Computing process capability indices for non‐normal data: a review and comparative study

When the distribution of a process characteristic is non‐normal, C_p and C_pk calculated using conventional methods often lead to erroneous interpretation of the process's capability. Though various methods have been proposed for computing surrogate process capability indices (PCIs) under non‐normality, there is a lack of literature that covers a comprehensive evaluation and comparison of these methods. In particular, under mild and severe departures from normality, do these surrogate PCIs adequately capture process capability, and which is the best method(s) in reflecting the true capability under each of these circumstances? In this paper we review seven methods that are chosen for performance comparison in their ability to handle non‐normality in PCIs. For illustration purposes the comparison is done through simulating Weibull and lognormal data, and the results are presented using box plots. Simulation results show that the performance of a method is dependent on its capability to capture the tail behaviour of the underlying distributions. Finally we give a practitioner's guide that suggests applicable methods for each defined range of skewness and kurtosis under mild and severe departures from normality. Copyright © 1999 John Wiley & Sons, Ltd.     

基于工序成本的多元过程能力指数分析

单一的多元过程能力指数在反映企业制造多元质量特性产品能力时存在缺陷,无法反映由于各工序成本的差异而造成的工序能力重要性的差别.基于此,从企业工序质量控制和工序成本关系的角度来分析,构建了基于工序成本修正的多元过程能力指数,反映企业的成本控制能力和产品竞争能力.最后,通过修正得到的多元过程能力指数,一方面优选企业的质量改进或控制方案,另一方面判定企业在同类产品制造中的实际质量控制能力和过程成本损失差异.     

Tables of a Truncated Standard Normal Distribution: A Singly Truncated Case

A normal distribution has a unique position in many engineering fields, and the standard normal distribution table has been widely used for more than a century. There are many situations, however, in which a truncated normal distribution needs to be considered. Although the theoretical foundations of the truncated normal distribution are well established, there has been little work on tabulating the characteristics associated with the truncated normal distribution, such as a cumulative probability, a truncated mean, and a truncated variance. In this article, we provide tables for a singly truncated normal distribution, which may be useful for quality practitioners.     

工序能力分析与评价中的几个问题

在工序质量分析与控制中,计算与评价工序能力指数是一项非常重要的工作,也是计算机辅助质量系统的一个重要模块。文章针对目前在工序能力计算与分析中出现的问题,提出了如何合理地进行抽样、样本数据的正态性检验以及对非正态性数据的处理、Ｃｐ的置信区间以及与样本含量的关系,旨在为实际生产过程中质量工程师进行工序能力分析和评价提供指导。     

Evaluation of Nonnormal Process Capability Indices using Generalized Lambda Distribution

Process capability indices (PCIs) are used to describe a manufacturing process expressing its ability to produce items within the specified limits. These indices are developed under the assumption that the underlying process distribution is normal. In industries, there are many manufacturing processes where process distribution can not be described by a normal distribution. In such cases, those PCIs will give misleading results about the process. The most commonly used approach for analysing a nonnormal process data is to fit a standard nonnormal distribution (e.g., weibull, gamma) or a family of distribution curves (e.g., Pearson, Johnson) to the process data and then to estimate the percentile points from the fitted distribution that can be used to compute generalized PCIs. In this article, we outline the procedure using the generalized lambda distribution (GLD) curve for modeling a set of process data and for estimating percentile points in order to compute generalized PCIs. The four-parameter GLD can assume a wide variety of curve shapes and hence it is very useful for the representation of data when the underlying model is unknown. Compared to the Pearson and Johnson family of distributions, the GLD is computationally simpler and more flexible. The article provides all necessary formulas for fitting a GLD curve, estimating its parameters, performing goodness-of-fit tests, and computing generalized PCIs. An example is used to illustrate the calculations that can be easily performed using spreadsheets.     

Algebraic Relationship Between Symmetrical and Asymmetrical Capability Indices

There have been many investigations on the capability indices C_p, C_pk, C_pm, and C_pmk, for the common situation in which the target is the midpoint of the tolerance interval. However, only a few investigations deal with the specific case of asymmetrical tolerances. In that particular case, a number of symmetrical and asymmetrical indices are put forward, but there is no full literature treatment or synthesis showing the similarity between those indices and the common ones. We intend here, to demonstrate that the algebraic links between the indices C_p, C_pk, C_pm, and C_pmk, are similar to the ones which relate the symmetrical indices proposed in the case of asymmetrical tolerances. In that case, the algebraic structure allows us to propose asymmetrical indices families. An example based on a pharmaceutical filling operation is used to illustrate the application.