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研究生： 呂亦宸
Yi-Chen Lu
論文名稱： 新失效模型於可修復系統之研究
New Failure Intensity Models for Repairable Systems
指導教授： 王福琨
Fu-Kwun Wang
口試委員: 林則孟
none
杜志挺
none
陳鴻基
none
歐陽超
none
學位類別： 博士
Doctor
系所名稱： 管理學院 - 工業管理系
Department of Industrial Management
論文出版年： 2016
畢業學年度： 104
語文別： 英文
論文頁數： 55
中文關鍵詞： 可靠度 、可維修系統 、浴缸曲線 、最大概似估計法
外文關鍵詞： reliability, repairable system, bathtub curve, maximum likelihood estimation
相關次數： 點閱：296  下載：0 
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    • 本研究主要是探討具有浴缸曲線特性之系統可靠度問題，一般而言，可靠度為特定產品在給定之操作環境及條件下，能成功的發揮其應有功能至一給定時間之機率，透過系統成功運作的機率，將可以判斷系統是否穩定。然而，在判斷系統是否穩定之前，首先須針對系統失效數據進行可靠度模型之建構，此時模型對數據的配適程度將會影響到可靠度的推估結果。本論文提出一個新的可靠度模型並應用於具有浴缸曲線特性之可維修系統，在研究過程中模型持續不斷地改良與精進。本研究主要是第二章至第四章為具有浴缸曲線特性之可維修系統的可靠度模型提出。第五章則是結論以及對於未來研究的建議。

    The main purpose of this study is to discuss the problem of reliability in repairable systems with bathtub-shaped failure intensity characteristic. In general, reliability is often defined as the probability that system can successful performs its required function under specified operating conditions for a stated period of time. The probability can determine the system is stable or unstable. However, the appropriate reliability model that matches the system failure data should be constructed before we evaluate the system. Because of the goodness-of-fit that matches failure data will affect the result of assessment. In this dissertation, we will propose a new reliability model to apply in repairable system with bathtub-shaped failure intensity characteristic. Chapter 2 - chapter 4 developed new failure intensity models for repairable systems. Finally, the contributions of this dissertation are summarized in Chapter 5.

    Table of Content 摘要 i Abstract ii Acknowledgement iii Table of Content iv List of Figures vi List of Tables vii Chapter 1 1 Introduction 1 1.1 Quality and Reliability 1 1.2 Repairable and Non-Repairable Systems 1 1.3 Bathtub-shaped and Generalized Bathtub-shaped Curve 2 1.4 Research Objectives 4 1.5 Organization 4 References 6 Chapter 2 7 A New Bounded Intensity Function for Repairable Systems 7 2.1 Introduction 7 2.2 New Model 8 2.3 Parameters Estimation 10 2.4 Illustrative Example 12 2.5 Conclusions 14 References 14 Chapter 3 16 A New Model for Repairable Systems with Nonmonotone Intensity Function 16 3.1 Introduction 16 3.2 New Model 18 3.3 Parameters Estimation 20 3.4 Illustrative Examples 22 3.5 Conclusions 30 References 30 Appendix 33 Chapter 4 35 A New Four-Parameter Failure Intensity Model for Repairable Systems 35 4.1 Introduction 35 4.2 Proposed Model 38 4.3 Parameters Estimation 40 4.4 Optimal Preventive Maintenance for the Proposed Model 42 4.5 Numerical Examples 43 4.6 Conclusions 48 References 49 Appendix 51 Chapter 5 54 Conclusions and Future Study 54 5.1 Conclusions 54 5.2 Future Study 55 List of Figures Figure 1.1 Bathtub-shaped curve 3 Figure 1.2 Generalized bathtub-shaped curve 4 Figure 1.3 Dissertation flow diagram 5 Figure 2.1 Changes in parameter c and θ of the intensity function for proposed model. 10 Figure 2.2 TTT plot on the 36 observations in Table 2.1 12 Figure 2.3 Plot of against time t for example 13 Figure 3.1 The failure intensities for various values of c and with a=0.1, k=0.02, =0.21 for the proposed model 19 Figure 3.2 TTT plot on the 55 observations for example 1 23 Figure 3.3 Plots of against time t for example 1 25 Figure 3.4 TTT plot on the 35 observations for example 2 25 Figure 3.5 Plots of against time t for example 2 27 Figure 3.6 TTT plot on the 50 observations for example 3 28 Figure 3.7 Plots of against time t for example 3 30 Figure 4.1 The failure intensities of the proposed model for 39 Figure 4.2 TTT plot for each failure data 44 Figure 4.3 Observed cumulative number of failures and ML estimate of the expected number of failures under different failure intensity functions for all data 47 Figure 4.4 Plots of against operating time t for each failure data 47 List of Tables Table 2.1 Failure times data of the 36 observations 12 Table 2.2 Comparison of different models for example 13 Table 3.1 Failure times (1000 km) of the powertrain system of bus 510 22 Table 3.2 Comparison of different models for example 1 24 Table 3.3 Failure times (1000 km) of the powertrain system of bus 514 25 Table 3.4 Comparison results of different models for example 2 27 Table 3.5 Simulation data with a failure truncated case based on an exponential-Weibull model with α=1.64,θ=0.0261, and σ=65.6363. 28 Table 3.6 Comparison results of different models for example 4. 29 Table 4.1 The development of failure intensity function for bathtub shaped behavior 37 Table 4.2 Simulated data 1 with a failure truncated case based on an exponential Weibull model with α = 1.2656,θ= 0.1183, and σ = 40.7382. 44 Table 4.3 Simulated data 2 with a failure truncated case based on an exponential Weibull model with α = 1.5129,θ= 0.1232, and σ = 67.0377 44 Table 4.4 Comparison results of different models for the two datasets 46 Table 4.5 The turning point and the PM interval under a variety of repair costs of different models for the two datasets 48

    CH1
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    6. Jensen F, and Petersen NE. Bum-in models for non-repairable and repairable equipment based upon bi-model component lifetime. Quality Assurance 1979; 5(4):l03-107.
    7. Jensen F, Petersen NE. Bum-in. John Wiley & Sons: New York, 1982.
    8. Alexanian IT, Brodie DE. A method for estimating the reliability of ICS. IEEE Transactions on Reliability 1997; R-26(5):359-361.
    9. Juran JM. Juranis Quality Control Handbook. McGraw-Hill: London, 1988.

    CH2
    1. Liu J, and Wang Y. On Crevecoeur’s bathtub-shaped failure rate model. Computational Statistics and Data Analysis 2013; 57(1):645-660.
    2. Meeker WQ, Escobar LA. Statistical Methods for Reliability Data. John Wiley & Sons: New York, 1998.
    3. Chen Z. A new two-parameter lifetime distribution with bathtub shape or increasing failure rate function. Statistics and Probability Letters 2000; 49(2):155-161.
    4. Crevecoeur GU. A model for the integrity assessment of ageing repairable systems. IEEE Transactions on Reliability 1993; 42(1):148-155.
    5. Hjorth U. A reliability distribution with increasing decreasing, constant and bathtub- shaped failure rates. Technometrics 1980; 22(1):99-107.
    6. Lai CD, Xie M, Murthy DNP. A modified Weibull distribution., IEEE Transactions on Reliability 2003; 52(1):33-37.
    7. Smith RM, and Bain LJ. An exponential power life-test distribution. Communications in Statistics-Simulation and Computation 1975; 4(5):469-481.
    8. Xie M, Lai CD. Reliability analysis using an additive Weibull model with bathtub-shaped failure function. Reliability Engineering and System Safety 1996; 52(1):87-93.
    9. Xie M, Tang Y, Goh TN. A modified Weibull extension with bathtub-shaped failure rate function. Reliability Engineering and System Safety 2002; 76(3):279-285.
    10. Guida M, Pulcini G. Reliability analysis of mechanical systems with bounded and bathtub shaped intensity function. IEEE Transactions on Reliability 2009; 58(3):432-443.
    11. Lee L. Testing adequacy of the Weibull and log linear rate models for a poisson process. Technometrics 1980; 22(2):195-199.
    12. Pulcini G. Modeling the failure data of repairable equipment with bathtub failure intensity. Reliability Engineering and System Safety 2001; 71(2):209-218.
    13. Block HW, Li Y, Savits TH, Wang J. Continuous mixtures with bathtub-shaped failure rates. Journal of the Applied Probability 2008; 45(1):260-270.
    14. Block HW, Langberg NA, Savits TH. Continuous mixtures of exponentials and IFR gammas having bathtub-shaped failure rates. Journal of the Applied Probability 2010; 47(4):899-907.
    15. Jiang R, Murthy DNP. Mixture of Weibull distribution-parameteric characterization of failure rate function. Applied Stochastic Models and Data Analysis 1998; 14(1):47-65.
    16. Ascher H. Reliability Models for Repairable Systems. Reliability Technology: Theory & Applications. Elsevier Science: North-Holland, 1986.
    17. Krivtsov VV. Practical extensions to NHPP application in repairable system reliability. Reliability Engineering and System Safety 2007; 92(5):560-562.
    18. Attardi L, Pulcini G. New model for repairable systems with bounded failure intensity. IEEE Transactions on Reliability 2005; 54(4):572-582.
    19. Vaurio JK. Identification of process and distribution characteristics by testing monotonic and non-monotonic trends in failure intensities and hazard rates. Reliability Engineering and System Safety 1999; 64(3):345-357.
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    22. Kvaly JT, Lindqvist B. TTT-based tests for trend in repairable system data. Reliability Engineering and System Safety 1998; 60(1):13-28.
    23. Glaser RE. Bathtub and related failure rate characterizations. Journal of the American Statistical Association 1980; 75(371):667-672.
    24. Barlow RE, Campo RA. Total time on test processes and applications to failure data analysis. In Reliability and Fault Tree Analysis, Society for Industrial and Applied Mathematics, 1975.
    25. Bergman B, Klefsjo B. A graphical method applicable to age replacement problems. IEEE Transactions on Reliability 1982; R-31(5):478-481.
    26. Aarset MV. How to identity a bathtub hazard rate. IEEE Transactions on Reliability 1987; 36(1):106-108.
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    28. Cid JER, Achcar JA. Bayesian inference for nonhomogeneous Poisson processes in software reliability models assuming nonmonotonic intensity functions. Computational Statistics and Data Analysis 1999; 32(2):147-159.
    29. R Development Core Team, R: A language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria, 2013.

    CH3
    1. Meeker WQ, Escobar LA. Statistical Methods for Reliability Data. John Wiley & Sons: New York, 1998.
    2. Dijoux Y. A virtual age model based on a bathtub shaped initial intensity. Reliability Engineering and System Safety 2009; 94(5):982-989.
    3. Dijoux Y, Idee E. New virtual age models for bathtub shaped failure intensities. Safety, Reliability and Risk Analysis: Theory, Methods and Applications 2009; 1:1901-1908.
    4. Veber B, Nagode M, Fajdiga M. Generalized renewal process for repairable systems based on finite Weibull mixture. Reliability Engineering and System Safety 2008; 93(10): 1461-1472.
    5. Mun BM, Bae SJ, Kvam PH. A superposed log-linear failure intensity model for repairable artillery systems. Journal of Quality Technology 2013; 45(1):100-115.
    6. Ascher HE, Feingold H. Repairable Systems Reliability: Modeling, Inference, Misconceptions and Their Causes. Marcel Dekker: New York, 1984.
    7. Cox DR, Lewis PAW. The Statistical Analysis of Series of Events. John Wiley & Sons: New York, 1966.
    8. Crow LH. Reliability analysis for complex repairable systems. In Reliability and Biometry, Proschan F, Serfling RJ (eds). Society for Industrial and Applied Mathematics: Philadelphia, 1974; 379-410.
    9. Pulcini G. Modeling the failure data of repairable equipment with bathtub failure intensity. Reliability Engineering and System Safety 2001; 71(2):209-218.
    10. Guida M, Pulcini G. Reliability analysis of mechanical systems with bounded and bathtub shaped intensity function. IEEE Transactions on Reliability 2009; 58(3):432-443.
    11. Pulcini G. A bounded intensity process for the reliability of repairable equipment. Journal of Quality Technology 2001; 33(4):480-492.
    12. Krivtsov VV. Practical extensions to NHPP application in repairable system reliability. Reliability Engineering and System Safety 2007; 92(5):560-562.
    13. Glaser RE. Bathtub and related failure rate characterizations. Journal of the American Statistical Association 1980; 75(371):667-672.
    14. Rajarshi S, Rajarshi MB. Bathtub distributions: a review. Communications in Statistics-Theory and Methods 1988; 17(8):2597-2621.
    15. Pham H, Lai CD. On recent generalizations of the Weibull distribution. IEEE Transactions on Reliability 2007; 56(3):454-458.
    16. Crevecoeur GU. A model for the integrity assessment of ageing repairable systems. IEEE Transactions on Reliability 1993; 42(1):148-155.
    17. Chen Z. A new two-parameter lifetime distribution with bathtub shape or increasing failure rate function. Statistics and Probability Letters 2000; 49(2):155-161.
    18. Wang KS, Hsu FS, Liu PP. Modeling the bathtub shape hazard rate function in terms of reliability. Reliability Engineering and System Safety 2002; 75(3):397-406.
    19. Xie M, Tang Y, Goh TN. A modified Weibull extension with bathtub-shaped failure rate function. Reliability Engineering and System Safety 2002; 76(3):279-285.
    20. Liu J, Wang Y. On Crevecoeur’s bathtub-shaped failure rate model. Computational Statistics and Data Analysis 2013; 57(1):645-660.
    21. Mudholkar GS, Srivastava DK, Freimer M. The exponentiated Weibull family: reanalysis of the bus-motor-failure data. Technometrics 1995; 37(4):436-445.
    22. Mudholkar GS, Srivastava DK, Kollia GD. A generalization of the Weibull distribution with application to analysis of survival data. Journal of the American Statistical Association 1996; 91(436):1575-1583.
    23. Saha A, Hilton L. Expo-power: A flexible hazard function for duration data models. Economics Letters 1997; 54(3):227-233.
    24. Silva GO, Ortega EMM, Cordeiro GM. The beta modified Weibull distribution. Lifetime Data Analysis 2010; 16(3):409-430.
    25. Cordeiro GM, Ortega EMM, Silva GO. The beta extended Weibull family. Journal of Probability and Statistical Science 2012; 10(10):15-40.
    26. Cordeiro GM, Gomes AE, da-Sivla CQ, Ortega EMM. The beta exponentiated Weibull distribution. Journal of Statistical Computation and Simulation 2013; 83(1):114-138.
    27. Domma F, Condino F. The Beta-Dagum distribution: definition and properties. Communications in Statistics-Theory and Methods 2013; 42(22):4070-4090.
    28. Hjorth U. A reliability distribution with increasing decreasing, constant and bathtub- shaped failure rates. Technometrics 1980; 22(1):99-107.
    29. Xie M, Lai CD. Reliability analysis using an additive Weibull model with bathtub-shaped failure function. Reliability Engineering and System Safety 1996; 52(1):87-93.
    30. Wang FK. A new model with bathtub-shaped failure rate using an additive Burr XII distribution. Reliability Engineering and System Safety 2000; 70(3):305-312.
    31. Jiang R, Murthy DNP, Ji P. Models involving two inverse Weibull distributions. Reliability Engineering and System Safety 2001; 73(1):73-81.
    32. Jiang R, Murthy DNP. Mixture of Weibull distribution-parameteric characterization of failure rate function. Applied Stochastic Models and Data Analysis 1998; 14(1):47-65.
    33. Block HW, Li Y, Savits TH, Wang J. Continuous mixtures with bathtub-shaped failure rates. Journal of the Applied Probability 2008; 45(1):260-270.
    34. Block HW, Langberg NA, Savits TH. Continuous mixtures of exponentials and IFR gammas having bathtub-shaped failure rates. Journal of the Applied Probability 2010; 47(4):899-907.
    35. Slymen DJ, Lanchenbruch PA. Survival distributions arising from two families and generated by transformations. Communications in Statistics-Theory and Methods 1984; 13(10):1179-1201.
    36. Louzada-Neto F. Polyhazard models for lifetime data. Biometrics 1999; 55(4):1281-1285.
    37. Louzada-Neto F. Modelling lifetime data: a graphical approach. Applied Stochastic Models in Business and Industry 1999; 15(2):123-129.
    38. Louzada-Neto F, Mazucheli J, Achcar JA. Mixture hazard models for lifetime data. Biometrical Journal 2002; 44(1):3-14.
    39. Barreto-Souza W, Santos AHS, Cordeiro GM. The beta generalized exponential distribution. Journal of Statistical Computation and Simulation 2010; 80(2):159-172.
    40. R Development Core Team, R: A language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria, 2013.
    41. Nelson W. Accelerated Testing: Statistical Models, Test Plan. John Wiley & Sons: New York, 1990.
    42. Begman B, Klefsjo B. A graphical method applicable to age replacement problems. IEEE Transactions on Reliability 1982; R-31(5):478-481.
    43. Kvaly JT, Lindqvist B. An area based test for trend in repairable systems data, 1996. http://www.math.ntnu.no/preprint/statistics/1996, accessed Nov. 12. 2013.
    44. Kvaly JT, Lindqvist B. TTT-based tests for trend in repairable system data. Reliability Engineering and System Safety 1998; 60(1):13-28.
    45. Vaurio JK. Identification of process and distribution characteristics by testing monotonic and non-monotonic trends in failure intensities and hazard rates. Reliability Engineering and System Safety 1999; 64(3):345-357.
    46. Kvaly JT, Lindqvist BH. A class of tests for renewal process versus monotonic and nonmonotonic trend in repairable systems data. Mathematical and Statistical Methods in Reliability 2003; 7:401-414.

    CH4
    1. Meeker WQ, Escobar LA. Statistical Methods for Reliability Data. John Wiley & Sons: New York, 1998.
    2. KuoW, Kuo Y. Facing the headaches of early failures: a state-of-the-art review of burn-in decisions. Proceedings of the IEEE 1983; 71(11):1257-1266.
    3. Jensen F, Petersen NE. Bum-in models for non-repairable and repairable equipment based upon bi-model component lifetime. Quality Assurance 1979; 5(4):l03-107.
    4. Alexanian IT, Brodie DE. A method for estimating the reliability of ICS. IEEE Transactions on Reliability 1997; R-26(5):359-361.
    5. Lee L. Testing adequacy of the Weibull and log linear rate models for a poisson process. Technometrics 1980; 22(2):195-199.
    6. Pulcini G. Modeling the failure data of repairable equipment with bathtub failure intensity. Reliability Engineering and System Safety 2001; 71(2):209-218.
    7. Guida M, Pulcini G. Reliability analysis of mechanical systems with bounded and bathtub shaped intensity function. IEEE Transactions on Reliability 2009; 58(3):432-443.
    8. Mun BM, Bae SJ, Kvam PH. A superposed log-linear failure intensity model for repairable artillery systems. Journal of Quality Technology 2013; 45(1):100-115.
    9. Wang FK, Lu YC. A new model for repairable systems with nonmonotone intensity function. Quality and Reliability Engineering International 2015; 31(8):1553-1563.
    10. Cox DR, Lewis PAW. The Statistical Analysis of Series of Events. John Wiley & Sons: New York, 1966.
    11. Crow LH. Reliability analysis for complex repairable systems. In Reliability and Biometry, Proschan F, Serfling RJ (eds). Society for Industrial and Applied Mathematics: Philadelphia, 1974; 379-410.
    12. Pulcini G. A bounded intensity process for the reliability of repairable equipment. Journal of Quality Technology 2001; 33(4):480-492.
    13. Krivtsov VV. Practical extensions to NHPP application in repairable system reliability. Reliability Engineering and System Safety 2007; 92(5):560-562.
    14. Glaser RE. Bathtub and related failure rate characterizations. Journal of the American Statistical Association 1980; 75(371):667-672.
    15. Rajarshi S, Rajarshi MB. Bathtub distributions: a review. Communications in Statistics-Theory and Methods 1988; 17(8):2597-2621.
    16. Pham H, Lai CD. On recent generalizations of the Weibull distribution. IEEE Transactions on Reliability 2007; 56(3):454-458.
    17. Crevecoeur GU. A model for the integrity assessment of ageing repairable systems. IEEE Transactions on Reliability 1993; 42(1):148-155.
    18. Chen Z. A new two-parameter lifetime distribution with bathtub shape or increasing failure rate function. Statistics and Probability Letters 2000; 49(2):155-161.
    19. Wang KS, Hsu FS, Liu PP. Modeling the bathtub shape hazard rate function in terms of reliability. Reliability Engineering and System Safety 2002; 75(3):397-406.
    20. Xie M, Tang Y, Goh TN. A modified Weibull extension with bathtub-shaped failure rate function. Reliability Engineering and System Safety 2002; 76(3):279-285.
    21. Liu J, Wang Y. On Crevecoeur’s bathtub-shaped failure rate model. Computational Statistics and Data Analysis 2013; 57(1):645-660.
    22. Mudholkar GS, Srivastava DK, Freimer M. The exponentiated Weibull family: reanalysis of the bus-motor-failure data. Technometrics 1995; 37(4):436-445.
    23. Mudholkar GS, Srivastava DK, Kollia GD. A generalization of the Weibull distribution with application to analysis of survival data. Journal of the American Statistical Association 1996; 91(436):1575-1583.
    24. Saha A, Hilton L. Expo-power: A flexible hazard function for duration data models. Economics Letters 1997; 54(3):227-233.
    25. Silva GO, Ortega EMM, Cordeiro GM. The beta modified Weibull distribution. Lifetime Data Analysis 2010; 16(3):409-430.
    26. Cordeiro GM, Ortega EMM, Silva GO. The beta extended Weibull family. Journal of Probability and Statistical Science 2012; 10(10):15-40.
    27. Cordeiro GM, Gomes AE, da-Sivla CQ, Ortega EMM. The beta exponentiated Weibull distribution. Journal of Statistical Computation and Simulation 2013; 83(1):114-138.
    28. Domma F, Condino F. The Beta-Dagum distribution: definition and properties. Communications in Statistics-Theory and Methods 2013; 42(22):4070-4090.
    29. Xie M, Lai CD. Reliability analysis using an additive Weibull model with bathtub-shaped failure function. Reliability Engineering and System Safety 1996; 52(1):87-93.
    30. Wang FK. A new model with bathtub-shaped failure rate using an additive Burr XII distribution. Reliability Engineering and System Safety 2000; 70(3):305-312.
    31. Jiang R, Murthy DNP, Ji P. Models involving two inverse Weibull distributions. Reliability Engineering and System Safety 2001; 73(1):73-81.
    32. Slymen DJ, Lanchenbruch PA. Survival distributions arising from two families and generated by transformations. Communications in Statistics-Theory and Methods 1984; 13(10):1179-1201.
    33. Louzada-Neto F. Polyhazard models for lifetime data. Biometrics 1999; 55(4):1281-1285.
    34. Louzada-Neto F. Modelling lifetime data: a graphical approach. Applied Stochastic Models in Business and Industry 1999; 15(2):123-129.
    35. Louzada-Neto F, Mazucheli J, Achcar JA. Mixture hazard models for lifetime data. Biometrical Journal 2002; 44(1):3-14.
    36. Barreto-Souza W, Santos AHS, Cordeiro GM. The beta generalized exponential distribution. Journal of Statistical Computation and Simulation 2010; 80(2):159-172.
    37. Domma F, Condino F. A new class of distribution functions of lifetime data. Reliability Engineering and System Safety 2014; 129:36-45.
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    43. Begman B, Klefsjo B. A graphical method applicable to age replacement problems. IEEE Transactions on Reliability 1982; R-31(5):478-481.
    44. Kvaly JT, Lindqvist B. An area based test for trend in repairable systems data, 1996. http://www.math.ntnu.no/preprint/statistics/1996, accessed Nov. 12. 2013.
    45. Kvaly JT, Lindqvist B. TTT-based tests for trend in repairable system data. Reliability Engineering and System Safety 1998; 60(1):13-28.
    46. Vaurio JK. Identification of process and distribution characteristics by testing monotonic and non-monotonic trends in failure intensities and hazard rates. Reliability Engineering and System Safety 1999; 64(3):345-357.
    47. Kvaly JT, Lindqvist BH. A class of tests for renewal process versus monotonic and nonmonotonic trend in repairable systems data. Mathematical and Statistical Methods in Reliability 2003; 7:401-414.
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