Probability-Based Process Capability Indices

This article deals with alternative process capability indices (PCIs) to traditional basic PCIs C _p , C _pk , and C _pm based on different fraction conforming type of probabilities. In view of various problems of constructing capability indices for univariate as well as multivariate set up, these alternative PCIs are very useful as compared to C _p , C _pk , and C _pm . Computing aspects of proposed PCIs are discussed for normal and non normal processes when process tolerance is symmetric as well as asymmetric. Generalization of these PCIs for multivariate set up is also discussed. Some simulation study results and real life problems are given for applications of proposed PCIs.     

Construction of closed interval for process capability indices Cpu,Cpl, and Spk based on Boole’s inequality and de Morgan’s laws

ABSTRACT

Considerable effort has been spent on the development of confidence intervals for process capability indices (PCIs) based on the sampling distribution of the PCI or the transferred PCI. However, there is still no definitive way to construct a closed interval for a PCI. The aim of this study is to develop closed intervals for the PCIs C_pu, C_pl, and S_pk based on Boole's inequality and de Morgan's laws. The relationships between different sample sizes, the significance levels, and the confidence intervals of the PCIs C_pu, C_pl, and S_pk are investigated. Then, a testing model for interval estimation for the PCIs C_pu, C_pl, and S_pk is built as a powerful tool for measuring the quality performance of a product. Finally, an applied example is given to demonstrate the effectiveness and applicability of the proposed method and the testing model.     

A general class of capability indices in the case of asymmetric tolerances

Vannman has earlier studied a class of capability indices, containing the indices C _p , C _pk , C _pm and C _pmk , when the tolerances are symmetric. We study the properties of this class when the tolerances are asymmetric and suggest a new enlargened class of indices. Under the assumption of normality an explicit form of the distribution of the new class of the estimated indices is provided. Numerical investigations are made to explore the behavior of the estimators of the indices for different values of the parameters. Based on the estimator a decision rule that can be used to determine whether the process can be considered capable or not is provided and suitable criteria for choosing an index from the family are suggested.     

A new generation of process capability indices based on fuzzy measurements

Process capability indices (PCIs) provide numerical measures on whether a process conforms to the defined manufacturing capability prerequisite. These have been successfully applied by companies to compete with and to lead high-profit markets by evaluating the quality and productivity performance. The PCI C_p compares the output of a process to the specification limits (SLs) by forming the ratio of the width between the process SLs with the width of the natural tolerance limits which is measured by six process standard deviation units. As another common PCI, C_pm incorporates two variation components which are variation to the process mean and deviation of the process mean from the target. A meaningful generalized version of above PCIs is introduced in this paper which is able to handle in a fuzzy environment. These generalized PCIs are able to measure the capability of a fuzzy-valued process in producing products on the basis of a fuzzy quality. Fast computing formulas for the generalized PCIs are computed for normal and symmetric triangular fuzzy observations, where the fuzzy quality is defined by linear and exponential fuzzy SLs. A practical example is presented to show the performance of proposed indices.     

Developing and designing of an efficient variables sampling system based on the process capability index

Process capability indices (PCIs) are extensively used in the manufacturing industries in order to confirm whether the manufactured products meet their specifications or not. PCIs can be used to judge the process precision, process accuracy, and the process performance. So developing of sampling plans based on PCIs is inevitable and those plans will be very much useful for maintaining and improving the product quality in the manufacturing industries. In view of this, we propose a variables sampling system based on the process capability index C_pmk, which takes into account of process yield and process loss, when the quality characteristic under study will have double specification limits. The proposed sampling system will be effective in compliance testing. The advantages of this system over the existing sampling plans are also discussed. In order to determine the optimal parameters, tables are also constructed by formulating the problem as a nonlinear programming in which the average sample number is minimized by satisfying the producer and consumer risks.     

Statistical analysis of process capability indices with measurement errors: The case ofC p

Process capability indices (PCIs) have been widely used in manufacturing industries to previde 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 errors effects on the performance of PCIs. The analysis is illustrated with reference toC _p , i.e. the simplest and most common measure suggested to evaluate process capability. The authors would like to thank two anonymous referees for their comments and suggestion that were useful in the preparation and improvement of this paper. This work was partially supported by a MURST research grant.     

An asymptotic confidence interval for the process capability index Cpm

Abstract

The use of indices as an estimation tool of process capability is long-established among the statistical quality professionals. Numerous capability indices have been proposed in last few years. C_pm constitutes one of the most widely used capability indices and its estimation has attracted much interest. In this paper, we propose a new method for constructing an approximate confidence interval for the index C_pm. The proposed method is based on the asymptotic distribution of the index C_pm obtained by the Delta Method. Under some regularity conditions, the distribution of an estimator of the process capability index C_pm is asymptotically normal.     

Estimating capability index cpk for processes with asymmetric tolerances

Pearn and Chen (1996) considered the process capability index C_pk, and investigated the statistical properties of its natural estimator under various process conditions. Their investigation, however, was restricted to processes with symmetric tolerances. Recently, Pearn and Chen (1998) considered a generalization of C_pk, referred to as C^? _pk, to cover processes with asymmetric tolerances. They investigated the statistical properties of the natural estimator of C^? _pk, and obtained the exact formulae for the expected value and variance. In this paper, we consider a new estimator of C^? _pk, assuming the knowledge on P(LI > T) = p is available, where 0 > p > 1, which can be obtained from historical information of a stable process. We obtain the exact distribution of the new estimator assuming the process characteristic follows the normal distribution. We show that the new estimator is consistent, asymptotically unbiased, which converges to a mixture of two normal distributions. We also show that by adding suitable correction factors to the new estimator, we may obtain the UMVUE and the MLE of the generalization C^? _pk.     

Parametric inference of the process capability index for exponentiated exponential distribution

Process capability indices (PCIs) are most effective devices/techniques used in industries for determining the quality of products and performance of manufacturing processes. In this article, we consider the PCI C_pc which is based on the proportion of conformance and is applicable to normally as well as non-normally and continuous as well as discrete distributed processes. In order to estimate the PCI C_pc when the process follows exponentiated exponential distribution, we have used five classical methods of estimation. The performances of these classical estimators are compared with respect to their biases and mean squared errors (MSEs) of the index C_pc through simulation study. Also, the confidence intervals for the index C_pc are constructed using five bootstrap confidence interval (BCIs) methods. Monte Carlo simulation study has been carried out to compare the performances of these five BCIs in terms of their average width and coverage probabilities. Besides, net sensitivity (NS) analysis for the given PCI C_pc is considered. We use two data sets related to electronic and food industries and two failure time data sets to illustrate the performance of the proposed methods of estimation and BCIs. Additionally, we have developed PCI C_pc using aforementioned methods for generalized Rayleigh distribution.     

A robust process capability index

The existing process capability indices (PCI's) assume that the distribution of the process being investigated is normal. For non-normal distributions, PCI's become unreliable in that PCI's may indicate the process is capable when in fact it is not. In this paper, we propose a new index which can be applied to any distribution. The proposed indexC_f:, is directly related to the probability of non-conformance of the process. For a given random sample, the estimation of C_f boils down to estimating non-parametrically the tail probabilities of an unknown distribution. The approach discussed in this paper is based on the works by Pickands (1975) and Smith (1987). We also discuss the construction of bootstrap confidence intervals of C_f: based on the so-called accelerated bias correction method (BC _a:). Several simulations are carried out to demonstrate the flexibility and applicability of C_f:. Two real life data sets are analyzed using the proposed index.     

A note on the asymptotic distribution of the process capability index C pmk

Process capability indices have been widely used to evaluate the process performance to the continuous improvement of quality and productivity. The distribution of the estimator of the process capability index C _pmk is very complicated and the asymptotic distribution is proposed by Chen and Hsu [The asymptotic distribution of the processes capability index C _pmk , Comm. Statist. Theory Methods 24(5) (1995), pp. 1279–1291]. However, we found a critical error for the asymptotic distribution when the population mean is not equal to the midpoint of the specification limits. In this paper, a correct version of the asymptotic distribution is given. An asymptotic confidence interval of C _pmk by using the correct version of asymptotic distribution is proposed and the lower bound can be used to test if the process is capable. A simulation study of the coverage probability of the proposed confidence interval is shown to be satisfactory. The relation of six sigma technique and the index C _pmk is also discussed in this paper. An asymptotic testing procedure to determine if a process is capable based on the index of C _pmk is also given in this paper.     

An asymptotic distribution of wright’s process capability index sensitive to skewness

In a recent paper (J. Statist. Comput. Simul., 1995, Vol.53, pp. 195–203) P. A. Wright proposed a new process capability index C_s which generalizes the Pearn-Kotz-Johnson’s index C_pmk by taking into account the skewness (in addition to deviation of the mean from tliCrntarget already incorporated in C_pmk ). The purpose of this article is to study the consistency and asymptotics of an estimate ?_s of C_s The asymptotic distribution provides an insight into some desirable properties of the estimate which are not apparent from its original definition     

Three New Multivariate Process Capability Indices

Different multivariate process capability indices are developed by researchers to evaluate process capability when vectors of quality characteristics are considered in a study. This article presents three indices referred to as NCpM, MCpM, and NMC _PM in order to evaluate process capability in multivariate environment. The performance of the proposed indices is investigated numerically. Simulation results indicate that the proposed indices have descended estimation error and improved performance compared to the existing ones. These results can be important to researchers and practitioners who are interested in evaluating process capability in multivariate domain.     

Capability indices for Birnbaum–Saunders processes applied to electronic and food industries

Process capability indices (PCIs) are tools widely used by the industries to determine the quality of their products and the performance of their manufacturing processes. Classic versions of these indices were constructed for processes whose quality characteristics have a normal distribution. In practice, many of these characteristics do not follow this distribution. In such a case, the classic PCIs must be modified to take into account the non-normality. Ignoring the effect of this non-normality can lead to misinterpretation of the process capability and ill-advised business decisions. An asymmetric non-normal model that is receiving considerable attention due to its good properties is the Birnbaum–Saunders (BS) distribution. We propose, develop, implement and apply a methodology based on PCIs for BS processes considering estimation, parametric inference, bootstrap and optimization tools. This methodology is implemented in the statistical software {\tt R}. A simulation study is conducted to evaluate its performance. Real-world case studies with applications for three data sets are carried out to illustrate its potentiality. One of these data sets was already published and is associated with the electronic industry, whereas the other two are unpublished and associated with the food industry.     

On a Process Capability Index for Asymmetric Specifications

Abstract

Among the process capability indices considered in the literature C _pm is one of the most widely used, despite the fact that its performance often becomes unsatisfactory in the case of processes with asymmetric specifications, i.e., processes whose target value is not located at the midpoint of the specification area. In this article, a new index that is a variant of C _pm , is introduced and shown to overcome this deficiency. In particular, the proposed index performs satisfactorily for processes with symmetric or asymmetric specifications. Moreover, the article compares the suggested index with the existing indices for asymmetric specifications, investigates the distributional properties of its estimator under the assumption of normality and deals with the assessment of confidence intervals using three bootstrap methods. The coverage achieved by each of these methods is investigated via simulation.     

The Asymptotic Distributions of Unified Process Capability Indices

Abstract

This paper derives the asymptotic distributions of the estimators of the unified process capability indices C _p (u, v) and C _pa (u, v) for arbitrary population under general, regularity conditions, assuming that the fourth moment about the mean exists.     

New sensitivity analysis subordinated to a contrast

ABSTRACT

In a model of the form Y = h(X₁, …, X_d) where the goal is to estimate a parameter of the probability distribution of Y, we define new sensitivity indices which quantify the importance of each variable X_i with respect to this parameter of interest. The aim of this paper is to define goal oriented sensitivity indices and we will show that Sobol indices are sensitivity indices associated to a particular characteristic of the distribution Y. We name the framework we present as Goal Oriented Sensitivity Analysis (GOSA).     

Estimation of the process incapability index

Greenwich and Jahr-Schaffrath (1995) introduced a new index C _pp a simple transformation of the index C _pm , which provides an uncontaminated separation between information concerning process accuracy and process precision. Under the assumption of normality, we first show that the estimators of C _pp proposed by Greenwich and Jahr-Schaffrath (1995) are UMVU estimators. We also show that for the inaccuracy index, the variance of the unbiased estimator is smaller than the mean squared error (MSE) of the natural (biased) estimator for n > 3. In addition, we obtain the r-th moment and the probability density function of these estimators.