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.     

On properties of probability-based multivariate process capability indices

Multivariate process capability indices (MPCIs) have been proposed to measure multivariate process capability in real-world application over the past three decades. For the practitioner's point of view, the intention of this paper is to examine the performances and distributional properties of probability-based MPCIs. Considering issues of construction of capability indices in multivariate setup and computation with performance, we found that probability-based MPCIs are a proper generalization of univariate basic process capability indices (PCIs). In the beginning of this decade, computation of probability-based indices was a difficult and time-consuming task, but in the computer age statistics, computation of probability-based MPCIs is simple and quick. Recent work on the performance of MPCI NMC_pm and distributional properties of its estimator reasonably recommended this index, for use in practical situations. To study distributional properties of natural estimators of probability-based MPCIs and recommended index estimator, we conducted simulation study. Though natural estimators of probability-based indices are negatively biased, they are better with respect to mean, relative bias, mean square error. Probability-based MPCI MC_pm is better as compared with NMC_pm with respect to performance and as its estimator quality. Hence, in real-world practice, we recommend probability-based MPCIs as a multivariate analogue of basic PCIs.     

New capability indices for evaluating the performance of multivariate manufacturing processes

Generally, an industrial product has more than one quality characteristic. In order to establish performance measures for evaluating the capability of a multivariate manufacturing process, several multivariate process capability indices have been developed in the past few years. Among them, Taam's MC_p and MC_pm indices have the drawback of overestimation and Hubele's three‐component capability vector lacks simplicity in practice. In this article, taking the correlation among multiple quality characteristics into account, we develop two novel indices; NMC_p and NMC_pm. Using two numerical examples we demonstrate that the true performance of multivariate processes are accurately reflected in our NMC_p and NMC_pm indices and in their associated interval estimates. Finally, simulation results show that our indices outperform both those of Taam and Hubele. Copyright © 2009 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.     

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。     

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.     

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.     

Distributional Properties of Multivariate Process Capability Indices under Normal and Non‐normal Distributions

Multivariate capability analysis has been the focus of study in recent years, during which many authors have proposed different multivariate capability indices. In the operative context, capability indices are used as measures of the ability of the process to operate according to specifications. Because the numerical value of the index is used to conclude about the capability of the process, it is essential to bear in mind that almost always that value is obtained from a sample of process units. Therefore, it is really necessary to know the properties that the indices have when they are calculated on sampling information, in order to assess the goodness of the inferences made from them. In this work, we conduct a simulation study to investigate distributional properties of two existing indices: NMCpm index based on ratio of volumes and Mp₂ index based on principal component analysis. We analyze the relative bias and the mean square error of the estimators of the indices, and we also obtain their empirical distributions that are used to estimate the probability that the indices classify correctly a process as capable or as incapable. The results allow us to recommend the use of one of these indices, as it has shown better properties. Copyright © 2016 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.     

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。     

Integrating Customer Perception into Process Capability Measures

Process capability indices provide a measure of the output of an in‐control process that conforms to a set of specification limits. These measures, which assume that process output is approximately normally distributed, are intended for measuring process capability for manufacturing systems. When the performance of a system results in a product that fails to fall within a given specification range, however, the product is typically scrapped or reworked, and the actual distribution that the customer perceives after inspection is truncated. In this paper, the concept of a truncated measure for three types of quality characteristics is introduced as the key to linking customer perception to process capability. Subsequently, a set of customer‐perceived process capability indices is presented as an extension of traditional manufacturer‐based counterparts. Finally, data transformation‐based process capability indices are also discussed. A comparative study and numerical example reveal considerable differences among the traditional and proposed process capability indices. It is believed that the proposed process capability index for various quality characteristics may more aptly lead to process improvement by facilitating a better understanding of the integrated effects found in engineering design problems. Copyright © 2015 John Wiley & Sons, Ltd.     

A Multivariate Process Capability Index Based on the First Principal Component Only

Often the quality of a process is determined by several correlated univariate variables. In such cases, the considered quality characteristic should be treated as a vector. Several different multivariate process capability indices (MPCIs) have been developed for such a situation, but confidence intervals or tests have been derived for only a handful of these. In practice, the conclusion about process capability needs to be drawn from a random sample, making confidence intervals or tests for the MPCIs important. Principal component analysis (PCA) is a well‐known tool to use in multivariate situations. We present, under the assumption of multivariate normality, a new MPCI by applying PCA to a set of suitably transformed variables. We also propose a decision procedure, based on a test of this new index, to be used to decide whether a process can be claimed capable or not at a stated significance level. This new MPCI and its accompanying decision procedure avoid drawbacks found for previously published MPCIs with confidence intervals. By transforming the original variables, we need to consider the first principal component only. Hence, a multivariate situation can be converted into a familiar univariate process capability index. Furthermore, the proposed new MPCI has the property that if the index exceeds a given threshold value the probability of non‐conformance is bounded by a known value. Properties, like significance level and power, of the proposed decision procedure is evaluated through a simulation study in the two‐dimensional case. A comparative simulation study between our new MPCI and an MPCI previously suggested in the literature is also performed. These studies show that our proposed MPCI with accompanying decision procedure has desirable properties and is worth to study further. Copyright © 2012 John Wiley & Sons, Ltd.     

Bootstrap Confidence Limits for Short-Run Capability Indices

The capability indices are widely used by quality professionals as an estimate of process capability. Recently, techniques and tables were developed to construct confidence limits for each index. These techniques are based on the assumption that the underlying process is normally distributed. In industries it may not be possible to get large samples, and, hence, the normality assumption is often violated. For the short-run production processes where sample size is small, appropriate indices can be developed. In this article, the capability indices such as C_p, C_pk, and C_pm are modified, and appropriate capability indices are constructed. A further bootstrap technique is used to define the confidence intervals. A simulation using two distributions (one normal and the other nonnormal) is conducted, and a comparison is made to show the performances of the three nonparametric confidence intervals.     

Robust process capability indices for multiple linear profiles

In some statistical process control applications, the quality of a process is described by a linear relationship between the response variable(s) and the independent variable(s), which is called a linear profile. Process capability is a significant issue in statistical process control. The ability of a process to meet customer specifications or standards is measured by the process capability indices (PCIs). There are several attempts for studying the process capability in linear profiles. In this research, two robust PCIs for multiple linear profiles are proposed. In the suggested robust PCIs, the process capability is estimated using the M-estimator and the Fast-τ-estimator. Performances of the proposed robust PCIs in comparison with the classical PCIs in the absence and presence of contamination are evaluated. The results show that the robust PCIs proposed in this research perform as well as the classical PCIs in the absence of contamination and much better in the presence of contamination. The proposed PCIs, using Fast-τ-estimator, perform better in small shifts, and the proposed PCIs, using M-estimator, perform better in large shifts. Introduction of robust indices for multivariate multiple linear profiles is an area for further research.     

Select better suppliers based on manufacturing precision for processes with multivariate data

Process capability indices have been widely used in the manufacturing industry for measuring process reproduction capability according to manufacturing specifications. Processes with univariate data have been investigated extensively, but are comparatively neglected for processes with multivariate data. Chou (Chou, Y.M., 1994. Selecting a better supplier by testing process capability indices. Quality Engineering, 6, 427–438) developed a procedure using univariate C_p to determine whether or not two processes are equally capable, which allows one to select the supplier with better quality. However, for processes with multiple characteristics, no methods are available for comparing two processes with multivariate data. In this paper, we consider the supplier selection problem based on manufacturing precision in which the processes involve multiple quality characteristics. We derive the distribution of the corresponding test statistic, and calculate critical values required for the comparison purpose. A real-world application is presented for justification.     

Model-based process capability indices: The dry-etching semiconductor case study

Process capability indices are widely used to check quality standards both at the production level and for business activity. They consider the location and the deviation from specification limits and targets. The literature contains many contributions on this topic both in the univariate and the multivariate context. Motivated by a real semiconductor case study, we discuss the role of rational subgroups and the challenge they present in the computation of capability indices, especially when data refer to lots of products. In addition, our context involves a mix of problems: unilateral specification limit, nonsymmetric distribution of the data, evidence of data from a mixture of distributions, and the need to filter one component of the mixture. After solving the previous issues and because of the peculiar characteristics of semiconductor processes based on the so called “wafers,” we contribute to the literature a proposal on how to compute capability indices in the case of heteroscedastic spatial processes. With a generalized additive model, we show that it is possible to estimate a capability surface that allows the identification of regions expected to not be fully compliant with the desired quality standards.     

MEWMA Control Chart and Process Capability Indices for Simple Linear Profiles with Within‐profile Autocorrelation

A multivariate exponentially weighted moving average (MEWMA) control chart is proposed for detecting process shifts during the phase II monitoring of simple linear profiles (SLPs) in the presence of within‐profile autocorrelation. The proposed control chart is called MEWMA‐SLP. Furthermore, two process capability indices are proposed for evaluating the capability of in‐control SLP processes, and their utilization is demonstrated through examples. Intensive simulations reveal that the MEWMA‐SLP chart is more sensitive than existing control charts in detecting profile shifts. Copyright © 2016 John Wiley & Sons, Ltd.     

Yield‐Related Process Capability Indices for Processes of Multiple Quality Characteristics

Process capability indices (PCIs) have been widely used in industries for assessing the capability of manufacturing processes. Castagliola and Castellanos (Quality Technology and Quantitative Management 2005, 2(2):201–220), viewing that there were no clear links between the definition of the existing multivariate PCIs and theoretical proportion of nonconforming product items, defined a bivariate C_pk and C_p (denoted by BC_pk and BC_p, respectively) based on the proportions of nonconforming product items over four convex polygons for bivariate normal processes with a rectangular specification region. In this paper, we extend their definitions to MC_pk and MC_p for multivariate normal processes with flexible specification regions. To link the index to the yield, we establish a ‘reachable’ lower bound for the process yield as a function of MC_pk. An algorithm suitable for such processes is developed to compute the natural estimate of MC_pk from process data. Furthermore, we construct via the bootstrap approach the lower confidence bound of MC_pk, a measure often used by producers for quality assurance to consumers. As for BC_p, we first modify the original definition with a simple preprocessing step to make BC_p scale‐invariant. A very efficient algorithm is developed for computing a natural estimator of BC_p. This new approach of BC_p can be easily extended to MC_p for multivariate processes. For BC_p, we further derive an approximate normal distribution for , which enables us to construct procedures for making statistical inferences about process capability based on data, including the hypothesis testing, confidence interval, and lower confidence bound. Finally, the proposed procedures are demonstrated with three real data sets. Copyright © 2012 John Wiley & Sons, Ltd.     

Performance of Multivariate Process Capability Indices Under Normal and Non‐Normal Distributions

In the context of process capability analysis, the results of most processes are dominated by two or even more quality characteristics, so that the assessment of process capability requires that all of them are considered simultaneously. In recent years, many researchers have developed different alternatives of multivariate capability indices using different approaches of construction. In this paper, four of them are compared through the study of their ability to correctly distinguish capable processes from incapable processes under a diversity of simulated scenarios, defining suitable minimum desirable values that allow to decide whether the process meets or does not meet specifications. In this sense, properties analyzed can be seen as sensitivity and specificity, assuming that a measure is sensitive if it can detect the lack of capability when it actually exists and specific if it correctly identifies capable processes. Two indices based on ratios of regions and two based on the principal component analysis have been selected for the study. The scenarios take into account several joint distributions for the quality variables, normal and non‐normal, several numbers of variables, and different levels of correlation between them, covering a wide range of possible situations. The results showed that one of the indices has better properties across most scenarios, leading to right conclusions about the state of capability of processes and making it a recommendable option for its use in real‐world practice. Copyright © 2015 John Wiley & Sons, Ltd.