多變量製程監控之研究

Abstract

本論文內容分成三個主題。 在第一個主題，我們提出一多變量管制圖來偵測多變量製程變異降低的方法，此為根據多變量單邊檢定 所建立之管制圖，其中 和 分別為所監控品質特性目前的和在控制狀態下的共變異數矩陣，考慮 已知或未知兩種情形，我們分別導出概似比檢定統計量，並以此建立管制圖。透過統計模擬對幾種 的變化來比較平均連串長度，證實所提出的管制圖對多變量製程變異降低的問題比現有基於雙邊概似比檢定所建立的管制圖，在偵測能力上有相當不錯的效率。並以一個實例和模擬例子，證實所提出的管制圖具有應用性及有效性。 在第二個主題，我們結合第一個主題所提出的多變量管制圖及先前Yen and Shiau (2008) 所提的一個偵測多變量製程變異增加的管制圖建立一結合性多變量管制圖來偵測多變量製程變異增加或降低的方法。並且考慮 為已知或未知兩種情形。透過統計模擬對幾種 的變化來比較平均連串長度，說明所提出的基於不均等尾端機率管制界限所建立之結合管制圖，對多變量製程變異增加或降低的問題，比現有基於雙邊檢定所建立的管制圖在偵測能力上也有相當不錯的效率。並以兩個實例和模擬例子，證實所提出的管制圖具有應用性及有效性。 此外，對監控多變量常態製程平均值向量， 管制圖是一被廣泛使用的統計製程管制工具，有一個主要缺點：當 管制圖偵測到製程為失控狀態時並無法直接提供那一個品質特性或那幾個品質特性是造成製程失控原因的資訊。第三個主題的目的則為提出一個根據概似比原理的方法，當 管制圖發出失控訊號時，來找出那一個個別的品質特性平均值最有可能發生改變而不是試著決定那一個個別的品質特性是否失控。此方法對現行所使用的 管制圖方法為一個診斷輔助工具而不是替代工具。

The contents of this dissertation are divided into three main subjects. In the first subject, a multivariate control chart for detecting decreases in process dispersion is proposed. The proposed chart is constructed based on the one-sided likelihood ratio test (LRT) for testing , where and are respectively the current and the in-control process covariance matrix of the distribution of the quality characteristic vector of interest. Both cases of known and unknown are considered. For each case, the LRT statistic is derived and then used to construct the control chart. A comparative simulation study is conducted and shows that the proposed control chart outperforms the existing two-sided-test-based control charts in terms of the average run length. The applicability and effectiveness of the proposed control chart are demonstrated through two real examples and two simulated examples. By combining the above mentioned one-sided LRT-based control chart and the one-sided LRT-based control chart for detecting dispersion increases proposed by Yen and Shiau (2008), we propose a combined chart scheme for detecting both cases of dispersion increases and decreases. Both cases of known and unknown are considered. It is found that a combined chart using an equal tail probability to construct a control limit is biased. By simulation studies, the proposed combined chart scheme when using a set of unequal tail probabilities for the two charts outperforms the existing two-sided-test-based control charts in terms of the average run length, when the process dispersion increases or decreases. Two real examples and two simulated examples are used to illustrate the applicability and effectiveness of our proposed combined chart. About the third subject, Hotelling's chart is a well-known statistical process control tool for simultaneously monitoring elements of the mean vector of a multivariate normal pro¬cess. But it has a drawback that an out-of-control (or a significant) value does not gives us direct information as to which variables in are likely to have caused the out-of-control condition. We propose a method, based on likelihood principle, for identifying a variable or a group of variables in a multivariate normal process with an unknown covariance matrix □□that is likely to be responsible for the out-of-control condition signaled by a significant value. Unlike certain existing methods, our method is not a control/monitoring but a diagnostic tool. Two examples from earlier literatures and one based on simulation are used to illustrate the proposed method. Finally, we compare our results with that of other existing methods for these three examples.