利用無母數Bootstrap法建構對數常態製程之全距管制圖

Abstract

利用 X ̅-R管制圖監控製程時，需要先決定製程內的變異是否在管制下，然後再檢視製程的平均數是否失控。Shewhart 所發展之R管制圖即是用來管制製程內的變異，R管制圖的管制界限公式是假設品質特性彼此獨立且呈常態分佈推導而得，但某些製程資料是呈非常態分佈(如：lognormal分佈)。針對非常態資料，過往有學者提出以Bootstrap法來建構管制圖，但僅使用百分位數複式(Percentile Bootstrap, PB)信賴區間來建構管制界限，並未考慮其他的Bootstrap信賴區間，如：標準複式(Standard Bootstrap, SB)、修正偏度百分位數(Biased-Corrected Percentile Bootstrap, BCPB)及加速修正偏誤(Bias-Corrected and Accelerated, BCa)複式信賴區間等。由於一些文獻提出BCa信賴區間估計法較其他的Bootstrap信賴區間準確，因此本研究針對lognormal製程的全距資料，利用無母數Bootstrap的PB及BCa兩種信賴區間估計法來建構R管制圖的管制界限，並模擬有不同參數值之對數常態資料進行敏感度分析，以驗證無母數Bootstrap R管制圖之有效性。本研究敏感度分析的結果顯示，當偵測能力以平均連串長度(ARL)表示時，在lognormal分佈不同之位置參數(location parameter)與尺度參數(scale parameter)下，當製程穩定時，以PB信賴區間所建構之R圖管制界限之ARL(即ARL_0)表現最佳，BCa次佳，傳統R管制圖最差；當製程發生失控時，在lognormal之尺度參數≤5時，PB、BCa與傳統R管制圖之ARL(即ARL_1)之表現均非常接近，其中傳統R管制圖僅略優於PB及BCa，PB又略優於BCa；而在尺度參數>5時，PB、BCa及傳統R管制圖的表現皆會隨著參數值上升而下降，雖然傳統R管制圖優於PB及BCa，BCa優於PB，但相對應其ARL_0的偵測能力相當不良。因此，整體而言，當製程資料呈lognormal分佈時，不論其參數值大小，可利用本研究建議的無母數Bootstrap抽樣方法及 PB信賴區間來建構R管制圖之管制界限，會比傳統Shewhart R管制圖效果為佳。

When using X ̅-R control chart to monitor the production process, the process variation must be under control before reviewing the mean of the process. Shewhart’s R control chart is typically utilized to monitor the process variation. The control limits of Shewhart control chart are derived under the assumption that the process data are independent and follow a normal distribution. However, the process data sometimes follow non-normal distributions (e.g., lognormal). Previous studies have proposed the Bootstrap methods to construct the control charts for some non-normal process, but these studies only consider utilizing Percentile Bootstrap (PB) confidence interval to construct the control limits. However, the Bias-Corrected and Accelerated (BCa) Bootstrap confidence interval is recommended as it is proved to be more precise than other Bootstrap confidence intervals (e.g., PB, Standard Bootstrap, Biased-Corrected Percentile Bootstrap). Hence, the objective of this study is to utilize non-parametric Bootstrap sampling method and two popular Bootstrap confidence intervals (PB, BCa) to construct the R charts, and the sensitivity analysis is conducted to verify the effectiveness of the proposed Bootstrap R chart for lognormal distribution. The simulation result indicated that based on the average run length (ARL) for stable process (ARL_0) under various values of location and scale parameters, the Bootstrap R chart constructed by PB method performs better than that of BCa method and traditional R chart. Moreover, based on the ARL for unstable process (ARL_1), when the scale parameters is less than or equal to 5, the R control chart developed using PB, BCa confidence interval and Shewhart’s R chart are very close, although the traditional R charts performs slightly better than PB and BCa, and PB performs slightly better than BCa. When scale parameters more than 5, the R control chart developed using PB, BCa confidence interval and Shewhart’s R chart will decrease, although the traditional R charts performs better than PB and BCa, and BCa performs slightly better than PB. In general, PB is recommend to construct the R chart when the data follow a lognormal distribution.