多變量製程能力指標之研究

Abstract

製程能力指標近年來在業界廣泛的被使用於，衡量製程製造產品的能力。衡量一維製程的指標已經被研究得相當完備，然而有關衡量二維製程能力的指標的研究雖然相當多，但指標和產品良率間的關係卻很少被強調，有鑑於此Castagliola和Castellanos[4] 提出了兩個指標BCpk以及BCp。這兩個指標是以計算超出凸多邊形規格區內不良品所占比例為基礎的指標。我們在本文章中將此指標推廣至多個產品特性的情形，並且提出估計這個指標在二維甚至多維製程的演算法，不像Castagliola和Castellanos[4]所使用的方法僅能運用在二維製程上。另外，我們利用四種拔靴法分別估計出BCpk的信賴下限。至於BCp，由於原來的定義在不同的製程特性上給予不同的比例縮放時無法保持不變性，所以我們提出了一個預先修正的方法，解決原本指標沒有不變性的問題，另外，我們也推導出BCp的自然估計量的近似分佈，進而推導出信賴區間、信賴下限、以及假設檢定。最後我們用一個實際的例子來說明我們在文中所提出的估計方法。

Process capability indices (PCIs) have been widely used in the industries for assessing the capability of manufacturing processes. The research of PCIs for univariate processes has been well developed. However, in the bivariate case, the PCI research may be plenty, but links between the index and the product yield are seldom emphasized. For this, by assuming a bivariate normal distribution and a rectangular specification region, Castagliola and Castellanos [4] proposed two indices BCpk and BCp. These two indices are defined based on the proportions of non-conforming products over convex polygons. We extend these indices to multivariate processes of more than two quality characteristics. We develop an algorithm for computing estimates of these indices, which is suitable for general multivariate processes, not like the algorithm in Castagliola and Castellanos [4] can only be used for bivariate processes. In addition, we estimate the lower confidence bound by bootstrap methods. As for BCp, we find the original definition is not scale invariant, meaning that the BCp value will vary with different scales on quality characteristics. We propose a pre-processing step to solve this problem. Moreover, we find an approximate distribution of the natural estimator of BCp, which enables us to develop statistical procedures for making inferences on process capability based on data, including hypothesis testing, confidence interval, and lower confidence bound. The latter is directly linked to the quality assurance. Finally, a real data set is used as an application example.