Process capability indices for skewed populations

This paper proposes a new method of constructing process capability indices (PCIs) for skewed populations. It is based on a weighted standard deviation method which decomposes the standard deviation of a quality characteristic into upper and lower deviations and adjusts the value of the PCI using decomposed deviations in accordance with the skewness estimated from sample data. For symmetric populations, the proposed PCIs reduce to standard PCIs. The performance of the proposed PCIs is compared with those of standard and other PCIs, and finite sample properties of the estimates are investigated using Monte Carlo simulation. Numerical studies indicate that considerable improvements over existing methods can be achieved by the use of the weighted standard deviation method when the underlying distribution is skewed. Copyright © 2002 John Wiley & Sons, Ltd.     

Generalized confidence intervals for process capability indices

The concept of generalized confidence intervals is used to derive lower confidence limits for some of the commonly used process capability indices. For the cases where approximate lower confidence limits are already available, numerical comparisons are made among the available approximations and the generalized lower confidence limit. The numerical results indicate that the generalized confidence interval does provide coverage probabilities very close to the nominal confidence level. Two examples are given to illustrate the results. Copyright © 2006 John Wiley & Sons, Ltd.     

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.     

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.     

A class of process capability indices for asymmetric tolerances

A process capability index is a measure that evaluates the behavior of a product or process characteristics related to the engineering specification. In the literature, many articles can be found about capability indices for both symmetric and asymmetric tolerance. In this article, we introduce a new class of indices, C^′′′_p(u, v), for the processes with asymmetric tolerance. In addition, we study the relation between this index and the departure ratio of the process centering, as well as, the relation between this index and the upper bound of the percentage of non-conforming products. Based on these results we suggest a criteria for choosing an index from the under investigation class. To guide practitioners, numerical examples are provided.     

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.     

The stochastic control of process capability indices

In manufacturing science, process capability indices play a role analogous to economic indices in government statistics. The existing capability indices are passive devices whose main role is to retroactively monitor process capability. The have been developed under the restrictive assumption of process stability, and the procedures for using them are based on ad hoc rules. Using the normative point of view for decision making, it can be shown that some of the indices are, at best, convoluted special cases of a more general strategy; they can be justified only under special assumptions, and the manner in which they are currently used could lead to incoherent actions. The available process capability indices should therefore be abandoned and replaced by procedures that are normative, and also proactive with respect to both, prediction and control. An approach towards achieving this goal is proposed. Research Sponsored by The National Institute of Standards and Technology Gaithersburg, Maryland 20899-0001 (Under Purchase Order No. 43NANB610868), The U.S. Army Research Office Grant DAAG-55-97-1-0323, and The Air Force Office of Scientific Research Grant AFOSR-F-49620-95-0107     

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.     

Process capability indices for Weibull distributions and upper specification limits

We consider a previously proposed class of capability indices that are useful when the quality characteristic of interest has a skewed, zero‐bound distribution with a long tail towards large values and there is an upper specification with a pre‐specified target value, T=0. We investigate this class of process capability indices when the underlying distribution is a Weibull distribution and focus on the situation when the Weibull distribution is highly skewed. We propose an estimator of the index in the studied class, based on the maximum likelihood estimators of the parameters in the Weibull distribution, and derive an asymptotic distribution for this estimator. Furthermore, we suggest a decision rule based on the estimated index and its asymptotic distribution and present a power comparison between the proposed estimator and a previously studied estimator. A simulation study is also performed to investigate the true significance level when the sample size is small or moderate. An example from a Swedish industry is presented. Copyright © 2008 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.     

Statistical analysis of process capability indices with measurement errors

Process capability indices (PCIs) have been widely used in manufacturing industries to provide a quantitative measure of process potential and performance. While some efforts have been dedicated in the literature to the statistical properties of PCIs estimators, scarce attention has been given to the evaluation of these properties when sample data are affected by measurement errors. In this work we deal with the problem of measurement‐error effects on the performance of PCIs. The analysis is illustrated with reference to and , i.e. the two most common measures suggested to evaluate process capability. Copyright © 2002 John Wiley & Sons, Ltd.     

On the monitoring of trended and regularly adjusted processes

Trended and regularly adjusted processes are common in manufacturing industries. Such processes are, for example, related to tool wear, material replenishment or some regular maintenance. When the process has a slow trend or is frequently adjusted, the Shewhart chart can be interpreted in the same way as for a stable process. To facilitate comparison between such a trended and adjusted process to a stable case, and to estimate further the loss of effectiveness when the traditional Shewhart chart is applied to trended and adjusted process, this paper provides a statistical interpretation of traditional Shewhart charts for this type of processes. Formulas are derived for the calculation of alarm rate and average run length (ARL). This study is useful when deciding if a traditional Shewhart chart is sufficient or if a more advanced Statistical Process Control method is necessary. Furthermore, given the in-control and out-of-control ARL, a combined decision with regard to the control limits setting and the adjustment interval can be made. The general formulation is described and a simple linear trend model with an actual data set is used as an illustration.     

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.     

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.     

多批次小批量生产过程能力指数的贝叶斯估计

针对过程能力指数(PCIs)的传统计算方法应用于多批次小批量生产过程出现的失真问题,以多批次小批量生产过程的总体均值和总体方差分别服从共轭正态分布和共轭逆伽马分布,利用共轭贝叶斯估计理论提出了多批次小批量生产过程分布参数估计方法、过程能力指数计算方法及其通用表达式,这种贝叶斯估计方法减小仅用当前批小样本容量估计过程参数...     

Process capability: a criterion for optimizing multiple response product and process design

In this research, we consider the maximization of process capability as the criterion in product/process design that is used for selecting preferred design factor levels and propose several approaches for single and multiple response performance measure designs. All of these approaches assume that the relationship between a process performance measure and a set of design factors is represented via an estimate of a response surface function. In particular, we develop; (i) criteria for selecting an optimal design, which we call MCpk and MCpm; (h) mathematical programming formulations for maximizing MCpk and MCpm, including formulations for maximizing the desirability index (Harrington, 1965) and for maximizing the standardized performance criteria (Barton and Tsui, 1991) as special cases of the formulation for maximizing MCpk, (iii) formulations for considering cost when maximizing MCpk and MCpm, (iv) a means for assessing propagation of error; (v) a robust design method for assessing design factor effects on residual variance; (vi) a means for assessing the optimality of a proposed solution: and (vii) an original application in the screening of printed circuit board panels.     

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.     

Distributional and inferential properties of some new multivariate process capability indices for symmetric specification region

Statistical quality control is used to improve performance of processes. Since most of the processes are multivariate in nature, multivariate process capability indices (MPCIs) have been developed by many researchers depending on the context. However, it is generally difficult to understand and calculate MPCIs, compared to their univariate counterparts like $C_p$, $C_{pk}$, and so on. This paper discusses a relatively new development in MPCIs, namely, $C_G(u,v)$, which is a multivariate analogue of $C_p(u,v)$—the celebrated superstructure of univariate process capability indices . Some statistical properties of $C_G(u,v)$ are studied, particularly of $C_G(0,0)$, a member MPCI of the superstructure, which measures potential capability of a multivariate process. A threshold value of $C_G(0,0)$ is computed, and this can be considered as a logical cut-off for other member indices of $C_G(u,v)$ as well. The expression for the upper limit of the proportion of nonconformance is derived as a function of $C_G(0,0)$. Density plots of asymptotic distributions of four major member indices of $C_G(u,v)$, namely, $C_G(0,0)$, $C_G(1,0)$, $C_G(0,1)$, and $C_G(1,1)$, are made. Finally, a numerical example is discussed to supplement the theory developed in this paper.     

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.