應用無母數Bootstrap法建構新製程之Q管制圖

Abstract

傳統Shewhart管制圖建構時製程資料須服從常態分配並且需要有足夠的歷史資料才能有效建構管制圖之管制界線。然而為因應廠商、顧客日新月異之要求，不斷推出新的產品，使的生產週期縮短，導致一些產品缺少足夠歷史資料來建構傳統Shewhart管制圖。針對此問題，Quesenberry (1991) 提出Q管制圖，以在僅有少量資料的情況下有效建構管制圖。在利用Q管制圖管制新製程資料，需假設製程資料彼此獨立且呈常態分佈；若製程資料分布呈現非常態分佈時，使用Q管制圖可能會增加型一誤差(Type I error)和型二誤差(Type II error)發生的機率，而無法準確地偵測出變異。因此，本研究利用無母數複式模擬法(Bootstrap methods)增生資料，並利用兩種複式信賴區間(Percentile Bootstrap,PB; Bias-corrected and accelerated bootstrap,BCa)來建構Q管制圖之管制界線。最後本研究模擬製程資料呈Weibull分布的各種參數情況下，進行敏感度分析以驗證本研究提出之無母數複式Q管制圖在不同樣本組數以及不同偏態下之有效性。本研究結果顯示，在多數情況下以PB信賴區間所建構之Q管制圖之管制界線在製程穩定的狀態下，平均連串長度(〖ARL〗_0)最佳，BCa和傳統Q管制圖較差；當製程平均數產生偏移時，在大多數情況下利用PB信賴區間所建立之Q管制圖其管制效果皆能優於傳統Q管制圖，因此，整體而言，當新製程資料呈非常態分佈(如:Weibull分布)時，在Weibull分佈不同的參數組合下，和傳統Q管制圖相比，本研究提出之無母數複式Q管制圖有較佳的監控能力。

Traditional Shewhart control chart reguires that the process data follow a normal distribution and also need sufficiently large data to accurately construct the control chart. However, in response to the ever-changing requirements from customers, the design of new products, the shortening of the production cycle, resulting in lack of sufficient historical data to build the traditional Shewhart control chart. Previous study has proposed a Q chart to effectively construct a control chart with only a small amount of data. In applying the Q control chart to control the new process data, it is still necessary to require that the process data are independent of each other and are normally distributed. If the distribution of the process data is non-normal, utilizing the Q chart may increase the probability of a type 1 error and type 2 error. Conseguently the Q chart can not accurately detect the process variation. Therefore, this study utilizes the Bootstrap Methods to generate data and that constructs the Q-chart using two types of Boostrap confidence intervals (Percentile Bootstrap (PB) and Bias-corrected and accelerated bootstrap (BCa)). The sensitivity analysis was carried out to verify the effectiveness of the proposed method of Boostrap Q-chart under various sample sizes and various values of parameter combinations of a Weibull distribution. The results of this study show that, in most cases, the average run length 〖ARL〗_(0 )of PB method has the largest values. When the process averages are shifted, in most cases the Q-control chart established using the PB confidence interval is superior to the traditional Q-control chart. Therefore, in general, when the new process data is presented in non-normal process (such as: Weibull distribution), the proposed non-parametric PB method is recommended to construct the Q-control chart.