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.     

Optimal process capability analysis for process design

Conventional process capability analysis is used to measure and control the quality level of a production process in real exercises for on-line quality management. There has been a deficiency in this type of management; namely, the defects which occur in the production process are only passively detected and modified afterwards. Additionally, conventional process capability expression has difficulty distinguishing between alternatives for process selection among possible candidates before process realisation. There is, therefore, considerable motivation for developing a process capability expression which can be used to evaluate alternatives at the beginning of the process design, i.e., off-line application. The conventional C_pm expression is built up by measuring mean deviation and process variances for on-line application. However, if C_pm is used for the process capability analysis for process design, an erroneous C_pm value is found and an inappropriate process design is ended. Thus, the proposed process capability expression revised from the conventional C_pm in consideration of the balance between tolerance cost and quality loss has been developed. This development is the main contribution of this research and, with this development, the appropriate mean and tolerance values can be determined simultaneously prior to the real production process so as to maximise the proposed process capability value. The production is then processed with the pre-determined mean and tolerance values in a real production process. The expectation after process realisation is that the produced responses will be the best of all the alternatives in terms of quality and cost, and that the process capability value obtained after the real production process will be close to the proposed process capability value maximised prior to the real production process.     

An improved measure of quality loss for notching processes

Nowadays, electronic products are progressively becoming thinner, lighter, and more convenient for people to use. Printed circuit boards, and especially integrated circuit (IC) substrates, are among the essential component of these products. The IC substrate not only protects circuits, fixes lines, and conducts heat, but is also the critical component that provides signal connectivity between the chip, the printed circuit boards, and other crucial parts during the packaging process. The process capability index C_pm is commonly used to assess the product quality loss for decision making in modern semiconductor packaging manufacturing. For high-definition products, packaging processes often have very strict quality requirements and thus the quality inspection procedure is time-consuming and complicated. Therefore, because of the limitation of manpower and capacity of the inspection instruments, the collected sample for quality assessment may be with small to moderate sample sizes. In this paper, we introduce an unbiased estimator for C_pm and provide a step-by-step parametric bootstrap procedure for obtaining a composite lower confidence bound on C_pm . To compare with the approaches discussed in the literature, numerical simulations are conducted under various process parameter settings. The results show that for small to moderate sample sizes, the proposed method applying the unbiased estimator has more accurate coverage rates than the existing methods. At the end of this paper, an application of quality loss assessment in notching processes is demonstrated.     

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.     

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.     

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.     

Process capability analysis chart with the application of Cpm

Process capability analysis (PCA) is a highly effective means of assessing the process ability of product that meets specifications. The process capability analysis chart (PCAC/C_pk ) evaluates the capabilities of multiprocess products together with nominal-the-best specifications, larger-the-better and smaller-the-better specifications. This study proposes process capability analysis chart (PCAC/C_pm ) to consider process yield and expected process loss. A new generated estimator for C_pm is proposed and the properties of statistical estimator and hypothesis test are discussed. A practical example was given for application.     

Capability Testing Based on Cpm with Multiple Samples

Numerous process capability indices have been proposed in the manufacturing industry to provide unitless measures on process performance, which are effective tools for quality improvement and assurance. Most existing methods for capability testing are based on the distribution frequency approaches. Recently, Bayesian approaches have been proposed for testing capability indices C_p and C_pm but restricted to cases with one single sample. In this paper, we consider estimating and testing capability index C_pm based on multiple samples. We propose accordingly a Bayesian procedure for testing C_pm. Based on the Bayesian procedure, we develop a simple but practical procedure for practitioners to use in determining whether their manufacturing processes are capable of reproducing products satisfying the preset capability requirement. A process is capable if all the points in the credible interval are greater than the pre‐specified capability level. To make the proposed Bayesian approach practical for in‐plant applications, we tabulate the minimum values of for which the posterior probability p reaches various desirable confidence levels. Copyright © 2004 John Wiley & Sons, Ltd.     

Lower confidence bounds with sample size information for Cpm applied to production yield assurance

The process capability index C _pm , sometimes called the Taguchi index, has been proposed to the manufacturing industry as providing numerical measures on process performance. A lower confidence bound estimates the minimum process capability, conveying critical information regarding product quality, which is essential to quality assurance. The sample size determination is directly related to the cost of the data collection plan. The purpose of this paper is to provide explicit formulas with efficient algorithms to obtain the lower confidence bounds and sample sizes required for specified precision of the estimation on C _pm using the maximum likelihood estimator (MLE) of C _pm . We also provide tables for the engineers/practitioners to use for their in-plant applications. A real-world example taken from a microelectronics manufacturing process is investigated to illustrate the applicability of the proposed approach. The implementation of existing statistical theory for capability assessment bridges the gap between the theoretical development and factory applications.     

Comparing Two Process Capability Indices under Balanced One‐Way Random Effect Model

Process capability indices such as C_p, C_pk, C_pmk and C_pm are widely used in manufacturing industries to provide a quantitative measurement of the performance of the products. In this article, we derived generalized confidence intervals for the difference between process capability indices for two processes under one‐way random effect model. Our study provides coverage probability close to the nominal value in almost all cases as shown via simulation. An example from industrial contexts is given to illustrate the results. Copyright © 2012 John Wiley & Sons, Ltd.     

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.     

Combining process capability indices from a sequence of independent samples

Process capability indices such as C_p, C_pk and C_pm have been widely used in manufacturing for process assessment and for evaluation of purchasing decisions. Quality engineers and process operators are often asked to summarize the quality of a process by combining a sequence of C_pk or C_pm. This paper compares three different methods for combining the estimates of the process capability indices C_pk, C_pm and C_pmk from a sequence of independent samples. The criterion of minimum mean-squared-error (MSE) is applied. We find that, in general, methods for the combination of sample process capability indices based on the overall sample mean and pooled sample variance give smaller MSE than those based on weighted averages of the estimates of process capability indices.     

Procedure for supplier selection based on Cpm applied to super twisted nematic liquid crystal display processes

The loss-based process capability index C _pm , sometimes called the Taguchi index, has been proposed to the manufacturing industry as a method to measure process performance. The index C _pm takes into account the targeting degree of the process, which essentially measures process performance based on average process loss. Based on the C _pm index, a mathematically complicated approximation method was developed previously for selecting a subset of processes containing the best supplier from a given set of processes. The present paper implements this method and develops a practical step-by-step procedure to aid supplier selection decisions. The accuracy of the selection method is investigated using a simulation technique. The accuracy study provides useful information about the sample size required for a designated selection power. A two-phase selection procedure is developed to select a better supplier and to examine the magnitude of the difference between the two suppliers. Also investigated is a real-world case on the super twisted nematic liquid crystal display manufacturing process, and it is applied to the selection procedure using actual data collected from the factories to reach a decision in supplier selection.     

A Bayesian procedure for process capability assessment

The usual practice of judging process capability by evaluating point estimates of some process capability indices has a flaw that there is no assessment on the error distributions of these estimates. However, the distributions of these estimates are usually so complicated that it is very difficult to obtain good interval estimates. In this paper we adopt a Bayesian approach to obtain an interval estimation, particularly for the index C_pm. The posterior probability p that the process under investigation is capable is derived; then the credible interval, a Bayesian analogue of the classical confidence interval, can be obtained. We claim that the process is capable if all the points in the credible interval are greater than the pre‐specified capability level ω, say 1.33. To make this Bayesian procedure very easy for practitioners to implement on manufacturing floors, we tabulate the minimum values of Ĉ_pm/ω, for which the posterior probability p reaches the desirable level, say 95%. For the special cases where the process mean equals the target value for C_pm and equals the midpoint of the two specification limits for C_pk, the procedure is even simpler; only chi‐square tables are needed. Copyright © 1999 John Wiley & Sons, Ltd.     

Efficient Technique for Assessing Actual Non‐normal Quality Loss: Markov Chain Monte Carlo

Markov chain Monte Carlo (MCMC) techniques have been extensively developed and are accepted for solving various real‐world problems. However, process capabilities are rarely analyzed with the means of MCMC. This study integrates the MCMC technique into Bayesian models for assessing the well‐known quality loss index C_pm for gamma and Weibull process distributions. After the MCMC iterations are completed, the quality manager can make reliable decisions via the proposed credible intervals. Furthermore, this study provides performance comparisons of the estimators of C_pm obtained by the MCMC and bootstrap techniques. Simulations show that the MCMC technique performs better than the bootstrap technique in most of the cases that were considered. Copyright © 2016 John Wiley & Sons, Ltd.     

Quality Evaluation of a Manufactured Product with Multiple Characteristics

A majority of manufactured products have many quality characteristics that are important to the customer. To assess or evaluate the capability of a manufactured product with multiple characteristics, the quality characteristics need to be determined in advance and some characteristics can be correlated with each other. Also, the specification limits of the quality characteristics can be one‐sided or two‐sided. In order to deal with such multivariate data, there is a need to develop a new approach. In this study, the geometric distance variable is used to combine the correlated or uncorrelated quality characteristics. Then, the Luceño capability index is used to summarize the performance for each geometric distance variable. Finally, a composite capability index, MC_pc, composed of several univariate capability indices, is proposed to analyze the capability of a manufactured product with multiple characteristics. In addition, the probability of the product non‐conforming is also proposed. The application of the proposed methodology to a real‐life case study is presented. Copyright © 2005 John Wiley & Sons, Ltd.     

An application of non-normal process capability indices

Numerous process capability indices, including C_p, C_pk, C_pm, and C_pmk, have been proposed to provide measures of process potential and performance. In this paper, we consider some generalizations of these four basic indices to cover non-normal distributions. The proposed generalizations are compared with the four basic indices. The results show that the proposed generalizations are more accurate than those basic indices and other generalizations in measuring process capability. We also consider an estimation method based on sample percentiles to calculate the proposed generalizations, and give an example to illustrate how we apply the proposed generalizations to actual data collected from the factory. © 1997 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.     

Confidence intervals and sample size determination for Cpm

The capability index, C_pm, sometimes called the Taguchi index, has the desirable characteristic of being sensitive to both dispersion and deviation of the process average from the engineering target. As a result, the proposed estimators of C_pm have a sampling distribution that is dependent on the non‐central chi‐square distribution. Hence, constructing confidence intervals, performing hypothesis testing or estimating sample size requirements necessitates manipulation of a rather complex functional expression, typically beyond the capabilities of practitioners who need readily available tools. Here, a simple graphical procedure is proposed and illustrated for obtaining exact confidence intervals for C_pm. The graphical procedure allows the user to simply enter the graph with an estimate of the index and a value of the non‐centrality parameter for a given sample size to arrive at end‐points of 90%, 95% or 99% one‐sided or two‐sided confidence intervals. Detailed tables are also provided to assist the user for a wider range of sample values and sample sizes. In addition, a procedure is also presented for determining the minimum sample size required for attaining a pre‐specified level of accuracy of the C_pm. Extensive tables are provided for the user with a simple example illustrating the facility of the technique. Copyright © 2001 John Wiley & Sons, Ltd.