利用Bootstrap管制圖監控對數常態製程平均數之有效性評估

Abstract

傳統X-bar管制圖是假設品質特性值在穩定的製程中為常態分佈，故管制界限公式是基於常態分佈下推導而得，若使用X-bar管制圖來監控非常態分佈的製程資料，會使管制圖之型一或型二誤差增加而誤判製程狀況，降低管制圖的偵測績效。過去雖有研究針對ㄧ些非常態製程資料，提出以無母數bootstrap法來建構X-bar管制圖之管制界限，但這些文獻僅使用PB信賴區間方法來建構X-bar管制圖之管制界限，也未完整的驗證bootstrap管制圖對非常態製程之有效性。因此本研究針對常見的非常態製程中之Lognormal分佈，以無母數bootstrap的兩種信賴區間方法(PB和BCa)來建構X-bar管制圖之管制界限，並以本研究所提出的敏感度分析流程來驗證其有效性。根據一些文獻顯示，BCa信賴區間方法表現常較其他三種bootstrap信賴區間方法佳，但由本研究模擬驗證的結果顯示，以無母數bootstrap法建構X-bar管制圖界限時，PB信賴區間方法所建構之管制界限其偵測能力較BCa佳；且以PB信賴區間方法所建構之X-bar管制圖界限其偵測能力雖然不是在任何情況下都非常優越，但整體而言，確實較傳統X-bar管制圖更適用於監控Lognormal分佈的製程資料。

The control limits of a traditional X-bar control chart are derived under the assumption that the process data follow a normal distribution. However, the Type I and Type II errors may have a higher chance to occur in using the X-bar chart to monitor the process when data follow a non-normal distribution. In the past, there are studies utilized non-parametric bootstrap methods to construct control limits based on non-normal distributions. These studies have applied the Percentile Bootstrap (PB) confidence interval in constructing the X-bar control limits. However, they did not verify the effectiveness of bootstrap control chart in monitoring the process mean of non-normal distributions. This study utilizes two non-parametric bootstrap confidence intervals (i.e., PB, Bias-Corrected and Accelerated Percentile Bootstrap (BCa)) in constructing the X-bar control limits and the effectiveness was verified using the sensitivity analysis based on Lognormal distribution. Although some studies indicated that BCa confidence interval performs better than the other three bootstrap confidence intervals, the simulation results of this study indicated that when control limits are constructed using non-parametric bootstrap method, PB confidence intervals performs better than BCa. PB confidence interval does not grant the best detecting ability in terms of type I and type II errors, but as a whole, it outperforms the traditional X-bar control chart in monitoring data that follow a Lognormal distribution.