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Extended Weibull Distributions in Reliability Engineering Are Highlighted

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Extended Weibull Distributions in Reliability Engineering Are Highlighted EXTENDED WEIBULL DISTRIBUTIONS IN RELIABILITY ENGINEERING TANG YONG NATIONAL UNIVERSITY OF SINGAPORE 2004 EXTENDED WEIBULL DISTRIBUTIONS IN RELIABILITY ENGINEERING TANG YONG (Bachelor of Economics, University of Science & Technology of China) A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT OF INDUSTRIAL & SYSTEMS ENGINEERING NATIONAL UNIVERSITY OF SINGAPORE 2004 ACKNOWLEDGEMENT I would like to express my heartfelt gratitude to my supervisors, Associate Professor Xie Min and Professor Goh Thong Ngee, for both their academic guidance of my research work and their kind concern of my life during the full period of my study in National University of Singapore. My special thankfulness also goes to Professor Ang Beng Wah, Associate Professor Tang Loon Ching, Dr. Lee Loo Hay and all the other faculty members in the Department of Industrial and Systems Engineering, from whom I have learnt a lot through the coursework, discussions and research seminars. Their enthusiasm in the research work and their personalities will always be remembered. I am also indebted to Ms Ow Lai Chun and many classmates who really made my stay in the department so enjoyable and fruitful. Particularly, I would like to give thanks to Aldy, Fengling, Guiyu, He Bin, Li Yanni, Jiying, Pei Chin, Priya, Shubin, Tingting, Wee Tat, Zeng Ling, Zhang Yu and the other friends for their help in one way or another. Last but never the least, I want to thank my parents for their continuous encouragement and support on my life and study in Singapore. TANG YONG i TABLE OF CONTENTS ACKNOWLEDGEMENT............................................................................................. i TABLE OF CONTENTS ............................................................................................. ii SUMMARY .................................................................................................................. iv NOMENCLATURE.................................................................................................... vii LIST OF FIGURES...................................................................................................viii LIST OF TABLES........................................................................................................ x Chapter 1 Introduction ................................................................................................ 1 1.1 Reliability........................................................................................................ 1 1.2 Reliability Engineering................................................................................... 3 1.3 Reliability Modeling....................................................................................... 6 1.4 Prospect of the Study .................................................................................... 11 1.5 Organization of the Dissertation................................................................... 14 Chapter 2 Literature Review..................................................................................... 18 2.1 Standard Weibull Distribution...................................................................... 18 2.2 Bathtub Failure Rate Distributions ............................................................... 27 2.3 Comments ..................................................................................................... 35 Chapter 3 Model Validity: Goodness of Fit Tests of Weibull Distribution........... 36 3.1 Introduction................................................................................................... 36 3.2 Goodness-of-fit Tests for the Weibull Distribution...................................... 38 3.3 Modified Weibull Distribution and its Properties......................................... 45 3.4 Tests of the Weibull Distribution Against a Modified Weibull Distribution ... ...................................................................................................................... 50 3.5 A Numerical Example .................................................................................. 53 3.6 Study of Power of Test for the Weibull Distribution Against the Modified Weibull Distribution ..................................................................................... 60 3.7 Conclusion .................................................................................................... 64 Chapter 4 A Weibull Extension Model with Bathtub-Shaped Failure Rate Function ....................................................................................................................... 67 4.1 Introduction................................................................................................... 67 4.2 A New Weibull Extension ............................................................................ 69 4.3 Relationship of Weibull Distribution and Other Distributions..................... 73 ii 4.4 Parametric Estimation of the Weibull Extension Model .............................. 76 4.5 Numerical studies......................................................................................... 80 4.6 Model Application in Decision Making ....................................................... 87 4.7 Conclusions................................................................................................... 89 Chapter 5 Further Analysis on the Weibull Extension Model ............................... 91 5.1 Introduction................................................................................................... 91 5.2 Model Properties........................................................................................... 92 5.3 Density Function and Extreme Values ......................................................... 97 5.4 Statistical Inference Based on Weibull Extension Model........................... 101 5.5 An Application Study ................................................................................. 106 5.6 Conclusion .................................................................................................. 108 Chapter 6 Hypothesis Tests of Goodness-of-fit for Weibull Extension Model ... 110 6.1 Introduction................................................................................................. 110 6.2 Parameter Estimation and EDF GoF tests .................................................. 113 6.3 Simulation Methodology............................................................................ 116 6.4 Comparison Study on MLE and Graphical Method ................................... 117 6.5 Quantiles of the EDF GoF Test Statistics................................................... 126 6.6 Study of Power of the GoF Tests................................................................ 136 6.7 Conclusion .................................................................................................. 143 Chapter 7 Change Points of Mean Residual Life and Failure Rate Functions for BFR Models............................................................................................................... 144 7.1 Introduction................................................................................................. 145 7.2 Failure Rate and Mean Residual Life for some BFR Distributions............ 149 7.3 Numerical Studies on the Change Points.................................................... 155 7.4 Study on the Flatness of Bathtub Curve ..................................................... 160 7.5 Conclusion .................................................................................................. 172 Chapter 8 Applications and Extensions of the Weibull Family Distributions .... 174 8.1 A Case Study of Anisotropic Conductive Adhesive Flip-Chip Joint with a New General Weibull Distribution............................................................. 174 8.2 Case Study on Electronic Products with Grouped Failure Data................. 188 8.3 Extension to Discrete Weibull Distribution................................................ 198 Chapter 9 Conclusion ............................................................................................... 202 9.1 Major Findings and Contributions.............................................................. 202 9.2 Future Research Areas................................................................................ 206 REFERENCE............................................................................................................ 208 APPENDIX................................................................................................................ 218 iii SUMMARY The well-known Weibull distribution has been widely studied and applied in various aspects. The assumption of a monotonic failure rate of the product, however, may not be suitable and accurate in reality. It is common that a product exhibits a bathtub failure rate property. The research of extending traditional Weibull distribution to estimate lifetime data with non-monotonic failure rate, especially those with bathtub- shaped failure rate property, is meaningful. Hence, this dissertation has been concentrated on the study and application of the extended Weibull distributions with bathtub shaped failure rate properties. A new bathtub shaped failure rate distribution, namely Weibull extension distribution, is proposed. This distribution, which is viewed as an extension of Weibull distribution, is closely related to the Weibull distribution and some other distributions. A full examination
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