先行順序管制圖的敏感度分析

The Sensitivity Analysis of Precedence Charts

陳煜翔(Yu-Hsiang Chen)

指導教授 ： 陳慧芬

中原大學/電機資訊學院/工業與系統工程研究所/碩士(2014年)

https://doi.org/10.6840/cycu201401022

摘要

本論文研究問題是先行順序管制圖的敏感度分析以及先行順序管制圖與修華特 管制圖的比較。
Chakraborti, S., van der Laan, P.與van der Wiel, M. A.於西元2004年提出先行順序管制圖，先行順序管制圖是用於偵測品質特性觀察值的中心位置發生偏移的工具。當品質特性觀察值的分配未知時，管制圖的管制界限必須用估計的。假設給定一組管制狀態下的品質特性觀察值的資料(稱之為Phase-I資料)，為了監控製程中心是否偏移，而蒐集的品質特性觀察值資料(稱之為Phase-II資料)。先行順序管制圖的監控統計量是採用中位數，管制界限是Phase-I資料量的順序統計量。令 表示當期望值發生偏移 個標準差時的平均連串長度，下列參數會影響先行順序管制圖的平均連串長度: Phase-I資料量m、Phase-II資料的樣本數n、品質特性觀察值在管制狀態時的累積分配函數 及 。先行順序管制圖具有下述優點: 因為不論品質特性觀察值分配為何，在相同的m、n以及 下，都可以得到相同的管制狀態下平均連串長度。
本論文針對先行順序管制圖進行敏感度分析，並將管制圖的績效指標設定為平均連串長度。敏感度分析為探討m、n、 以及 對 的影響，尤其是觀察在不同的 值下(本研究只考慮正值)， 曲線的變化。
在針對先行順序管制圖進行敏感度分析時可以得到以下結論，在使用先行順序管制圖時m值越大， 的曲線會下降。當m值很小時，實際的管制狀態下平均連串長度ARL0值與指定ARL0可能差異大。在使用先行順序管制圖時，n值越大， 的曲線會越低。而樣本特性觀察值的分配會影響 曲線，若品質特性觀察值的分配為對稱分配， 的最大值發生在當 等於0時。若品質特性觀察值分配為不對稱分配時， 的最大值不一定發生在 等於0時。
接著我們進行先行順序管制圖與修華特 管制圖的比較，假設品質特性觀察值的分配 為常態分配但期望值與標準差未知，在相同m值、n值與指定的管制狀態下平均連串長度ARL0值的條件之下進行修華特管制圖與先行順序管制圖的比較，兩管制圖的管制界限須滿足指定的ARL0值。品質特性觀察值服從常態分配，且n值設定為5，m值設定為50、500及1000。實驗的結果如下: 當m分別為50及500時，先行順序管制圖的 曲線在在修華特管制圖的 曲線下方。m為1000時，修華特管制圖的 曲線在先行順序管制圖的 曲線下方。因此可以推斷當m值提高時，修華特管制圖的 曲線下降速度較先行順序管制圖快速。

關鍵字

敏感度分析 ； 平均連串長度 ； 先行順序管制圖

並列摘要

Abstract
We conduct a sensitivity analysis for precedence control charts and comparisons between precedence charts and Shewhart charts.
Chakraborti, S., Laan, P. and Wiel, M. A proposed the precedence control chart in A.D. 2004. The precedence control chart is a tool to detect the mean shift of the quality characteristics measurement of interest. The control limit of a precedence control chart need to be estimated when the distribution of the quality characteristics measurement is unknown. Assume that a set of in-control data is given which called Phase-I data, and a sample for detecting the mean shift, (called Phase-II data) is given. The precedence control chart method use the median to control statics, and the control limit is the order statistic from Phase-I. Let is the average run length when the expected value shift standard deviation. The parameter of Phase-I date m, the sample size n of Phase-II, and the cumulative distribution function of in-control quality characteristics measurement observation would influence in-control average run length of precedence control chart. Precedence control chart has advantage on following information. No matter what the distribution of quality characteristics measurement are, precedence control can get the same in-control average run length when m, n, and is the same.
This study will start at the Sensitivity Analysis Method of precedence control chart, and the Key Performance Indicators of control chart is the average run length. Discuss the average run length under the influence of the Sensitivity Analysis Method parameter of m, n, and .
The conclusion of the Sensitivity Analysis Method of precedence control chart: The curve of is decreased when the parameter m of is bigger in precedence control chart. When parameter m is small the setup in-control average run length and the actually average run length would be different. The curve of is decreased when the parameter n of precedence control chart is bigger. If the cumulative distribution function of the quality characteristics measurement of precedence control chart is symmetrical, the would have maximum value when is happened on 0. If the cumulative distribution function of the quality characteristics measurement of precedence control chart is asymmetrical, the would have maximum value when is not happened on 0.
We compare the precedence control chart and Shewhart chart. The quality characteristics measurement of distribution is normal distribution but the expect value and the standard deviation are unknown. The ARL0 of control limits must be satisfied when precedence control chart and Shewhart chart have the same condition parameter m, n. Quality characteristics measurement observation values are following normal distribution, and the n value is set to 5, m value is set to 50,500 and 1000. The experimental results are as follows: When the m are 50 and 500 , the curve of precedence control chart below the curve of Shewhart control chart. When m is 1000, the curve of Shewhart control chart below the curve of precedence control chart. It can be inferred that the rate of decline curve of Shewhart control chart is more quickly than the precedence control chart when the m value is increased.

並列關鍵字

sensitivity analysis ； Precedence control charts ； average run length

參考文獻

4. Bickel, P., and Doksum, K. A. (2001), Mathematical Statistics: Basic Ideas and Selected Topics, Volume I, second edition, Prentice Hall.

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