利用近似之最大概似法預測重覆性實驗計畫中型II受限資料及構建相關之最佳化演算法

Abstract

工業界在研發新產品或是改善產品品質時常應用實驗設計(Design of Experiments, D.O.E.)或田口方法(Taguchi Method)來規畫實驗及分析實驗數據，以有效找出最佳之因子水準組合。目前科技產品之可靠度已達到某種水準之上，在應用實驗設計或田口方法找出最佳因子水準組合的過程中，若將與可靠度有關的變數(如：產品壽命)當作反應變數(response)，則執行實驗所需花費的時間會非常冗長，若能在執行實驗的過程中，先設定實驗中受限資料的個數，而將無法觀察到的實驗數據當成受限資料(censored data)來處理，則可節省實驗時間及加快產品上市。當一組實驗數據包含受限資料時，其統計模式不再具直交性(orthogonality)，因此無法使用一般之變異數分析方法來找出實驗之最佳因子水準組合。中、外文獻所提出分析受限資料的方法，大多是針對型I受限資料，這些方法中有些使用不易或假設條件嚴格，故實用價值有限。另外，型I受限資料之中止時間亦不易設定，若中止時間設定太小，則收集到的資料就會太少，使得分析結果難以令人信服；若中止時間設定太大，雖可收集到較多的資料，甚至可收集到完整的資料，但實驗成本會因時間消耗過長而增加。因此，本研究針對型II受限資料，在製程資料服從常態分配及製程之平均數與變異數均未知的假設下，提出四種近似最大概似預測式(approximate maximum likelihood predictors; AMLPs)來預測受限資料。首先，本研究利用兩種形式之概似函數(likelihood functions)來推導出受限資料之近似最大概似預測式，由於概似函數中包含了故障函數(hazard function)，本研究採用兩種方式近似故障函數之值以求其封閉解。第一種方式為將第一種概似函數中的故障函數以其期望值(expected value)代之，再推導出受限資料之預測式，稱此預測式為第I型近似最大概似估計式(簡稱Model I AMLP)；第二種方式是將第二種概似函數中的故障函數以其期望值(expected value)代之，再推導出受限資料之預測式，稱此預測式為第II型近似最大概似估計式(簡稱Model II AMLP)；第三種近似最大概似預測式則是利用泰勒展開式(Taylor series)來近似第一種概似函數中的故障函數，再推導出受限資料之預測式，稱此預測式為第III型近似最大概似估計式(簡稱Model III AMLP)；第四種近似最大概似預測式則是利用泰勒展開式來近似第二種概似函數中的故障函數，再推導出受限資料之預測式，稱此預測式為第IV型近似最大概似估計式(簡稱Model IV AMLP)。然後本研究利用蒙地卡羅模擬法與變異數分析法來比較此四種預測式之準確性與有效性，結果發現Model II AMLP與Model IV AMLP較為準確與有效。本研究最後再利用Model II AMLP與Model IV AMLP發展出一套分析具受限資料之實驗數據最佳化演算法，並分別利用傳統實驗計畫與田口計畫之實例，證實本研究所提的受限資料分析演算法確實有效。

Design of experiments and Taguchi methods are widely employed in industry to develop new product or enhance product quality and reliability. Because the high technology products are often required to have high reliability, the lifetime or reliability of the high technology products is often considered as a response variable in the reliability experiments. Such experiments are usually time-consuming. In order to shorten the time for bringing the new product to the market, the experiments must be terminated before all the experiment runs are completed. In these situations, the incomplete data are called censored data. When the censored data are arisen in the experimental data, the orthogonality of the statistical models no longer exists and the usual analysis of variance methods of analyzing experimental data cannot be used to determine the optimal factor–level combinations. Many studies proposed various methods to analyze the censored data; especially for type I censored data. However, these methods are either computationally complex or have little practical use. Besides, the time of terminating the experiments to obtain type I censored data cannot be determined easily. If the time of terminating the experiments is too short, the number of uncensored data may be too few; if the time of terminating the experiments is too long, the number of uncensored data may be too many, and consequently, the experimental cost increases. Since the population mean and variance of experimental data are usually unknown in practice, this study proposes four approximate maximum likelihood predictors (AMLPs) to predict the type II censored data under the assumption that process data followed a normal distribution with unknown mean variance. This study utilizes two types of likelihood function to derive AMLPs. Both of two types of likelihood functions involve hazard functions. The first alternative is to replace the hazard functions by their expected values in the first type of likelihood function and then derive the predictor (which is designated as Model I AMLP). The second alternative is to replace the hazard functions by their expected values in the second type of likelihood function and then derive the predictor (which is designated as Model II AMLP). The third alternative is to use the Taylor series expressions of hazard functions to approximate the hazard functions to obtain the predictor (which is designated as Model III AMLP). The fourth alternative is to use the Taylor series expressions of hazard functions to approximate the hazard functions to obtain the predictor (which is designated as Model IV AMLP). Monte Carlo simulation and analysis of variance (ANOVA) method are used to compare the bias and effectiveness of these AMLPs. The results indicate that Model II AMLP and Model IV AMLP are more effective. Finally, the algorithms of optimizing the response from repetitious experiments are developed. Two cases are also given to demonstrate the effectiveness of the proposed optimization algorithms.