Re-evaluating the process capability indices for non-normal distributions

Classical process capability indices (PCIs) C _p , C _pu , C _pl and C _pk can indicate the potential process capability accurately when the quality characteristic of the product is normally distributed. When the process has a non-normal distribution, classical PCIs will be inappropriate and can misled the assessment of process capability. Zwick (1995), Schneider and Pruett (1995-96) and Tong and Chen (1998) proposed various PCIs for non-normal distributions. This paper compares the accuracy of these indices for several selected non-normal distributions based on the proportion of non-conformity of manufactured product. The results indicate that these PCIs lead to a larger number of errors in various combinations of shape parameters and specification limits. Therefore, this paper proposes three indices, S _pu , S _pl and S _pk , which can reflect accurately the proportion of nonconformity in either normal or non-normal distributions.     

Non‐normal Capability Indices for the Weibull and Lognormal Distributions

Because the normal process capability indices (PCIs) C_p, C_pu, C_pl, and C_pk represent the times that the process standard deviation is within the specification limits; then, based on and by using the direct relations among the parameters of the Weibull, Gumbel (minimum extreme value type I) and lognormal distributions, the Weibull and lognormal PCIs are derived in this paper. On the other hand, because the proposed PCIs P_p, P_pu, P_pl, and P_pk were derived as a function of the mean and standard deviation of the analyzed process, they have the same practical meaning with those of the normal distribution. Results show that the proposed PCIs could be used as the standard C_p, C_pu, C_pl, and C_pk if a short‐term variance is analyzed. An application to a set of simulated data is presented. Copyright © 2015 John Wiley & Sons, Ltd.     

Assessing the actual Gamma process quality – a curve-fitting approach for modifying the non-normal flexible index

Process capability indices (PCIs) have been widely adopted for quality assurance activities. By analysing PCIs, a production department can trace and improve a poor process to enhance product quality level and satisfy customer requirements. Among these indices, C_pk remains the most prevalent for facilitating managerial decisions because it can provide bounds on the process yield for normally distributed processes. However, processes are often non-normal in practice, and C_pk may quite likely misrepresent the actual product quality. Hence, the flexible index C_jkp, which considers possible differences in the variability above and below the target value, has been developed for practical use. However, C_jkp continues to suffer from serious bias in assessing actual capability, especially when the process distribution is highly skewed. In this paper, we modify C_jkp for assessing the actual process quality of a Gamma process. A correction factor is obtained by the curve-fitting method. The results show that our proposed method can significantly reduce the bias for calculation of actual nonconformities. Moreover, we introduce a sample estimator for our modified index. The ratio of this estimator’s average value and the modified index is approximately 1. This implies that our proposed estimator can provide an appropriate estimation for assessing the actual Gamma process quality.     

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.     

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.     

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.     

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.     

Process capability analysis—a robustness study

The robustness of two popular process capability ratios, C_p and C_pk, when the random process being observed departs from normality is analysed. The distributions of estimated process capability ratios are derived and used as a basis for validation of large-scale simulation studies in an examination of departures from normality. The simulation studies and analytical results provide a basis for recommended procedures. It is recommended that the impact of process distributions be considered before using popular process capability indexes, due to the lack of robustness when departures from normality are ecountered. As an extension of our findings, we consider the Taguchi loss function as a generalization to process capability analysis regardless of the underlying population distribution.     

The development of a target‐focused process capability index with multiple characteristics

Within an industrial manufacturing environment, Process Capability Indices (PCIs) enable engineers to assess the process performance and ultimately improve the product quality. Despite the fact that most industrial products manufactured today possess multiple quality characteristics, the vast majority of the literature within this area primarily focuses on univariate measures to assess process capability. One particular univariate index, C_pm, is widely used to account for deviations between the location of the process mean and the target value of a process. While some researchers have sought to develop multivariate analogues of C_pm, modeling the loss in quality associated with multiple quality characteristics continues to remain a challenge. This paper proposes a multivariate PCI that more appropriately estimates quality loss, while offering greater flexibility in conforming to various industrial applications, and maintaining a more realistic approach to assessing process capability. Copyright © 2010 John Wiley & Sons, Ltd.     

Effect of Non-Normality on Tolerance Limits Which Control Percentages in Both Tails of Normal Distribution

Effect of non-normality on Owen's tolerance limits, which control percentages p/2 in each of the tails of the normal distribution, is investigated. The limits are insensitive to departures from normality for p ≥ 0.20, but the effect of non-normality is increasingly felt as p decreases. Factors for tolerance limits in the non-normal case, which require at least a rough knowledge of parental skewness β₁ and kurtosis β₂, are given.     

Tool replacement for production with a low fraction of defectives

In manufacturing industry, the tool replacement cost is, in many cases, a significant portion of the production cost. Early tool replacement increases the production cost. Overdue tool replacement, however, results in poor production quality. Accordingly, improving production quality while maintaining a low production cost is essential. The index C_pk is regarded as a yield-based index. For a fixed C_pk value, the production yield and fraction of defectives can be calculated. In this paper, we present an analytical approach using C_pk to determine the optimal tool replacement time. An accurate process capability must be calculated, particularly when the data contain assignable cause variation. Tool wear is a dominant and inseparable component in many machining processes (a systematic assignable cause), and ordinary capability measures become inaccurate because process data are contaminated by the assignable cause variation. Considering process capability changes dynamically, an estimator of C_pk is investigated. The closed form of the exact sampling distribution is derived. An effective tool management procedure for determining the optimal tool replacement time is presented for processes with a low fraction of defectives. For illustrative purposes, an application example involving tool wear is presented.     

Monitoring manufacturing quality for multiple Li-BPIC processes based on capability index C pmk

The multi-process performance analysis chart (MPPAC) based on process capability indices has been developed to analyse the manufacturing performance for multiple processes, which conveys critical information regarding the departure of the process mean from the target value, process variability, capability levels, which provides a guideline of directions for capability improvement. Existing MPPAC researches have plotted the sample estimates of the process indices on the chart. Conclusions were then made on whether processes meet the capability requirement and directions need to be taken for further quality improvement. Such an approach is highly unreliable since the sample point estimate is a random variable with no assessment of the sampling errors. Further, existing MPPAC researches only considered one single sample. Current quality control practice is to estimate process capability using multiple groups of control chart samples rather than one single sample. In this paper, we propose the C _pmk MPPAC combining the accuracy index C _a to access the performance of multiple manufacturing processes. Distributions of the estimated C _pmk and C _a are derived based on multiple control chart samples, and accurate lower confidence bounds are calculated. The lower confidence bounds of the estimated C _pmk and C _a are then employed to the MPPAC to provide reliable capability grouping for those multiple processes. A real-world example is presented to illustrate the applicability of the proposed MPPAC.     

Efficient bivariate EWMA charts for monitoring process dispersion

To ensure high quality standards of a process, the application of control charts to monitor process performance has become a regular routine. Multivariate charts are a preferred choice in the presence of more than one process variable. In this article, we proposed a set of bivariate exponentially weighted moving average (EWMA) charts for monitoring the process dispersion. These charts are formulated based on a variety of dispersion statistics considering normal and non-normal bivariate parent distributions. The performance of the different bivariate EWMA dispersion charts is evaluated and compared using the average run length and extra quadratic loss criteria. For the bivariate normal process, the comparisons revealed that the EWMA chart based on the maximum standard deviation (SMAX_E) was the most efficient chart when the shift occurred in one quality variable. It also performed well when the sample size is small and the shift occurred in both quality variables. The EWMA chart based on the maximum average absolute deviation from median (MDMAX_E) performed better than the other charts in most situations when the shift occurred in the covariance matrix for the bivariate non-normal processes. An illustrative example is also presented to show the working of the charts.     

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.     

A smooth nonparametric approach to process capability

The determination of the capability of a stable process using the standard process capability indices requires that the quality characteristic of interest be normally distributed. Departures from normality can result in erroneous conclusions when using these indices. In this paper we propose assessing the capability of a process using a nonparametric estimator. This estimator is based on kernel estimation of the distribution function. Bandwidth selection for this method can be based either on a normal reference distribution or on a nonparametric estimate. An example is presented that applies this proposed method to non-normal process data. The performance of the resulting estimator is then compared with the sample proportion and a normal-based estimate in a simulation study. © 1998 John Wiley & Sons, Ltd.     

On developing sensitive nonparametric mixed control charts with application to manufacturing industry

Control charts are designed under the normality assumption of the quality characteristic of the process. However, the normality assumption rarely holds in practice. In non-normal conditions, parametric charts tend to display more false alarm rates and invalid out-of-control comparisons. The exponentially weighted moving average chart is a frequently used memory-type control chart for monitoring the process target that only performs effectively under the smoothing parameter's small choices. This study proposes a nonparametric mixed exponentially weighted moving average-progressive mean chart based on sign statistic (NPMEP_SN) under simple and ranked set sampling schemes to address this said drawback. Normal and non-normal distributions are included in this study to observe the proposed chart's in-control behavior and out-of-control efficacy. The prominent feature of the proposed schemes is that it works efficiently in detecting small and persistent shifts in the process location corresponding to the given values of the smoothing parameter. The proposed scheme is also tested under the ranked set sampling scheme to enhance the NPMEP_SN chart's performance (hereafter named “NPMEP_RSN”). The performance of the proposed charts is investigated through simulations using run-length profiles. The proposed schemes were seen to outperform other alternatives, specifically under the ranked set sampling scheme. A real data-set related to the diameter of a piston ring is included as a demonstration of the proposal.     

The Performance of Some Rough Tests for Bivariate Normality Before and After Coordinate Transformations to Normality

Some rough tests for bivariate normality are employed in an attempt to quantify the intuitive notion that coordinate transformations to normality produce distributions which are “more bivariate normal” than the original variables. These tests are not rigorous procedures but are intuitively satisfying, based on natural statistics, and provide numerical measures of the “distance” of a bivariate distribution from the normal model. It is shown that, for a wide class of non-normal (X, Y) distributions, coordinate transformations to normality decrease this distance as measured by these tests. It is indicated how one may estimate the coordinate transformations and applications to correlation theory are explored.     

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.     

非正态过程能力指数研究中的几个问题

非正态分布条件下过程能力指数的研究是目前关于过程能力分析中的热点之一。针对该领域中存在的若干问题,分析了正态性假设的重要性,指出导致质量特性呈现非正态分布的原因及其后果,以及如何进行质量特性分布的一维和多维正态性检验。对非正态过程能力指数的估计方法进行了分析和归类,简要指出在非正态过程能力指数研究中需要继续深入的几个方向。