多重型 I 設限下串聯系統之可靠度分析與最佳化設計

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許琮明(Tsung-Ming Hsu) 查詢紙本館藏

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論文名稱

多重型 I 設限下串聯系統之可靠度分析與最佳化設計
(Reliability Analysis and Optimal Design for Series Systems under Multiple Type-I Censoring)

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摘要(中)

在串聯系統內，只要系統中任一個物件失效就會導致系統失效。由於這些物件來自於同一個系統，所以物件間可能是相關的。過去已有的研究大多探討在物件壽命分佈互相獨立時之可靠度分析，本文首先考慮在多重型一設限下，以二元對數分佈去配適具兩物件的系統壽命。在傳統最大概似方法中，由於數值方法在求模型參數之最大概似估計時對於起始值相當敏感，因此我們將模型參數重新參數化後，以貝氏經由馬可夫鏈蒙地卡羅方法(MCMC)進行可靠度推論。模擬結果驗証貝氏方法可提供較準確地估計結果。另一方面，由於考量多元分布模型會限制物件壽命的分布，本文進一步考慮具一般化的 Clayton 關聯結構(copula)模型，其中具相關性的物件壽命之各邊際分佈皆為對數位置尺度族(log-location scale family)。而在串聯系統中，時常無法觀測到造成系統失效的物件，此時該系統為隱蔽資料。當系統壽命被設限或隱蔽時，我們使用期望值-最大化演算法去求得模型中未知參數的最大概似估計。同時，我們也利用遺失資訊法則去計算最大概似估計的費雪訊息矩陣，以近似最大概似估計的標準誤差。模擬結果顯示當物件間存在正相關時，物件之可靠度函數具高估的現象。最後，本文將壽命試驗推廣至恆定應力加速壽命實驗，考慮物件壽命服從獨立的指數分配且其平均壽命與應力具對數多項式關係，分別在D-準則、A-準則與V-準則下，以數值模擬探討多應力水準恆定應力加速壽命試驗之最佳化問題。

摘要(英)

In a series system, the components are connected within the same system, so their life time distributions may not be independent. In this study, we first consider reliability analysis of multiple Type-I censored life tests of series systems of bivariate log-normal life time distribution for the components. However, location of the MLEs highly relies on the initial values in executing the computation numerically given only the minimum life time of the components. Alternatively, we apply the Bayesian approach after a reparametrization of the parameters of interest. As a result, the Bayesian approach provides considerably accurate inference via simulation study. Moreover, to release the restriction that the joint distribution is constructed from the same family of marginal distributions, we consider series systems in which each component has marginal life time distribution in the family of log-location-scale distributions and the dependence among the life times is generated by the Clayton copula with unknown copula parameter. The MLEs and the Fisher information under masked data are derived via EM algorithm as well as missing information principle. The effect due to misspecification by independent models is investigated. Furthermore, optimal design problems for series systems are also discussed to improve the precision of estimators. Under independently exponential life time distributions, we investigate the problem of optimal sample size allocations for multi-level CSALTs when the mean of each component is a log-polynomial function in the stress level. The behavior of the optimal plans of test systems based on D-optimality, V-optimality, and A-optimality are discussed numerically.

關鍵字(中)

★ 二元對數分佈
★ 馬可夫鏈蒙地卡羅方法
★ Clayton 關聯結構模型
★ 對數位置尺度族
★ 隱蔽資料
★ 恆定應力加速壽命實驗

關鍵字(英)

論文目次

Contents
1 Introduction 1
1.1 Motivation and Background . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Objective and Preview of Results . . . . . . . . . . . . . . . . . . . . . . . . 5
1.4 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2 Two-Component Series Systems with Bivariate Log-Normal Life Time Distribution 9
2.1 Maximum Likelihood Inference . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.2 Bayesian Analysis under Re-parametrized Model . . . . . . . . . . . . . . . . 12
2.3 Data Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.3.1 Simulation Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.3.2 Real Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3 Multiple-Component Series Systems with Copula-Based model 25
3.1 Model Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.2 Maximum Likelihood Inference . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.3 Numerical Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.3.1 Simulation Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.3.2 Real data analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
i
4 Constant-Stress Accelerated Life time Test Plans for a Series System of
Independent Components with Type-I Censoring 44
4.1 Model Assumption and Likelihood Inference . . . . . . . . . . . . . . . . . . 44
4.2 Optimal Allocation Plans under CSALT . . . . . . . . . . . . . . . . . . . . 49
4.2.1 D-optimality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4.2.2 A-optimality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4.2.3 V-optimality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.2.4 Data Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
4.3 Log-Linear Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.3.1 D-optimality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
4.3.2 A-optimality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
4.3.3 V-optimality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
4.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
5 Conclusion 85
Appendices 87
References 93

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指導教授

樊采虹

審核日期

2016-1-28

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