管制圖為品質管理常用手法之一，可用於監控製程是否發生變異，常見管制圖有：修華特管制圖、累積和管制圖(cumulative sum；CUSUM)、指數加權移動平均管制圖(exponentially weighted moving average；EWMA)，修華特管制圖常用於監測平均數及變異數，缺點為無法快速偵測參數的小位移，而CUSUM及EWMA可快速偵測參數小位移，上述三者皆為須針對欲偵測的製程設定參數，其設定之參數與實際情況不一定相符。因此後續提出一般化概似比管制圖(generalized likelihood ratio；GLR)，一般的GLR管制圖不須設定參數，對於小位移或大位移的製程皆有良好的偵測績效，但在本篇所比較的Bernoulli GLR管制圖需設立參數，當連續出現不合格品時，在計算統計量會產生無定義的情況，因此其設立不合格率參數估計的上限，此為設定參數的一種情況。本研究使用Kullback-Leibler information建構管制圖，利用伯努利分配監測伯努利過程中的不合格率，為不須設立參數之管制圖，分別監測不合格率上升、不合格率下降，以及同時監測不合格率上升及下降，並探討管制圖在不同不合格率下的監測情況。提出兩種方法解決管制圖在特例點無定義的情況，第一種為假設檢定，將不合格品出現機率與型一誤差比較(α法)，第二種為極限近似的方法(limit法)，透過模擬結果，當不合格率較大時，α法監測績效會優於limit法，當不合格率較小時，則兩者監測績效將相同。在監控不合格率上升時，與其他管制圖比較時本管制圖監測績效並非最佳，；監控不合格率下降時，本管制圖α法為最佳管制圖；在同時監控不合格率上升、下降時，使用兩種方法監控，第一種為對稱管制界限(two-sided)、第二種為非對稱管制界限(combined)，最佳管制圖為本管制圖combined α法。

關鍵字:不合格率管制圖、伯努利過程、CUSUM、EWMA、GLR、資訊理論、Kullback-Leibler information

This study considers the problem of monitoring the proportion p of nonconforming items when all items from the Bernoulli process are classified into two categories. The aim is to effectively detect a wide range of increases and decreases in p. The proposed control chart is based on Kullback-Leibler information, which measures the information lost of two distributions. We name this control chart Kullback-Leibler information control chart (KLI control chart), differently from other control charts; we construct a parameter-free control chat. This study uses a backward sequential approach to construct the KLI control chart. For some special observations, such as all conforming items or all nonconforming items, the KLI control chart statistics for monitoring the process are undefined. Therefore, we show two methods to let statistic is always well defined, hypothesis test (α) and limit, respectively. The Phase Ⅱ performance of this chart for detecting a shift in a proportion is evaluated by the average number of observations to signal (ANOS). Comparison of the Bernoulli CUSUM, the Bernoulli EWMA, and the Bernoulli GLR shows that the performance of the KLI control chart is better than its competitors when detecting a decrease in p or both increase p and decrease in p. It is effective in detecting a wide range of shifts.

Keywords: p-chart, Bernoulli process, CUSUM, EWMA, GLR, Information theory, Kullback-Leibler information

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