非對稱管制界限設定方式之討論與比較-以 管制圖為例-

管制界限的制定方式有許多種，通常可概略分為兩類：對稱管制界限與非對稱管制界限。其中，對稱管制界限又可稱作是非對稱管制界限中的一種特例。在現今的製程中，Shewhart 管制圖中的管制界限的設定是最常被拿來應用在統計製程分析上。其中Shewhart 管制圖假設母體觀察值皆服從常態母體，以 3倍標準差建立對稱的管制界限。如果此假設未成立，設立非對稱管制界限來監控觀察值，應該能夠更快發現製程的偏移。通常在製程偏移時，向上偏移的機率和向下偏移的機率不一定相同。且在偏移發生後，也可能會有不同的變異情形。本研究主要探討非對稱管制界限設立方式應用在管制圖上的比較。本實驗的衡量方法有兩種，其一為預先給定製程在管制狀態下的平均串聯長度(ARL0)的特定值，在比較當製程失控時的平均串聯長度(ARL1)值，ARL1愈小愈好。另一種為預先給定製程失控時的平均串聯長度(ARL1)值，比較製程在管制狀態下的平均串聯長度(ARL0)的特定值，ARL0愈大愈好。對於管制圖的幾種管制界限設定方式，本研究分別選定了五種設定方式—第一種為Shewhart 管制界限：和。第二種類似前一種，但將中心線變更為中位數：和。上述兩種為對稱管制界限，其中。第三種則是落在管制上界及管制下界外的機率相等：。第四種為和，其中和須滿足下列目標式及限制式：，而為預先給定製程在管制狀態下的平均串聯長度(ARL0)的特定值。第五種和第四種類似：和，其中和滿足的目標式及限制式改為：。而為預先給定製程在失控狀態下的平均串聯長度(ARL1)的特定值。而用來作為五種管制界限比較的品質特性分別有兩種不同的分配：第一種為假設品質特性獨立服從gamma分配，第二種為假設品質特性服從Johnson家族分配。而Johnson家族分配分別有三種—Johnson bounded、Johnson unbounded和Johnson Lognormal。實驗結果發現，績效指標為ARL1時，第四種管制界限設定方式為最好，而績效指標為ARL0時，第五種為最好。在實驗結果中可以發現，如果品質特性為有界的分配(Johnson bounded)，使用對稱的管制界限其績效指標為ARL1時，在製程失控後，反而使品質特性落在管制界限內的面積增加，會出現ARL1較大的情形。所以當資訊不充足時(製程失控後，偏移的量未知)，選擇第三種，也就是落在管制上下界外的面積相等，會有較好的績效。

關鍵字

非常態分配；平均串聯長度；管制圖；非對稱管制界限

並列摘要

There are many methods to construct the control limits. In general, there are two kinds of control limits : one is symmetric control limits, the other is asymmetric control limits. Also symmetric control limits is a special case in asymmetric control limits. The Shewart 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 control limits, e.g., 3-sigma control limits, called Shewhart control limits. It is only under the assumption which is quality measurement has normally distributed. When the process is not normally distributed, use asymmetric control limits may get better performance. When the process is out-of-control for charts, the probability of process shift to upward and downward maybe not equal. If shift occurs, the variance of upward and downward maybe change. This work compares the charts performance for the different setting of control limits, include symmetric and asymmetric control limits. The performance measurement we used here are two kinds : one is out-of-control average run length (ARL1) while keeping their in-control run length (ARL0) values the same. The other one is ARL0 while keeping their ARL1 values the same. We choose five different control limits to compare their performance for control charts. The sample mean of quality characteristic X has gamma distribution and Johnson family distribution. The Johnson family distribution includes three types: Johnson bounded、Johnson unbounded and Johnson Lognormal distribution. In our experiment results, the performance of asymmetric control limits is better than symmetric control limits. When the process has bounded distribution (Johnson bounded distribution), the ARL1 of symmetric control limits will become extreme large than asymmetric control limits in some situations. If we don’t have enough information, like the unit of mean shift, the probability symmetric (when the probability below lower control limit and above upper control limits are equal) will more preferable.

並列關鍵字

Asymmetric Control Limits ； Average Run Length ； Non-normally Distribution ； control charts

參考文獻

[1] Bai D.S. and I.S. Choi. (1995). ¯X and R control charts for skewed population.

Journal of Quality Technology 21, 954-957.

[2] Chan L.K. and H.J.Cui. (2003). Skewness Correction ¯X and R for skewed distribution.

Naval Research Logistics 50, 555-573.

School of Industrial Engineering, Purdue University, West Lafayette, Indiana,

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非對稱管制界限設定方式之討論與比較-以管制圖為例-

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