Comparing Confidence Intervals for Multivariate Process Capability Indices

Multivariate process capability indices (MPCIs) are needed for process capability analysis when the quality of a process is determined by several univariate quality characteristics that are correlated. There are several different MPCIs described in the literature, but confidence intervals have been derived for only a handful of these. In practice, the conclusion about process capability must be drawn from a random sample. Hence, confidence intervals or tests for MPCIs are important. With a case study as a start and under the assumption of multivariate normality, we review and compare four different available methods for calculating confidence intervals of MPCIs that generalize the univariate index C_p. Two of the methods are based on the ratio of a tolerance region to a process region, and two are based on the principal component analysis. For two of the methods, we derive approximate confidence intervals, which are easy to calculate and can be used for moderate sample sizes. We discuss issues that need to be solved before the studied methods can be applied more generally in practice. For instance, three of the methods have approximate confidence levels only, but no investigation has been carried out on how good these approximations are. Furthermore, we highlight the problem with the correspondence between the index value and the probability of nonconformance. We also elucidate a major drawback with the existing MPCIs on the basis of the principal component analysis. Our investigation shows the need for more research to obtain an MPCI with confidence interval such that conclusions about the process capability can be drawn at a known confidence level and that a stated value of the MPCI limits the probability of nonconformance in a known way. Copyright © 2011 John Wiley & Sons, Ltd.     

Yield‐Based Capability Index for Evaluating the Performance of Multivariate Manufacturing Process

Process capability indices (PCIs) have been widely used in the manufacturing industry providing numerical measures on process precision, accuracy and performance. Capability indices measures for processes with a single characteristic have been investigated extensively. However, an industrial product may have more than one quality characteristic. In order to establish performance measures for evaluating the capability of a multivariate manufacturing process, multivariate PCIs should be introduced. In this paper, we analyze the relationship between PCI and process yield. The PCI EC_pk is proposed based on the idea of six sigma strategy, and there is a one‐to‐one relationship between EC_pk index and process yield. Following the same, idea we propose a PCI MEC_pk to measure processes with multiple characteristics. MEC_pk index can evaluate the overall process yield of both one‐sided and two‐sided processes. We also analyze the effect of covariance matrix on overall process yield and suggest a solution for improving overall process yield. Copyright © 2013 John Wiley & Sons, Ltd.     

On the Implementation of the Principal Component Analysis–Based Approach in Measuring Process Capability

The use of principal component analysis in measuring the capability of a multivariate process is an issue initially considered by Wang and Chen (1998). In this article, we extend their initial idea by proposing new indices that can be used in situations where the specification limits of the multivariate process are unilateral. Moreover, some new indices for multivariate processes are suggested. These indices have been developed so as to take into account the proportion of variance explained by each principal component, thus making the measurement of process capability more effective. Copyright © 2011 John Wiley & Sons, Ltd.     

Multivariate Process Capability Index Based on Data Depth Concept

The purpose of this paper is to provide a multivariate process capability index, which could be used regardless on data distribution and also on data correlation. Such an index could be defined because of application of non‐parametric methodology that utilizes a data depth concept. Based on this concept, a two‐phase methodology was developed. In the first phase the modified tolerance region is estimated, while in the second one, a current process is assessed using the proposed three‐component index. Estimation of a modified tolerance region on the basis on historical data allows applying the methodology not only for bilateral quality characteristics but also for unilateral ones, where often in practice, the modified tolerance region could be defined as a closed region. The performance of the proposed index was evaluated using bilateral and unilateral examples. The obtained results showed that the proposed index performs satisfactorily for all the considered cases. Copyright © 2016 John Wiley & Sons, Ltd.     

多元质量特性的过程能力指数

多元质量特性过程能力指数是一个尚未得到很好解决的问题。本文利用主成分分析，给出了计算多元质量特性过程能力指数的一种新方法，实证分析表明这种方法是可行的。     

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.     

An Approach to Build Process‐Oriented Basis for Causation‐Based Multivariate SPC and Process Capability Analysis

Although various multivariate process monitoring techniques have been developed, they do not diagnose the process for finding the root causes of irregularities during production. There have been recent studies on a new method that involves process‐oriented basis representation, which links the process variation to its causes, and thus helps in monitoring and diagnosing a process. However, all the studies done so far focused on its application. In this paper, a method is proposed to build the process‐oriented basis for a process irrespective of the number of variables characterizing it. Along with various other statistical techniques, factor analysis and cluster analysis, with customized distance function, are used in developing the method. The built in process‐oriented basis is further used for multivariate statistical process control and process capability analysis. Multivariate solder‐paste problem from electronics industry is used for illustration. Copyright © 2012 John Wiley & Sons, Ltd.     

Multivariate Statistical Process Control Based on Principal Component Analysis (MSPC-PCA): Some Reflections and a Case Study in an Autobody Assembly Process

In modern manufacturing processes, massive amounts of multivariate data are routinely collected through automated in-process sensing. These data often exhibit high correlation, rank deficiency, low signal-to-noise ratio and missing values. Conventional univariate and multivariate statistical process control techniques are not suitable to be used in these environments. This article discusses these issues and advocates the use of multivariate statistical process control based on principal component analysis (MSPC-PCA) as an efficient statistical tool for process understanding, monitoring and diagnosing assignable causes for special events in these contexts. Data from an autobody assembly process are used to illustrate the practical benefits of using MSPC-PCA rather than conventional SPC in manufacturing processes.     

A New Multivariate Process Capability Index Under Both Unilateral and Bilateral Quality Characteristics

Process capability indices (PCIs) are used in statistical process control to evaluate the capability of the processes in satisfying the customer's needs. In the past two decades varieties of PCI are introduced by researchers to analyze the process capability with univariate or multivariate quality characteristics. To the best of our knowledge, most famous multivariate capability indices are proposed when the quality characteristics have both upper and lower specification limits. These indices are incapable to assess the multivariate processes capability with unilateral specification. In this article, we propose a new multivariate PCI to analyze the processes with one or more unilateral specification limits. This new index also accounts for all problems in the best PCIs of the literature. The performance of the proposed index is evaluated by real cases under different situations. The results show that the proposed index performs satisfactorily in all cases considered. Copyright © 2012 John Wiley & Sons, Ltd.     

基于工序能力指标的工序公差优化设计研究

在分析了现有的工序公差优化设计模型基础上,提出了一种基于工序能力指标最大化的工序公差优化设计模型,并采用功效系数法将多目标规划转换成单目标规划,对于实现工序公差的稳健设计、提高CAPP 的质量,有重要意义。     

Multivariate process incapability vector

Process capability indices evaluate the capability of the processes in satisfying customer's requirements. This paper introduces a superstructure multivariate process incapability vector for multivariate normal processes and then, compares it with four recently proposed multivariate process capability indices to show its better performance. In addition, the effects of two modification factors are investigated. Also, bootstrap confidence intervals for the first component of the proposed vector are obtained. Furthermore, real manufacturing data sets are presented to demonstrate the applicability of the proposed vector.     

Multivariate Process Capability Vector Based on One‐Sided Model

To evaluate the capability of manufacturing processes in satisfying the customer's needs, a variety of indices has been developed. Some of them are introduced by researchers to analyse the processes with multivariate quality characteristics. Most of the proposed in the literature multivariate capability indices are defined under assumption of normality distribution of the quality characteristics. Thus, the process region describing the variation of the data has an elliptical shape. In this paper, a multivariate process capability vector with three components is introduced, which allows to access the capability of a process with both normally and non‐normally quality characteristics due to application of a pair of one‐sided models as the process region shape. At the beginning, one‐sided models are defined, next the proposed vector components are proposed and the methodology of their evaluation is presented. The methodology (which in fact could be also applied to both the correlated and non‐correlated characteristics) is verified by applying simulation and real problems. The obtained results show that the proposed methodology performs satisfactorily in all considered cases. Copyright © 2014 John Wiley & Sons, Ltd.     

基于过程能力指数的抽样风险研究

如果将生产过程中上一道工序的加工过程（设备）视为生产者,而紧邻的下一道工序的加工过程（设备）视为使用者,则上道工序加工的零部件在通过质量抽样检验进入下道工序的过程中．就将产生抽样风险。而由抽样风险确定的抽样方案直接关系到加工过程质量控制和检验成本控制等加工效率问题。考虑到许多大批量自动生产线的检验设备已经具备在线计算过程能力指数的能力．提出了一个基于过程能力指数的抽样风险分析方法,通过计算过程能力指数,得到相应的抽样风险．进而确定合理的抽样检验方案,从而将加工过程的重要指标——过程能力指数与抽样检验方案之间建立了联系。最后,以某汽车发动机曲轴生产线加工过程为例进行了抽样风险分析。     

基于熵权和模糊物元的改进多元过程能力分析

为解决多品种小批量产品数据采集困难及各指标间存在交互影响致使传统方法难以奏效的现状,基于主成分分析法的多元过程能力指数会造成信息量缺损。该方法将工序能力影响因素采用模糊物元方法构造工序相似条件,选出相似工序将小批量问题变成大批量处理,应用公差系数法将原始数据转化为相对数据。提出基于熵权理论和模糊物元的改良多元过程能力指数。该论文结合某产品进行实例分析,验证了该方法的实用性和有效性。     

基于粗糙集与四分位法的多元质量特性过程能力指数

多元质量特性过程能力指数的求解问题一直未得到很好的解决。本文简述了当前的研究现状,并利用粗糙集与四分位法,给出了计算多元质量特性过程能力指数的一种新方法,实例证明该方法是有效可行的。     

A Method of Estimating the Process Capability Index from the First Four Moments of Non‐normal Data

A method is presented to estimate the process capability index (PCI) for a set of non‐normal data from its first four moments. It is assumed that these four moments, i.e. mean, standard deviation, skewness, and kurtosis, are suitable to approximately characterize the data distribution properties. The probability density function of non‐normal data is expressed in Chebyshev–Hermite polynomials up to tenth order from the first four moments. An effective range, defined as the value for which a pre‐determined percentage of data falls within the range, is solved numerically from the derived cumulative distribution function. The PCI with a specified limit is hence obtained from the effective range. Compared with some other existing methods, the present method gives a more accurate PCI estimation and shows less sensitivity to sample size. A simple algebraic equation for the effective range, derived from the least‐square fitting to the numerically solved results, is also proposed for PCI estimation. Copyright © 2004 John Wiley & Sons, Ltd.     

Developing New Multivariate Process Capability Indices for Autocorrelated Data

Traditionally, the process capability index is developed assuming that the process output data are independent and follow normal distribution. However, in most environmental cases, the process data have more than one quality characteristic and exhibit property of autocorrelation. We propose two novel multivariate process capability indices for autocorrelated data, NMAC_p and NMAC_pm, for the nominal‐the‐best case. For the smaller‐the‐better case, Γ(0) is used to modify the ND index and a new multivariate autocorrelated process capability index NMAC_pu is derived. Furthermore, a simulation study is conducted to compare the performance of the various multivariate autocorrelated indices. The simulation results show that the actual nonconforming rates can be correctly reflected by our proposed indices, which outperform the previous C_p^m, MC_p, MC_pm, NMC_p, NMC_pm, and ND indices under different time series models. Thus, our proposed capability indices can be used in evaluating the performance of multivariate autocorrelated processes. Finally, a realistic example in hydrological application further demonstrates the usefulness of our proposed capability indices. Copyright © 2013 John Wiley & Sons, Ltd.     

基于主成分层次分析的产品设计语义差分评价研究

目的 以外观设计评价为研究对象,提出一种语义差分评价方法。方法 运用主成分分析法,对初始多维语义进行主成分提取,得出综合语义指标和分值,再通过层次分析法构建综合语义指标的主观评价矩阵,最终得出综合语义分值。结论 针对46款热销女士洗发水瓶的外观设计样本,实现了语义差分评价。运用主成分—层次分析—语义差分评价的方法,实现了近似语义的归类。依据综合语义权值,得出了语义差分分值,并适用于较大样本的外观设计评价。     

A Note on the Multivariate Process Capability Index MCpm

Important features of multivariate process capability indices are comparability, interpretability and ease of implementation. When poor process capability is indicated by an index, the user should determine why the process is incapable (e.g. excessive variability or off‐target process mean). One of the most used multivariate process capability indices is MC_pm because it provides assessments of process precision and accuracy. In this work, we study and discuss a peculiarity of MC_pm: processes that are equivalent in terms of precision, accuracy and MC_pm index, after the occurrence of the same increase in the process variability, can have different values of the index. Because MC_pm is often used for comparing processes, this behaviour may cause comparability difficulties. Therefore, we suggest how to take into account this specific behaviour for avoiding erroneous conclusions. Copyright © 2013 John Wiley & Sons, Ltd.     

Multivariate process capability using principal component analysis

Wang and Chen (Qual. Eng. 1998; 11:21–27) have defined process capability indices (PCIs) for multivariate normal processes data using principal component analysis (PCA). Veevers (Statistical Process Monitoring and Optimization. Marcel Dekker: New York, NY, 1999; 241–256) has suggested a multivariate capability index based on the first principal component (PC). In this paper we demonstrate the problem in the definition of PCIs given by Wang and Chen (Qual. Eng. 1998; 11:21–27) and the non‐suitability of PCI given by Veevers (Statistical Process Monitoring and Optimization. Marcel Dekker: New York, NY, 1999; 241–256) through some examples. We also suggest an alternative method for assessing multivariate process capability based on the empirical probability distribution of PCs. This method has been performed on industrial and simulated data. Copyright © 2008 John Wiley & Sons, Ltd.