Abstract
在統計製程管制中，管制圖是一個基本且應用相當廣泛的統計工具；本研究主要為探討西雅耶夫-羅柏管制圖（Shiryayev-Roberts Charts）的經濟設計及經濟統計設計。除此之外，文中將假設製程變異機制服從一種新的分布，以取代傳統討論的指數分布中的不合理性；而這個新的分布為加上預防保養的韋伯分布，我們稱之為韋伯PM分布。經由數值方法運算之後，所求得的最佳解結果顯示，如果預防保養的費用不高的話，採用預防保養策略的平均成本會比不用預防保養策略來得低。從文中提供的數據還可以看出在經濟統計設計之下所得到的平均成本比在經濟設計之下來得高一些，不過這樣可以得到比較合理及可接受的ARL或ATS值。文中針對幾個製程參數（包括韋伯分布的兩個參數及預防保養的費用參數）做敏感度分析，發現平均成本會隨這些參數增加而增加。而針對假設的變異時的平均數（Delta）和真正的變異時的平均數（Mu）所做的敏感度分析中可以看出，當所選擇的Delta值剛好與Mu值一樣時，可以得到較低的平均成本。在最後我們除了將兩個製程變異機制的分布（韋伯分布及韋伯PM分布）做了比較之外，還比較了Shiryayev-Roberts Chart和幾個常見的管制圖（如EWMA Chart或Cusum Chart）的最佳解的平均成本。發現Shiryayev-Roberts Chart的平均成本比起其他幾個管制圖還來得低。Control chart is a primary tool for statistical process control.In this article, we present some numerical results for theoptimal economic and economic statistical designs of Shiryayev-Roberts control chart. Constraints on average run length (ARL)or average time to signal (ATS) are added to the economic modelsto meet industry's desire for low-process variability and long-term product quality. Furthermore, instead of the traditionalexponential distribution, the process-failure mechanism weconsidered in this paper is assumed to obey Weibull distributionwith the preventive maintenance (PM) policy. We call theproposed model the Weibull-PM shock model. The computationalresults indicate that the Weibull-PM model provides a lower costthan Weibull model if the cost of the preventive maintenancepolicy is not too high or too close to the cost of repair. Fromthe results, we also see that, although the economic statisticaldesign has a higher cost than the pure economic design, itprovides a reasonable/acceptable ARL or ATS value. Sensitivityanalyses of the process parameters such as scale and shapeparameters of the Weibull distribution show that expected costincreases as any of the parameters values increases. Thesensitivity analysis of the assumed mean shift,Delta, versus theactual mean shift, Mu, shows that we will get a lower cost ifthe assumed mean shift is selected exactly equal to the actualmean shift. Finally, we compare the Shiryayev-Roberts chart withtwo common used charts: Cusum and EWMA charts, and the proposedShiryayev-Roberts chart yields a lowest cost. Computer codes aregiven to help evaluate the ARL's for the proposed designs.