應用無母數Bootstrap法建構Burr製程之全距管制圖

Abstract

蕭華特(Shewhart)所發展之X ̅-R管制圖可以監控製程平均數與製程變異，使用X ̅-R管制圖時，必須先決定R管制圖是否在管制內，才能檢視X ̅管制圖是否失控。傳統R管制圖是假設品質特性値在穩定的製程中呈常態分佈，故其管制界限公式是基於常態分佈下推導而得，但在實務上，某些製程資料是呈非常態分佈，若仍使用傳統R管制圖，會使型一或型二誤差增加，降低管制圖的偵測績效。過去雖有研究針對非常態資料提出以Bootstrap法來建構管制圖，但這些研究大多僅使用PB信賴區間來建構R管制圖之管制界限，並未考慮其他Bootstrap信賴區間(如:SB、BCa等)。因有一些文獻認為BCa信賴區間法較其他的Bootstrap信賴區間準確，但用BCa信賴區間來建構管制圖是否較PB準確，並未見有文獻探討，因此，本研究針對Burr製程資料，利用無母數Bootstrap的PB及BCa兩種信賴區間方法來建構R管制圖的管制界限，並模擬不同參數組合之Burr分佈資料進行敏感度分析來驗證本研究方法之有效性。模擬驗證的結果顯示，若偵測能力以平均串連長度(ARL)表示時，在不同參數組合之Burr分佈下，當製程穩定時，無論分佈呈右偏、左偏或大致對稱，以PB信賴區間法所建構R管制圖之管制界限的ARL_0表現皆優於BCa及傳統Shewhart’ s R管制圖；當製程失控時，傳統R管制圖之ARL_1表現優於PB及BCa，其中當標準差變動量越大時，傳統Shewhart’ s R、BCa與PB的表現非常接近，雖然傳統R管制圖優於PB及BCa，但傳統R管制圖相對應其ARL_0的偵測能力相當不佳，且PB與BCa之ARL_1偵測能力非常接近。因此，整體而言，當製程資料呈Burr分佈時(亦即無論分佈呈右偏、左偏或大致對稱)，可利用本研究建議的無母數Bootstrap抽樣方法及PB信賴區間法來建構全距管制圖之管制界限，會比傳統R管制圖之效果為佳。

Shewhart’s X ̅-R control chart is typically utilized to monitor the process mean and the variation in industry. In exploring the X ̅-R control charts, the R control chart must be in-control before reviewing the X ̅ control chart. The control limits of the traditional R control chart are derived under the assumption that the process data follow a normal distribution. If the process data follow a non-normal distribution, utilizing the R control chart to monitor the process variation may increase the chance of committing Type I and Type II errors. Previous studies have developed some bootstrap control charts for non-normal distributions, but these studies just employed the bootstrap re-sampling technique and Percentile Bootstrap (PB) confidence interval to construct the control limits of the R control chart. There are four types of Bootstrap confidence interval (namely, Standard Bootstrap (SB), Percentile Bootstrap (PB), Biased-Corrected Percentile bootstrap (BCPB), Bias-corrected and accelerated percentile bootstrap (BCa)). Some studies indicated that in many cases the BCa confidence interval is more precise than other Bootstrap confidence intervals (i.e., SB, PB, BCPB) in estimating the population parameter. However, it is rarely seen that the BCa bootstrap interval is employed in constructing the X ̅-R control charts. Moreover, the Burr distribution can be utilized to describe many non-normal distributions. Therefore, the objective of this study is to utilize the non-parametric bootstrap sampling technique and two popular bootstrap confidence intervals (PB, BCa) to construct the R control charts based on the Burr distribution. The sensitivity analysis is conducted to verify the effectiveness of the proposed Bootstrap R charts under various combinations of the parameters values of a Burr distribution. Based on the average run length (ARL) for the stable process (ARL_0), the simulation result indicated that the Bootstrap R chart constructed by the PB method performed significantly better than the BCa method and the Shewhart’s R chart. Furthermore, based on ARL for the unstable process (ARL_1), the traditional R chart outperformed the PB and BCa methods slighty. However, when the process has large shift, the Shewhart’s R chart, PB and BCa methods performed very closely. In general, PB is recommended to construct the R chart when the data follow a Burr distributions.