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.     

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.     

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.     

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.     

A robust multivariate measurement error model with skew-normal/independent distributions and Bayesian MCMC implementation

Skew-normal/independent distributions are a class of asymmetric thick-tailed distributions that include the skew-normal distribution as a special case. In this paper, we explore the use of Markov Chain Monte Carlo (MCMC) methods to develop a Bayesian analysis in multivariate measurement errors models. We propose the use of skew-normal/independent distributions to model the unobserved value of the covariates (latent variable) and symmetric normal/independent distributions for the random errors term, providing an appealing robust alternative to the usual symmetric process in multivariate measurement errors models. Among the distributions that belong to this class of distributions, we examine univariate and multivariate versions of the skew-normal, skew-t, skew-slash and skew-contaminated normal distributions. The results and methods are applied to a real data set.     

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.     

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.     

Inference on C pk for autocorrelated data in the presence of random measurement errors

The present paper examines the properties of the C _pk estimator when observations are autocorrelated and affected by measurement errors. The underlying reason for this choice of subject matter is that in industrial applications, process data are often autocorrelated, especially when sampling frequency is not particularly low, and even with the most advanced measuring instruments, gauge imprecision needs to be taken into consideration. In the case of a first-order stationary autoregressive process, we compare the statistical properties of the estimator in the error case with those of the estimator in the error-free case. Results indicate that the presence of gauge measurement errors leads the estimator to behave differently depending on the entity of error variability.     

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.     

A note on multivariate capability indices

In manufacturing industry, there is growing interest in quantitative measures of process variation under multivariate setting. This paper introduces a multivariate capability index and focuses its applications in geometric dimensioning and tolerancing. This index incorporates both the process variation and the process deviation from target. Two existing multivariate indices are compared with the proposed index.     

On C pk Assessment in the Presence of Tool Wear

Process capability indices are routinely used in manufacturing industries for process monitoring. A basic assumption while using process capability indices is that there are no assignable causes of variation present. However, when variation due to an assignable cause is present and is tolerated, the conventional methods of capability measurement become inaccurate. In this article, we suggest an estimate of C_pk assuming that the process capability changes dynamically. We obtain an exact form of the sampling distribution in the presence of a systematic assignable cause. We discuss the problem of testing whether a given process is capable. The critical values for different sample sizes are obtained based on the sampling distribution. An example involving tool wear problem is presented.     

Process Capability Assessment for Asymmetric Tolerances with Consideration of Gauge Measurement Errors

In recent years, the issue of process capability assessment in the presence of gauge measurement errors (GME) for cases with symmetric tolerances was investigated enthusiastically. However, even processes with symmetric tolerances are very common in practical situations, cases of asymmetric tolerances also occur in manufacturing industries. In this article, a novel approach, called the generalized confidence interval (GCI) approach, is applied to assess the capabilities of processes with asymmetric tolerances in the presence of the GME. To examine the performance of the proposed approach, an exhaustive simulation was conducted. The conclusion is that the proposed approach appears quite satisfactorily for assessing process performance with asymmetric tolerances in the presence of GME in terms of the coverage rate (CR) and the average value of the generalized lower confidence limits.     

On the distributions of the estimated process capability indices

The exact distributions of the estimated process capability indices are presented and their means, variances, and mean-squared errors are given. The basic assumption is that the process measurements are taken from a normal distribution. Theresults in this article are useful in evaluating process capability.     

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

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.     

Designing optimal double-sampling plan based on process capability index

Process capability analysis is applied to monitor the process quality. Process capability can be quantified by process capability index. These indices have wide application in quality control methods and acceptance sampling plans. In this paper, we introduce a double-sampling plan based on process capability index. In this type of scheme, under a decision rule and with the specified rejection and acceptance numbers, the second sample is selected and the decision of rejection or acceptance is made based on the information obtained from two samples. The purpose of this scheme is to reduce the average sample number in order to reduce the time and cost of sampling. A comparison study has been conducted in order to evaluate the performance of proposed method in comparison with classical single sampling plans.     

A Bayesian model for measurement and misclassification errors alongside missing data,with an application to higher education participation in Australia

In this paper we consider the impact of both missing data and measurement errors on a longitudinal analysis of participation in higher education in Australia. We develop a general method for handling both discrete and continuous measurement errors that also allows for the incorporation of missing values and random effects in both binary and continuous response multilevel models. Measurement errors are allowed to be mutually dependent and their distribution may depend on further covariates. We show that our methodology works via two simple simulation studies. We then consider the impact of our measurement error assumptions on the analysis of the real data set.     

Multivariate Process Capability Indices: A Directional Approach

We propose a unified, universal, natural, and very intuitive way how to obtain new multivariate and tool wear extensions of univariate process capability indices by means of projection pursuit. We also illustrate the methodology in detail of the popular precision and accuracy indices, generalize the latter in a few different ways in the same spirit, add some personal insight, discuss the computational issues involved, and demonstrate the advantages of our approach in a small data example.     

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.