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.     

Sample size determination for estimating multivariate process capability indices based on lower confidence limits

With the advent of modern technology, manufacturing processes have become very sophisticated; a single quality characteristic can no longer reflect a product's quality. In order to establish performance measures for evaluating the capability of a multivariate manufacturing process, several new multivariate capability (NMC) indices, such as NMC _p and NMC _pm , have been developed over the past few years. However, the sample size determination for multivariate process capability indices has not been thoroughly considered in previous studies. Generally, the larger the sample size, the more accurate an estimation will be. However, too large a sample size may result in excessive costs. Hence, the trade-off between sample size and precision in estimation is a critical issue. In this paper, the lower confidence limits of NMC _p and NMC _pm indices are used to determine the appropriate sample size. Moreover, a procedure for conducting the multivariate process capability study is provided. Finally, two numerical examples are given to demonstrate that the proper determination of sample size for multivariate process indices can achieve a good balance between sampling costs and estimation precision.     

A Bayesian procedure for assessing process performance based on the third-generation capability index

Capability indices that qualify process potential and process performance are practical tools for successful quality improvement activities and quality program implementation. Most existing methods to assess process capability were derived on the basis of the traditional frequentist point of view. This paper considers the problem of estimating and testing process capability based on the third-generation capability index C _pmk from the Bayesian point of view. We first derive the posterior probability p for the process under investigation is capable. The one-sided credible interval, a Bayesian analog of the classical lower confidence interval, can be obtained to assess process performance. To investigate the effectiveness of the derived results, a series of simulation was undertaken. The results indicate that the performance of the proposed Bayesian approach depends strongly on the value of ξ=(μ?T)/σ. It performs very well with the accurate coverage rate when μ is sufficiently far from T. In those cases, they have the same acceptable performance even though the sample size n is as small as 25.     

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 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.     

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.     

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.     

A General Class of Test Procedures for Left-Truncated and Right-Censored Data

When the distribution of one of the characteristics of a process is non normal, methods based on empirical percentiles suggest the use of several process capability indices (PCIs) which are similar to the usual C _p , C _pk , C _pm , and C _pmk indices. However most of these PCIs apply only to the case of symmetrical tolerances. To take into account the asymmetry of the tolerances as well as the asymmetry of the process distribution, new PCIs which improve the previous ones are proposed. In the end and in order to validate the method proposed here, we apply it to a real production case.     

Multivariate Capability Indices: Distributional and Inferential Properties

Process capability indices have been widely used in the manufacturing industry for measuring process reproduction capability according to manufacturing specifications. Properties of the univariate processes have been investigated extensively, but are comparatively neglected for multivariate processes where multiple dependent characteristics are involved in quality measurement. In this paper, we consider two commonly used multivariate capability indices MC_p and MC_pm, to evaluate multivariate process capability. We investigate the statistical properties of the estimated MC_p and obtain the lower confidence bound for MC_p. We also consider testing MC_p, and provide critical values for testing if a multivariate process meets the preset capability requirement. In addition, an approximate confidence interval for MC_pm is derived. A simulation study is conducted to ascertain the accuracy of the approximation. Three examples are presented to illustrate the applicability of the obtained results.     

Multivariate process capability a bayesian perspective

In this paper an attempt has been made to examine the multivariate versions of the common process capability indices (PCI's) denoted by C_p and C_pk . Markov chain Monte Carlo (MCMC) methods are used to generate sampling distributions for the various PCI's from where inference is performed. Some Bayesian model checking techniques are developed and implemented to examine how well our model fits the data. Finally the methods are exemplified on a historical aircraft data set collected by the Pratt and Whitney Company.     

Evaluation of process capability in multivariate nonlinear profiles

ABSTRACT

Process capability indices measure the ability of a process to provide products that meet certain specifications. Few references deal with the capability of a process characterized by a functional relationship between a response variable and one or more explanatory variables, which is called profile. Specifically, there is not any reference analysing the capability of processes characterized by multivariate nonlinear profiles. In this paper, we propose a method to measure the capability of these processes, based on principal components for multivariate functional data and the concept of functional depth. A simulation study is conducted to assess the performance of the proposed method. An example from the sugar production illustrates the applicability of this approach.     

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.     

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.     

On improvement of prediction in arma processes

Necessary and sufficient conditions are derived in the paper that enable to decide whether an additional multivariate process will improve the prediction in a given multivariate discrete stationary process. The both processes are assumed to form together a process ARMAm n Further it was investigated wnen one can asser t that the both processes are uncorrelated provided the additional process did not improve the prediction in the original process, Some hints for the actual construction of predictors in a multivariate ARMA. (m n) process can be found in the paper.     

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.     

Monitoring Multivariate Simple Linear Profiles in the Presence of between Profile Autocorrelation

In recent years, statistical profile monitoring has emerged as a relatively new and potentially useful subarea of statistical process control and has attracted attention of many researchers and practitioners. A profile, waveform, or signature is a function that relates a dependent or a response variable to one or more independent variables. Different statistical methods have been proposed by researchers to monitor profiles where each method requires its own assumptions. One of the common and implicit assumptions in most of the proposed procedures is the assumption of independent residuals. Violation of this assumption can affect the performance of control procedures and ultimately leading to misleading results. In this article, we study phase II analysis of monitoring multivariate simple linear profiles when the independency assumption is violated. Three time series based methods are proposed to eliminate the effect of correlation that exists between multivariate profiles. Performances of the proposed methods are evaluated using average run length (ARL) criterion. Numerical results indicate satisfactory performance for the proposed methods. A simulated example is also used to show the application of the proposed methods.     

Monitoring Variation in a Multivariate Process When the Dimension is Large Relative to the Sample Size

A control procedure is presented for monitoring changes in variation for a multivariate normal process in a Phase II operation where the subgroup size, m, is less than p, the number of variates. The methodology is based on a form of Wilk' statistic, which can be expressed as a function of the ratio of the determinants of two separate estimates of the covariance matrix. One estimate is based on the historical data set from Phase I and the other is based on an augmented data set including new data obtained in Phase II. The proposed statistic is shown to be distributed as the product of independent beta distributions that can be approximated using either a chi-square or F-distribution. An ARL study of the statistic is presented for a range of conditions for the population covariance matrix. Cases are considered where a p-variate process is being monitored using a sample of m observations per subgroup and m < p. Data from an industrial multivariate process is used to illustrate the proposed technique.     

Control charts for fraction nonconforming in a bivariate binomial process

Many multivariate quality control techniques are used for multivariate variable processes, but few work for multivariate attribute processes. To monitor multivariate attributes, controlling the false alarms (type I errors) and considering the correlation between attributes are two important issues. By taking into account these two issues, a new control chart is presented to monitor a bivariate binomial process. An example is illustrated for the proposed method. To evaluate the performance of the proposed method, a simulation study is conducted to compare the results with those using both the multivariate np chart and skewness reduction approaches. The results show that the correlation is taken into account in the designed chart and the overall false alarm is controlled at the nominal value. Moreover, the process shift can be quickly detected and the variable that is responsible for a signal can be determined.     

Multivariate Control Charts Based on Hybrid Novelty Scores

We propose a new nonparametric multivariate control chart that integrates a novelty score. The proposed control chart uses as its monitoring statistic a hybrid novelty score, calculated based on the distance to local observations as well as on the distance to the convex hull constructed by its neighbors. The control limits of the proposed control chart were established based on a bootstrap method. A rigorous simulation study was conducted to examine the properties of the proposed control chart under various scenarios and compare it with existing multivariate control charts in terms of average run length (ARL) performance. The simulation results showed that the proposed control chart outperformed both the parametric and nonparametric Hotelling's T ² control charts, especially in nonnormal situations. Moreover, experimental results with real semiconductor data demonstrated the applicability and effectiveness of the proposed control chart. To increase the capability to detect small mean shift, we propose an exponentially weighted hybrid novelty score control chart. Simulation results indicated that exponentially weighted hybrid score charts outperformed the hybrid novelty score based control charts.