POST-SALE COST MODELING AND OPTIMIZATION LINKING WARRANTY AND PREVENTIVE MAINTENANCE WU JUN (B ENG, TSINGHUA UNIVERSITY) A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT OF INDUSTRIAL AND SYSTEMS ENGINEERING NATIONAL UNIVERSITY OF SINGAPORE 2011 ACKNOWLEDGEMENTS First and foremost I would like to express my deepest gratitude to my supervisor Prof Xie Min for his precious advices, supervision and generous support through my entire doctoral study I have benefited a lot from his knowledge as well as attitude of dealing with work My gratitude also goes to Dr Ng Tsan Sheng Adam, my co-supervisor, for his valuable suggestions on my research as well as the time he spent reading and examining my academic papers This dissertation would not have been possible without their help Sincere thanks also go to other ISE faculty members and office staff who have assisted me and inspired me in one form or another More thanks are given to Dr Jaruphongsa Wikrom who provided useful guidance on my first year research in NUS I’m very grateful to my fellow graduate students in the ISE Computing Laboratory, and my deal colleagues at the NUS Graduate Students’ Society Thanks to each of them for their long-time support, friendship and trust The journey has been made much more enjoyable with their company Lastly, I would like to dedicate this dissertation to my parents in China, for their love, understanding and encouragement that I could never repay i TABLE OF CONTENTS ACKNOWLEDGEMENTS i  TABLE OF CONTENTS ii  SUMMARY vii  LIST OF TABLES x  LIST OF FIGURES xi  LIST OF SYMBOLS xii  CHAPTER 1  1.1  INTRODUCTION 1  Product Warranty 1  1.1.1   Role of Warranty 2  1.1.2   Warranty Policies 3  1.2   Maintenance 5  1.2.1   Classification on Maintenance 5  1.2.2   Preventive Maintenance Policies 8  1.3   Maintenance Cost Factor 9  1.3.1   Warranty and Preventive Maintenance Policies 10  1.3.2   Maintenance Cost Structure 11  1.3.3   Product Ageing Mechanism 12  1.3.4   Impact of Maintenance on the Product Reliability 12  1.3.5   Criterion for Cost Measurement 13  1.4   Problems and Research Objectives 14  1.5   Organization of the Thesis 17  ii CHAPTER 2  LITERATURE REVIEW 20  2.1   Methods for Modeling Imperfect Maintenance 20  2.2   Warranty Cost and Revenue Analysis 24  2.3   Preventive Maintenance Policies 32  2.3.1   Age-Dependent PM Policy 33  2.3.2   Periodic PM Policy 35  2.3.3   Condition-Based PM Policy 37  2.4   Literature Review Linking Warranty and Preventive Maintenance 39  2.4.1   Warranty Service Design Incorporating Preventive Maintenance 39  2.4.2   Life-Cycle Maintenance Service Design under Warranty Context 41  CHAPTER 3  WARRANTY COST ANALYSIS FOR COMPLEX SYSTEMS WITH FAILURE INTERACTION 45  3.1   Introduction 45  3.2  General Model and Assumptions 47  3.2.1  Renewing Free-Replacement Warranty (RFRW) 47  3.2.2  Failure Interaction 48  3.2.3  Definition of Ns(w), pi(w) and αi(w) 49  3.3  RFRW for Series Systems under Failure Interaction 51  3.3.1  Distribution of Nij 52  3.3.2  Warranty Cost Analysis 53  3.4  RFRW for Parallel Systems under Failure Interaction 54  3.4.1  Recursive Algorithms for the Calculation of FS(w) 55  3.4.2  Illustration Example for Memoryless System 58  3.5  Numerical Example and Sensitivity Analysis 59  iii 3.6  CHAPTER 4  Conclusion 65  CONDITION-BASED WARRANTY SERVICE DESIGN 67  4.1  Introduction 67  4.2  Problem Description 70  4.2.1  Assumption 70  4.2.2  Condition-based Warranty Servicing Policies 71  4.3  Cost Modeling for Renewing Warranty 73  4.3.1  Preliminary Results 74  4.3.2  Warranty Cost Model for Policy A1 75  4.3.3  Warranty Cost Model for Policy A2 76  4.4  Cost Modeling for Non-Renewing Warranty 79  4.4.1  Warranty Cost Model for Policy B1 80  4.4.2  Warranty Cost Model for Policy B2 81  4.4.3  Warranty Cost Model for Policy C1 83  4.4.4  Warranty Cost Model for Policy C2 84  4.5  Numerical Examples 85  4.6  Conclusion 90  CHAPTER 5  PERIODIC PM SERVICE DESIGN INCORPORATING VALUE OF MAINTENANCE 92  5.1  Introduction 92  5.2  Periodic Preventive Maintenance Model 95  5.3  Life-Cycle Cost Model Incorporating Ageing Losses 99  5.3.1  Total Life-Cycle Cost Model 99  5.3.2  Modeling of Ageing Losses 100  iv 5.3.3  Discussion on Optimal PM Strategies 102  5.4  Numerical Example 104  5.5  Conclusion 112  CHAPTER 6  MSS MAINTENANCE SERVICE DESIGN 114  6.1  Introduction 114  6.2  Model Formulation 117  6.2.1  System Description 117  6.2.2  System Replacement Policies 119  6.3  Model Development 120  6.3.1  Preliminary Results 120  6.3.2  The EDMC Model for Policy O 121  6.3.3  The EDMC Model for Policy A 123  6.3.4  The EDMC Model for Policy B 125  6.3.5  The EDMC Model with Warranty Incorporation 127  6.4  Optimization of the Maintenance Thresholds 131  6.4.1  Method 1: Optimizing (J,τ) Using Inverse LST 131  6.4.2  Method 2: Optimizing (J,τ) Using Discretization in the Time Domain 132  6.5  Numerical Example 137  6.6  Conclusion 143  CHAPTER 7  WARRANTY REVENUE ANALYSIS INTEGRATING USER'S MAINTENANCE DECISIONS 144  7.1  Introduction 144  7.2  Model Formulation 147  7.2.1  Product Warranty and Age Replacement 147  v 7.2.2  Sales Model 148  7.2.3  The Buyer’s Cost Model and His Decision Problem 149  7.2.4  The Seller’s Profit Model and His Decision Problem 151  7.3  Integrating the Buyer’s and Seller’s Optimal Strategies 152  7.3.1  The Buyer’s Optimal Strategy 153  7.3.2  The Seller’s Optimal Strategy 157  7.4  Special Case: Cd → or r(t) ≡ r(0) 159  7.4.1  Stationary Point for π0(w,Cp ) 159  7.4.2  Second-order Conditions 160  7.5  Illustrative Examples 161  7.5.1  Gamma Distribution: When Y(∞) < ∞ 161  7.5.2  Weibull Distribution: When Y(∞) = ∞ 163  7.5.3  Impact of Warranty Cost-sharing Ratio ρ on (Cp*,w*) and π1(Cp*,w*) 165  7.6  CHAPTER 8  Conclusion 167  CONCLUSION AND FUTURE WORK 169  BIBLIOGRAPHY 173  APPENDIX A.  Proof of Lemma 3.1 198  APPENDIX B.  Proof of Lemma 3.2 199  APPENDIX C.  Proof of Lemma 3.3 200  APPENDIX D.  Expressions of the EDMC for Scenario 201  APPENDIX E.  Proof of Theorem 7.1 203  vi SUMMARY This thesis investigates several important issues in post-sale cost modeling and optimization The costing of a maintenance program is analyzed under the warranty context from both the manufacturer’s and consumer’s point of view In particular, for the manufacturer, we study the issue of warranty cost analysis where preventive maintenance (PM) is an important planning tool in terms of service improvement and warranty cost reduction For the consumer, we investigate the PM scheduling problem during the product life cycle where warranty is an important factor in influencing the maintenance decisions In addition to costing analysis, warranty as an effective marketing instrument for enhancing the revenue is also discussed In the modeling of warranty expense for complex systems, the majority of researchers presume the system as a “black box” which does not utilize the information of inner structure Chapter studies two basic system structures, i.e series and parallel, and derives the respective warranty cost functions under renewable warranty policies Unique in this study is the incorporation of failure dependence factor between each two of the system components We investigate the impact of such factor on the total warranty expense and the risk of ignoring it Manufacturers usually rely on the age information of a product as the single metric for maintenance design under warranty Such policy may result in unnecessary maintenance operations for products with inadequate deteriorations In Chapter 4, we propose such a condition-based warranty policy that counts on the product state information for executing maintenance decisions We derive the warranty cost vii functions under both renewable and non-renewable warranty policies, based on which the optimal scheduling of inspection services is further analyzed For the owners of industrial equipments, investing in maintenance is widely modeled from the costing perspective while the value or return of maintenance investments is seldom emphasized Motivated by this, Chapter investigates the optimal design of maintenance servicing on revenue-generating equipments by integrating both the cost and value aspects of maintenance The study assumes imperfect maintenance operations and generalizes the existing periodic PM models The influence of warranty as well as many other models parameters on the optimal PM decisions is illustrated In the design of maintenance policies, two major assumptions are commonly adopted: binary system (i.e functioning or failed) and infinite planning horizon In contrast, almost all systems are operating under finite life times and many of them exhibit multiple performance levels Therefore, in Chapter 6, we investigate the repairreplacement policies for multi-state systems (MSS) under finite life cycles Corrective and preventive replacement decisions are modeled and compared via two control parameters – a threshold on the current system state and a threshold on the residual life cycle Value of time is taken into account for the maintenance cost modeling Extension is further made to generalize the cost functions under the warranty context Warranty as an effective product marketing tool has been extensively studied in the literature Majority of the researchers focus on the joint determination of selling price and warranty length that maximizes the seller’s profit (rate) function In Chapter 7, we further extend this topic by integrating the factor of the buyer’s age-replacement decisions into the design phase of the seller’s product marketing strategy For such viii integration to be viable, a game theoretic model is formulated that allows the seller to foresee the buyer’s maintenance decisions and subsequently make his own moves ix Sheu, S.H., 1990 Periodic replacement when minimal repair costs depend on the age and the number of minimal repairs for a multi-unit system Microelectronics Reliability 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Tian, Z.G., 2006 Performance evaluation of generalized multi-state k-Outof-n systems IEEE Transactions on Reliability 55, 319–327 197 APPENDIX A Proof of Lemma 3.1 To prove Lemma 3.1, we have Pr | , Pr d d Pr ( are independent ) ( d and and are independent ) Ω, d    (since   and  Ω, ) The definition of d is straightforward As under series structure, Pr , Ω Pr  Ω ∑ , Ω Pr  , we have             198 APPENDIX B Proof of Lemma 3.2 The proof of (3.4) is presented first Define  , According to the discussion in Section 3.1, the number of natural failures are independent with each other even failure interaction is considered Using the definition in Lemma 3.1, the probability that the system failure is caused by component  is simply  | ~Binomial , , We have Ω, and thus ∞ Pr Pr |   see Section 3.2 ∞ ∞   The last two steps utilize the result ∑∞ Equation (3.2) can be proved in a similar way, based on which (3.3) and (3.5) simply follow ∞ Pr Pr | Pr   (using the results in 3.4 )                  ∞ 199 APPENDIX C Proof of Lemma 3.3 Ω For any 0,1,2, … , , let | | denote the number of elements (i.e components) in Using the method of Mathematical Induction (MI): Prove that the result holds for any single-component system, i.e | | , 1 , Then for any subsystem with | | , 1, the result holds Assume that (3.14) is true for any subsystem with | | the expression of   , subsystem with | | Ω Then according to , it is easily verified that, for any 1, , , According to the definition of , , we have ∑ , , ∑ , , ∑ , which exactly means that (3.14) is true for all subsystems of Ω 200 APPENDIX D Expressions of the EDMC for Scenario The de facto “analytical” form of the EDMC under Policies O, A and B for Scenario without warranty incorporation is given as: Policy O 5098.8 100.0 226.5 0.91 201.6 0.91 4772.3   5098.8 105.6 198.2 0.91 235.2 0.91 5006.3   5098.8 138.7 299.1 0.91 189.4 0.91 5259.2   5098.8 146.6 183.8 0.91 308.4 0.91 5429.1 Policy A |1, |2, 0.53 0.68 0.47 93.2 |3, 4880.0 0.32 0.36 0.53 4880.0 3821.2 0.53 3728.0 0.36 0.32 0.32 196.2 4224 0.75 201 0.02 0.58 0.56 109.2 0.50 0.32 0.13 0.75 4027.8 Policy B |1, 3822.8 52.2 78.7 0.13 1.00 135.8 0.38 0.36 158.4 |3, 192.9 0.26 0.96 169.8 0.96 0.22 0.22 3663.0 0.28 0.06 0.26 0.09 0.10 0.21 0.32 0.60 0.07 0.36 1.02 0.11 0.21 0.05 0.20 0.28 4346.0 0.17 0.54 118.2 0.18 0.05 0.23 0.04 45.2 0.40 1.02 0.06 3691.8 0.10 47.6 0.12 0.20 0.24 |2, 0.07 0.24 0.15 1.00 3828.8 0.08 0.05 0.11 0.03 0.31 0.47 0.08 0.03 0.30 4107.9 202 APPENDIX E Proof of Theorem 7.1 Denote and as the optimal replacement ages for and respetively / 1) From the assumption, we know that Given   and , it is easy to verify that optimal replacement age for is obtained at To derive the optimal replacement age for When Given ′ for Thus, the , we discuss the following cases ∞ ′ 0, we have ∞ ∞ Comparing and this results in ∞ Consequently, Thus ∞ and ′ for any , we have ∞ When For is an increasing function of , there is a finite replacement age is minimized is thus obtained from Again, comparing thus and ( ′ and ∞ and ), it is easy to verify that , and On the other hand, let function of and such that Since is an increasing is a decreasing function of , we simply have ′ 203 and Furthermore, There is a unique ∞ ∞ ,∞ solution ∞ that , satisfies Combining the definition of , we have Consequently, that provides a lower bound for that is tight when i.e Note , i.e lim When ′ Note that for any achieved at , and Hence, the optimal solution is / 2) ∞ When ∞ is minimized at for and for ∞ Hence, Thus , we have ′ and Again, ∞ When is minimized at Again, , which follows the same logic as 1) Since ′ and , and and , which follows the same logic as 1) Comparing and , it is easy to verify that When 204 We have ′ for and ′ for Thus When Note that Thus, On the other hand, Comparing when and for and and thus , we have , and 205 ... information for warranty and preventive maintenance (PM) cost modeling and optimization Reviews of existing literature linking warranty and PM modeling are presented in detail 2.1 Methods for Modeling. .. Review Linking Warranty and Preventive Maintenance 39  2.4.1   Warranty Service Design Incorporating Preventive Maintenance 39  2.4.2   Life-Cycle Maintenance Service Design under Warranty. .. several important issues in post- sale cost modeling and optimization The costing of a maintenance program is analyzed under the warranty context from both the manufacturer’s and consumer’s point of