Contributions to planning and analysis of accelerated testing

Founded 1905 CONTRIBUTIONS TO PLANNING AND ANALYSIS OF ACCELERATED TESTING YANG GUIYU (B. Eng., XJTU) A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT OF INDUSTRIAL AND SYSTEMS ENGINEERING NATIONAL UNIVERSITY OF SINGAPORE 2004 Acknowledgements I would like to express my profound gratitude to my supervisors, A/Prof Tang Loon Ching and A/Prof Xie Min, for their invaluable advice and guidance throughout the whole work. I have learnt tremendously from their experience and expertise, and am truly indebted to them. My sincere thanks are conveyed to the National University of Singapore for offering me a Research Scholarship and the Department of Industrial & Systems Engineering for use of its facilities, without any of which it would be impossible for me to complete the work reported in this dissertation. I also wish to thank the ISE Quality & Reliability laboratory technician Mr. Lau Pak Kai for his kind assistance in rendering me logistic support. And to members of the ISE department, who have provided their help and contributed in one way or another towards the fulfillment of the dissertation. Last but not the least, I want to thank my parents, parents-in-law and my husband Deng Bin for giving me their unwavering support. Their understanding, patience and encouragement have been a great source of motivation for me. i Table of Contents ACKNOWLEDGEMENTS I TABLE OF CONTENTS II SUMMARY . VII ACRONYMS . IX NOTATIONS . XI LIST OF TABLES .XVI LIST OF FIGURES XVIII CHAPTER INTRODUCTION AND LITERATURE SURVEY 1.1 INTRODUCTION 1.2 BASICS OF AT . 1.2.1 The Commonly Used Lifetime Distributions 1.2.1.1 The Exponential Distribution 1.2.1.2 The Normal Distribution . 1.2.1.3 The Lognormal Distribution . 10 1.2.1.4 The Weibull Distribution 10 1.2.1.5 The Extreme Value Distribution . 11 1.2.1.6 The Inverse Gaussian Distribution & The Birnbaum-Saunders Distribution . 12 1.2.2 The Commonly Used Acceleration Models . 13 1.2.2.1 The Arrhenius Model 13 1.2.2.2 The Inverse Power Law Model . 14 1.2.2.3 The Eyring Model and the Generalized Eyring Model . 14 1.2.3 Modeling of Degradation Processes 16 1.2.3.1 Deterministic Degradation Models . 16 1.2.3.2 Stochastic Degradation Models 17 1.2.4 Parameter Estimation Methods 18 1.2.4.1 Parametric Methods 18 1.2.4.2 Non-parametric Methods 20 1.2.5 Failure Mechanism Validation . 20 1.2.6 Destructive Testing and Non-destructive Testing . 23 1.3 ANALYSIS OF ALT DATA AND PLANNING OF ALT TEST . 24 1.3.1 Analysis of ALT Data 24 1.3.2 Planning of ALT Test 25 1.3.3 Objectives of Our Proposed CSALT Planning Approach . 30 ii 1.3.4 Value of Our Proposed CSALT Planning Approach . 31 1.4 DATA ANALYSIS AND PLANNING OF ADT TEST . 32 1.4.1 Analysis of ADT Data . 33 1.4.2 Planning of ADT Test 36 1.4.3 Objectives of Our Proposed ADT Analysis and Planning Approach 38 1.4.4 Value of Our Proposed ADT Analysis and Planning Approach . 39 1.5 SCOPE OF THE STUDY . 40 CHAPTER PLANNING OF MULTIPLE-STRESS CSALT 42 2.1 INTRODUCTION 42 2.2 THE EXPERIMENT DESCRIPTION AND MODEL ASSUMPTIONS . 44 2.3 THE GRAPHICAL REPRESENTATION OF NEAR OPTIMAL TWOSTRESS CSALT PLANS . 46 2.4 THE SOLUTION SPACE FOR THREE-STRESS CSALT PLANS . 48 2.5 CONNECTIONS OF TWO-STRESS AND THREE-STRESS CSALT PLANS 51 2.6 ALTERNATIVE PROCEDURES FOR THREE-STRESS CSALT PLANNING . 54 2.6.1 Approach . 54 2.6.2 Approach . 55 2.6.3 Approach . 55 2.6.4 Numerical Examples 56 2.7 CONCLUSIONS . 58 CHAPTER ANALYSIS OF SSADT DATA 60 3.1 INTRODUCTION 60 3.2 THE EXPERIMENT DESCRIPTION AND MODEL ASSUMPTIONS . 62 3.3 PARAMETER ESTIMATION 66 3.3.1 Estimation of b and η 67 3.3.2 Estimation of σ 02 . 69 3.4 THE MEAN LIFETIME AND ITS CONFIDENCE INTERVAL . 69 3.4.1 Modeling the Failure Time with an IGD . 69 3.4.2 Modeling the Failure Time with a BSD . 71 3.5 A NUMERICAL EXAMPLE 72 3.6 SIMULATIONS 74 iii 3.7 CONCLUSIONS . 78 CHAPTER A GENERAL FORMULATION FOR PLANNING OF ADT .79 4.1 INTRODUCTION 79 4.2 THE EXPERIMENT DESCRIPTION AND MODEL ASSUMPTIONS . 81 4.3 A GENERAL FORMULATION FOR PLANNING OF CSADT AND SSADT 84 4.3.1 The Cost Functions 86 4.3.2 The Precision Constraint 87 4.4 NUMERICAL EXAMPLES 93 4.5 SIMULATIONS 98 4.5.1 Simulation Study of the Optimal CSADT Plan 98 4.5.2 Simulation Study of the Optimal SSADT Plan 102 4.5 CONCLUSIONS . 104 CHAPTER OPTIMAL CSADT PLANS 107 5.1 INTRODUCTION 107 5.2 OPTIMAL TWO-STRESS CSADT PLANS 109 5.3 SENSITIVITY ANALYSIS . 113 5.4 CONCLUSIONS . 120 CHAPTER OPTIMAL SSADT PLANS 121 6.1 INTRODUCTION 121 6.2 OPTIMAL TWO-STRESS SSADT PLANS 122 6.2.1 Determination of the Lower Stress X and the Inspection Time Interval ∆t 127 6.2.2 Determination of the Precision Parameters c and p 129 6.2.3 Sensitivity Analysis . 132 6.3 OPTIMAL THREE-STRESS SSADT PLANS 137 6.3.1 Introduction 137 6.3.2 Three-stress SSADT Plans 139 6.3.2.1 Approach 140 6.3.2.2 Approach 141 6.4 CONCLUSIONS . 142 CHAPTER PLANNING OF DESTRUCTIVE CSADT 144 7.1 INTRODUCTION 144 iv 7.2 PLANNING OF THE DESTRUCTIVE CSADT . 146 7.2.1 Experiment Description & Model Assumptions 146 7.2.2 Planning Policy 147 7.3 OPTIMAL DESTRUCTIVE CSADT PLANS . 149 7.3.1 Simulations 149 7.3.2 A Numerical Example . 152 7. DETERMINATION OF THE LOWER STESS X1 154 7.4.1 Determination of the Optimal Lower Stress X1 without Constraints 154 7.4.2 Determination of the Optimal Lower Stress X1 with the Test Time Constraint 155 7.4.3 Determination of the Optimal Lower Stress X1 with the Sample Size Constraint 157 7.4.4 Determination of the Optimal Lower Stress X1 with Both Test Time and Sample Size Constraints . 158 7.5 ROBUSTNESS ANALYSIS . 158 7.5.1 Sensitivity of n to σ . 159 a 7.5.2 Sensitivity of π to σ 160 a 7.5.3 Sensitivity of T1 and T2 to σ 161 a 7.6 CONCLUSIONS . 162 CHAPTER CONCLUSIONS AND FUTURE RESEARCH 164 REFERENCES 170 APPENDIX A: A MATLAB PROGRAM FOR ANALYSING SSADT DATA .187 APPENDIX B1: FIRST AND SECOND ORDER PARTIAL DERIVATIONS OF LnLi , j ,k .189 APPENDIX B2: A VBA PROGRAM TO OPTIMISE CSADT AND SSADT PLANS WITH A INTERACTIVE DIALOG WINDOW 190 APPENDIX C: OPTIMAL CSADT PLANS WITH MIS-SPECIFIED σ a .196 APPENDIX D: OPTIMAL SSADT PLANS WITH MIS-SPECIFIED σ 200 a APPENDIX E1: DERIVATION OF ESTIMATE PRECISION CONSTRAINT FOR DESTRUCTIVE CSADT PLANNING 205 APPENDIX E2: DESTRUCTIVE CSADT PLANS .207 v PUBLICATIONS .214 vi Summary Accelerated Life Testing (ALT) and Accelerated Degradation Testing (ADT) have become attractive alternatives for reliability assessments as they distinctly save the testing time and testing cost. They are employed when specimens are tested at high stresses to induce early failures or degradations. Through an assumed stress-life or stress-degradation relationship, failure information is extrapolated from the test stress to that at design stress. Although such practice saves time and expense, estimates obtained via extrapolation are inevitably less precise. Hence, a systematic and in-depth study on ALT and ADT data analysis and experiment planning is in demand. This dissertation involves three parts. The first part addresses the planning of Constant Stress ALT (CSALT), in which we propose a method to quantify the departure from the usual optimality criterion. A contour plot is developed to provide the solution space for sample allocations at high and low stress levels in two-stress and three-stress CSADT plans. Based on the output from the contour plot, three related approaches to planning CSALT are then presented. The results show that our plans are: (1) capable of providing sufficient failures at middle stress to detect non-linearity in the stress-life model if it exists; (2) able to serve as follow-up tests during product development; (3) flexible in setting stress levels and sample allocations. The second part addresses the analysis of Step Stress ADT (SSADT) data. We monitor the degradation path with stochastic processes and finally obtain a closed form estimation for unknown parameters. The mean lifetime and its confidence intervals are also derived when failure time follows the Inverse-Gaussian distribution (IGD) or vii Birnbaum-Saunders distribution (BSD). Compared the existing approaches, our method alleviates the difficulty in determining the particular deterministic degradation functions. The third part deals with the planning of ADT. Motivated by the successful application of stochastic model in ADT data analysis, we present a general formulation to design both CSADT and SSADT by considering the tradeoff between the total experiment cost and the attainable estimate precision level. Decision variables such as the sample size, the test-stopping time or the stress-changing time in a CSDAT or a SSADT are optimized. Influence of the lower stress and inspection time interval on optimal plans is analyzed. Effect of precision parameters on optimal SSADT plans is also studied. The results imply that our formulation is easily coded, and our plans require fewer test samples and less test duration. Hence, testing cost is reduced. Compared with CSADT, SSADT is more powerful in this aspect. Thus implementation of SSADT is highly recommended in real case. This dissertation also contains numerical examples and simulation studies to demonstrate the validity and efficiency of each approach developed. We highlight the important findings and discuss the comparisons with existing methods. Finally, we point out some possible research directions. Since our current research focuses on single accelerated environment, the planning strategies proposed in this dissertation can be extended to multi-component multi-acceleration environment. viii Acronyms AF Acceleration Factor AT Accelerated Testing ADT Accelerated Degradation Testing ALT Accelerated Life Testing BSD Birnbaum-Saunders Distribution c.d. f cumulative density function CC, PC Cost Constraint and estimate Precision Constraint CE Cumulative Exposure model CST Constant Stress Testing CSADT Constant Stress ADT CSALT Constant Stress ALT Dev Deviation DT Degradation Testing DM Deterministic Model ED plan Plans with Equalized Degradation EL plan Plans with Equalized Log(degradation) IGD Inverse Gaussian Distribution LED Light Emitting Diode LS Lease Square method LSE Lease Square Estimate ML Maximum Likelihood method MLE Maximum Likelihood Estimate MMLE Modified Maximum Likelihood Estimate ND Normal Distribution p.d.f probability density function PSADT Progressive Stress ADT PSALT Progressive Stress ADT PST Progressive Stress Testing SM Stochastic Model SST Step Stress Testing SSADT Step Stress ADT ix Appendix Continued σ o a 120 o X1 n 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 28 34 42 50 59 71 89 112 143 185 17 20 25 28 34 42 50 61 78 100 132 178 15 16 20 24 28 33 41 51 61 79 104 141 195 T1 T2o To n1o n2o L10 L2o 4800 4800 4800 4800 4800 4800 4800 4800 4800 4800 4560 4800 4800 4800 4800 4800 4800 4800 4800 4800 4800 4800 4080 4560 4800 4800 4800 4800 4800 4800 4800 4800 4800 4800 4800 1200 1920 2400 3120 4080 4800 4800 4800 4800 4800 720 1200 1440 2640 3120 3360 4080 4800 4800 4800 4800 4800 480 1440 1440 1920 2640 3360 3600 3840 4800 4800 4800 4800 4800 4800 4800 4800 4800 4800 4800 4800 4800 4800 4800 4560 4800 4800 4800 4800 4800 4800 4800 4800 4800 4800 4800 4080 4560 4800 4800 4800 4800 4800 4800 4800 4800 4800 4800 4800 24 29 33 39 46 55 67 82 101 126 14 16 19 22 25 30 36 45 56 68 88 115 13 13 15 17 20 24 28 35 42 53 68 92 121 11 13 16 22 30 42 59 6 12 14 16 22 32 44 63 13 16 19 26 36 49 74 20 20 20 20 20 20 20 20 20 20 19 20 20 20 20 20 20 20 20 20 20 20 17 19 20 20 20 20 20 20 20 20 20 20 20 10 13 17 20 20 20 20 20 11 13 14 17 20 20 20 20 20 6 11 14 15 16 20 20 20 20 20 o 199 Appendix Appendix D: Optimal SSADT plans with mis-specified σ /a c 80 X1 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 >=0.75 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 >=0.7 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 >=0.8 0.05 0.1 n T1 T2 T 2880 240 3120 10 2400 240 2640 10 2640 240 2880 10 2640 480 3120 2880 720 3600 11 2640 480 3120 10 2880 720 3600 10 3360 480 3840 10 2880 960 3840 12 2880 480 3360 10 3120 960 4080 11 3360 480 3840 12 3360 240 3600 12 3120 480 3600 13 3120 240 3360 1680 240 1920 2160 240 2400 1680 240 1920 2160 240 2400 2400 240 2640 2160 480 2640 1680 480 2160 2160 720 2880 2160 480 2640 1920 480 2400 2640 240 2880 2640 240 2880 2400 480 2880 2400 240 2640 1920 240 2160 1680 240 1920 1920 240 2160 1680 240 1920 2160 240 2400 1920 240 2160 1440 480 1920 1920 720 2640 1920 480 2400 1680 480 2160 1440 720 2160 2400 240 2640 2400 240 2640 2400 240 2640 2160 480 2640 2160 240 2400 1680 240 1920 1440 240 1680 To be continued L 13 11 12 13 15 13 15 16 16 14 17 16 15 15 14 10 10 11 11 12 11 10 12 12 12 11 9 10 11 10 9 11 11 11 11 10 L1 12 10 11 11 12 11 12 14 12 12 13 14 14 13 13 9 10 9 11 11 10 10 8 8 10 10 10 9 σ a L2 1 3 4 2 1 1 1 2 2 1 1 1 1 2 1 1 200 Appendix Continued σ /a 80 c 90 X1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 >=0.7 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 >=0.75 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 >=0.7 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 n T1 T2 T 1920 240 2160 1680 240 1920 1920 240 2160 1680 480 2160 1440 480 1920 1920 480 2400 1920 240 2160 1680 480 2160 2400 240 2640 2160 480 2640 1920 720 2640 2160 240 2400 10 2880 240 3120 10 3120 240 3360 10 3120 480 3600 10 3120 720 3840 10 3360 720 4080 12 2880 720 3600 10 3840 720 4560 11 3600 720 4320 13 3360 480 3840 14 2880 720 3600 13 3600 480 4080 13 3360 720 4080 12 4320 240 4560 12 4080 480 4560 11 4800 240 5040 2160 240 2400 2400 240 2640 1920 240 2160 2160 480 2640 2640 240 2880 2160 480 2640 2640 480 3120 2400 480 2880 2880 480 3360 2640 480 3120 10 2640 240 2880 10 2400 480 2880 10 2160 720 2880 3120 240 3360 1680 240 1920 2160 240 2400 1680 240 1920 2160 240 2400 2400 240 2640 2160 480 2640 1680 480 2160 2160 720 2880 2160 480 2640 1920 480 2400 To be continued L 9 10 9 11 11 11 10 13 14 15 16 17 15 19 18 16 15 17 17 19 19 21 10 11 11 12 11 13 12 14 13 12 12 12 14 10 10 11 11 12 11 10 L1 8 8 10 9 12 13 13 13 14 12 16 15 14 12 15 14 18 17 20 10 11 11 10 12 11 11 10 13 9 10 9 L2 1 2 2 1 3 3 3 1 1 2 2 2 1 1 1 2 2 201 Appendix Continued σ /a c 90 110 X1 0.55 0.6 0.65 >=0.7 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 >=0.7 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 >=0.75 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 >=0.7 0.05 0.1 0.15 0.2 n T1 T2 T 2640 240 2880 2640 240 2880 2400 480 2880 2400 240 2640 2160 240 2400 1920 240 2160 2160 240 2400 1920 240 2160 2400 240 2640 2160 240 2400 1680 480 2160 2400 240 2640 2160 480 2640 1920 480 2400 1680 720 2400 2400 480 2880 2400 480 2880 2400 240 2640 12 3600 240 3840 13 3600 240 3840 14 3360 480 3840 15 3360 480 3840 15 3360 720 4080 15 3600 720 4320 15 3840 720 4560 14 4080 960 5040 16 3600 960 4560 15 4080 960 5040 15 4800 480 5280 16 4560 480 5040 16 4320 720 5040 17 4320 480 4800 15 5280 240 5520 2880 240 3120 3120 240 3360 3120 240 3360 10 2640 480 3120 10 2880 480 3360 10 3120 480 3600 11 2640 720 3360 10 3120 720 3840 11 2880 720 3600 12 2400 960 3360 12 3360 240 3600 12 3120 480 3600 12 2880 720 3600 11 3840 240 4080 2640 240 2880 2160 240 2400 2400 480 2880 2640 480 3120 To be continued L 12 12 12 11 10 10 11 10 11 11 10 10 12 12 11 16 16 16 16 17 18 19 21 19 21 22 21 21 20 23 13 14 14 13 14 15 14 16 15 14 15 15 15 17 12 10 12 13 L1 11 11 10 10 9 10 10 10 10 10 15 15 14 14 14 15 16 17 15 17 20 19 18 18 22 12 13 13 11 12 13 11 13 12 10 14 13 12 16 11 10 11 L2 1 1 1 1 2 2 1 2 3 4 2 1 1 2 3 1 2 202 Appendix Continued σ /a c 110 120 X1 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 >=0.75 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 >=0.7 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 >=0.8 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 n T1 T2 T 10 2160 480 2640 2640 480 3120 10 2160 720 2880 2640 720 3360 10 2400 720 3120 2880 720 3600 10 2880 480 3360 10 2640 720 3360 11 2640 480 3120 3120 720 3840 10 3360 240 3600 2640 240 2880 2160 240 2400 2400 480 2880 1920 480 2400 2400 480 2880 2640 480 3120 2400 480 2880 2880 480 3360 2640 480 3120 10 2400 480 2880 10 2160 720 2880 2880 480 3360 2880 480 3360 10 2880 240 3120 12 4320 240 4560 13 4080 480 4560 14 4080 480 4560 15 3840 720 4560 16 3840 720 4560 16 3840 960 4800 16 4080 960 5040 16 4320 960 5280 18 3600 1200 4800 18 4320 720 5040 17 4800 720 5520 16 5520 480 6000 16 5280 720 6000 17 5520 240 5760 17 5280 480 5760 18 5280 240 5520 3120 240 3360 3360 240 3600 3360 480 3840 11 2880 480 3360 11 3120 480 3600 11 3360 480 3840 13 2640 720 3360 12 3120 720 3840 11 3600 720 4320 12 3360 720 4080 To be continued L 11 13 12 14 13 15 14 14 13 16 15 12 10 12 10 12 13 12 14 13 12 12 14 14 13 19 19 19 19 19 20 21 22 20 21 23 25 25 24 24 23 14 15 16 14 15 16 14 16 18 17 L1 11 11 10 12 12 11 11 13 14 11 10 10 11 10 12 11 10 12 12 12 18 17 17 16 16 16 17 18 15 18 20 23 22 23 22 22 13 14 14 12 13 14 11 13 15 14 L2 2 3 3 3 1 2 2 2 2 2 1 2 3 4 3 1 2 2 3 3 203 Appendix Continued σ /a c 120 X1 n T1 T2 T 0.55 0.6 0.65 0.7 0.75 0.8 >=0.85 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 >=0.7 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 >=0.7 13 12 13 13 13 13 14 9 9 10 10 11 10 11 10 11 12 10 11 8 10 9 10 11 10 11 10 10 11 3120 3840 3840 3840 3840 3600 3600 2400 2640 2880 3120 2640 2880 2400 2880 2400 3120 2880 2880 3360 3600 2400 2640 2160 2400 2640 2160 2640 2400 2640 2640 2400 3120 3120 3120 720 480 240 240 240 480 240 240 240 240 240 480 480 720 720 960 720 720 480 720 240 240 240 240 480 480 720 720 480 960 720 720 480 480 240 3840 4320 4080 4080 4080 4080 3840 2640 2880 3120 3360 3120 3360 3120 3600 3360 3840 3600 3360 4080 3840 2640 2880 2400 2880 3120 2880 3360 2880 3600 3360 3120 3600 3600 3360 L 16 18 17 17 17 17 16 11 12 13 14 13 14 13 15 14 16 15 14 17 16 11 12 10 12 13 12 14 12 15 14 13 15 15 14 L1 13 16 16 16 16 15 15 10 11 12 13 11 12 10 12 10 13 12 12 14 15 10 11 10 11 11 10 11 11 10 13 13 13 L2 1 1 1 2 3 3 1 1 2 3 3 2 204 Appendix Appendix E1: Derivation of estimate precision constraint for destructive CSADT planning From equation (7.1), the log-likelihood of each increment is U2 1 ⎛ ⎞ ln Li = − ln(2π ) − ln⎜ ∑ Tk ⎟ − ln σ − i 2 ⎝ k =1 ⎠ (∆Di − Tk (a + bX k )) where U i = Tk σ The first partial derivates of U i and LnL are respectively: T ∂U i =− k ∂a σ Tk ∂U i T X =− k k ∂b σTk ∂U i U =− i ∂σ σ ⎛ T ∂LnLi = (− U i ) ⋅ ⎜ − k ⎜ ∂a ⎝ σ Tk ⎞ U i Tk ⎟= ⎟ σ T k ⎠ ⎛ T X ∂LnLi = −U i * ⎜ − k k ⎜ σ T ∂b k ⎝ ⎞ ⎟ = U i Tk X k ⎟ σ Tk ⎠ ∂LnLi 1 U2 ⎛ U ⎞ = − −Ui ⋅⎜− i ⎟ = − + i σ σ σ ∂σ ⎝ σ ⎠ The second order partial derivates of LnL are: ⎛ ⎞ ⎜ Tk ⎟ ∑ ⎜ ⎟ T ∂Ln Li = −⎜ k =1 = − k2 ⎟ 2 σ ∂a ⎜σ Tk ⎟⎟ ∑ ⎜ k =1 ⎝ ⎠ 2 (Tk X k ) Tk X k2 ∂Ln Li =− =− σ 2Tk σ2 ∂b ∂Ln Li 2U i ⎛ U i ⎞ U i2 − 3U i2 = + = ⎜− ⎟− σ ⎝ σ ⎠ σ2 σ σ2 ∂σ ⎛ T X ⎞ Tk T X ∂Ln Li = ⋅⎜− k k ⎟ = − k k ∂a∂b σ σ Tk ⎜⎝ σ Tk ⎟⎠ ∂Ln Li Tk 2U T ⎛ U ⎞ = ⋅⎜− i ⎟⋅ = − i k ∂a∂σ σ Tk ⎝ σ ⎠ σ Tk 2U T X ∂LnLi = − 2i k k ∂b∂σ σ Tk 205 Appendix Hence, the fisher information matrix is derived as: ⎡ ⎛ n ∂ ln Li ⎞ ⎛ n ∂ ln Li ⎞ ⎤ ⎛ n ∂ ln Li ⎞ ⎜− ∑ ⎟⎥ ⎜ ⎟ ⎜ ⎟ , E E E − − ⎢ ⎜ ∑ ⎜ ⎟ ⎜ ∑ ∂a∂b ⎟ ⎟ ⎝ i =1 ∂a∂σ ⎠ ⎥ ⎝ i =1 ⎠ ⎢ ⎝ i =1 ∂a ⎠ ⎥ ⎢ ⎛ n ∂ ln Li ⎞ ⎛ n ∂ ln Li ⎞ ⎥ ⎟ ⎜ ⎟ E ⎜⎜ − ∑ E − F =⎢ ⎟ ⎜ ∑ ∂b∂σ ⎟ ⎥ ⎢ ⎝ i =1 ∂b ⎠ ⎝ i =1 ⎠ ⎥ ⎢ ⎢ ⎛ n ∂ ln Li ⎞ ⎥ ⎟⎥ symmetric E ⎜⎜ − ∑ ⎢ ⎟ ⎝ i =1 ∂b ⎠ ⎦ ⎣ 0⎤ ⎡π 1T1 + (1 − π )T2 , π 1T1 X + (1 − π )T2 X , n ⎢ ⎥ = ⎢ 0⎥ π 1T1 X 12 + (1 − π )T2 X 22 , σ ⎢ symmetric ⎥⎦ ⎣ and Let [ ( ) F = (π 1T1 + (1 − π )T2 ) π 1T1 X 12 + (1 − π )T2 X 22 − (π 1T1 X + (1 − π )T2 X ) Q = F11 = ∑π k ] Tk X k2 k =1 ⎛ ⎞⎛ ⎞ ⎛ ⎞ ⎜ ∑ π k Tk ⎟⎜ ∑ π k Tk X k2 ⎟ − ⎜ ∑ π k Tk X k ⎟ ⎝ k =1 ⎠⎝ k =1 ⎠ ⎝ k =1 ⎠ π 1T1 X 12 + (1 − π )T2 X 22 = (π 1T1 + (1 − π )T2 )(π 1T1 X 12 + (1 − π )T2 X 22 ) − (π 1T1 X + (1 − π )T2 X )2 π 1T1 X 12 + (1 − π )T2 X 22 = π (1 − π )T1T2 ( X − X )2 Then the asymptotic variance of MLE of µ ( X ) is calculated as: ) ) ) ) ) ) ⎛ ∂µ ( X ) ∂µ ( X ) ∂µ ( X ) ⎞ −1 ⎛ ∂µ ( X ) ∂µ ( X ) ∂µ ( X ) ⎞ ) A var(µ ( X )) = ⎜ , , , , ⎟ ⎟' F ⎜ ∂b ∂σ ⎠ ∂b ∂σ ⎠ ⎝ ∂a ⎝ ∂a ) σ Dc2 = ⋅ ) ⋅Q n a4 206 Appendix Appendix E2: Destructive CSADT plans σ /a c 80 X1 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 >0.85 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 >0.9 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 >0.9 n π1 19 0.789474 21 0.761905 24 0.708333 28 0.714286 31 0.677419 36 0.638889 41 0.634146 47 0.617021 54 0.592593 63 0.587302 80 0.5875 101 0.594059 142 0.598592 216 0.592593 357 0.57423 >500 14 0.785714 16 0.75 18 0.722222 20 0.7 24 0.666667 26 0.653846 30 0.633333 34 0.617647 39 0.589744 46 0.565217 53 0.566038 64 0.546875 84 0.559524 117 0.581197 192 0.557292 362 0.546961 >500 12 0.833333 14 0.785714 16 0.75 19 0.684211 21 0.666667 23 0.652174 27 0.62963 31 0.612903 35 0.6 41 0.585366 47 0.574468 56 0.553571 69 0.550725 95 0.568421 153 0.568627 286 0.552448 >500 To be continued T1 T2 4512 4848 5520 5760 6336 6816 7488 8304 9216 9888 9984 9984 9960 9960 9960 672 1248 1440 1872 2448 2880 3456 4080 4896 6144 7152 9120 9832 9928 9928 3264 3600 3888 4320 4464 5040 5424 6240 6864 7728 8784 9936 9984 9984 9984 9960 528 768 1104 1440 1680 2208 2640 3024 3696 4224 5280 6336 7920 9840 9984 9928 2832 3168 3312 3600 4080 4560 4896 5376 6096 6720 7824 8976 9936 9984 9960 9960 624 768 1104 1200 1488 1920 2208 2640 3216 3840 4704 5712 7296 9216 9832 9928 207 Appendix Continued σ /a 80 c 90 X1 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 >0.9 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 >0.85 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 >0.9 0.1 0.15 0.2 n π1 11 0.818182 13 0.769231 15 0.733333 17 0.705882 19 0.684211 23 0.652174 25 0.64 28 0.607143 33 0.606061 38 0.578947 44 0.568182 53 0.54717 63 0.539683 87 0.551724 134 0.567164 252 0.551587 >500 21 0.809524 25 0.76 27 0.740741 32 0.6875 36 0.666667 40 0.65 46 0.630435 53 0.603774 62 0.596774 75 0.613333 96 0.604167 124 0.604839 179 0.603352 273 0.582418 452 0.577434 >500 15 0.8 18 0.777778 21 0.714286 23 0.695652 26 0.653846 30 0.633333 34 0.617647 38 0.605263 44 0.590909 52 0.576923 60 0.566667 77 0.571429 99 0.585859 147 0.571429 243 0.559671 459 0.562092 >500 14 0.785714 16 0.75 18 0.722222 To be continued T1 T2 2736 2928 3264 3504 3840 4080 4512 5328 5520 6432 7344 8304 9984 9984 9984 9960 576 768 912 1200 1536 1632 2208 2496 3120 3600 4416 5328 6672 8400 9888 9880 4896 5376 6000 6384 7008 7776 8400 9216 9888 9984 9984 9984 9960 9960 9960 960 1152 1776 2064 2592 3264 3936 4656 5616 7056 8160 9840 9928 9928 9928 3840 3984 4224 4704 5280 5616 6336 6960 7728 8496 9696 9936 9984 9984 9984 9960 624 912 1200 1632 1920 2352 2736 3504 4128 4848 6000 7104 9360 9984 9984 9928 3264 3600 3888 528 768 1104 208 Appendix Continued σ /a c 90 100 X1 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 >0.9 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 >0.9 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 >0.8 0.1 0.15 0.2 0.25 0.3 0.35 0.4 n π1 20 0.7 24 0.666667 26 0.653846 30 0.633333 34 0.617647 39 0.589744 45 0.577778 54 0.555556 64 0.546875 84 0.571429 117 0.589744 193 0.57513 362 0.552486 >500 13 0.769231 15 0.8 17 0.705882 19 0.684211 22 0.681818 25 0.64 29 0.62069 33 0.606061 37 0.594595 43 0.581395 50 0.56 60 0.55 75 0.56 104 0.567308 170 0.576471 318 0.54717 >500 23 0.826087 26 0.769231 30 0.733333 34 0.705882 39 0.666667 45 0.644444 51 0.627451 60 0.616667 72 0.625 90 0.611111 111 0.621622 153 0.620915 221 0.60181 337 0.581602 >500 0.581602 17 0.823529 20 0.75 22 0.727273 26 0.692308 29 0.655172 33 0.6363 38 0.631579 To be continued T1 T2 4320 4464 5088 5424 6048 6864 7824 8736 9984 9984 9984 9960 9960 1440 1680 2160 2640 3168 3696 4368 5088 6288 7920 9840 9928 9928 3072 3264 3648 3984 4320 4608 4992 5712 6384 7056 8208 9408 9936 9984 9960 9960 528 816 1008 1344 1584 2016 2352 2688 3408 4128 4896 5856 7680 9552 9880 9928 5568 6288 6672 7344 8016 8640 9552 9984 9984 9984 9984 9960 9960 9960 9960 4080 4368 4992 5088 5856 6432 6864 1056 1440 1968 2448 2928 3504 4224 5136 6480 7584 9648 9928 9928 9928 9928 768 1056 1392 1824 2112 2544 3120 209 Appendix Continued σ /a c 100 110 X1 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 >0.85 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 >0.9 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 >0.9 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 n π1 43 0.604651 49 0.591837 57 0.578947 69 0.57971 90 0.577778 120 0.6 183 0.579235 302 0.566225 >500 16 0.8125 17 0.764706 20 0.75 23 0.695652 26 0.692308 29 0.655172 34 0.617647 38 0.605263 44 0.590909 50 0.58 59 0.559322 75 0.56 98 0.581633 144 0.583333 238 0.567227 447 0.55481 >500 15 0.8 16 0.75 19 0.736842 22 0.681818 24 0.666667 28 0.642857 32 0.625 36 0.611111 42 0.595238 48 0.583333 56 0.571429 67 0.567164 89 0.573034 127 0.574803 209 0.555024 393 0.557252 >500 25 0.8 30 0.766667 34 0.735294 39 0.692308 44 0.681818 50 0.64 58 0.637931 70 0.642857 85 0.635294 103 0.631068 To be continued T1 T2 7632 8640 9552 9984 9984 9960 9960 9960 3792 4512 5472 6816 8016 9832 9832 9880 3504 4080 4176 4656 5088 5568 6144 6864 7584 8640 9792 9960 9960 9960 9960 9960 576 960 1344 1536 1920 2448 2688 3360 3984 4896 5808 7096 9064 9928 9928 9928 3312 3792 3984 4368 4848 5280 5760 6240 7008 7872 9072 9984 9960 9960 9960 9960 528 912 1152 1344 1824 2064 2496 3216 3648 4512 5376 6816 8392 9832 9928 9928 6336 6432 7104 7680 8496 9312 9840 9984 9984 9984 1056 1632 2112 2640 3216 3888 4752 5664 6912 8784 210 Appendix To be continued σ /a c 110 X1 0.6 0.65 0.7 0.75 >0.8 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 >0.85 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 >0.85 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 n π1 133 0.639098 185 0.616216 268 0.604478 408 0.585784 >500 19 0.789474 22 0.772727 25 0.72 29 0.689655 33 0.666667 37 0.648649 41 0.634146 47 0.617021 54 0.592593 64 0.59375 81 0.604938 102 0.598039 144 0.597222 221 0.579186 365 0.575342 >500 17 0.823529 20 0.75 22 0.727273 26 0.692308 29 0.655172 32 0.65625 37 0.648649 42 0.619048 48 0.604167 56 0.571429 68 0.573529 86 0.581395 114 0.596491 175 0.582857 288 0.569444 >500 16 0.8125 18 0.777778 21 0.714286 23 0.695652 27 0.666667 30 0.633333 34 0.617647 40 0.6 45 0.6 53 0.584906 62 0.564516 77 0.584416 102 0.588235 153 0.575163 253 0.565217 475 0.545263 To be continued T1 T2 9984 9960 9960 9960 9936 9928 9880 9928 4560 4752 5376 5712 6144 6816 7632 8544 9504 9888 9984 9984 9984 9960 9960 720 1200 1440 1824 2304 2832 3552 4128 4944 6240 7344 9408 9984 9880 9928 3792 4176 4752 4992 5664 6096 6672 7488 8352 9408 9984 9984 9984 9960 9960 816 1008 1344 1632 1968 2688 3120 3696 4464 5232 6336 8016 9936 9832 9928 3648 3984 4368 4944 5136 6000 6576 6912 7776 8688 9936 9984 9984 9960 9960 9960 672 1056 1248 1632 2016 2304 2832 3408 4224 4896 5856 7680 9504 9928 9928 9928 211 Appendix Continued σ /a c X1 110 >0.9 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.7 0.75 0.8 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 >0.85 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 >0.85 0.1 0.15 0.2 0.25 0.3 120 n π1 >500 27 0.814815 33 0.757576 37 0.72973 42 0.690476 48 0.666667 55 0.654545 65 0.646154 79 0.64557 96 0.645833 120 0.641667 158 0.639241 318 0.597484 486 0.592593 >500 21 0.809524 24 0.75 27 0.740741 30 0.7 35 0.685714 40 0.65 45 0.622222 51 0.607843 61 0.590164 74 0.608108 91 0.604396 119 0.605042 318 0.597484 263 0.581749 434 0.569124 >500 19 0.789474 21 0.761905 24 0.708333 27 0.703704 31 0.677419 35 0.657143 40 0.625 46 0.608696 52 0.596154 62 0.580645 78 0.589744 99 0.59596 136 0.602941 208 0.591346 343 0.574344 >500 17 0.823529 20 0.75 22 0.727273 26 0.692308 29 0.655172 To be continued T1 T2 6864 7008 7776 8592 9216 9936 9984 9984 9984 9984 9984 9960 9960 1248 1728 2304 2832 3552 4320 5472 6528 7968 9312 9984 9928 9928 4848 5328 5808 6480 6768 7440 8352 9264 9984 9984 9984 9984 9960 9960 9960 816 1200 1680 2160 2688 3168 3792 4608 5232 6624 8448 9888 9928 9880 9928 4224 4656 5232 5664 6096 6816 7440 8112 9120 9936 9984 9984 9984 9960 9960 720 1200 1440 1920 2352 2784 3360 4032 4944 5760 6912 8688 9888 9880 9928 4080 4368 4992 5136 5856 768 1056 1392 1776 2112 212 Appendix Continued σ /a 120 c X1 n π1 T1 T2 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 >0.9 34 38 43 49 56 70 87 119 183 301 566 >500 0.647059 0.631579 0.604651 0.591837 0.571429 0.571429 0.586207 0.605042 0.584699 0.55814 0.558304 6192 6864 7632 8640 9840 9984 9984 9984 9960 9960 9960 2496 3120 3792 4512 5472 6576 8688 9984 9832 9928 9928 213 Appendix Publications Tang LC, Yang GY & Xie M. Planning step-stress accelerated degradation tests, Proceedings of the Annual Reliability and Maintainability Symposium, pp287-292, 2004, (Stan Ofsthun Award). Tang LC, Yang GY. Planning multiple levels constant stress accelerated life tests, Proceedings of the Annual Reliability and Maintainability Symposium, pp338-342, 2002. 214 [...]... related to the testing time, definition similar to n * (∆t ) , n * , SN n (∆t ) U i , j ,k a transformation of ∆Di , j ,k xiii wk ⎧q k ⎪ =⎨ ⎪π k ⎩ for CSADT for SSADT , an indicator in ADT planning Xk the standardized testing stress X L× 2 a L by 2 matrix related to testing stresses X'×X the multiplication of X ' and X L×2 L× 2 2 X m, p the pth quantile of the X 2 distribution with m degree of freedom... Simulation of degradation paths in a SSADT experiment (X1=0.3 and X2=1) Table 5.1 A summary of the existing DT and CSADT plans Table 5.2 Optimal two-stress CSADT plans (c=5, p=0.9) Table 5.3 Influence of ∆t on n and T in optimal two-stress CSADT plans Table 5.4 Sensitivity of Rn to σ Table 5.5 Sensitivity of Rπ 1 to σ a Table 5.6 Sensitivity of RT to σ in two-stress CSADT plans Table 5.7 Sensitivity of RT2 to. .. the potential acceleration model; and (3) random effects of individual product characteristics There are three types of deterministic models, i.e linear, convex and concave models To determine the format of a model, one needs to comprehensively understand the failure mechanisms of the product under test Historical data, previous testing experience and engineering handbooks will be exactly useful in... Realizations of the simulated SSADT plan Figure 5.1 Main effect plot of sensitivity of n to mis-specified σ a in two-stress CSADT plans Figure 6.1 Main effect plot of optimal stopping time n in SSADT planning Figure 6.2 Main effect plot of optimal stopping time T in SSADT planning Figure 6.3 Plot of L2/L1 Vs X1 in two-stress SSADT plans Figure 6.4 Boundaries of {c, p}, the precision constraint in SSADT planning. .. not impossible, to collect enough failure data to estimate the time -to- failure under normal test condition In order to shorten the testing time and reduce the testing cost, Accelerated Testing (AT) is promoted in such circumstances AT can be conducted in two ways One is the Accelerated Life Testing (ALT), which is employed at higher than usual stresses to induce early failures Physical failures are... be given on ALT and ADT data analysis and test planning 1.2 BASICS OF AT 1.2.1 The Commonly Used Lifetime Distributions AT is a quick way to assess reliability inferences on the performance of devices at a lower stress level and at operation time far beyond the length of experiments These inferences are obtained through extrapolations in two dimensions, i.e time and stress Effect of increased stress... Table 7.3 Numerical comparisons of our proposed plans with the existing plans Table 7.4 Sensitivity of n to σ Table 7.5 Sensitivity of π 1 to σ a Table 7.6 Sensitivity of T1 to σ a Table 7.7 Sensitivity of T2 to σ a a in destructive CSADT plans in destructive CSADT plans in destructive CSADT plans in destructive CSADT plans xvii List of Figures Figure 1.1 An example of the stress-loading pattern in... modeling the failure time of electronic components when they are assigned to high temperature, high electric field, or a combination of both temperature and electric field It is used for calculating the failure rates due to electromigration in discrete and integrated devices The lognormal distribution is also powerful to model failures of the fracture of substrate The p.d.f of the lognormal distribution... but it is difficult to control the stress changing rate and to model its effect Thus PST is not commonly adopted in real world Therefore, in this dissertation, we put our emphasis on data analysis and experiment design of CST and SST We will not cover details of PST in the following chapters 5 Introduction and Literature Survey Stress levels Chapter 1 Time Figure 1.5.An example of the stress-loading... with correct value of σ n 0 , π 10 , T10 , T20 optimal n, π 1 , T1, T2 with incorrect value of σ p a probability bound pk the expected proportion of failures at Xk p1, p2 the probability related to confidence interval Pr(*) probability of (*) q the quantile of a distribution qk ⎧Tk ⎪T ⎪ =⎨ ⎪ nk ⎪n ⎩ a a for CSADT for SSADT Q a derivation with respect to asymptotic variance r speed of reaction in the . Destructive Testing and Non-destructive Testing 23 1.3 ANALYSIS OF ALT DATA AND PLANNING OF ALT TEST 24 1.3.1 Analysis of ALT Data 24 1.3.2 Planning of ALT Test 25 1.3.3 Objectives of Our Proposed. 1905 CONTRIBUTIONS TO PLANNING AND ANALYSIS OF ACCELERATED TESTING YANG GUIYU (B. Eng., XJTU) A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY. CSALT Planning Approach 30 iii 1.3.4 Value of Our Proposed CSALT Planning Approach 31 1.4 DATA ANALYSIS AND PLANNING OF ADT TEST 32 1.4.1 Analysis of ADT Data 33 1.4.2 Planning of ADT