A new generation of process capability indices

In quality control, we may confront imprecise concepts. One case is a situation in which upper and lower specification limits (SLs) are imprecise. If we introduce vagueness into SLs, we face quite new, reasonable and interesting processes, and the ordinary capability indices are not appropriate for measuring the capability of these processes. In this paper, similar to the traditional process capability indices (PCIs), we develop a fuzzy analogue by a distance defined on a fuzzy limit space and introduce PCIs, where instead of precise SLs we have two membership functions for upper and lower SLs. These indices are necessary when SLs are fuzzy, and they are helpful for comparing manufacturing process with fuzzy SLs. Some interesting relations among these introduced indices are proved. Numerical examples are given to clarify the method.     

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.     

Process capability analysis for serially dependent processes of Poisson counts

Process capability indices evaluate the actual compliance of a process with given external specifications in a single number. For the case of a process of independent and identically distributed Poisson counts, two types of index have been proposed and investigated in the literature. The assumption of serial independence, however, is quite unrealistic for practice. We consider the case of an underlying Poisson INAR(1) process which has an AR(1)-like autocorrelation structure. We show that the performance of the estimated indices is degraded heavily if serial dependence is ignored. Therefore, we develop approaches for estimating the process capability (both for the observation and innovation process), which explicitly consider the observed degree of autocorrelation. For this purpose, we introduce a new unbiased estimator of the innovations’ mean of a Poisson INAR(1) process and derive its exact as well as asymptotic stochastic properties. In this context, we also present new explicit expressions for the third- and fourth-order moments of a Poisson INAR(1) process. Then the capability indices and the performance of their estimators are analysed and recommendations for practice are given.     

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 total and specific process capability indices in three-stage processes with measurement errors

In multistage processes, two kinds of process capability indices (specific and total process capability indices) are defined in each stage. The total process capability index calculates the capability of each stage when it is affected by the previous stages and the specific process capability index calculates the capability of the stage when the effects of the previous stages are eliminated. Process capability indices in multistage processes are proposed under the assumption of no measurement errors. However, sometimes this assumption may be violated and leads to misleading interpretations. In this paper, the effects of measurement errors on the specific and total process capability indices in the second and third stages of the three-stage processes are statistically analysed. In addition, the effects of the measurement errors on the bias and the mean square error value of the total and specific process capability indices in the second and third stages are studied. Finally, the effects of the measurement errors on the specific and total process capability indices are shown through a numerical example.     

Impact of measurement errors on the performance and distributional properties of the multivariate capability index $$\mathbf{NMC }_\mathbf{pm }$$

Current industrial processes are sophisticated enough to be tied to only one quality variable to describe the process result. Instead, many process variables need to be analyze together to assess the process performance. In particular, multivariate process capability analysis (MPCIs) has been the focus of study during the last few decades, during which many authors proposed alternatives to build the indices. These measures are extremely attractive to people in charge of industrial processes, because they provide a single measure that summarizes the whole process performance regarding its specifications. In most practical applications, these indices are estimated from sampling information collected by measuring the variables of interest on the process outcome. This activity introduces an additional source of variation to data, that needs to be considered, regarding its effect on the properties of the indices. Unfortunately, this problem has received scarce attention, at least in the multivariate domain. In this paper, we study how the presence of measurement errors affects the properties of one of the MPCIs recommended in previous researches. The results indicate that even little measurement errors can induce distortions on the index value, leading to wrong conclusions about the process performance.     

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.     

Skewed zero-bound distributions and process capability indices for upper specifications

A common practical situation in process capability analysis, which is not well developed theoretically, is when the quality characteristic of interest has a skewed distribution with a long tail towards relatively large values and an upper specification limit only exists. In such situations, it is not uncommon that the smallest possible value of the characteristic is 0 and this is also the best value to obtain. Hence a target value 0 is assumed to exist. We investigate a new class of process capability indices for this situation. Two estimators of the proposed index are studied and the asymptotic distributions of these estimators are derived. Furthermore, we suggest a decision procedure useful when drawing conclusions about the capability at a given significance level, based on the estimated indices and their asymptotic distributions. A simulation study is also performed, assuming that the quality characteristic is Weibull-distributed, to investigate the true significance level when the sample size is finite.     

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 using principal component analysis in the presence of measurement errors

Various different definitions of multivariate process capability indices have been proposed in the literature. Most of the research works related to multivariate process capability indices assume no gauge measurement errors. However, in industrial applications, despite the use of highly advanced measuring instruments, account needs to be taken of gauge imprecision. In this paper we are going to examine the effects of measurement errors on multivariate process capability indices computed using the principal components analysis. We show that measurement errors alter the results of a multivariate process capability analysis, resulting in either a decrease or an increase in the capability of the process. In order to achieve accurate process capability assessments, we propose a method useful for overcoming the effects of gauge measurement errors.     

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.     

Testing process capability based on Cpm in the presence of random measurement errors

Process capability indices have been widely used in the manufacturing industry providing numerical measures on process performance. The index C_p provides measures on process precision (or product consistency). The index C_pm, sometimes called the Taguchi index, meditates on process centring ability and process loss. Most research work related to C_p and C_pm assumes no gauge measurement errors. This assumption insufficiently reflects real situations even with highly advanced measuring instruments. Conclusions drawn from process capability analysis are therefore unreliable and misleading. In this paper, we conduct sensitivity investigation on process capability C_p and C_pm in the presence of gauge measurement errors. Due to the randomness of variations in the data, we consider capability testing for C_p and C_pm to obtain lower confidence bounds and critical values for true process capability when gauge measurement errors are unavoidable. The results show that the estimator with sample data contaminated by the measurement errors severely underestimates the true capability, resulting in imperceptible smaller test power. To obtain the true process capability, adjusted confidence bounds and critical values are presented to practitioners for their factory applications.     

Bootstrap confidence intervals of CNpk for inverse Rayleigh and log-logistic distributions

In this article bootstrap confidence intervals of process capability index as suggested by Chen and Pearn [An application of non-normal process capability indices. Qual Reliab Eng Int. 1997;13:355–360] are studied through simulation when the underlying distributions are inverse Rayleigh and log-logistic distributions. The well-known maximum likelihood estimator is used to estimate the parameter. The bootstrap confidence intervals considered in this paper consists of various confidence intervals. A Monte Carlo simulation has been used to investigate the estimated coverage probabilities and average widths of the bootstrap confidence intervals. Application examples on two distributions for process capability indices are provided for practical use.     

An entropic structure in capability indices

Abstract

Analysis capability indices for symmetric process in normal case is obtained via maximum entropy approach of distribution function of the data. In view of it, we have perused on production processes to be in statistical control. Generally a process is capable based on capability indices when its reasonable index was more than a known threshold value. Thus by conditioning on indices, the most general distribution is found out whose parameters can be approximated by using the data of process. Also analysis via Kullback-Leibler information measure based on the above arguments is obtained in the last part of the paper.     

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.     

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.     

Conjugacy as a Distinctive Feature of the Dirichlet Process

Abstract.  Recently the class of normalized random measures with independent increments, which contains the Dirichlet process as a particular case, has been introduced. Here a new technique for deriving moments of these random probability measures is proposed. It is shown that, a priori , most of the appealing properties featured by the Dirichlet process are preserved. When passing to posterior computations, we obtain a characterization of the Dirichlet process as the only conjugate member of the whole class of normalized random measures with independent increments.     

CAPABILITY MEASURES FOR m-DEPENDENT STATIONARY PROCESSES

Process capability indices, providing numerical measures on process potential and process performance, have received substantial research attention. Most research assumes that the process is normally distributed and the process data are independent. In real-world applications such as chemical, soft drinks, or tobacco/cigaratte manufacturing processes, process data are often auto-correlated. In this paper, we consider the capability indices Cp, Cpk, Cpm, Cpmk for strictly m-dependent stationary processes. We investigate the statistical properties of their natural estimators. We derive the asymptotic distributions, and establish confidence intervals so that capability testing can be performed.     

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.     

Elliptical safety region plots for C pk

The process capability index C _pk is widely used when measuring the capability of a manufacturing process. A process is defined to be capable if the capability index exceeds a stated threshold value, e.g. C _pk >4/3. This inequality can be expressed graphically using a process capability plot, which is a plot in the plane defined by the process mean and the process standard deviation, showing the region for a capable process. In the process capability plot, a safety region can be plotted to obtain a simple graphical decision rule to assess process capability at a given significance level. We consider safety regions to be used for the index C _pk . Under the assumption of normality, we derive elliptical safety regions so that, using a random sample, conclusions about the process capability can be drawn at a given significance level. This simple graphical tool is helpful when trying to understand whether it is the variability, the deviation from target, or both that need to be reduced to improve the capability. Furthermore, using safety regions, several characteristics with different specification limits and different sample sizes can be monitored in the same plot. The proposed graphical decision rule is also investigated with respect to power.