具伽瑪壽命分配之遮蔽資料的可靠度分析

The Reliability Analysis of Masked Data When Lifetime of Components Follow Gamma Distributions

楊健宏(YANG CHIEN HUNG)

指導教授 ： 陳雲岫博士

元智大學/工程學院/工業工程與管理學系/碩士(2001年)

摘要

近年來，隨著科技的進步，生活水準日益提高，各類產品的性能不斷的創新與提升，促使廠商間的競爭越來越激烈，產品壽命的品質要求日漸重要。產品之可靠度與其壽命就成為消費者衡量產品品質的指標之一，因此，工業界已日益重視具高可靠度的產品。傳統的可靠度觀念為產品的可靠度是經由製程的管制及檢驗即可達成，現今的高可靠度為產品在設計階段已賦予其品質。
遮蔽資料系統（Masked data system）為觀測到系統之失效時間時，可能無法確定系統失效是由那個零件損壞所造成。因此，我們在觀測系統失效資料時，也會觀測到遮蔽型資料。
在可靠度工程的分析中，大部分的研究方向是從零件的失效資料，來求得零件的壽命分配，進而推論系統之壽命分配。本論文的研究方向是探討如何從系統的失效時間與導致系統失效之零件，來討論各零件壽命分配之參數值。首先，假設由串聯J個零件所構成的系統，各零件之壽命分配均遵從伽瑪分配且相互獨立。隨機得到N組失效資料並根據此N組之觀測值(失效時間與失效原因)，利用概似函數(Likelihood Function)，以最大概似估計法(Method of Maximum Likelihood)來求得各零件壽命分配之參數估計值。在求參數估計的過程中，我們利用迭代法(Iterative Methods)來求聯立方程式之MLE解，經由我們求出之參數估計值，即可推論出零件之壽命。
本論文並以兩個零件為例，在參數相同與不相同時，分別解出MLE之估計值，進而估計可靠度值。在可靠度之估計結果方面，雖然零件一在不同遮蔽率下之壽命誤差百分比會略高於零件二，但兩零件真實可靠度均同落於95％信心水準之信賴區間內。資料顯示兩零件參數值相同時，MLE有良好的估計能力，在兩零件參數值不相同，MLE會因遮蔽資料的增加而降低其估計能力。

關鍵字

壽命分配 ； 失效時間 ； 遮蔽資料 ； 最大概似估計法 ； 迭代法

並列摘要

High quality and multi-functionality products are designed to meet people’s needs recently. High reliability is one index to measure the quality of products. Traditionally, the concept of high reliability is achieved by well-control the manufacturing process and sampling examination. In Contrast to the traditional methods, controlling the quality from the designing stage is more reliable method nowadays. Therefore, determination of design parameters plays an important rule in the system engineering. In other hand, making good estimation of design parameters is very important in reliability analysis.
Masked data system for observe the failure time of system, the exact cause of system failure might not be known .We refer to such observations as being masked. In this research, We estimate the design parameters of components by method of maximum likelihood and then predict the reliability function of a system composed of J components for the case of masked data system. The system considered in the research is connected by J components in series and lifetime distributions of J components follow Gamma distributions independently. N sets of masked data are observed, the derivation of design parameters for each components by method of maximum likelihood is given. The closed forms of MLEs of parameters are intrackable, thus, We use iterative method to approximate the MLEs. A numerical analysis is performed for the case of J=2. We consider two situations when the parameters are equal or unequal for the components. Numerical analysis shows that the performance of MLE is very good when parameters are equal, and the precise of estimation will reduce as the masked rate increases when parameters are unequal. Reliability function of the system is estimated and compared with theoretic results. It shows that the ability of predict the reliability is increasing as the masked rate decreases.

並列關鍵字

Life Distribution ； Failure Time ； Masked data ； Method of Maximum Likelihood ； Iterative Method

參考文獻

6. Guess, F.M.,Lin ,K.J. and Usher ,J.S.(1993),“ Exact Maximum likelihood Estimation using Masked system Data, ” IEEE Trans.Reliability,vol.42,pp631-635 .

7. Reiser, B.,Guttman, I. And Lin , D.K.J.et al(1995),“Bayesian Inference for Masked system lifetime Data, ” Applied Statistics ,vol 44,pp79-90 .

8. Usher, J.S. and Hodgson, T.J(1988),“Maximum likelihood analysis of component Reliability using Masked System life-Test Data, ” IEEE Trans.Reliability,vol.37 ,pp550-555 .

9.Greenwood, J.A.,and Durand , D.(1960) ,“Aids for fitting the gamma distribution by maximum likelihood, ” Technometrics ,2 , pp55-65

10.Harter, H.L.and Moore , A.H.(1965) ,“Maximum likelihood estimation of the parameters of gamma and wibull populations for complete and from censored samples, ” Technometrics ,7 , pp639-643 .