監控多變量製程變異之研究

Abstract

製程監控在品管上是很重要的一環。文獻上也有許多SPC方法來因應不同情形的製程監控問題。工程統計大師Woodall和Montgomery (1999, JQT)曾言，監控製程變異和監控製程平均一樣重要，或許更重要。然而絕大部分的研究均著重於品質特性之平均值的監控，相對的，如何監控製程變異之研究有如鳳毛麟角。當然這是研究的難易所致。也因此在變異監控上也留下了較多的研究空間。 本計劃為二年期的研究，主旨即在於研究探討如何有效地監控多變量製程的變異。我們發現一般監控變異的方法往往在變異增加或減少時之偵測能力有相當不相同的表現。例如，我們初步的研究發現真正的雙邊概似比檢定(LRT)為一有偏檢定(biased test)；在偵測變異增加時輸給了一個修正成不偏的雙邊概似比檢定，但在偵測變異減少時却又顛倒過來了。因此我們想尋找一個更佳的監控方法，在變異增加或減少兩種狀況下偵測力都能比上述兩個雙邊概似比檢定好。 ， 在第一年的研究中將針對多變量合理子群型資料提出一個利用合併控制圖(combined chart)方式的監控方法，亦即，我們希望能靠unbiased LRT控制圖來偵測變異變大，而靠biased LRT控制圖來偵測變異變小。另外，我們之前的研究採用單邊概似比檢定來偵测變異變大是因為它會比雙邊概似比檢定有較好之偵測力，因此我們亦將發展另一邊的單邊概似比檢定來偵测變異變小的情形。如此我們亦可合併此二單邊控制圖來處理双邊檢定的問題。目前已有初步想法，並將用電腦模擬來研究此監控方法之表現。 工業上有許多製程只能產生個別型的觀察值(individual observations)資料，而單一個別觀察值是無法對變異作估計的，因此監控變異有其困難。第二年我們將挑戰這個難題。初步想法是先利用指數加權平均(EWMA)的方式來取得共變異矩陣估計量，然後採用上述類概似比檢定的方式來發展監控方法。目前文獻上針對此問題所提出的方法大都假設控制下之共變異矩陣是已知的情形，我們的方法可以處理未知的情形。 本計劃之研究成果將能補SPC技術上之不足，對業界之品質管制與改進應有相當大的助益。

Process monitoring is one of the key components in quality control. There are many SPC methods designed for monitoring processes with various situations. It was mentioned in Woodall and Montgomery (1999, JQT) that most of the research works on SPC methods have been focused on monitoring the process mean, and yet the process variation is just as, or more, important. Apparently, it is a consequence of the fact that research on mean is a lot easier than on variation. Consequently, there still are rooms for research on process variation monitoring. This is a two-year project focusing on developing effective process variation monitoring schemes. It is found that many variation monitoring schemes behave quite differently when detecting increase or decrease of process variation. For example, the two-sided likelihood ratio test (LRT) is a biased test and has smaller detecting power than its unbiased counterpart when detecting increases in variation; and the situation is reversed when detecting decreases in variation. The objective of this research is to find a monitoring scheme that performs better than both of the two-sided LRT charts when detecting either direction of changes. For the first year of the project, we propose to study a combined-chart scheme by combining the biased two-sided and unbiased two-sided LRT charts. Also, since the one-sided LRT chart developed in the previous research has better power than that of the two two-sided LRT chart, we propose to develop the one-sided LRT chart for detecting decreases in variation and then combining the two one-sided LRT chart to form a monitoring scheme. A preliminary study indicates the approach is promising. We will further conduct a simulation study to study the performance of the proposed schemes. In many industrial applications, process data are individual observations instead of rational subgroups. It is impossible to have an estimator of the covariance matrix with a single individual observation. Thus monitoring variation is even more difficult. For the second year of the project, we will focus our study on developing good schemes for monitoring process variation when data are individual observations. The preliminary idea is: first taking the EWMA of the product of each observation and its transpose as the estimator of the covariance matrix and then apply the techniques we learn from the first year project to develop good schemes. We propose to develop new LRT-like control charts for monitoring (i) only increase, (ii) only decrease, and (iii) either direction in process variation respectively. Unlike the existing research works that assume the in-control covariance matrix 0Σ is known, we will be able to handle both cases of known and unknown. 0Σ The results of this study, when finished, will provide some new process monitoring schemes that can be useful in quality control and improvement.