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  • 學位論文

Xbar及R管制圖的對稱與非對稱管制界限之比較

Comparisons of Symmetric and Asymmetric Control Limits for Xbar and R Charts

指導教授 : 陳慧芬
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摘要


本研究主要探討Xbar和R管制圖的對稱與非對稱管制界限比較。Shewhart Xbar和R管制圖在統計製程管制領域中經常被使用,傳統上,這兩種管制圖均假設測量值的母體服從常態分配,並以3倍標準差來建立管制圖的上下界限,稱為對稱管制界限。多數研究者在研究非對稱管制界限的制定時,是針對品質特性服從非常態分配,但從未在相同基準下去比較對稱與非對稱管制界限。有鑑於此,我們將去探討一個問題,當品質特性服從非常態分配時,非對稱管制界限真的比對稱管制界限佳嗎?本實驗的衡量方法,預先給定製程在管制狀態下的平均串聯長度(ARL0)的特定值,再比較當製程失控時的平均串聯長度(ARL1)值。另外本研究針對有相關性的品質特性提出一個新的非對稱管制界限制定方法,並同樣與對稱管制界限做比較。 對於Xbar管制圖中的對稱和非對稱管制界限,本研究使用三種不同的品質特性做比較。前兩個例子假設品質特性分別為獨立服從gamma分配和Johnson unbounded分配。第三個例子假設品質特性為相關性服從autoregressive to anything (ARTA) with exponential分配。實驗結果發現,品質特性為右偏時,當製程平均向右偏移,對稱管制界限較非對稱管制界限為佳;反之,當製程平均向左偏移,非對稱管制界限較對稱管制界限為佳。同理,品質特性為左偏時,當製程平均向右偏移,非對稱管制界限較對稱管制界限為佳;當製程平均向左偏移,對稱管制界限較非對稱管制界限為佳。 針對R管制圖的分析,本研究使用五種不同的品質特性做比較。前三個例子假設品質特性分別為獨立服從uniform分配、exponential分配、Johnson unbounded分配。第四和第五個例子分別假設品質特性為相關性服從ARTA process with exponential分配及ARTA process with uniform分配。實驗結果發現,全距分配為右偏時,當製程標準差增大,對稱管制界限較非對稱管制界限為佳;反之,當製程標準差降低,非對稱管制界限較對稱管制界限為佳。同理,全距分配為左偏時,當製程標準差增大,非對稱管制界限較對稱管制界限為佳;反之,當製程標準差降低,對稱管制界限較非對稱管制界限為佳。從實驗結果發現 管制圖和R管制圖的結果是相同的。

並列摘要


This work compares the Xbar and R charts performance for the symmetric and asymmetric limits. The Shewhart X and R control charts are control schemes used commonly in statistical process control. A conventional way of setting the control limits is to choose a set of symmetric limits, e.g., the 3-sigma control limits. Despite literature of constructing asymmetric control limits for skewed distributions exists, none has compared these two kinds of limits on an equal basis. In addition, this thesis proposes a new method for constructing the asymmetric control limits for quality measurements are autocorrelated. We compare the out-of-control ARL (average run length) for symmetric and asymmetric limits while keeping their in-control ARL values the same. In our experiments, we use three and five testing examples for the X and R charts, respectively. Three testing examples are used for X control chart. The first two examples assume that the quality measurements are independent with gamma, and Johnson unbounded distributions, respectively. The thrid assumes that the quality measurements are autocorrelated with a exponential marginal distribution. Five testing examples are used for R control chart. The first three examples assume that the quality measurements are independent with uniform, exponential, and Johnson unbounded distributions, respectively. The forth and fifth assumes that the quality measurements are autocorrelated with a exponential and uniform marginal distribution, respectively. Our empirical study that the X chart performance for the symmetric and asymmetric control limits depends on the shift direction and the skewness of the X distribution (as well as the population skewness). When the quality characteristic has a rightly skewed distribution, the symmetric control limits perform better than the asymmetric control limits when the process mean shifts to the right and worse when the process mean shifts to the left. Analogously, when the skewness is negative, the asymmetric control limits perform better than the symmetric limits when the process mean shifts to the right and worse when the process mean shifts to the left. For the R chart, our empirical study shows similar results as for the X chart. If the range has a rightly skewed distribution, the symmetric limits are more powerful in detecting an increase in the process standard deviation. Analogously, if the range has a left skewed distribution, the asymmetric limits perform better than the symmetric.

參考文獻


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