A general approach for testing the process capability index

This paper presents a generalized approach to the problem of testing with the process capability index C?_p. It allows for testing at any α-level and any sample size. For small sample sizes n < 30, n odd, tables are included with αs incremented by 0.025, which are generated by a computer program. This program is written in BASIC and appears in an Appendix. For different α levels (say α = 0·1) the program can be used to generate the appropriate value to use in determining the critical values. For larger sample sizes an approximation method is supplied. Additionally, the method to derive an associated operating characteristic curve is examined.     

Lower confidence limits for process capability indices Cp and Cpk when data are autocorrelated

Many organizations use a single estimate of C_p and/or C_pk for process benchmarking, without considering the sampling variability of the estimators and how that impacts the probability of meeting minimum index requirements. Lower confidence limits have previously been determined for the C_p and C_pk indices under the standard assumption of independent data, which are based on the sampling distributions of the index estimators. In this paper, lower 100(1‐α)% confidence limits for C_p and C_pk were developed for autocorrelated processes. Simulation was used to generate the empirical sampling distribution of each estimator for various combinations of sample size (n), autoregressive parameter (?), true index value (C_p or C_pk), and confidence level. In addition, the minimum values of the estimators required in order to meet quality requirements with 100(1‐α)% certainty were also determined from these empirical sampling distributions. These tables may be used by practitioners to set minimum capability requirements for index estimators, rather than true values, for the autocorrelated case. The implications of these results for practitioners will be discussed. Copyright © 2008 John Wiley & Sons, Ltd.     

A Close Form Solution for the Product Acceptance Determination Based on the Popular Index Cpk

Product acceptance determinations are practical tools for quality control applications involving quality contract on product orders between the vendor and the buyer. It provides the vendor and the buyer rules for product acceptance to meet the preset product quality requirement. As the rapid advancement of manufacturing technology, more than one quality characteristic must be simultaneously considered to improve the product quality because of the product design. In this article, we introduce an efficient product acceptance procedure on the basis of the generalization index, to deal with lot sentencing problem with very low fraction of defectives. We tabulate the required sample size n and the corresponding critical acceptance value c₀ for various α‐risk, β‐risk, and the levels of the lot fraction of defectives that correspond to acceptance and rejecting quality levels. Practitioners can use the proposed method to make reliable decisions in product acceptance. Copyright © 2012 John Wiley & Sons, Ltd.     

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.     

A model‐free approach to eliminate autocorrelation when testing for process capability

There is an increasing use of on‐line data acquisition systems in industry. This usually leads to autocorrelated data and implies that the assumption of independent observations has to be re‐examined. Most decision procedures for capability analysis assume independent data. In this article we present a new way of performing capability analysis when data are autocorrelated. This method is based on what can be called the ‘iterative skipping’ strategy. In that, by skipping a pre‐determined number of observations, e.g. considering every fifth observation, the data set is divided into subsamples for which the independence assumption may be valid. For each such subsample of the data we estimate a capability index. Then traditional tests, assuming independence, can be performed based on each estimated capability index from the subsamples. By combining the information from each test statistic based on the subsamples in a suitable way, a new and efficient decision procedure is obtained. We discuss different ways of combining the information from these individual tests. A main appeal of our proposed method is that no time‐series model is needed. Copyright © 2007 John Wiley & Sons, Ltd.     

Capability assessment for processes with multiple characteristics: A generalization of the popular index Cpk

Process capability index C_pk is the most popular capability index widely used in the manufacturing industry. Existing research on the yield‐based measure index C_pk to date is restricted to processes with single characteristics. However, many manufacturing processes are commonly described with multiple characteristics, for example, the gold bumping process in the TFT‐LCD (thin film transistor‐liquid crystal display) manufacturing industry. In the gold bumping process, gold bumps have multiple characteristics all having effects on the process yield. Obtaining accurate gold bumping manufacturing yield is very important for quality assurance and in providing guidance toward process improvement. To obtain accurate yield assessment for processes with multiple characteristics, we propose a new overall yield‐measure index C, which is a generalization of the index C_pk, and a natural estimator of C. For the purpose of making inferences on the process capability, we derive a quite accurate approximation of the distribution of since the distribution is analytically intractable. With this distribution, we tabulate the lower confidence bounds of the new index under various sample sizes for in‐plant applications. In addition, we construct a statistical test on the new yield‐measure index in order to examine whether the yield meets the customers' requirements. For illustration purpose, a real case in a gold bumping factory located in the Science‐based Industrial Park at Hsinchu, Taiwan is presented. Copyright © 2011 John Wiley & Sons, Ltd.     

A population balance model for the chocolate roller refining process

A study was performed to analyze the evolution of particle size distribution in the chocolate roller refining process. A mathematical model based on the continuity and population balance equations was developed to interpret industrial data. The industrial data were provided for chocolate pastes with powder/crystalline sugar particles at low/high shear rates. The parameters of the breakage frequency and fragment distribution function were estimated using industrial data. After parameter fitting, the model shows good agreement with the experimental results for varying conditions with a single consistently chosen set of parameters. This provides confidence that the general model structure is suitable for process evaluation. The population balance model was used to analyze the influence of changing some process variables on the efficiency of grinding. The results show that there might exist optimum values of the roll’s diameter and rotational velocity for a specific configuration of the equipment.     

带随机干扰更新风险模型破产概率的局部定理

为推广经典风险过程以研究各种风险引发的破产的可能性,本文研究了保险金融领域中一个更为现实的模型:带随机干扰的更新风险模型的破产概率的渐近估计的局部化形式。在相对安全负荷条件下,采用纯概率的方法,得出了当索赔额为重尾索赔时破产概率的局部渐近等价式,它与原更新风险模型相应的破产概率的局部渐近等价式一致,说明在重尾索赔下,Wiener过程对破产概率的影响可以忽略。     

A new adaptive CUSUM control chart for detecting the multivariate process mean

We propose a new multivariate CUSUM control chart, which is based on self adaption of its reference value according to the information from current process readings, to quickly detect the multivariate process mean shifts. By specifying the minimum magnitude of the process mean shift in terms of its non‐centrality parameter, our proposed control chart can achieve an overall performance for detecting a particular range of shifts. This adaptive feature of our method is based on two EWMA operators to estimate the current process mean level and make the detection at each step be approximately optimal. Moreover, we compare our chart with the conventional multivariate CUSUM chart. The advantages of our control chart detection for range shifts over the existing charts are greatly improved. The Markovian chain method, through which the average run length can be computed, is also presented. Copyright © 2010 John Wiley & Sons, Ltd.     

A Control Chart for the Multivariate Coefficient of Variation

Existing charts in the literature usually monitor either the mean or the variance of the process. However, in certain scenarios, the practitioner is not interested in the changes in the mean or the variance but is instead interested in monitoring the relative variability compared with the mean. This relative variability is called the coefficient of variation (CV). In the existing literature, none of the control charts that monitor the CV are applied for multivariate data. To fill this gap in research, this paper proposes a CV chart that monitors the CV for multivariate data. To the best of the authors' knowledge, this proposed chart is the first control chart for this purpose. The distributional properties of the sample CV for multivariate data and the procedures to implement the chart are presented in this paper. Formulae to compute the control limits, the average run length, the standard deviation of the run length, and the expected average run length for the case of unknown shift size are derived. From the numerical examples provided, the effects of the number of variables, the sample size, the shift size and the in‐control value of the CV are studied. Finally, we demonstrate the usefulness and applicability of the proposed chart on real data. Copyright © 2015 John Wiley & Sons, Ltd.     

A control chart for monitoring the lognormal process variation using repetitive sampling

Control charts are developed to make the specific quality measures for a successful production process and follow normal distribution behaviors. But some real-life practices do not match such practices and exhibit some positively skewed behavior like lognormal distribution. The present study has considered this situation and proposed a monitoring control chart based on lognormal process variation using a repetitive sampling scheme. This concept proved better for detecting shifts as quickly as possible, and compared with the existing concept, results are elaborated through extensive tables. The average run lengths and standard deviations of the run lengths are being used as a performance evaluation measures and computed by using Monte Carlo simulations performed in R language. A real-life situation has been discussed in the example section to strengthen the proposed control chart concept in a real-life situation.     

An EWMA chart for monitoring the covariance matrix of a multivariate process based on dissimilarity index

In this article, we propose an exponentially weighted moving average (EWMA) control chart for monitoring the covariance matrix of a multivariate process based on the dissimilarity index of 2 matrices. The proposed control chart essentially monitors the covariance matrix by comparing the individual eigenvalues of the estimated EWMA covariance matrix with those of the estimated covariance matrix from the in‐control (IC) phase I data. It is different from the conventional EWMA charts for monitoring the covariance matrix, which are either based on comparing the sum or product or both of the eigenvalues of the estimated EWMA covariance matrix with those of the IC covariance matrix. We compare the performance of the proposed chart with that of the best existing chart under the multivariate normal process. Furthermore, to prevent the control limit of the proposed EWMA chart developed using the limited IC phase I data from having extensively excessive false alarms, we use a bootstrap resampling method to adjust the control limit to guarantee that the proposed chart has the actual IC ARL(average run length) not less than the nominal level with a certain probability. Finally, we use an example to demonstrate the applicability and implementation of the proposed EWMA chart.     

A new double EWMA-t chart with auxiliary information for the process mean

In this paper, we propose an auxiliary-information-based (AIB) double EWMA-$t$ (AIB-DEWMA-$t$) chart for monitoring the process mean. The DEWMA-$t$ chart encompasses the EWMA-$t$ and AIB-EWMA-$t$ charts. The Monte Carlo simulations are used to compute the run length characteristics of the AIB-DEWMA-$t$ chart. Based on detailed run length comparisons, it is found that the AIB-DEWMA-$t$ chart may uniformly and substantially outperform the AIB-EWMA-$t$ chart when detecting different shifts in the process mean. In addition, the AIB-DEWMA-$t$ chart is uniformly more sensitive than the DEWMA-$t$ chart. Similar trends are observed when comparing these control charts with the variable sampling interval feature. A real dataset is also considered to demonstrate the implementation of the proposed chart.     

Enhanced EWMA charts for monitoring the process coefficient of variation

The coefficient of variation (CV) is an important quality characteristic when the process variance is a function of the process mean for a production process. In this paper, we develop an auxiliary information–based (AIB) estimator for estimating the squared CV, along with its approximated mean and variance. This estimator is then used to devise new one-sided EWMA charts for monitoring the increases or decreases in the squared CV of a normal process, named the AIB-EWMA CV charts. In addition, the sensitivities of these control charts are also enhanced with the fast initial response feature. The Monte Carlo simulation method is used to compute the run length characteristics of the proposed CV charts. Based on detailed run length comparisons, it is found that the proposed AIB-EWMA CV charts are uniformly and substantially better than the existing EWMA CV charts when detecting different kinds of upward/downward shifts in the squared CV. The proposed charts are also applied to a real dataset to support the proposed theory.