for rectification on the manufacture s charge for warranty price. Figure 5.13: Effect of buyer s cost of repair over the manufacture/dealer s charge for warranty price. Analysis of Figure 5.13 shows that the manufacturer s charge for warranty price decreases with the increase of the rate of buyer s return to the original manufacturer/dealer for rectification. For some technical reasons such as supply logistics, it is economical for the manufacturer to keep more buyers within warranty (Murthy et al (2004)). Therefore, when the rate of return of buyers is higher it is an opportunity for the manufacturer to keep the buyers in warranty by lowering the warranty price CONCLUSIONS In this chapter, a study and analysis of both buyers and manufacturer/dealers risk attitudes towards lifetime warranty policies using utility functions, and certainty equivalent are carried out. Applying the exponential utility function, the decision models that maximise the manufacturer/dealer s certainty profit equivalent are derived. Risk models are developed for products with time dependent failure intensity (rate). These models have been proposed with Non homogeneous Poisson s 143

144 process for failure intensity function, constant repair costs, and concave utility function. These warranty models incorporate risk preferences in finding the optimal warranty price through the use of the manufacturer s utility function for manufacturer s cost and the buyer s utility function for repair cost. Furthermore, the sensitivity of the optimal warranty price is analysed with a numerical example with respect to the factors such as the buyer s and the manufacturer/dealer s risk preferences, buyer s anticipated and manufacturer s estimated product failure intensity, the buyer s loyalty to the original manufacturer/dealer in repairing the failed product and the buyer s repair costs for unwarranted products. Here, it is ignored the discounting of the repair costs for present value and corrections for inflation in modelling warranty cost which is important for products with a long service life. There is scope for future research with other forms of failure distributions, impact of preventive maintenance and scope for replacements during lifetime with possible trade-ins. These models can be extended for products with constant failure rate or bath tub type failure intensity. In application, the specific forms of the prior distribution of the product failure and the utility functions should be supplied, and their parameter should be estimated for particular products and monitory asset. These issues remain subjects for future research. Lifetime warranty is one of the important types of Long-term warranty policies. In this chapter and in the previous chapters, different lifetime warranty policies, cost models, and buyer s and manufacturer s risk attitudes towards lifetime warranties were developed and discussed. Another important type of Long-term policy is service contract which has not yet been properly worked on. The next chapter (Chapter 6) focuses on developing policies and conceptual cost models for various service contracts policies considering both corrective and preventive maintenance as servicing strategies over the contract period. 144

145 CHAPTER 6 MODELLING POLICIES AND COSTS FOR SERVICE CONTRACTS 6.1. INTRODUCTION Complex and expensive equipment/assets require special tools and personnel to carry out rectifications (repairs and replacements) when they fail. Often it is uneconomical for the owner/user of the equipment/asset to have such specialist tools and personnel in house. In such situations, it is more economical to contract out (to outsource) the maintenance management (preventive and corrective) of such equipment/asset to external agents. The external agent could be the manufacturer of the asset or an independent third party interested in investing in that asset (Murthy and Ashgarizadeh, 1995). Estimation of costs for these type of contracts is complex and it is important for both the users/owners and the service provider. There is a growing demand to develop mathematical models for understanding future costs to build it into the contract price. Failure to do so may result in loss to the service provider or the user/owner because of uncertainties associated with failures and their implication on business. These costs depend on failure mode and distribution of the equipment/asset failure and the servicing strategies (e.g. corrective maintenance, planned preventive maintenance, and/or inspection procedures and reliability) to be carried out during the contract period. Long-term warranties linked to after sales service were discussed in Chapter 2 which included extended warranty, warranty for used products, lifetime warranties and 145

146 service contracts. A reasonable academic research work has conducted on the extended warranty and warranty for used products. In the previous chapters, lifetime warranty policies were discussed together with the development and analysis of taxonomy and cost models for those policies. In this chapter, policies and conceptual cost models of another important type of long-term warranty, called service contract are developed. Since, the demand for servicing occurs due to product (preferably complex asset/equipment) failure, the starting point for such an analysis is the modelling of the failures over the contract period. A brief background of service contracts is presented in Section 6.2. Section 6.3 discusses different servicing strategies that involve corrective maintenance and planned preventive actions. Service contracts together with various servicing policies are developed and discussed in Section 6.4. In section 6.5, conceptual cost models for service contracts are proposed. Finally, a conclusion is drawn to see the contributions and future scope of this chapter SERVICE CONTRACT - BACKGROUND A service contract is the outsourcing of maintenance actions where equipment/asset failures are rectified by an external agent for an agreed period of time. The service provider s profit is influenced by many factors such as the terms of the contract, reliability of the asset, servicing strategies, costs of resources needed to carry out maintenance and to provide such services. Blischke and Murthy (2000) proposed a policy for service contract with the scope for negotiation. In recent years service contract has received significant attention from the practitioners due to increased profit through selling those services and reduction of risk from owners due to better maintainability provided by the experts in the trade. Murthy and Yeung (1995) proposed stochastic models for expected profit. Murthy and Ashgarizadeh (1995) developed a model to characterise the optimal strategies for a single customer and service provider. Ashgarizadeh and Murthy (2000) extended this to multiple customers. Rinsaka and Sandoh (2006) proposed mathematical models for setting suitable charge of service contracts in the case where a manufacturer offers an additional warranty service under which the failed system is replaced by a new one for its first failure, but minimal repairs are carried out to the system for its succeeding failures before the contract expires. Detailed literature reviews can be seen in Chapter 2. All the above 146

147 models only considered corrective maintenance (CM) as the servicing strategy during the contract period. These models considered rectification (repair and/or replacement) actions only on failure of the item. They ignored or failed to include planned preventive maintenance (PM) actions during the contract period. But in a real life situation, a cost effective servicing policy for a repairable system should take into account corrective maintenance actions (repair/replacements) in case of failure or fault, as well as a well planned preventive maintenance actions to prolong the service life by retaining the reliability, and/or to reduce the risks of failures SERVICING STRATEGIES DURING THE CONTRACTS Servicing strategies during the contract period can be developed by understanding the reliability and its analysis. Failure data are in many cases time or usage dependent for certain conditions. In a probabilistic sense, asset/system failures are functions of usage and/or age. In real life situations, servicing strategies involve both corrective maintenance (CM) and planned preventive maintenance (PM) actions. Corrective maintenances are unscheduled actions intended to restore the system/asset to its operational state through corrective actions after the occurrence of failures or detection of fault in the system. This usually involves replacement or repairing the component that is responsible for the failure of the overall system. Corrective maintenance is performed at unpredictable intervals because a component's failure time is not known a priori. The objective of corrective maintenance is to restore the system to satisfactory operation within the shortest possible time. Corrective maintenance is typically carried out in three steps: Diagnosis of the problem: The maintenance technician must take time to locate the failed parts or otherwise satisfactorily assess the cause of the system failure. Repair and/or replacement of faulty component(s): Once the cause of system failure has been determined, action must be taken to address the cause, usually by replacing or repairing the components that caused the system to fail. 147

148 Verification of the repair action: Once the components in question have been repaired or replaced, the maintenance technician must verify that the system is again successfully operating. In contrast, planned preventive maintenance actions are carried out to reduce the likelihood of failures or prolong the life of the asset/system (Murthy and Jack, 2003). This type of maintenance is the schedule of planned maintenance actions aimed at the prevention of breakdowns and failures of an item before it actually occurs. It is designed to preserve and enhance equipment reliability by replacing worn components before they actually fail. Preventive maintenance activities include equipment checks, partial or complete overhauls at specified periods, oil changes, lubrication and so on. Long-term effects and cost comparisons usually favour preventive maintenance over performing maintenance actions only when the system fails. Long-term benefits of preventive maintenance include: Improved system reliability. Decreased cost of replacement. Decreased system downtime. Better spares inventory management. Normally, preventive maintenance actions are carried out at discrete time instances. But when PM actions are carried out fairly frequently, it can be treated as occurring continuously over time. Preventive maintenance actions can be classified into different categories such as: Time based maintenance: Here the preventive maintenance actions are carried out at fixed interval of time. Block replacement policy is one example of this type of PM. Age based maintenance: Age replacement is an example of PM whereby replacement is made on the basis of the age of the item. Usage based maintenance: Similar to the Age based policy, PM actions are taken based on the usage of the item. Example: rail grinding based on the Million Gross Tonnes of usage. 148

149 Condition based maintenance: Iin this type of PM, actions are based on the condition of the item being maintained which involves inspection and monitoring of the wear or deterioration process of the components of the item such as crack growth in a mechanical component. In some cases, it is difficult to detect the variable of interest directly, and in such cases some other variables may be used to obtain estimates of the variable of interest. For example, the wear of journal bearing can be traced by measuring the vibration, noise, or the temperature of the bearing case, since there is a strong correlation these variables and the bearing wear. Opportunity based maintenance: This is applicable for multi-component items, where PM or CM actions for one component create an opportunity to carry out PM actions on some other or remaining components of the item. Design out maintenance: this involves performing modifications through redesigning the component. As a result, the new component has better reliability characteristics. Many different types of model formulations have been proposed to study the effect of preventive maintenance on process deteriorations and occurrence of item failures and to derive optimal preventive maintenance strategies. In the case of a repairable system, both corrective maintenance and planned preventive maintenance take into account different types of servicing actions depending on the failure modes and types. These actions are classified as per degree of restorability as shown in Figure 6.1. The item/system fails at point t due to the malfunctioning of one or more components. Various servicing actions can be adopted at this point to restore the functionality of the item/system. 149

150 t Hazard rate λ(t) c b a Age/usage Figure 6.1: Failure rate with effect of various maintenance actions The probable servicing strategies applicable for service contracts are: (i) (ii) Replacement: the failed system can be replaced with a new identical system or with a used but good item/system. This turns failure rate of the item to its original state if replaced with a new one (see curve a in Figure 6.1). This implies that a replacement with new restores the full reliability and turns the hazard rate to zero. If replaced with a used, good one, it restores a part reliability and the hazard rate falls any point between as good as new and as bad as old depending on the age and condition of the replaced item/system. Overhauling or perfect repair is a restorative maintenance action that enables the system to be in a as good as new condition as it turns the hazard rate almost to zero (See curve a or close to curve a ). Here the failure time distribution of the repaired item is almost identical to that of a new one and one can model successive failures using an ordinary renewal process. This type of maintenance action is normally taken before an item/asset has reached a defined failed state (Jardine, 1973). (iii) Imperfect repair restores a substantial portion of wear/damage and like replacement with a used, good item, it improves reliability of the failed item/asset to a certain level whereby the hazard/failure rate falls in between as good as new and as bad as old (see curve b ) depending on the type and quality of the repair works. 150

151 (iv) A minimal repair is the repair/replacement only of the failed component/s of a complex item/system and other components of that item/system remain untouched. This makes insignificant improvement of reliability and the condition after maintenance is called as bad as old (curve c in), since the hazard rate of other components remain unchanged. Here the failed item is returned to operation with the same effectiveness as it possessed immediately prior to failure. Failures then occur according to a non-homogeneous Poisson process with the intensity function having the same form as the hazard rate of the time to first failure distribution MODELLING POLICIES FOR SERVICE CONTRACT In modelling cost of service contracts, three different policies are proposed where the rectification of an item takes into account both corrective maintenance and planned preventive maintenances. The corrective maintenance could be replacement of the whole item in case of complete failure or minimal repairs (repair replacement of a part/components/s). Preventive maintenance actions are done at constant intervals which retain the item/asset reliability to some extent. Three service contract policies are: Service contract policy 1: Under this policy, the contract terminates when contract period reaches to a time/ usage level L or a renewal of the system is essential due to the complete failure of the system whichever comes first. Under this policy renewal of the system is not included. Here, the system life is normally considered longer than the contract period. Planned preventive maintenance actions at constant intervals and corrective maintenances in the form of minimal or major repairs must be provided by the service provider. According to this policy replacement is not included during the contract period which implies that R L. where L and R are the contract period and the first replacement (renewal) time respectively (Figure 2 represents this policy graphically). Preventive maintenance (PM) is planned at constant intervals. Between two successive preventive maintenances there could be one or more minimal corrective actions. This type of service contract is mostly suitable for complex repairable systems with a longer life. Service contract policy 2: 151

152 Under this policy, the contract period is up to the first replacement (renewal) of the system due to complete failure of System or economically beyond any service. Like policy 1, replacement (renewal) of the system is excluded and constant interval preventive maintenances and corrective maintenances in the form of minimal repairs are included over the contract period. This implies that it has not any trade off or salvage value at the end of the contract period. In this case, L = R, (Figure 3 represents this policy graphically). Note that here, the contract period is a random variable and ceases when the item is economically beyond any service. This policy is suitable for a system or items with a shorter life such as mobile phones. Service contract policy 3: Under this policy, the servicing strategy covers any rectification action including replacement (renewals)/s of the System due to complete failure during the contract period (Figure 4 represents this policy graphically). In bet,en two consecutive replacements, there may be one or more constant interval PM actions and there may also be a number of minimal repairs in between two successive PMs. Replacement or renewal time R is a random variable This policy is applicable for long-term service contracts MODELLING COSTS OF SERVICE CONTRACT FOR DIFFERENT POLICIES Total costs of service contract depends on the type of policies defined in the previous section and may include the cost of planned preventive maintenance and corrective maintenance in the form of minimal repairs and failure replacement, cost of inspections and condition monitoring, cost of risks, and penalties for failure to meet agreed safety, reliability and availability of requirements. Therefore, The total expected costs of service contract C T = C s + C i + C d + C r + C p Where, C s = total expected costs of maintenance services 152

153 C i = expected costs of inspection C d = costs of downtime C r = cost of risks associated C p = penalties for failure to meet contract agreements. Here, it is aimed to develop conceptual cost models for service contracts that focus only on the maintenance services during the contract period. These models can be further extended by adding inspection, downtime, risks associated with such contracts and incurred penalties due to failure to meet contract agreement. Formulation of Models: In modelling service contracts, a number of assumptions are considered Assumptions The following assumptions are made for all the cases. Failure rate increases with accumulated MGT. Servicing actions restore life to some extent. The level of restoration depends on the type and quality of the maintenance performed. Age restoration, after each preventive maintenance (PM) is constant. Preventive maintenance actions are taken at constant interval (x) before a replacement is made throughout the contract period. An item is replaced only when it fails completely otherwise minimal repairs are preferred. All replacement is made with new and identical items. All cost factors are constant over the contract period. Money discount is 1 throughout the contract period (this assumption is true when contract period is short) Notations and Reliability Preliminaries Failure intensity () t = Λ( t kτ ) Λ (6.1) pm where, 153

154 Λ pm (t): Failure intensity at time t, with maintenance. Λ(t): original failure intensity at t when no maintenance is performed. N: Number of times the planned servicing is performed during the contract period N i : Number of times the planned servicing is performed during the ith replacement i = 1, 2, 3,. M: Number of replacements corrective actions. L: Duration (length) of service contract k: Number of times PM is carried up to t. τ α C re C mr C pm C cl Age restoration after each PM. τ = αx, Quality or effectiveness of preventive maintenance action Cost of replacement Cost for each minimal repair. Cost for each PM Expected cost for the last cycle. Let the failure intensity function of the item/asset of life distribution F(t) having [ ( )] ( ) () d F t f t density function f () t = be defined by Λ() t = dt 1 F t When it follows Weibull distribution this can be modelled as: F β ( t) = 1 exp( ( λt) ) (6.2) and f β 1 β () t = λβ ( λt) exp( ( λt) ) (6.3) with the parameters β (Known as shape parameter of the distribution) > 0 and λ (Known as inverse of characteristic function for the distribution) > 0 β greater than 1 indicates an increasing failure rate of the item under study and ageing is predominant in failure mechanism. Then the failure intensity function Λ(m) derived from (1) and 2 can be given by 154

155 Λ( f ( t) 1 F( t) λβ ( λt) exp( ( λt) ) 1 (1 exp( ( λt) ) β 1 β β 1 t ) = = = λβ ( λt) (6.4) β Policy wise cost models for service contracts are proposed here Modelling Cost for Service Contract Policy 1 Under this policy, the contract terminates at a predetermined time L or at a time when renewal is essential due to the complete failure of the item/asset, whichever comes first. This implies that replacement is not covered in this policy. That is M = 0. The Figure 6.2 is the graphical representation of the model for the policy 1. Here, the expected cost of service contract per unit time for repairable item/asset with servicing and preventive maintenance plan is given by the total sum of expected costs of all minimal repairs and the expected costs of all the planned preventive maintenance over the contract period divided by the length of service contract (length of L or R which comes first). That is Expected cost of service contract per unit time = ((Expected costs of all minimal repairs + Expected cost of all planned preventive maintenances) during the contract period) / Length of contract duration. 155

156 Failure intensity Λ(t) τ t Age/usage 0 x x x x Contract duration L Replacement due R Legends τ Λ 1 (t) Λ 2 (t) Λ 3 (t) time of restoration failure rate distribution after 1 st PM failure rate distribution after 2nd PM failure rate distribution after 3rd PM Figure 6.2: Graphical representation of the Service contract policy 1 Expected cost of all minimal repairs over the contract period can be given by N i ( ) k+ 1 x = C Λ () t mr kx k = 0 pm dt (6.5) Now substituting Equation (6.1) in Equation (6.5), the expected cost of minimal repair is given by Expected cost of minimal repair = C N i mr kx k = 0 ( k + 1) x Λ( t kτ ) dt (6.6) 156

157 When failures are modelled as per Weibull distribution from the Equation (6.4) the failure rate is given by: Λ β 1 () t = λ β βt Therefore, from Equation (6.4) and Equation (6.4), expected cost of minimal repair strategy can be expressed as ( k + 1) x β = Cmr λ β ( t kτ ) N i k= 0 kx β 1 dt β [ ] λ (6.7) β β β = Cmr x ( k kα + 1) ( k kα ) N i k= 0 where, τ = α x, and α is the quality of the maintenance or effectiveness of preventive maintenance, α ranges from 0 to 1. α = 1 signifies as good as new and in case of minimal repair α = 0, here the condition after maintenance is called as bad as old since the failure intensity of the whole system remains unchanged. Effectiveness of maintenance on the reliability for other actions such as a major repair or replacement with a used one can fall any point between 0 and 1 depending on the quality of repair action or age of the used system. Expected cost of preventive maintenance during the contract = NC pm (6.8) The total expected cost per unit time C(L,x, N, ) can therefore be expressed as C N 1 = mr L k = 0 β [ ] + N C β β β ( L, x, N ) C λ x ( k kα + 1) ( k kα ) pm (6.9) For policy 1, L= x (N+1). Now an optimal preventive maintenance interval x *, optimal number of PM (N * ) and minimal total expected cost per unit time can be obtained by differentiating equation 6.9 with respect to x and equating it to zero. These optimal values can easily be obtained by programming in any mathematical software such as MATLAB, MAPLE. 157

158 There could be excess age for the item with some trade off or salvage value shown in Figure 6.2. This could be a subject of interest in future studies Modelling Cost for Service Contract Policy 2 Under the conditions of this policy, duration of contract is randomly variable and the contract is terminated any time at which a renewal is mandatory due the complete failure of the item/asset. This implies L = R. Figure 6.3 is the graphical representation of the model for policy 2. t Failure intensity Λ(t) τ A replacement is essential at this point due to complete failure time 0 x x x x Contract duration, L Replacement due, R Figure 6.3: Failure intensity for various servicing strategies for Service contract policy 2. Here, expected total cost per unit time of service contract can be estimated as follows Expected total cost per unit time = (expected total cost of all the minimal repairs + expected total cost of all planned preventive maintenance)/contract duration Which is given by Equation (6.9) with randomly variable L and an optimal contract period L * is the product of optimal number of preventive maintenance (N * +1) and optimal interval of PM (x * ). That is ( N + ) L = 1 x 158

159 Modelling Cost for Service Contract Policy 3 This policy is appropriate for long-term service contract. Under the conditions of this policy, one or more replacements or renewals due to complete failure of the item/asset are covered over the prefixed contract period. Let the total number of replacements be M. In between two successive renewals there could be number constant interval preventive maintenances and there could also be some minimal repairs as corrective actions in between two successive PMs because of the failure of components. In between the last replacement (renewal) and the end of contract duration there may be one or more preventive maintenances as well as some minimal repairs. This excess period is named as the last cycle. It is assumed that the nature and characteristics of the last cycle is similar to those of policy 1. Figure 6.4 is the graphical representation of the model for policy 3. Let R 1, R 2, represent the renewal times respectively. Λ(t) R 1 R 2 A replacement due here 0 x x x x N 1 N 2 N i Last Cycle Contract duration (L) Figure 6.4: Failure intensity for various servicing strategies for Service contract policy 3 The total expected cost per unit time = (expected total costs of all minimal repairs up to the last cycle + expected total cost of all preventive maintenances up the last cycle + expected total cost of replacements + expected cost of last cycle) / Length of service contract. 159

160 Expected total cost of minimal repairs up to the last cycle M 1 β [ ] λ (6.10) β β β = C x ( k kα + 1) ( k kα ) mr i= 0 N i k = 0 Expected total cost of PM up to the last cycle M N i C = pm i=0 where i = 1,2,3,.. (6.11) Total cost of replacement = MC re (6.12) Therefore, the total cost of service contract up to the last cycle can be given by adding equations = C mr M Ni i= 0k= 0 β β [( k kα + 1) ( k kα ) ] M β β λ x + NiC pm + MCre (6.13) i= 0 After the last replacement there may be still some period to complete the contract period. Expected total cost of last cycle (C cl ) can be expressed as C cl 1 λ ( t kτ ) dt + ( Ncl ) C pm (6.14) Ncl β ( kcl + 1) x β = Cmr β k cl x k= 0 Ncl cl = mr kcl =0 β [ cl cl cl cl ] + NclC pm β β β or, C C x ( k k α + 1) ( k k α ) λ (6.16) N cl is the number of PMs in the last cycle Therefore, the total expected cost per unit time C(L, x, N i, M) can be expressed by adding equations 6.13 and 6.16 and is given by equation 6.17 ( L, x, N, M ) C i M N M 1 i β β β β = Cmr λ x [( k kα + 1) ( k kα ) ] + NiC L i= 0 k= 0 i= 0 pm + C cl + MC re (6.17) 160

161 Parameters Estimation Different methods such as method of Least square, method of Moments, and method of Maximum likelihood can be used to estimate the failure parameters. If the data requires, non-parametric analysis can also be applied. An example for estimating parameters by using Maximum likelihood method is presented here in line with Chattopadhyay (1999), where it is considered a time dependent failure distribution which implies that failure rate increases with time. The following section demonstrates the estimation of item/asset failure parameters when failure follows Weibull distribution and the parameters are given by Weibull inverse characteristic life parameter λ and Weibull shape parameter β. Let t i time to its ith failure n T where, i =1, 2, 3, number of failures over contract period is the observation period and 0< t 1 < t 2.< T This implies that there is no failure in (0, t 1 ], one failure in (t 1, t 1 + δt 1 ], one failure in (t 2, t 2 + δt 2 ] for small time intervals of δt 1, δt 2. Suppose failures occur independently and according to a non-homogeneous Poisson process with intensity function Λ(t). Prob.{no failures in (0, t 1 ]} = 1 exp Λ() t t t0 dt Prob.{one of failures in (t 1, t 1 + δt 1 ]} = Λ ( t ) δt 1 2 Prob.{no failures in (t 1 + δt 1, t 2,,] } = exp Λ () t t + δt 1 t 1 dt ( ) t Prob.{one of failures in (t 2, t 2 + δt 2 ]} = Λ t δ 2 161

162 The probabilities for other failures can be derived similarly. As a result, the likelihood function for failures the product to occur is (Crowder et al 1991) L { dt} n ti ( θ ) = Λ( t ) exp Λ( t) i= 1 i t i 1 { } n t0 + T = Λ( t ) exp Λ( t) = i= i dt 1 t n T { Λ( t )} Λ() t 0 { } i= i dt 1 0 exp (6.18) Let l(θ) be the log transformation of the likelihood function n T ( ) = log{ L( θ )} = logλ( t ) Λ( t) i= 1 i l θ dt (6.19) 0 Where θ is the model parameter for set (λ, β). For Non-Homogeneous Poisson Process Λ(t) is given by λβ ( λt ) Then, n β ( ) = log λ β ( ) i= 1 β 1 β { ( t i )} λ ( T ) β l θ (6.20) β 1 The maximum likelihood estimates are the values of the parameters that maximise the log likelihood function. These are given by the solution to the following equations (obtained from the first order necessary condition for maximisation): ( θ ) δl δλ = 0 (6.21) and ( θ ) δl δβ = 0 (6.22) 162

163 By taking the first derivative of the Equation (6.19) and Equation (6.20) with respect to λ and β respectively and equating these to zero, and by solving the new equations the estimators λˆ and βˆ can be obtained as follows. ˆ n λ = β T 1 β (6.23) and n ˆβ (6.24) = n nlnt i= 1 ln ( ) t i Now by applying the equations (6.23) and (6.24), inverse characteristic parameter λ and shape parameter β of a distribution set can be estimated when the failures of the item increase or decrease with time (i.e. failures follow non-homogeneous Poisson process) CONCLUSIONS There is a growing trend for asset intensive industries to outsource maintenance services of their supporting systems/assets using service contracts since outsourcing reduces upfront investments in infrastructure, expertise and specialised maintenance facilities. Three policies for service contracts (short-term and long-term) are proposed in this chapter considering the concepts of outsourcing assets (or complex products such as heavy equipment) to the service providers. The proposed policies for service contracts are summarised in Table 6.1. The service contract policies developed here are currently offered and used by some of the service providers and these policies would be of potential interest to both the service provider and the user/owner of the asset. 163

164 Table 6.1: Service Contract Policies Policy Features Service contract 1 Fixed term contract. The contract terminates when the item/asset reaches the end of the prefixed contract duration or complete failure (first renewal) of the item/asset, whichever comes first. The item/asset life is considered longer than the contracted period. This can have some salvage value or trade off at the end of contract period No replacement actions are applicable. Constant interval preventive maintenance and minimal repair in between two successive PMs in case of fault or defects Can be applicable for short-term contract Service contract 2 Contract terminates at first replacement (renewal) of item/asset due to complete failure of the item/asset. Contract period is randomly variable. No trade off or salvage value at the end of the contract period No replacement actions are applicable. Constant interval preventive maintenance and minimal repair in between two successive PMs in case of fault or defects Can be applicable for short term contract Service contract 3 Fixed time contract Replacement in case of complete failure Constant interval preventive maintenance and minimal repair in between two successive PMs in case of fault or defects. Long-term contract Conceptual models are developed in estimating costs of various service contracts based on the policies. These models can be applicable in estimating the cost of outsourcing maintenance services through service contracts for any complex and expensive repairable assets/equipment used in different asset intensive industries. Total costs of alternative strategies and cost per unit of service provided is considered for managerial decisions. Analytical solutions for service contract 164

165 policies 1 and 2 can be carried out easily by using modern mathematical software such as MAPLE and MATLAB. The model developed for service contract policy 3 is complex and appears analytically intractable. Search algorithms and simulation can be used to explore the various preventive maintenance and corrective maintenance combinations for a particular contract period, attractive to the service provider and the user/owner. This is influenced by the perception of the service provider and the user/owner towards failure intensity during the contract period and costs associated with rectifications. However, the cost models developed in this chapter consider a number of assumptions. These models can be made more realistic by removing some of the assumptions. These models can be further extended by including a discount rate, provisions for used items, and utility functions for linking customer/manufacturers risk preferences. More complex models could be developed linking risks, downtime and penalties for failure to meet agreed safety, reliability and availability standards. A case study and analysis of outsourcing rail maintenance actions through service contract are aimed to carry out in Chapter 7. Real life rail maintenance data are used to estimate the cost of outsourcing rail maintenance. 165

166 CHAPTER 7 A CASE STUDY OUTSOURCING RAIL MAINTENANCE THROUGH SERVICE CONTRACTS 7.1 INTRODUCTION Rail networks are generally spread over wide geographically distant areas. It is expensive and complex for non-railway industries such as mining and other heavy industries to install and manage the maintenance of such huge networks. It needs investment in infrastructure, experts and specialised facilities to provide the services and carry out maintenance works. Outsourcing the maintenance through service contracts reduces upfront investments in infrastructure, expertise and specialised maintenance facilities (Murthy and Ashgarizadeh, 1995). This results in a growing trend for the owners of these asset intensive industries to outsource the management of maintenance activities of these huge network services to external agencies (service providers). The service providers for such services can be one of the rail operators or manufacturers of the rails or independent third parties, interested in investment for rail infrastructure. The service provider in turn charges a price or premium for such services. The estimation of the cost for these contracts is complex and it is important to the user and the service providers for economic variability. The service provider s profit is influenced by factors such as the terms of the contract, reliability of rails, the servicing 166

167 strategies, costs of resources needed to carry out maintenance and to provide such services. Servicing strategy for a rail network can be developed by understanding the reliability of rails used in the rail track system. Reliability analysis of rails can be carried out by understanding the failure mechanism of rail through modelling and analysis of failure data. These failure data are, in most cases, usage dependent. In a probabilistic sense, rail failure is a function of its usage in terms of Million Gross Tones (MGT) for certain conditions. This chapter will analyse real life rail industry data, deal with the limitations of available data and use these data in the predictive models for maintenance and replacement decisions. Parameters of the models are estimated using real world data with an application of non-homogeneous Poisson process. In Chapter 6, three policies for long-term service contracts were proposed and cost models for different policies were also developed. In this chapter, these conceptual models were applied to predict failure /break of rail and to estimate costs of outsourcing rail maintenance through service contract by using real life rail failure/break data obtained from Swedish Rail as a case study for this research. The outline of Chapter 7 is: in Section 7.1, the importance and complexities in modelling costs of outsourcing of rail maintenance is discussed. Section 7.2 deals with rail degradations and resulting rail failure/breaks. Probabilistic rail break/failure models are developed in Section 7.3. In Section 7.4, costs of outsourcing rail maintenance are estimated by applying the cost models developed in Chapter 6 for different service contract policies. Section 7.5 analyses the proposed cost models with numerical examples by using real world rail failure data for different service contract policies. In the final section, the summaries and scope for future work are discussed. 7.2 DEGRADATION OR FAILURE OF RAIL TRACK Degradation or failure of rail track is a complex process and it depends on the rail materials, traffic density, speed, curve radius, axle loads, Million Gross Tonnes (MGT), wheel rail contact, rail track geometries and importantly, the servicing strategies. The rail profile and curves make large contributions to rail degradation. 167

168 Rail tracks are designed to reduce the contact stresses and the twisting effect of the wheel load. Wheel loads produce bending moment and shear forces in the rail causing longitudinal compressive and tensile stresses concentrated mainly in the head and foot of the rail and shear stresses in the web. Gauge Rail Head Rail and Wheel contact area Rail Web Rail Foot Figure 7.1: Rail Profile and wear area It is important to provide adequate resistance against the bending moment which determines the area of the head and foot of the rail. Furthermore, corrosion and surface cracks have significant effects on the rail breaks. Traffic wear, rolling contact and plastic deformation are the growing problems for modern railways (for details see Chattopadhyay et. al., 2003 and Larsson et al., 2003). Factors influencing rail degradation are: Traffic wear, which is caused due to the wheel-rail contact, primarily in curves. Wear generally occurs at the gauge face in curves. Vertical head wear is caused by wheel contact and rail grinding. Wear of wheels and rails can be directly measured by the use of profilometers. Wear is influenced by material response to combined tangential and normal stresses and slippage. Rolling contact fatigue, which results in various defect types such as transverse defects and shells. When the rail reaches its fatigue limit, these defects occur more frequently. 168

169 Plastic deformation, which is often found in the form of corrugations in rails, together with mushrooming of the railhead and wheel burns. Plastic deformation of material causes damage of the surface layer, and eventually can cause the formation of cracks. A crack grows under the influence of mechanical loading and trapped fluid and results in a break if not maintained regularly. Rail and wheel material, wheel flange welding and wheel profile. High strengths rail steels are achieved by making the spacing between the pearlite lamellae finer by controlling the growth of pearlite. Alloying elements such as chromium and nickel are added to improve material properties. Hardness and composition of rail steel plays an important role in reducing wear rate. Figure 7.2 shows hardness of rail steel and its resistance to wear curve (Yates, 1996). Figure 7.2: The wear rate (mg m -1 ) vs hardness (HV) of rail steel Train speed, traffic loads in MGT; 169

170 Environmental and weather conditions: temperature and humidity, cyclic drying and wetting, snowfall, oxidation etc have some effect on accelerating the rail degradation process. Servicing strategies: Servicing strategies for rail during the contract period can be a combination of both corrective action (involves replacement or repair of rail in case of break/failure) and planned preventive maintenance actions (to prolong the rail life through inspection or monitoring condition of track, rail grinding, lubrication etc). Grinding is used to remove corrugation and surface crack and to reduce internal defects and improve the rail profile to give better vehicle steering control (Sawley and Reiff, 1999). Rail grinding has been demonstrated to have improved rail life in rail curves compared to other strategies (Judge, 2000). It is important to remove just enough metal to prevent the initiation of rolling contact fatigue defects. Rail wear caused by friction at the interface can be reduced by lubrication of the side of rail or the gauge face primarily on curves (Sawley and Reiff, 1999). However, large lateral forces still occur as the train goes around the curve causing degradation in the track structure (DeGaspari, 2001). Larsson (2000) found in field measurements that rail head wear on the main line in the north of Sweden indicate that wear rates on the flange can have an average 0.82 mm/month during the winter when no lubrication was used. To predict the rail failures and to decide on maintenance strategies for the rail track it is necessary to model the rail break/failures and associated costs related to maintenance actions of rail. 7.3 MODELLING RAIL BREAK/FAILURES Ageing takes place in the line due to tonnage accumulation on the track resulting from traffic movement leading to defect. It is realistic to assume that initiated defects left in the system will continue to grow with increases in cumulative passes in Million Gross Tonnes (Chattopadhyay et al (2005). Rail failures/breaks can be modelled as a point process with an intensity function Λ(m) where m represents Millions of Gross Tonnes (MGT) and Λ(m) is an increasing function of m indicating that the number of breaks/failures in a statistical sense increases with MGT. This 170

171 implies that older rails with higher cumulative MGT passing through the section are expected to have more probability of initiating defects and if undetected then through further passing of traffic can lead to rail failures. As a result, the number of failures till an accumulated MGT is a function of usage MGT, m, and is a random variable and can be modelled using Weibull distribution. Let cumulative MGT of rail, m, be known. F(m) and f(m) denote the cumulative rail failure function and density function respectively, Here, F ( m) df f ( m) = (7.1) dm ( m) P{ m m} = 1 where m 1 is the MGT to rail failure This can be modelled as Non-homogeneous Poisson process with shape parameters β and inverse of characteristic life λ and can be given by: β F ( m) = 1 exp( ( λm) ) (7.2) The density function can be obtained by differentiating equation (7.2) with respect to m as in equation (7.3) f β 1 β ( m) = λβ ( λm) exp( ( λm) ) (7.3) β Greater than 1 indicates an increasing failure rate of the item under study and ageing is predominant in failure mechanism. Then failure intensity function Λ(m) is derived from (7.2) and (7.3) and is given by the equation (7.4) Λ( β 1 t ) = λβ ( λm) (7.4) Rail track is normally made operational through repair or rectification of the failed segment and no action is taken with regards to the remaining length of the rail in case of detected defects and rail breaks. Since the length of failed segment replaced at each failure is very small relative to the whole track, the rectification action have negligible impact on the failure rate of the track as a whole (Chattopadhyay, et al, 2003). Based on these rail failure/break models, in the following sections we discuss 171

172 the potential servicing strategies of outsourcing rail network and it is also proposed servicing strategies and cost models for those service contracts. 7.4 ESTIMATING COSTS OF OUTSOURCING RAIL MAINTENANCE The three service contract policies and cost models developed in Chapter 6 now can be applied in estimating costs of outsourching rail maintenance through service contracts. In doing so the policies and the cost models need a little modification since rail failure/breaks are usage dependent in terms of Million gross tonnes of passing. This modification is made by capturing usage (MGT) instead of time. For this purpose we are using real life rail failure data obtained from the Swedish rail as a case study. For model simplification purposes, we consider one segment of rail of 110 meters long. Rail maintenance can be outsourced in section wise where each rail section consists of several kilometres of rail lines. Here, planned grinding and lubrication activities are viewed as preventive maintenance for rail and unplanned repair and replacement of the cracked or broken portion/s of rail segment considered as minimal repair action since the repair replacement of such small portion can not improve the overall reliability of the whole rail segment (Barlow and Hunter, 1960). Examples of minimal repair action are replacing a small damaged portion of rail segment or welding the cracks etc. It is assumed that a replacement or renewal of the whole segment is essential only when the segment is out of order (unusable) due to complete break or failure of the segment. Replacements are made only with a new and identical rail (same material and geometry). The three policies of service contracts defined in the Chapter 6 can now be modified here for the purpose of application for outsourcing rail maintenance. Rail Service Contract Policy1: Under this policy the contract terminates when the total usage reaches a pre-agreed Million Gross Tonnes of usage (L) or the state at which renewal of the rail segment is essential due to the complete failure of the segment (when it is beyond economic repair) whichever comes first. According to the conditions of this policy, renewal or replacement of the whole segment is excluded. 172

173 Under this policy, preventive maintenance actions are considered at constant intervals. This policy is represented graphically in Figure 6.2. Here the time dimension can be replaced by the usage in MGT and the failure intensity can be seen as λ(m), where m is any accumulated MGT. It is assumed that the preventive maintenance actions are taken at constant intervals of MGT passing and each PM restores the reliability of rail to some extent depending on the quality of maintenance action. Let L and R be the contract period and the first replacement (renewal) of rail due to complete failure respectively in terms of MGT of usage. Rail failures are usage dependent and can be considered to follow non-homogeneous Poisson process (NHPP). Let λ and β be the inverse characteristic life parameter and shape parameter for rail failure. Then from Figure 6.2, the total expected cost per unit MGT, C(L,x, N i, ) can be obtained by modifying the Equation (6.9) as: C N 1 = mr L k= 0 β [ ] β β β ( L, x, N ) C λ x ( k kα + 1) ( k kα ) + C Where, k is the number of PM is carried up to an MGT, m and τ is the MGT restoration after each PM. τ = αx, where, α is the quality of the maintenance, that ranges from 0 to 1. pm N N C mr C pm Nnumber of times the planned PMs are performed during the contract period. Cost for each minimal repairs Cost for each PM Rail Service Contract Policy2: Under this Policy, the contract terminates when a renewal of whole segment is essential due to the complete failure of the segment. Notice that the contract period L is not fixed, rather it is a random variable. This can be represented graphically by modifying the Figure 6.3 where time axis can be replaced by the usage in terms of MGT. Since rail failure is MGT dependent, the 173

174 failure intensity can be expressed as Λ(m) and the failure or break can be modelled a non-homogeneous Poisson s process (NHPP). The total expected cost for this policy is similar to that of Policy 1. In this case, however, the contract period is considered uncertain and is with randomly variable and an optimal contract period L * is the product of optimal number of preventive maintenance (N * ) and optimal interval of PM (x * * * ). That is ( ) L = N + 1 x Rail Service Contract Policy 3: Under this policy, replacement/s of the rail segment due to complete failure of the segment (renewals) is covered during the contract period. This can be represented by Figure 6.4 by replacing the time axis with usage in million gross tonnes. Under the condition of policy 3, there may be one or more PM actions before finally a replacement is made and there may also be a number of minimal repairs in between two successive PM. Let R be the usage level between two renewals and R is random variable and R R *, where R * is the optimal replacement interval and if it fails completely before R * it is replaced. This policy is applicable for long-term service contract. In this case there may be one or more replacements (due to complete failure) needed during the contract period Here, the expected total cost per unit time = (expected cost of minimal repairs + expected cost of preventive maintenance + cost of replacement + expected cost of last cycle up to the end of contract) / Length of service contract. This can be expressed by modifying the service contract policy 3 model as shown in Equation (6.7) in Chapter 6 as C ( L, x, N, M ) i β β [( k kα + 1) ( k kα ) ] M M 1 Ni β β = Cmr λ x + NiC L i= 0 k= 0 i= 0 pm + C cl + MC The conceptual models developed here for complex long-term service contracts applicable to outsourcing rail maintenance are now analysed with numerical examples. Real life rail industry data are used to analyse the models with a view to re 174

175 estimate the cost of service contracts. Total costs of servicing, cost per MGT of service provided, optimal interval between PMs, and optimal number of PMs can be considered for managerial decisions. 7.5 ANALYSIS OF THE MODELS FOR RAIL For the purpose of estimating and analysing the models it used a set of real life rail failure data collected from Swedish Rail. A total of 208 rail break/failure data are used here. Table 7.1 represents 208 rail break data in Million gross tonnes (MGT) obtained from Swedish Rail. Table 7.1 Rail breaks in Million gross tonnes (MGT) A preliminary evaluation indicated several shortcomings in accuracy of these data such as the data indicates a rail break but does not indicate what type and how severe the break is, the type and accuracy of the inspection procedures, whether preventive maintenance actions are in use, what are the effectiveness of preventive maintenance actions, what minimal repair measures were applied, and the exact cost of rectification. These data also do not provide adequate information on the rail population. The population of rails involved is comprised of various makes (brands) and are made up of many different materials. However, these data are used for 175

176 modelling failure distribution of rails by making some assumptions and one of the assumptions is that the breaks occur according to a non-homogeneous Poisson process. Raw data were first censored and rectified to make it useable. It is obvious by inspecting Table 7.1 that there is marked sad trend present which can seen clearly in the Figure Rail Break Number Operating MGT Figure 7.3: Cumulated Rail break vs. accumulated MGT. The failure or breakage MGT in the analysis is generated as follows: Usage span is considered as 720 MGT. A plot of the accumulated number of rail break versus the accumulated breakage MGT is displayed in the Figure 7.5. The lack of linearity of the plotted data is an indication that the rate of rail break is not constant. Rather it is usage dependent. Increase of rail breaks with the increase of usage in terms of MGT implies the rail break or failure follows a First Weibull distribution or Non homogeneous Poisson process since Estimation of rail failure parameters In estimating the rail failure parameters, one can use different methods such as the method of Least square, method of Moments, regression analysis, and method of Maximum likelihood. The method of Maximum Likelihood (MLE) is applied here to 176

177 estimate the parameters λ and β. In this case, estimation of the parameters considering a Weibull distribution (two parameters) is carried out by using a MATLAB program (see Appendix D). The MATLAB expression provides Figure 7.6 with the Weibull graph for the rail failure/break data presented in Table 7.1. From Figure 7.6, it can be estimated the inverse characteristic life parameter λ = per MGT and the shape parameter β = at 95% confidence interval Cumlative rail break Probability β = λ= η = % CI Operational MGT Figure 7.4: MATLAB generated Weibull graph for rail failure data Estimating Costs of Different Service Contracts for Rail In this section estimated parameters are used in determining the cost of service contracts for three different policies. Here, ß = and λ = per MGT. It is assumed that each Preventive maintenance action involves only one pass of grinding and lubrication. 177

178 Policy 1: Let, Cost of minimal repair, cost of replacement/repair of one rail for any Segment due to worn out regulation C mr = $150 (Approximately) Cost of each preventive maintenance (rail grinding and Lubrication), C pm = ($4.00 per meter (approximately) 110m = $440 Cost of replacement, C re = $1700 Quality of each PM, α = 0.16, which implies that each PM restores only 16% of total reliability (we assume it is constant for each PM) Let the contracted usage in MGT, L = 300 MGT. Here, a MAPLE program has been used to determine the optimal interval and number of PMs. This gives us the following results Optimal interval between preventive maintenance x* = MGT Optimal number of PMs N* = 5 Expected total cost of Service Contract C*(L,x,N)= AUD 5.98 per MGT This implies that for a short-term contract with the presented rail failure distribution, each segment needs at least 5 preventive grinding and lubrication at maximum interval of MGT. However, the actual total costs can be varied based on the grinding process and lubricants to be used during the contract period. Policy 2: Cost of minimal repair, cost of replacement/repair of one rail for any Segment due to worn out regulation C mr = $150 (Approx) Cost of each preventive maintenance (rail grinding and Lubrication), C pm = ($4.00 per meter (approximately) 110m = $440 Cost of replacement, C re = $1700 Quality of each PM, α = 0.16, which implies that each PM restores only 16% of total reliability (we assume it is constant for each PM) Here, a MAPLE program has been used to determine the optimal interval and number of PMs. The result is: Optimal interval between preventive maintenance x* = MGT 178

179 Optimal number of PMs N* = 5 L * = MGT Servicing cost per MGT C*(L,x,N)=AUD This means that under policy 2, with this failure distribution of rail, the maximum life of a rail segment is MGT which needs at least five PM actions in the form of rail grinding and lubrication at a constant interval of MGT of usage. Note that, here we assumed single pass grinding and lubrication for each PM action. In real life, however, multiple grinding and lubrication may be required depending on the rail condition and the decision maker s perception on rail failure. This might result in higher cost per MGT. The actual cost per MGT can also be varied based on the grinding process and lubricants and lubricators to be used during the contract period Policy 3: The model developed for this policy is very complex and is difficult to solve analytically. Therefore a simulation approach is needed to solve this. A flow diagram of the proposed model is presented in Figure

180 Figure 7.5: Framework for service contract cost model Solution of this model applying simulation approach can be scope for future work 7.6 CONCLUSIONS In this chapter, the service contract policies and cost models developed in the Chapter 6 were applied for estimating the cost of outsourcing rail maintenance. Real life rail failure/break data were collected and analysed. Models developed for the first two service contract policies were validated successfully. Parameters were estimated and applied to analyse the costs. Total costs of servicing and cost per unit of service provided can be considered for managerial decisions. Validation of the model for policy 3 is mathematically intractable and a simulation approach was 180

181 suggested in this chapter which can be a scope for future research. These models can also be applicable to outsource the maintenance of major infrastructures belonging to asset intensive industries. For simplification purposes, a number of assumptions were made. Some of these assumptions can be minimised to obtain more realistic decision models. This creates a further research opportunity in extending the developed models and their application. There is huge scope for I) Developing integrated decision models for long-term service contracts applicable to rail and other asset intensive industries. II) Development of penalty rates for train operators and infrastructure providers not complying with maintenance standards. III) Inclusion of other maintenance actions such as Reliability centred maintenance, Condition based maintenance, Optimal inspection policies as servicing strategies during the contract period. 181

182 CHAPTER 8 CONCLUSIONS AND SUGGESTIONS FOR FUTURE RESEARCH 8.1 INTRODUCTION The scope of this research was to define useful life, develop new policies for lifetime warranties and long-term service contracts, develop reliability and cost models for those policies, develop models for manufacturer s and buyer s risks associated with offering and buying those policies. This Chapter summarises the contributions of this thesis and discusses some scope for future research. The main contributions of this thesis are (i) definition for lifetime and formulation of taxonomy for new lifetime warranty policies (ii) mathematical models for product failures, expected costs for those policies, (iii) models for pricing decisions based on the manufacturer s and buyer s risk preferences (iv) formulation of new long-term service contract policies, and (v) mathematical modelling and analysis for the expected costs of those service contracts. vi) analysis of rail sector data for possible application to outsourcing maintenance. A summary of the contributions are provided in Section 8.2. Scope for future research is discussed in Section CONTRIBUTION OF THIS THESIS Chapter 1 of this thesis provides the scope of this research and modelling reliability and costs of lifetime warranties and service contracts. It discusses the problems associated with studying and modelling lifetime warranties and service contracts, and the need for the development of new policies, modelling product failures, and 182

183 expected costs for offering those policies and analysing risks to manufacturer and buyers. In Chapter 2, a brief review of the literature on product warranties and long-term warranty policies and failure cost models was carried out to provide a background for this research. The main contributions of this research are: Development of new policies and taxonomy for lifetime warranties: Even though a large number of products are currently being sold with lifetime warranties, there was no acceptable definition of lifetime or the useful life of product in the literature. Different manufacturers/dealers are using different conditions of life measures of period of coverage. It is often difficult to tell whose life measures are applicable to the period of coverage (The Magnuson-Moss Warranty Act, 1975). In Chapter 3, lifetime or the useful life is defined. This definition provides a transparency for the useful life of products to both manufacturers and the customers. This is also helpful to the public policy makers in developing mechanisms for handling the lifetime warranty related claims and settling disputes between the manufacturer and the consumer. In Chapter 3, taxonomy is developed for lifetime warranty policies. Eight lifetime warranty policies are developed and are grouped into three major categories: A. Free rectification lifetime warranty policies (FRLTW), B. Cost Sharing Lifetime Warranty policies (CSLTW), and C. Trade in policies (TLTW). Most of these policies are currently being considered by the manufacturers/dealers and the others would be of potential interest to industries. Development of failure and cost model capturing the uncertainties in decision making over the longer period: In Chapter 4, the mathematical models for predicting failures and expected costs for lifetime warranty policies were developed and analysed. Claims over the uncertain lifetime coverage period are used for modelling failures of products at system level. These failure and cost models were used to analyse the expected warranty costs to 183

184 the manufacturer and the buyers. Various rectification policies were proposed and analysed over the lifetime. In this thesis one dimensional policy were analysed and illustrated by numerical examples for estimating costs to manufacturer and buyers. These are important for the manufacturer or dealer when making decisions about warranty pricing and buyers for understanding whether these policies are worth buying or not. Modelling manufacturer and buyer s risk preference for lifetime warranty policies: In Chapter 5, decision models are developed to maximise the manufacturer/dealer s certainty profit equivalent. Manufacturer and buyer risk models have been developed for optimal warranty price using manufacturer s utility function for manufacturer s profit and the buyer s utility function for repair cost. These models considered: 1) the manufacturer s and buyer s risk attitude towards lifetime warranties; 2) their perceptions about the product failure intensity; 3) the manufacturer s technical monopoly in repairing failed products; 4) the buyer s repair costs; and 5) buyers rate of return to the original manufacturer for repairing failed products. Developing policies and cost models for service contracts considering both corrective maintenance and preventive maintenance: In Chapter 6, three policies for short-term and long-term service contracts were developed for outsourcing maintenance. In modelling product failures and costs of service contracts, corrective maintenance (rectification on failure) and preventive maintenance actions (to prolong the reliability of the asset/equipment) were considered. The service contract policies developed can be offered by the manufacturer or dealer and/or used by service providers. Application of service contracts policies and cost models in estimating the cost of outsourcing rail maintenance: In Chapter 7, a case study was presented for estimating the costs of outsourcing maintenance of rails through service contracts. Rail failure/break data were collected 184

185 from Swedish Rail and analysed for predicting failures. Total costs of different service contracts per Millions Gross Tonnes (MGT) were estimated. 8.3 SCOPE FOR FUTURE RESEARCH There is a huge scope for future research in this and related areas. Some of these are Lifetime warranty policies and cost models [A] [B] [C] [D] New lifetime warranty policies can be developed by combining two or more policies defined in Chapter 3. For example, new policies can be developed by combining cost limit and trade in policies. This thesis developed one dimensional trade in policies. One can develop trade in lifetime policies for two dimensional cases which consider both the age and usage. In Chapter 4 lifetime warranty cost models were developed at a system level. One can also developed cost models at a component level by applying renewal, modified renewal or other related processes and aggregating to the system level. For model simplification purposes a number of assumptions were made. To make the model more realistic some of these assumptions can be relaxed in future research. These are Discount factor is considered as 1 throughout the lifetime which implies that the future cost is not influenced by inflation. This assumption can be relaxed by including the discounting of the repair cost for present value and corrections for inflation. In modelling costs products were assumed to have a time dependent failure rate. The time dependent failure rate assumption made in this thesis can be left open and a general failure rate function could be considered in future research. In estimating warranty cost, it was assumed that an item failure results in an immediate claim and all claims are valid (that is the buyer always exercises his/her warranty). In practice, there are a number of instances when a warranty is not exercised, even though it is possible 185

186 to do so. The percentage of buyers who actually use their warranties is uncertain. Several factors are responsible for this uncertainty such as the point of time in the warranty when the failure occurs, buyer s ignorance about his or her rights and negligence on claims. [E] In this thesis, rectification cost per warranty claim was assumed to be uncertain and these rectification costs were modelled as random variables characterised by a distribution function. In real life, parameters could be functions of type of defects and corresponding rectification action Risk Preference Models I) The risk preference models developed in Chapter 5 can be further extended by relaxing some of the assumptions such as possible trade-ins. Use of second hand or used parts in rectifications is another possibility. II) In this thesis the developed risk models considered risk averse manufacturers and buyers. In practice, there may be risk seeker/risk neutral manufacturers or buyers. Inclusion of different risk attitude manufacturer or buyer could be an area for future research on risk analysis Service Contract Policies and Cost Models There is huge scope for I. Developing integrated decision models for long-term service contracts applicable to asset intensive industries. II. Development of penalty rates for service provider not complying with maintenance standards and penalty provisions for delaying the service action for infrastructures. III. Inclusion of other maintenance actions such as Reliability centred maintenance; Condition based maintenance, optimal inspection policies as servicing strategies during the contract period. IV. These models can be further extended by including discount rate, provisions for used items as replacement, and application of utility functions for linking service provider s and service receiver s risk preferences. 186

187 8.3.4 Other Scope Scope for future work in other related areas include decision models in logistics associated with warranty and service contracts e.g. availability of rectification facilities and spare parts. 187

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191 The Magnuson-Moss Warranty Act (1975) - United States federal law of warranties on consumer products, (15 U.S.C et seq.). Enacted in Malik, M. A. K. (1979), Reliable preventive maintenance policy, AIIE Transactions, 11, Manna, D.K., Surajit, P. and Sagnik, S., (2006) Optimal determination of warranty region for 2D policy: A customers perspective, Computers & Industrial Engineering, 50, Menezes, M.A.J. (1989), The Impact of Product warranties on consumer preferences, A draft paper submitted to Harvard Business School (Report No ) Menke, W. W. (1962), Determination of warranty reserves, Management Science 15, Mitra, A., Patnakar, J. G. (1997), Market Share and Warranty Cost for Renewable Warranty Programs, International Journal of Production Economics 50, Moskowitz, H., Chun, Y.H. (1988), A Bayesian Model for Two-Attribute Warranty Policies., Krannert Graduate School of Management, Purdue University: West Lafayette,. Moskowitz, H., Plante, R. (1984). "Effect of risk aversion on single sample attribute inspection plans." Management Science Vol.30: Murthy, D. N. P., Nguyen, D.G. (1985a), Study of two component system with failure interaction, Naval Research Logistics Quarterly, Murthy, D. N. P., Nguyen, D.G (1985b), Study of multi component system with failure interaction, European Journal of Operational Research 21, Murthy, D. N. P., Nguyen, D.G (1988), An optimal repair cost limit policy for servicing warranty, Mathematical and Computer Modelling 11, Murthy, D. N. P., Page, N. W., Rodin, E. Y. (1990), Mathematical modelling, A tool for problem solving, Pergamon Press, Oxford, UK. Murthy, D. N. P., Blischke, W. R. (1992a), Product warranty and management-ii: An integrated framework for study, European Journal of Operational Research, 62, Murthy, D. N. P. and Blischke, W. R. (1992b), Product warranty and management-iii: A review of mathematical models, European Journal of Operational Research 63, Murthy D. N. P., Iskander, B.P., and Wilson, R.J.,(1995), Two dimensional failure free warranties: two dimensional point process models, Operations Research, 43, Murthy, D.N.P., Djamaludin, I., Wilson, R.J. (1995), A Consumer Incentive Warranty Policy and Servicing Strategies for Products with Uncertain Quality. Quality and Reliability Engineering International, 11: p Murthy D. N. P., Yeung, (1995) Modelling and analysis of service contracts, Mathematical and Computer Modelling 22(10-12),

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194 Appendices 194

195 APPENDIX A: LIFETIME WARRANTY POLICIES 195

196 Magnuson-Moss Federal Warranty Act 1975 A Businessperson's Guide to Federal Warranty Law Table of Contents State law (the Uniform Commercial Code). Sections & Section Introduction Understanding Warranties Implied Warranties Express Warranties Understanding the Magnuson-Moss Warranty Act For the full legal texts listed below, consult the supplement to this manual. The FTC's Dispute Resolution Rule, 16 C.F.R. Part 703. REQUIREMENT I: The Magnuson-Moss Warranty Act, Section 102. REQUIREMENT II: The FTC's Disclosure Rule, 16 C.F.R. Part 701. What the Magnuson-Moss Act Does Not Require What the Magnuson-Moss Act Requires What the Magnuson-Moss Act Does Not Allow Disclaimer or Modification of Implied Warranties "Tie-in-Sales" Provisions Deceptive Warranty Terms How the Magnuson-Moss Act May Affect Warranty Disputes Consumer Lawsuits Alternatives to Consumer Lawsuits Titling Written Warranties as "Full" or "Limited" What the Terms "Full" and "Limited" Mean Examples of Full Warranties, Limited Warranties, and Multiple Warranties Stating Terms and Conditions of Your Written Warranty Basic Information Required for All Warranties Specific Information Required When Your Warranty Contains Certain Optional Terms and Conditions Making Warranties Available Prior to Sale What Retailers Must Do What Mail Order Companies Must Do What Door-to-Door Sales Companies Must Do What Manufacturers Must Do REQUIREMENT III: The FTC's Pre-Sale Availability Rule, 16 C.F.R. Part 702. Advertising Warranties How to Advertise Warranties Covered by the Pre-Sale Availability Rule 199

197 The FTC's Warranty Advertising Guides. How to Advertise a Satisfaction Guarantee How to Advertise a Lifetime Warranty or Guarantee Offering Service Contracts Statement of Terms and Conditions Disclaimer or Limitation of Implied Warranties The Magnuson-Moss Warranty Act, Section 106. Additional Sources of Information Understanding the Magnuson- Moss Warranty Act The Act improves consumers' access to warranty information. The Act enables consumers to comparison shop for warranties. The Act encourages warranty competition. The Act promotes timely and complete performance of warranty obligations. The Magnuson-Moss Warranty Act is the federal law that governs consumer product warranties. Passed by Congress in 1975, the Act requires manufacturers and sellers of consumer products to provide consumers with detailed information about warranty coverage. In addition, it affects both the rights of consumers and the obligations of warrantors under written warranties. To understand the Act, it is useful to be aware of Congress' intentions in passing it. First, Congress wanted to ensure that consumers could get complete information about warranty terms and conditions. By providing consumers with a way of learning what warranty coverage is offered on a product before they buy, the Act gives consumers a way to know what to expect if something goes wrong, and thus helps to increase customer satisfaction. Second, Congress wanted to ensure that consumers could compare warranty coverage before buying. By comparing, consumers can choose a product with the best combination of price, features, and warranty coverage to meet their individual needs. Third, Congress intended to promote competition on the basis of warranty coverage. By assuring that consumers can get warranty information, the Act encourages sales promotion on the basis of warranty coverage and competition among companies to meet consumer preferences through various levels of warranty coverage. Finally, Congress wanted to strengthen existing incentives for companies to perform their warranty obligations in a timely and thorough manner and to resolve any disputes with a minimum of delay and expense to consumers. Thus, the Act makes it easier for consumers to pursue a remedy for breach of warranty in the courts, but it also creates a framework for companies to set up procedures for resolving disputes inexpensively and informally, without litigation. What the Magnuson-Moss Act Does Not Require In order to understand how the Act affects you as a businessperson, it 200

198 is important first to understand what the Act does not require. First, the Act does not require any business to provide a written warranty. The Act allows businesses to determine whether to warrant their products in writing. However, once a business decides to offer a written warranty on a consumer product, it must comply with the Act. Second, the Act does not apply to oral warranties. Only written warranties are covered. The Act does not compel you to give a written warranty. Third, the Act does not apply to warranties on services. Only warranties on goods are covered. However, if your warranty covers both the parts provided for a repair and the workmanship in making that repair, the Act does apply to you. Finally, the Act does not apply to warranties on products sold for resale or for commercial purposes. The Act covers only warranties on consumer products. This means that only warranties on tangible property normally used for personal, family, or household purposes are covered. (This includes property attached to or installed on real property.) Note that applicability of the Act to a particular product does not, however, depend upon how an individual buyer will use it. The following section of this manual summarizes what the Magnuson- Moss Warranty Act requires warrantors to do, what it prohibits them from doing, and how it affects warranty disputes. What the Magnuson-Moss Act Requires In passing the Magnuson-Moss Warranty Act, Congress specified a number of requirements that warrantors must meet. Congress also directed the FTC to adopt rules to cover other requirements. The FTC adopted three Rules under the Act, the Rule on Disclosure of Written Consumer Product Warranty Terms and Conditions (the Disclosure Rule), the Rule on Pre-Sale Availability of Written Warranty Terms (the Pre-Sale Availability Rule), and the Rule on Informal Dispute Settlement Procedures (the Dispute Resolution Rule). In addition, the FTC has issued an interpretive rule that clarifies certain terms and explains some of the provisions of the Act. This section summarizes all the requirements under the Act and the Rules. The Act and the Rules establish three basic requirements that may apply to you, either as a warrantor or a seller. There are three FTC Rules under the Act. 1. As a warrantor, you must designate, or title, your written warranty as either "full" or "limited." 2. As a warrantor, you must state certain specified information about the coverage of your warranty in a single, clear, and easy-to-read document. 3. As a warrantor or a seller, you must ensure that warranties are available where your warranted consumer products are sold so that consumers can read them before buying. The titling requirement, established by the Act, applies to all written warranties on consumer products costing more than $10. However, the disclosure and pre-sale availability requirements, established by FTC Rules, apply to all written warranties on consumer products costing 201

199 more than $15. Each of these three general requirements is explained in greater detail in the following chapters. Section 102 of the Act directs how to title your warranty. The Disclosure Rule {16 C.F.R. Part 701} directs what you must include in your warranty. The Pre-Sale Availability Rule {16 C.F. R. Part 702} directs how to make your warranty available before sale. What the Magnuson-Moss Act Does Not Allow There are three prohibitions under the Magnuson-Moss Act. They involve implied warranties, so-called "tie-in sales" provisions, and deceptive or misleading warranty terms. Disclaimer or Modification of Implied Warranties The Act prohibits anyone who offers a written warranty from disclaiming or modifying implied warranties. This means that no matter how broad or narrow your written warranty is, your customers always will receive the basic protection of the implied warranty of merchantability. This is explained in Understanding Warranties. There is one permissible modification of implied warranties, however. If you offer a "limited" written warranty, the law allows you to include a provision that restricts the duration of implied warranties to the duration of your limited warranty. For example, if you offer a two-year limited warranty, you can limit implied warranties to two years. However, if you offer a "full" written warranty, you cannot limit the duration of implied warranties. This matter is explained in Titling Written Warranties as "Full" or "Limited". If you sell a consumer product with a written warranty from the product manufacturer, but you do not warrant the product in writing, you can disclaim your implied warranties. (These are the implied warranties under which the seller, not the manufacturer, would otherwise be responsible.) But, regardless of whether you warrant the products you sell, as a seller, you must give your customers copies of any written warranties from product manufacturers. If you give a written warranty on a consumer product, Section 108 of the Act prevents you from eliminating or restricting implied warranties. "Tie-In Sales" Provisions Generally, tie-in sales provisions are not allowed. Such a provision would require a purchaser of the warranted product to buy an item or service from a particular company to use with the warranted product in order to be eligible to receive a remedy under the warranty. The following are examples of prohibited tie-in sales provisions. In order to keep your new Plenum Brand Vacuum Cleaner warranty in effect, you must use genuine Plenum Brand Filter Bags. Failure to have scheduled maintenance performed, at your expense, by the Great American Maintenance Company, Inc., voids this warranty. While you cannot use a tie-in sales provision, your warranty need not cover use of replacement parts, repairs, or maintenance that is inappropriate for your product. The following is an example of a permissible provision that excludes coverage of such things. While necessary maintenance or repairs on your AudioMundo Stereo System can be performed by any company, we recommend that you use only authorized AudioMundo dealers. Improper or incorrectly performed maintenance or repair voids this warranty. Although tie-in sales provisions generally are not allowed, you can include such a provision in your warranty if you can demonstrate to the satisfaction of the FTC that your product will not work properly 202

200 without a specified item or service. If you believe that this is the case, you should contact the warranty staff of the FTC's Bureau of Consumer Protection for information on how to apply for a waiver of the tie-in sales prohibition. Deceptive Warranty Terms Obviously, warranties must not contain deceptive or misleading terms. You cannot offer a warranty that appears to provide coverage but, in fact, provides none. For example, a warranty covering only "moving parts" on an electronic product that has no moving parts would be deceptive and unlawful. Similarly, a warranty that promised service that the warrantor had no intention of providing or could not provide would be deceptive and unlawful. With some exceptions, Section 102 (c) of the Act prohibits you from including a tie-in sales provision in your warranty. These are examples of prohibited tie-in sales provisions. How the Magnuson Moss Act May Affect Warranty Disputes Two other features of the Magnuson-Moss Warranty Act are also important to warrantors. First, the Act makes it easier for consumers to take an unresolved warranty problem to court. Second, it encourages companies to use a less formal, and therefore less costly, alternative to legal proceedings. Such alternatives, known as dispute resolution mechanisms, often can be used to settle warranty complaints before they reach litigation. Consumer Lawsuits The Act makes it easier for purchasers to sue for breach of warranty by making breach of warranty a violation of federal law, and by allowing consumers to recover court costs and reasonable attorneys' fees. This means that if you lose a lawsuit for breach of either a written or an implied warranty, you may have to pay the customer's costs for bringing the suit, including lawyer's fees. Because of the stringent federal jurisdictional requirements under the Act, most Magnuson-Moss lawsuits are brought in state court. However, major cases involving many consumers can be brought in federal court as class action suits under the Act. Although the consumer lawsuit provisions may have little effect on your warranty or your business, they are important to remember if you are involved in warranty disputes. This is an example of a permissible warranty provision to use instead of a tie-in. Alternatives to Consumer Lawsuits Although the Act makes consumer lawsuits for breach of warranty easier to bring, its goal is not to promote more warranty litigation. On the contrary, the Act encourages companies to use informal dispute resolution mechanisms to settle warranty disputes with their customers. Basically, an informal dispute resolution mechanism is a system that works to resolve warranty problems that are at a stalemate. Such a mechanism may be run by an impartial third party, such as the Better Business Bureau, or by company employees whose only job is to administer the informal dispute resolution system. The impartial third party uses conciliation, mediation, or arbitration to settle warranty disputes. The Act allows warranties to include a provision that requires customers to try to resolve warranty disputes by means of the informal dispute resolution mechanism before going to court. (This provision applies only to cases based upon the Magnuson-Moss Act.) If you 203

201 include such a requirement in your warranty, your dispute resolution mechanism must meet the requirements stated in the FTC's Rule on Informal Dispute Settlement Procedures (the Dispute Resolution Rule). Briefly, the Rule requires that a mechanism must: Section 110(c) (2) of the Act prohibits deceptive warranties. Section 110(d) of the Act makes breach of warranty a violation of federal law, and enables consumers to recover attorneys' fees. Be adequately funded and staffed to resolve all disputes quickly; Be available free of charge to consumers; Be able to settle disputes independently, without influence from the parties involved; Follow written procedures; Inform both parties when it receives notice of a dispute; Gather, investigate, and organize all information necessary to decide each dispute fairly and quickly; Provide each party an opportunity to present its side, to submit supporting materials, and to rebut points made by the other party; (the mechanism may allow oral presentations, but only if both parties agree); Inform both parties of the decision and the reasons supporting it within 40 days of receiving notice of a dispute; Issue decisions that are not binding; either party must be free to take the dispute to court if dissatisfied with the decision (however, companies may, and often do, agree to be bound by the decision); Keep complete records on all disputes; and Be audited annually for compliance with the Rule. It is clear from these standards that informal dispute resolution mechanisms under the Dispute Resolution Rule are not "informal" in the sense of being unstructured. Rather, they are informal because they do not involve the technical rules of evidence, procedure, and precedents that a court of law must use. Currently, the FTC's staff is evaluating the Dispute Resolution Rule to determine if informal dispute resolution mechanisms can be made simpler and easier to use. To obtain more information about this review, contact the FTC's warranty staff. As stated previously, you do not have to comply with the Dispute Resolution Rule if you do not require consumers to use a mechanism before bringing suit under the Magnuson-Moss Act. You may want to consider establishing a mechanism that will make settling warranty disputes easier, even though it may not meet the standards of the Dispute Resolution Rule.. 204

202 Advertising Warranties Deceptive warranty advertising is unlawful. The FTC's Guides for the Advertising of Warranties and Guarantees {16 C.FR. Part 239} can advise you on how to advertise your warranty. The Magnuson-Moss Warranty Act does not cover the advertising of warranties. However, warranty advertising falls within the scope of the FTC Act, which generally prohibits "unfair or deceptive acts or practices in or affecting commerce." Therefore, it is a violation of the FTC Act to advertise a warranty deceptively. To help companies understand what the law requires, the FTC has issued guidelines called the Guides for Advertising Warranties and Guarantees. To obtain a copy, see Additional Sources of Information. However, the Guides do not cover every aspect of warranty advertising, and cannot substitute for consultation with your lawyer on warranty advertising matters. The Guides cover three principal topics: how to advertise a warranty that is covered by the Pre-Sale Availability Rule; how to advertise a satisfaction guarantee; and how to advertise a lifetime guarantee or warranty. How to Advertise Warranties Covered by the Pre-Sale Availability Rule In general, the Guides advise that if a print or broadcast ad for a consumer product mentions a warranty, and the advertised product is covered by the Pre Sale Availability Rule (that is, the product is sold in stores for more than $15) the ad should inform consumers that a copy of the warranty is available to read prior to sale at the place where the product is sold. Print or broadcast advertisements that mention a warranty on any consumer product that can be purchased through the mail or by telephone should inform consumers how to get a copy of the warranty. Advertisements for products covered by the Pre-Sale Availability Rule need only state that the warranty can be seen where the product is sold. {16 C.F.R }. For advertisements of consumer products costing $15 or less, the Guides do not call for the pre-sale availability disclosure. Instead, the Guides advise that the FTC's legal decisions and policy statements are the sole sources of guidance on how to avoid unfairness or deception in advertising warranties. Consult your attorney for assistance in researching and applying the FTC's case decisions and policy statements. How to Advertise a Satisfaction Guarantee The Guides advise that, regardless of the price of the product, advertising terms such as "satisfaction guaranteed" or "money back guarantee" should be used only if the advertiser is willing to provide full refunds to customers when, for any reason, they return the merchandise. The Guides further advise that an ad mentioning a satisfaction guarantee or similar offer should inform consumers of any material conditions or limitations on the offer. For example, a restriction on the offer to a specific time period, such as 30 days, is a material condition that should be disclosed. How to Advertise a Lifetime Warranty or Guarantee 205

203 "Satisfaction" and "Money back" guarantees constitute an offer of a full refund for any reason.{16 C.F.R }. "Lifetime" warranties or guarantees can be a source of confusion for consumers. This is because it is often difficult to tell just whose life measures the period of coverage. "Lifetime" can be used in at least three ways. For example, a warrantor of an auto muffler may intend his "lifetime" warranty's duration to be for the life of the car on which the muffler is installed. In this case, the muffler warranty would be transferable to subsequent owners of the car and would remain in effect throughout the car's useful life. Or the warrantor of the muffler might intend a "lifetime" warranty to last as long as the original purchaser of the muffler owns the car on which the muffler is installed. Although commonly used, this is an inaccurate application of the term "lifetime." Finally, "lifetime" can be used to describe a warranty that lasts as long as the original purchaser of the product lives. This is probably the least common usage of the term. The Guides advise that to avoid confusing consumers about the duration of a "lifetime" warranty or guarantee, ads should tell consumers which "life" measures the warranty's duration In that way, consumers will know which meaning of the term "lifetime" you intend. Clarify what you are talking about when you advertise a "lifetime" warranty. {16 C.F.R }. Offering Service Contracts Section 106 of the Act deals with service contracts. Service contracts, unlike warranties, are purchased separately from a product. A service contract is an optional agreement for product service that customers sometimes buy. It provides additional protection beyond what the warranty offers on the product. Service contracts are similar to warranties in that both concern service for a product. However, there are differences between warranties and service contracts. Warranties come with a product and are included in the purchase price. In the language of the Act, warranties are "part of the basis of the bargain" Service contracts, on the other hand, are agreements that are separate from the contract or sale of the product. They are separate either because they are made some time after the sale of the product, or because they cost the customer a fee beyond the purchase price of the product. The Act includes very broad provisions governing service contracts that are explained in the following sections. Statement of Terms and Conditions If you offer a service contract, the Act requires you to list 206

204 conspicuously all terms and conditions in simple and readily understood language. However, unlike warranties, service contracts are not required to be titled "full" or "limited,' or to contain the special standard disclosures. In fact, using warranty disclosures in service contracts could confuse customers about whether the agreement is a warranty or a service contract. The company that makes the service contract is responsible for ensuring that the terms and conditions are disclosed as required by law. This is not the responsibility of the seller of the service contract, unless the seller and the maker are the same company. Section 108 (a) of the Act prohibits you from disclaiming. warranties on a product if you sell a service contract on it. Disclaimer or Limitation of Implied Warranties Sellers of consumer products who make service contracts on their products are prohibited under the Act from disclaiming or limiting implied warranties. (Remember also that sellers who extend written warranties on consumer products cannot disclaim implied warranties, regardless of whether they make service contracts on their products.) However, sellers of consumer products that merely sell service contracts as agents of service contract companies and do not themselves extend written warranties can disclaim implied warranties on the products they sell. Additional Sources of Information For a supplement to this booklet containing texts of the Magnuson-Moss Warranty Act, the related FTC Rules, and the FTC Warranty Advertising Guides, write: Federal Trade Commission Consumer Response Center Washington, D.C For additional copies of this publication or for other FTC warranty related business manuals, contact the U.S. Government Printing Sales Office at (202) Your Opportunity to Comment The Small Business and Agriculture Regulatory Enforcement Ombudsman and 10 Regional Fairness Boards collect comments from small business about federal enforcement actions. Each year, the Ombudsman evaluates enforcement activities and rates each agency s responsiveness to small business. To comment on FTC actions, call [back to top] 207

205 APPENDIX B: PROGRAM FOR LIFETIME WARRANTY POLICIES 208

206 APPENDIX B EXPECTED LIFETIME WARRANTY COSTS ESTIMATION MAPLE PROGRAM FOR MANUFACTURER S COST FOR FRLTW > with(linalg): Warning, the protected names norm and trace have been redefined and unprotected > with(detools); [ DEnormal, DEplot, DEplot3d, DEplot_polygon, DFactor, DFactorLCLM, DFactorsols, Dchangevar, GCRD, LCLM, MeijerGsols, PDEchangecoords, RiemannPsols, Xchange, Xcommutator, Xgauge, abelsol, adjoint, autonomous, bernoullisol, buildsol, buildsym, canoni, caseplot, casesplit, checkrank, chinisol, clairautsol, constcoeffsols, convertalg, convertsys, dalembertsol, dcoeffs, de2diffop, dfieldplot, diffop2de, dpolyform, dsubs, eigenring, endomorphism_charpoly, equinv, eta_k, eulersols, exactsol, expsols, exterior_power, firint, firtest, formal_sol, gen_exp, generate_ic, genhomosol, gensys, hamilton_eqs, hypergeomsols, hyperode, indicialeq, infgen, initialdata, integrate_sols, intfactor, invariants, kovacicsols, leftdivision, liesol, line_int, linearsol, matrixde, matrix_riccati, maxdimsystems, moser_reduce, muchange, mult, mutest, newton_polygon, normalg2, odeadvisor, odepde, parametricsol, phaseportrait, poincare, polysols, ratsols, redode, reduceorder, reduce_order, regular_parts, regularsp, remove_rootof, riccati_system, riccatisol, rifread, rifsimp, rightdivision, rtaylor, separablesol, solve_group, super_reduce, symgen, symmetric_power, symmetric_product, symtest, transinv, translate, untranslate, varparam, zoom] > y:=(lambda^beta*x^beta)*(rho*exp(-rho*x))/(exp(-rho*l)- exp(-rho*u)); λ β x β ρ e ( ρ x ) y := e ( ρ l) e ( ρ u ) > z:=int(y,x=l..u); > eval(z); > eval(z); > lambda:=.443; > beta:=2; z := λ :=.443 β := 2 209

207 > rho:=.2; > l:=4; > u:=6; > cr:=100; > z:=int(y,x=2..5); > s:=z*cr; ρ :=.2 l := 4 u := 6 cr := 100 z := s := > 210

208 MAPLE PROGRAM FOR CUSTOMER S COST FOR SPELTW-APP1 > with(linalg): Warning, the protected names norm and trace have been redefined and unprotected > with(detools); Warning, the name adjoint has been redefined transinv, translate, untranslate, varparam, zoom] > y:=((lambda*l)^beta)*(rho*exp(-rho*l))/(exp(-rho*l)- exp(-rho*u)); ( λ L) β ρ e ( ρ L ) y := e ( ρ l) e ( ρ u ) > z:=int(y,l=l..u); l u z:= ( λ L) β ρ e ( ρ L ) e ( ρ l) e ( ρ u ) dl > lambda:=.443;beta:=2;rho:=.4;p:=.4;l:=1.5;u:=2;c:=100; λ :=.443 β := 2 ρ :=.4 p :=.4 l := 1.5 u := 2 c := 100 > s:=c*p*z; s := > 211

209 MAPLE PROGRAM FOR MANUFACTURER S COST FOR SPELTW-APP 1 > with(linalg): Warning, the protected names norm and trace have been redefined and unprotected > with(detools); Warning, the name adjoint has been redefined transinv, translate, untranslate, varparam, zoom] > y:=((lambda*l)^beta)*(rho*exp(-rho*l))/(exp(-rho*l)- exp(-rho*u)); ( λ L) β ρ e ( ρ L ) y := e ( ρ l) e ( ρ u ) > z:=int(y,l=l..u); l u z:= ( λ L) β ρ e ( ρ L ) e ( ρ l) e ( ρ u ) dl > lambda:=.443;beta:=2;rho:=.4;q:=.6;l:=4;u:=5;c:=100; λ :=.443 β := 2 ρ :=.4 q :=.6 l := 4 u := 5 c := 100 > s:=c*q*z; s := > 212

210 MAPLE PROGRAM FOR CUSTOMER S COST FOR SPELTW-APP 2 > with(linalg): Warning, the protected names norm and trace have been redefined and unprotected > with(detools): Warning, the name adjoint has been redefined > y:=(lambda*l)^beta*rho*exp(-rho*l)/(exp(-rho*l)-exp(- rho*u)); ( λ L) β ρ e ( ρ L ) y := e ( ρ l) e ( ρ u ) > z:=int(y,l=l..u); z := > lambda:=.241;beta:=2.31;rho:=.4;c:=70;l:=4;u:=6.5; λ :=.241 β := 2.31 ρ :=.4 c := 70 l := 4 u := 6.5 > k:=evalf(z); k := > k := ; > s:=k*c; s := > 213

211 MAPLE PROGRAM FOR MANUFACTURER S COST FOR SPELTW-APP 2 > with(linalg): Warning, the protected names norm and trace have been redefined and unprotected > with(detools); > y:=((lambda*l)^beta)*(rho*exp(-rho*l))/(exp(-rho*l)- exp(-rho*u)); ( λ L) β ρ e ( ρ L ) y := e ( ρ l) e ( ρ u ) > z:=int(y,l=l..u); l u z:= ( λ L) β ρ e ( ρ L ) e ( ρ l) e ( ρ u ) dl > lambda:=.443;beta:=2;rho:=.4;q:=.6;l:=4;u:=5;c:=100; λ :=.443 β := 2 ρ :=.4 q :=.6 l := 4 u := 5 c := 100 > s:=c*q*z; s := > 214

212 APPENDIX C: LIFETIME WARRANTY RISK CALCULATIONS (EXCEL) 215

213 APPENDIX C LIFETIME WARRANTY RISK CALCULATION USING EXCEL SIMULATIONS 216

214 217

215 218

216 219

217 220

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219 222

220 APPENDIX D: PROGRAMS FOR SERVICE CONTRACTS FOR RAIL MAINTENANCE 223

221 APPENDIX D OUT SOURCING RAIL MAINTENANCE THROUGH SERVICE CONTRACTS MATLAB PROGRAM FOR ESTIMATING RAIL FAILURE PARAMETERS function weibull_rail(d) %WEIBULL_RAIL Create plot of datasets and fits %WEIBULL_RAIL(D) %Created by Anisur Rahmana %Creates a plot, similar to the plot in the main distribution fitting window, using the data %that you provide as input. % This function was generated on 05-Sep :20:56 % Data from dataset "d data": % Y = d d = d(:); % Set up figure to receive datasets and fits f_ = clf; figure(f_); set(f_,'units','pixels','position',[ ]); legh_ = []; legt_ = {}; % handles and text for legend probplot('weibull'); ax_ = gca; title(ax_,''); set(ax_,'box','on'); grid(ax_,'on'); hold on; % --- Plot data originally in dataset "d data" t_ = ~isnan(d); Data_ = d(t_); h_ = probplot(ax_,data_,[],[],'noref'); % add to probability plot set(h_,'color',[ ],'Marker','o', 'MarkerSize',6); xlabel('data'); ylabel('probability') legh_(end+1) = h_; legt_{end+1} = 'd data'; % Nudge axis limits beyond data limits xlim_ = get(ax_,'xlim'); if all(isfinite(xlim_)) xlim_ = exp(log(xlim_) + [-1 1] * 0.01 * diff(log(xlim_))); set(ax_,'xlim',xlim_) end x_ = linspace(xlim_(1),xlim_(2),100); % --- Create fit "fit 2" % Fit this distribution to get parameter values 224

222 t_ = ~isnan(d); Data_ = d(t_); % To use parameter estimates from the original fit: p_ = [ , ]; p_ = wblfit(data_, 0.05); h_ = probplot(ax_,@wblcdf,p_); set(h_,'color',[1 0 0],'LineStyle','-', 'LineWidth',2); legh_(end+1) = h_; legt_{end+1} = 'fit 2'; hold off; h_ = legend(ax_,legh_,legt_,'location','northwest'); set(h_,'interpreter','none'); 225

223 ESTIMATION OF QUALITY OF MAINTENANCE Quality of maintenance can be measured by its effectiveness on the reliability or serviceability of the component that simply means how a maintenance action can extend the service life. If a maintenance action has a capability to extend the service life of a component with an expected service life of 30 years to 35 years, then the quality of maintenance can be determined as follows: Quality of maintenance Le L α = = = Le Where, L e = Extended service life L 0 = Expected service life without this type of maintenance Quality of maintenance ranges from 0 to 1. The quality of minimal repair action can be considered as almost 0, as this type of maintenance action cannot make a significant improvement of reliability of the component. As for example, a simple painting of a Power Supply wooden pole can extend the service life of the pole in an average 1-3 years which gives a value of α that falls in the range 0.02 to 0.08, if we consider the normal service life wooden pole to be between years. This range provides an almost zero restoration of reliability. On the other hand, a mechanical reinstatement action has an ability to extend the service life of wood pole ten to fifteen years that gives a value of α approximately equal to 0.3 or more. Once again the quality of maintenance is very much related to the cost of maintenance. The greater is the quality of maintenance, the higher is the cost of maintenance. This will be considered and included in deciding servicing strategies. 226

224 SERVICE CONTRACT POLICY 1 FOR RAIL MAINTENANCE MAPLE Program for the Model With Maintenance Quality alpha = 0.16, Beta = 2.78, λ = with(linalg): > x:='x':realn:='realn':realxmin:='realxmin':m:='m': > RealCmin:='RealCmin':a:='a':C:='C':N:='N': > cmr:=150.0: al:=0.16:cpm:=400:be:=2.78: lambda:=.189: ep:=0.1: > a:=(sum((k-k*al+1)^be-(k-k*al)^be,k=0..n)); N ( 65 / ) ( 65 / ) a := (( k + 1. ) k ) Sk = 0 > m:=400:xmin:=array(1..m):cmin=array(1..m): > for N from 1 to m do > C:=(1/N/x)*(cmr*(x^be*lambda^be)*a+(N)*cpm); > y1:=diff(c,x);xmin[n]:=fsolve(y1=0,x,0.1..m); > Cmin[N]:=subs( x=xmin[n], C ); > if xmin[n]<m/n then > Cmin[N]:=subs( x=xmin[n], C ); > else > xmin[n]:=m/n;cmin[n]:=subs( x=xmin[n], C );fi: > od: > > RealCmin:=Cmin[1]:RealXmin:=xmin[1]: > for N from 1 to m do > if Cmin[N]<RealCmin then > RealCmin:=Cmin[N]: > RealXmin:=xmin[N]: > RealN:=N: > fi: > od: > [RealN,RealXmin,RealCmin]; [5, , ] 227

225 SERVICE CONTRACT POLICY 2 FOR RAIL MAINTENANCE MAPLE Program for the Model With Maintenance Quality alpha = 0.16, Beta = 2.789, and Nu = 407 > with(linalg): Warning, the protected names norm and trace have been redefined and unprotected > x:='x':realn:='realn':realxmin:='realxmin':m:='m': > RealCmin:='RealCmin':a:='a':C:='C':N:='N': > cmr:=150.0: al:=0.16:cpm:=440:cre:=1700:be:=2.78: nu:=400: > a:=(sum((k-k*al+1)^be-(k-k*al)^be,k=0..n)); æ N æ139ö æ139ö ö ç è 50 ø è 50 ø a := ç Sk è ( k + 1. ) k ø = 0 > m:=300:xmin:=array(1..m):cmin=array(1..m): > for N from 1 to m do > C:=(1/N/x)*(cmr*(x^be/nu^be)*a+(N-1)*cpm+cre); > y1:=diff(c,x);xmin[n]:=fsolve(y1=0,x,0.1..m); > Cmin[N]:=subs( x=xmin[n], C ); > if xmin[n]<m/n then > Cmin[N]:=subs( x=xmin[n], C ); > else > xmin[n]:=m/n;cmin[n]:=subs( x=xmin[n], C );fi: > od: > > RealCmin:=Cmin[1]:RealXmin:=xmin[1]: > for N from 1 to m do > if Cmin[N]<RealCmin then > RealCmin:=Cmin[N]: > RealXmin:=xmin[N]: > RealN:=N: > fi: > od: > [RealN,RealXmin,RealCmin]; [ 5, , ] 228