監控非常態製程的平均數與標準差之管制圖

Abstract

統計製程管制 (SPC) 是利用統計分析的方法,對生產過程做及時的監控,使製程穩定和降低變異來改善製程能力。在統計製程管制的工具中,管制圖在監控過程中是最有效的。在假設特徵分佈是常態分配之下,對於監控過程的平均數跟標準差時,Shewhart 的平均數管制圖和標準差管制圖是最常用的統計管制圖。然而,在實際的應用裡,資料經常並非服從常態分 配的假設,而是遵循一種偏態分佈像是對數常態分配。因此,在本篇論文中,首先將針對監控對數常態分配的平均數,我們建構許多平均數管制圖和推導出一個近似平均連串長度 (Average run length) 的公式。接著,對於對數常態分配的標準差,我們提出一個新的標準差管制圖。在模擬研究中,我們使用平均連串長度準則去比較不同管制圖的優劣程度,而結果顯示當對數常態分配比較多偏態時,我們所提出的標準差管制圖比其他的標準差管制圖更好。更進一步,我們討論同時監控對數常態分配的平均數和標準差的管制圖。我們亦利用一個實際例子來展示我們所提出的對數常態管制圖表現得比其他管制圖好。在本篇論文的最後一部分,在單一觀察 值下,我們研究監控製程的變異數的無母數管制圖。我們根據經驗概似比檢定方法,對於監控尺度參數提出了一個無母數的管制圖。模擬結果顯示所提出無母數管制圖的偵測機率比其他無母數管制圖來的更高。我們亦把所提出的無母數管制圖應用到實際例子。

Among the statistical process control tools, the control chart has been proven to be effective in process monitoring. The Shewhart X-chart and S-chart are the most commonly used statistical control charts for monitoring the process mean and variability based on the assumption that the distribution of the quality characteristic is normal. However, in real applications, many process distributions may follow a positively-skewed distribution such as the lognormal distribution. In this dissertation, we first discuss the construction of several control charts for monitoring the mean of a lognormal distribution and derive the formula to approximate the average run length of a X-chart. Secondly, we propose a new control chart for monitoring the standard deviation of a lognormal distribution. The simulation results show that the proposed chart is more effective than the existing charts when the underlying lognormal distribution is more skewed. Moreover, we discuss the combined X- and S-charts for jointly monitoring the mean and the standard deviation of a lognormal distribution. The simulation results show that the combined lognormal X- and S-charts are more effective than the existing combined X- and S-charts when the underlying lognormal distribution is more skewed. A real example is used to demonstrate how these X-charts and the proposed S-chart including the X- and S-charts can be applied in practice. In the last of the dissertation, we study a nonparametric Phase I control chart for monitoring the process variability with individual observations which is based on the empirical likelihood ratio test. We propose a new nonparametric Phase I control chart for monitoring the scale parameter based on the empirical likelihood ratio test. The simulation results show that the proposed chart is more effective than the existing charts in terms of signal probability. A real example is used to demonstrate how the proposed chart can be applied in practice.