應用對稱分位數於製程能力評估之研究

Abstract

現今較常使用的製程能力指標有四種，分別為C(p)、 C(pk)、 C(pm)與 C(pmk)。對於製程是非常態分配的情形，Clements (1989)在Pearson family的假設下，提出以U(p)-L(p)取代6σ和用中位數M取代μ的方法，並且應用在C(p)與C(pk)這兩個指標上；Pearn and Kotz (1994)把此方法應用在指標C(pm)與C(pmk)上。本論文的主要目的便是應用Chen and Chiang (1996)所提出之對稱分位數（symmetric quantiles）於製程能力指標上。考慮分別使用以平均值為中心之對稱分位數的區間，及以目標值為中心之對稱分位數的區間取代Clements的U(p)-L(p)，重新定義指標 C(p)、 C(pk)、 C(pm)與 C(pmk)。由Chen and Chiang (1996)可得一般分位數與以平均值為中心之對稱分位數的樣本區間長度漸近變異，而本文中推導得以目標值為中心之對稱分位數的樣本區間長度漸近變異，比較三者結果顯示以目標值為中心之對稱分位數的樣本區間長度變異最小。經由模擬分析評估對稱分位數與一般分位數之製程能力指標的表現，結果顯示對稱分位數的表現比一般分位數好，其中在大部分的情況下，以目標值為中心的又較佳；而且以目標值為中心定義的指標C(p)與C(pk)，就可以表現其他方法在C(pm)與C(pmk)才能呈現之製程平均值與目標值偏差的影響。我們也建構製程能力指標之信賴區間。最後，並以Pearn and Kotz (1994)所提出的實際資料計算對稱分位數與一般分位數之製程能力指標，與Clements的方法做一比較，結果顯示對稱分位數的能力指標也可以達到同樣的效果。

C(p), C(pk), C(pm), and C(pmk) are the four commonly used process capability indices (PCIs) used in industrial applications. Clements (1989) proposed a method assuming that the process distribution can be characterized by a Pearsonian distribution. The main idea of Clements’ method is to replace 6σ by U(p)-L(p) and μ by M. Clements (1989) applied this method to C(p) and C(pk). Pearn and Kotz (1994) extended the method to C(pm) and C(pmk). In this paper, we assess process capability by symmetric quantiles proposed by Chen and Chiang (1996), more specifically, we define C(p), C(pk), C(pm), and C(pmk) in terms of symmetric quantiles. We consider both the mean and target as the center of the symmetric quantiles, respectively. From Chen and Chiang (1996), we can get asymptotic variances of the sample interval lengths of the ordinary quantiles and the mean-based symmetric quantiles respectively. In this paper, we also derive asymptotic variance of the sample interval length of the target-based symmetric quantiles. Comparing the above three estimators, we find that asymptotic variance of the sample interval length of the target-based symmetric quantiles is the smallest. We also compare the performance of quantile-based PCIs by simulation. We find that symmetric quantiles performs better than the ordinary quantiles, and for most cases, the target-based symmetric quantiles performs better than the mean-based estimator. We also construct a confidence interval of PCIs by bootstrap methods. Finally, as an illustrative example, we calculate PCIs for a real data set given in Pearn and Kotz (1994).