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Optimal Step-Stress Plans for Accelerated Life Testing Considering OPTIMAL STEP-STRESS PLANS FOR ACCELERATED LIFE TESTING CONSIDERING RELIABILITY/LIFE PREDICTION A Dissertation Presented by Chenhua Li to The Department of Mechanical and Industrial Engineering in partial fulfillment of the requirement for the degree of Doctor of Philosophy in Industrial Engineering Northeaster University Boston, Massachusetts April 2009 Abstract Products or materials are often tested at higher levels of stress than operational conditions to quickly obtain information on the life distribution or product performance under normal use. Such testing could save much time and money. In step-stress accelerated life test (SSALT), the stress for survival units is generally changed to a higher stress level at a pre-determined time. Determination of the stress change times is one of the most important design problems in SSALT. In this dissertation research, we focus on the SSALT design problem for Weibull failure data because of its broad application in industry. The optimal simple SSALT, which involves only two stress levels, is first derived. Log-linear life stress relationship is assumed. Two different types of optimization criteria are presented, considering life estimate and reliability estimate. Optimal SSALT plan is proposed by minimizing the Asymptotic Variance (AV) of the desired life/reliability estimate. In many applications and for various reasons, it is desirable to use more than one accelerating stress variable. Integration of Weibull failure data with multiple stress variables results in more complexity in the Fisher information matrix, and a more complicated problem to solve. Two stress variables are considered first, leading to the bivariate SSALT model. Bivariate SSALT model is then extended to a more generalized model: multi-variate SSALT, which includes k steps and m stress variables. i In addition to log-linear life-stress relationship, proportional hazards (PH) model is another widely used life-stress relationship for multiple stress variables. In this dissertation research, the baseline intensity function is defined at the highest stress levels to obtain a quick initial estimate of the parameters. PH model is assumed for all other stress levels. A simple SSALT design is considered first. The results are extended to multiple SSALT, which considers multiple steps, but only one stress variable. Optimal stress change times for each step are obtained. A more generalized case, multi-variate SSALT based on PH model is then proposed, including k steps and m stress variables. Fisher information matrix and AV of maximum likelihood estimation (MLE) are constructed. Optimal plan is designed to minimize the AV of MLE. ii Acknowledgement I would like to express my deep gratitude to my dissertation advisor, Professor Nasser S. Fard, for his guidance throughout the development of this dissertation. His patience, confidence, insights, and expertise were foundational throughout the uncertainty faced in the completion of this effort. Without him, I would not have been able to finish this dissertation. I offer my sincere thanks to Prof. Emanuel Melachrinoudis, and Prof. Ronald Mourant for serving on my advisory committee and for their guidance and cooperation as thesis committee member. I am also grateful for the assistance of the staff in the Mechanical and Industrial Engineering Department, including Mr. Noah Japhet, Ms. Joyce M. Crain, Ms. Mary Traboulsy, Mr. Jeff Doughty, Dr. Le Baron Briggs, and Mr. Richard Weston. Last but not least, I owe much gratitude to my family, including my parents Dewang Li and Zhuying Chen, my husband Bo Peng, my son Arvin Hexiang Peng, and my sisters Yanhua Li and Wenhua Li for their unwavering love, patience, and understanding throughout the process. Without their love and enormous support, this dissertation would not be possible. iii Table of Contents Abstract ............................................................................................................................... i Acknowledgement ............................................................................................................. ii List of Tables .................................................................................................................. viii List of Figures ................................................................................................................... ix List of Abbreviations ....................................................................................................... xi 1 Introduction ..............................................................................................................1 1.1 Problem Background .........................................................................................1 1.2 Accelerated Life Testing ...................................................................................3 1.2.1 Qualitative ALT vs. Quantitative ALT ........................................................4 1.2.2 Constant Stress ALT vs. Step Stress ALT ...................................................7 1.2.3 Complete Data & Censored Data .................................................................8 1.3 Step-Stress ALT Design Problem .....................................................................9 1.4 Main Objective ................................................................................................10 1.5 Dissertation Outline .........................................................................................12 2 General Design Steps and Literature Review ......................................................14 2.1 SSALT Design Steps .......................................................................................14 2.1.1 Choosing Appropriate Underlying Life Distribution .................................15 2.1.2 Selecting Proper Life-Stress Relationship .................................................15 2.1.3 Cumulative Exposure (CE) Model – SSALT Data Analysis Model .........19 2.2 Literature Review ............................................................................................21 2.2.1 SSALT Data Analysis Models ...................................................................21 iv 2.2.2 Derivation of Optimum Test Plans ............................................................23 3 Simple SSALT Analysis and Design .....................................................................27 3.1 Simple Exponential SSALT Analysis and Design ..........................................28 3.1.1 Model Assumptions ...................................................................................28 3.1.2 Optimization Criteria .................................................................................30 3.1.3 Numerical Example ...................................................................................33 3.1.4 Sensitivity Analysis ...................................................................................33 3.2 Simple Weibull SSALT Analysis and Design ................................................34 3.2.1 Model Assumptions ...................................................................................35 3.2.2 Standardized Model ...................................................................................36 3.2.3 Optimum Test Design ................................................................................40 3.2.4 Numerical Examples and Analysis ............................................................46 4 Bivariate SSALT Analysis and Design .................................................................55 4.1 Bivariate Exponential SSALT Analysis and Design .......................................57 4.1.1 Model Assumptions ...................................................................................57 4.1.2 Optimization Criteria .................................................................................60 4.1.3 Numerical Example ...................................................................................61 4.1.4 Sensitivity Analysis ...................................................................................62 4.2 Bivariate Weibull SSALT Analysis and Design .............................................66 4.2.1 Model Assumptions ...................................................................................66 4.2.2 MLE and Fisher Information Matrix .........................................................67 4.2.3 Optimality Criterion & Test Design ..........................................................74 4.2.4 Numerical Examples and Sensitivity Analysis ..........................................75 5 Multi-Variate SSALT Analysis and Design .........................................................80 5.1 Multi-Variate SSALT Model ..........................................................................80 5.1.1 Test Procedure ...........................................................................................81 v 5.1.2 Model Assumptions ...................................................................................82 5.1.3 Parameter Estimation .................................................................................83 5.2 Optimal Multi-Variate SSALT Design ...........................................................86 5.2.1 Fisher Information Matrix..........................................................................87 5.2.2 Optimization Criterion ...............................................................................90 5.3 Numerical Examples and Sensitivity Analysis ...............................................92
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