監控隨機效應線型品質特性之製程

Abstract

監控製程與產品之剖面(profile)資料在統計品質管制中是ㄧ個非常熱門且有前景的研究領域。我們的研究將以對具有隨機效應之線型資料(linear profile)的監控方法為主，目的在使模型能容許更多合理的變異來處理實際的問題。我們發現，若我們忽略應該被納入模型的變異，而用固定效應的模型來做製程監控，則假警報的比率會非常高。 對隨機效應的資料模型，我們用三個Shewhart-type 管制圖分別對線型資料的三個參數作監控。在製程監控之第二階段中，因為三個監控統計量是相互獨立的，我們分配同樣的假警報比率給三個統計量，從而控制整個製程的假警報率為事先設計的比率 。 在製程監控之第一階段中，不同資料所對應的監控統計量並不獨立，但是對任一條線型資料而言，三個監控統計量是相互獨立的。因此，在第一個階段中，我們分別用Bonferroni 方法和 Multiple FDR 方法來控制整個歷史資料的假警報率(overall false-alarm rate)，並且比較這兩種方法的優劣。模擬的結果顯示，若以偵測力為判斷方法優劣的標準，用Multiple FDR 方法較Bonferroni 方法好，特別是當歷史資料中超出管制界限的線型資料愈多時。用Multiple FDR 方法使偵測力增加的代價是將歷史資料中在管制狀態下的線型資料誤判為失控狀態的比率會略為提高。

The monitoring of process and product profiles is a very popular and promising area of research in statistical process control. This study proposes a monitoring scheme for linear profiles with a random-effect model to incorporate the subject-to-subject variation in some real-life problems. It is found that if we ignore the subject-to-subject variation and monitor the process by the existing scheme based on the fixed-effect model, the false-alarm rate could be incredibly high. For random-effect models, we use three Shewhart-type control charts to monitor linear profiles. In Phase II operation, since the three monitoring statistics are mutually independent, we give the same in-control false-alarm rate to each control chart such that the overall in-control false alarm rate of the combined-chart scheme is controlled at the prescribed level . In Phase I operation, the monitoring statistics across profiles are not independent, but the three monitoring statistics for the same profile are mutually independent. Therefore, to control the overall false-alarm rate in Phase I monitoring, the Bonferroni method and Multiple FDR method are implemented and compared. Simulation results show that the Multiple FDR method is better than the Bonferroni method in terms of detecting power, especially when more out-of-control profiles are in the historical data. The tradeoff of using the Multiple FDR method is the slightly larger “false-alarm rate”, which is defined as the proportion of the false alarms within the in-control profiles in the historical data.