Gamma 衰變過程之設計與分析

Design and Analysis of Gamma Degradation Process

蔡志群(Chih-Chun Tsai)

指導教授 ： 曾勝滄

國立清華大學/理學院/統計學研究所/博士(2009年)

https://doi.org/10.6843/NTHU.2009.00558

摘要

高可靠度的時代為了提升產品競爭力,製造商必須即時地提供顧客有關產品可靠度的訊息(如平均失效時間)。唯高可靠度產品之壽命推論,即使利用加速壽命試驗, 亦很難在有限的測試時間內獲得產品失效資料。此時, 若存在與壽命相關之品質特徵值(quality characteristics, QC)且會隨時間逐漸衰變,則可藉由衰變路徑來推估產品壽命相關資訊。因此, 對於生產製造商而言,如何建構衰變模
型及推論產品壽命分配,將顯得格外重要。文獻上有關衰變模型之建構,大都以隨機效應(random effect) 或Wiener 過程之衰變模型來描述之。然而對於產品的衰變路徑為單調遞增,如金屬疲勞(metal fatigue), 此時若採用隨時間遞增之gamma過程來描述產品衰變路徑將更為合理。本論文主要針對gamma衰變模型探討以下三個研究主題:
(1) 此研究主題以laser產品為動機例子,主要探討當gamma衰變模型被誤判為Wiener衰變模型時,對於產品平均壽命估計值之準確度(accuracy)與精確度(precision)的影響。具體來說,本文導出在大樣本下,當模型發生誤判時,產品平均壽命估計量的漸近分配,藉此結果可探討模型誤判對於產品平均壽命之影響。結果顯示臨界值(critical value)與gamma衰變模型中的尺度(scale)參數之比值,會影響產品平均壽命估計值之準確度。而當gamma衰變模型中的形狀(shape)或尺度參數大時,模型誤判對於產品平均壽命估計值之精確度的影響是不容忽視的。此外, 在小樣本或較短測試時間下,本主題以模擬方式探討誤判對於產品平均壽命估計值之影響, 其模擬結果與
大樣本下的結論是差異不大的。
(2) 逐步應力加速衰變試驗(step-stress accelerated degradation test, SSADT)是推估高可靠度產品壽命之一可行方法, 其優點是以較少的測試樣本及試驗設備, 在較短的實驗時間內, 可迅速推估高可靠度產品的壽命分配資訊。本主題首先建構出gamma過程之SSADT衰變模型,並在試驗總成本不超過事先給定的預算下,尋找一最佳試驗計畫(optimal test plan)。換言之,極小化產品平均失效時間估計值之近似變異數,以獲得最佳樣本數、每一應力水準下的量測次數與量測頻率。藉由敏感度分析(sensitivity analysis)可知
最佳試驗計畫的表現是相當穩健的。此外,在節省成本考量下,可對SSADT之最佳化問題增加限制條件, 以解得次最佳(suboptimal) 試驗計畫。本主題亦將上一主題的模型誤判問題推廣到SSADT上,探討誤判對於產品MTTF之壽命推估的影響。結果顯示在SSADT下,誤判gamma衰變模型為Wiener衰變模型對於產品壽命估計精確度的影響是相當嚴重的。
(3) 預燒試驗(burn-in test)可有效地在產品運送到顧客手中之前,將產品製造過程中所產生的潛在失效與不良產品加以剔除。對於現今高可靠度產品,可收集與產品可靠度相關的QC,藉由收集到的衰變資料來進行篩選試驗。本主題假設產品之衰變路徑來自混合gamma 衰變模型, 並訂定如何區別正常產品與不良產品的準則,接著推導出每個時間點下的最佳切斷點(cutoff point), 再以極小化總成本函數得到最佳之預燒時間。除此之外, 本文亦探討混合gamma衰變模型被誤判為混合Wiener衰變模型之誤判問題,模擬結果顯示模型誤判對於預燒試驗的錯誤分類機率將有很大的影響。

關鍵字

Gamma 衰變模型 ； Wiener 衰變模型 ； 逐步應力加速衰變試驗 ； 模型誤判分析 ； 最佳試驗計畫 ； 預燒測試 ； 混合分配

並列摘要

Nowadays, degradation analysis has been widely used to assess the lifetime information of highly reliable products. In the literature, degradation models are
mostly built by the randomeffect models and/orWiener process. However, for materials that lead to fatigue data, it is more appropriately modeled by a gamma process
which exhibits a monotonic increasing pattern. Hence, it is of great interest to design and analyze the degradation tests based on gamma process. In this thesis, we study the following three topics.
(1) Motivated by laser data,we discuss the effects ofmodelmis-specificationwhen the truemodel comes fromgamma degradationprocess, and iswrongly treated as Wiener degradation process. We derive the asymptotic distribution of quasi MLE (QMLE) so that the penalty formodelmis-specification can be addressed sequentially. The result demonstrates that the effect on the accuracy of the product’smean-time-to failure (MTTF) prediction strongly depends on the ratio of critical value and scale parameter of gamma degradation process. Besides, the effects on the precision of the product’s MTTF prediction are critical
when the shape and scale parameters of gamma degradation process are large enough. Furthermore,we use a simulationstudy to address the penalty of model mis-specification. The simulation result is quite close to the theoretical result even when the sample size and termination time are not large enough.
(2) Step-stress accelerated degradation test (SSADT) is a useful tool for assessing the lifetime distribution of highly reliable products when the available test items are very few. Firstly, we propose the SSADT model based on a gamma process with a linear degradation path. Under the constraint that the total experimental cost not exceeding a pre-specified budget, the optimal test plan can be obtained by minimizing the approximate variance of the estimated MTTF
of the lifetime distribution of the product. The sensitivity analysis reveals that the optimal test plan is quite robust to moderate departures from the model
parameters. Furthermore,we address the effect ofmodelmis-specification under SSADT. The results show that the effect on the accuracy of product’s MTTF prediction is not so critical, while the effect on the precision of product’s MTTF prediction is substantial when gamma degradation process ismis-specified as Wiener degradation process.
(3) Burn-in test has been widely used to eliminate latent failures or weak components in the factory before the products are shipped to the customers. The traditional burn-in test over a short period of time to collect time-to-failure data is rather inefficient. This decision problem can be solved by degradation models. We first propose amixture gamma process to describe the degradation path of the product. Next, a decision rule for classifying a unit as a normal or a weak unit is proposed. Finally, a cost model is used to determine the optimal burn-in time. Furthermore,we address themis-specification issue of treating a mixture gamma degradation process as amixtureWiener degradation process by a simulation study. The results reveal that the effect on the probabilities of
misclassification is significant.

並列關鍵字

Gamma degradation process ； Wiener degradation process ； step-stress accelerated degradation tests ； model mis-specification ； optimal test plan ； burn-in test ； mixture distribution

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