Multivariate Process Capability Research Articles

Multivariate Process Capability Index: A Review and Some Results

Process capability is defined as inherent variability of a process which is running under chance cause of variation only. Process capability index is measuring the ability of a process to meet the product specification limit. Generally Cp and Cpk have been widely used as statistical tools to assess the manufacturing process performance. These indices provide numerical measures on process precision, process accuracy and process performance. The multivariate process capability indices, which are used for evaluation of processes with correlated quality characteristics are considered as new emerging research area. In this paper, we describe an approach of estimating multivariate process capability indices assuming multivariate g-and-h distribution and illustrate their performance using simulated and practical data set.

An Approach to Build Process‐Oriented Basis for Causation‐Based Multivariate SPC and Process Capability Analysis

Although various multivariate process monitoring techniques have been developed, they do not diagnose the process for finding the root causes of irregularities during production. There have been recent studies on a new method that involves process‐oriented basis representation, which links the process variation to its causes, and thus helps in monitoring and diagnosing a process. However, all the studies done so far focused on its application. In this paper, a method is proposed to build the process‐oriented basis for a process irrespective of the number of variables characterizing it. Along with various other statistical techniques, factor analysis and cluster analysis, with customized distance function, are used in developing the method. The built in process‐oriented basis is further used for multivariate statistical process control and process capability analysis. Multivariate solder‐paste problem from electronics industry is used for illustration. Copyright © 2012 John Wiley & Sons, Ltd.

Non-normal multi-response optimization by multivariate process capability index

Most of the researches developed for single response and multi response optimization problems are based on the normality assumption of responses, while this assumption does not necessarily hold in real situations. In the real world processes, each product can contain correlated responses which follow different distributions. For instance, multivariate non-normal responses, multi-attribute responses or in some cases mixed continuous-discrete responses. In this paper a new approach is presented based on multivariate process capability index and NORTA inverse transformation for multi response optimization problem with mixed continuous-discrete responses. In the proposed approach, assuming distribution function of the responses is known in advance based on historical data; first we transform the multivariate mixed continuous-discrete responses using NORTA inverse transformation to obtain multivariate normal distributed responses. Then the multivariate process capability index is computed in each treatment. Finally, for determining the optimum treatment, the geometric mean value of multivariate Process Capability Index (PCI) is computed for each factor level and the most capable levels are selected as the optimum setting. The performance of the proposed method is verified through a real case study in a plastic molding process as well as simulation studies with numerical examples.

A new approach for assessing the correlated risk

PurposeThe purpose of this paper is to explore the relationship between multivariate process capability indices and loss functions for both nominal‐the‐best and smaller‐the‐better cases, so the likelihood and consequences resulting from the nonconforming of a manufacturing process or an environmental system can be evaluated simultaneously.Design/methodology/approachIn this paper, the authors present a new approach of correlated risk assessment by linking the multiple process capability indices and loss functions, in which the multivariate process capability indices and multivariate loss functions describe the likelihood and consequences as a result of nonconformities in multivariate manufacturing or environmental system, respectively. Then, the associated relationship equations are developed using multivariate methods. Moreover, a step‐by‐step procedure is provided to facilitate the implementation of the correlated risk assessment.FindingsGiven the multivariate process capability indices, the authors show that the expected loss can be estimated by developed relationship equations and two numerical examples are also given, to demonstrate how the correlated manufacturing and environmental risks can be properly assessed by linking the multivariate process capability indices and multivariate loss function.Practical implicationsThe risk information of likelihood and expected loss, classified in the four planning zones of a strategic planning matrix, provides practising managers and engineers with a decision‐making tool for prioritizing their quality improvement projects when conducting risk assessment for any multivariate process or environmental system.Originality/valueOnce the existing quality/environmental problems and their Key Performance Indicators are identified, one may conduct risk assessment by applying the relationship equations to evaluate the impact of correlated risk on manufacturing processes or multiple environmental emissions inside company; this can lead to the direction of continuous improvement for any industry.

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 Multivariate Process Capability Index Based on the First Principal Component Only

Often the quality of a process is determined by several correlated univariate variables. In such cases, the considered quality characteristic should be treated as a vector. Several different multivariate process capability indices (MPCIs) have been developed for such a situation, but confidence intervals or tests have been derived for only a handful of these. In practice, the conclusion about process capability needs to be drawn from a random sample, making confidence intervals or tests for the MPCIs important. Principal component analysis (PCA) is a well‐known tool to use in multivariate situations. We present, under the assumption of multivariate normality, a new MPCI by applying PCA to a set of suitably transformed variables. We also propose a decision procedure, based on a test of this new index, to be used to decide whether a process can be claimed capable or not at a stated significance level. This new MPCI and its accompanying decision procedure avoid drawbacks found for previously published MPCIs with confidence intervals. By transforming the original variables, we need to consider the first principal component only. Hence, a multivariate situation can be converted into a familiar univariate process capability index. Furthermore, the proposed new MPCI has the property that if the index exceeds a given threshold value the probability of non‐conformance is bounded by a known value. Properties, like significance level and power, of the proposed decision procedure is evaluated through a simulation study in the two‐dimensional case. A comparative simulation study between our new MPCI and an MPCI previously suggested in the literature is also performed. These studies show that our proposed MPCI with accompanying decision procedure has desirable properties and is worth to study further. Copyright © 2012 John Wiley & Sons, Ltd.

A process capability index for simple linear profile

In some cases, the quality of a process or a product is characterized by a linear regression model which is also called a linear profile. Process capability index is an important concept in statistical quality control and measures the ability of a process to manufacture products that meet certain specifications. There has been little attempt to study the process capability in linear profiles. In this paper, two methods are proposed for measuring the process capability in simple linear profiles. The first method uses the percentage of nonconforming parts produced at each level of the independent variable to introduce a process capability index. The second method is a multivariate process capability approach where a vector of three components is introduced. The components of the vector assess the process dispersion, its centrality, and its location within the upper and the lower specification limits. In comparison to the only existing method, numerical analysis based on simulation studies indicates that the suggested process capability measurements perform more accurately in evaluating the capability of a process generating a simple linear profile. When the error term variance is 1 and the actual percentage of nonconformities is 0.11 %, the proposed method estimates 0.11 % nonconforming percentage and a capability index of 1.06. While the existing method results in 0.27 % nonconforming percentage and a capability index of 0.9, the existing method underestimates the process capability.

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.

A New Multivariate Process Capability Index Under Both Unilateral and Bilateral Quality Characteristics

Process capability indices (PCIs) are used in statistical process control to evaluate the capability of the processes in satisfying the customer's needs. In the past two decades varieties of PCI are introduced by researchers to analyze the process capability with univariate or multivariate quality characteristics. To the best of our knowledge, most famous multivariate capability indices are proposed when the quality characteristics have both upper and lower specification limits. These indices are incapable to assess the multivariate processes capability with unilateral specification. In this article, we propose a new multivariate PCI to analyze the processes with one or more unilateral specification limits. This new index also accounts for all problems in the best PCIs of the literature. The performance of the proposed index is evaluated by real cases under different situations. The results show that the proposed index performs satisfactorily in all cases considered. Copyright © 2012 John Wiley & Sons, Ltd.

MPCI: AnRPackage for Computing Multivariate Process Capability Indices

Manufacturing processes are often based on more than one quality characteristic. When these variables are correlated the process capability analysis should be performed using multivariate statistical methodologies. Although there is a growing interest in methods for evaluating the capability of multivariate processes, little attention has been given to developing user friendly software for supporting multivariate capability analysis. In this work we introduce the package MPCI for R, which allows to compute multivariate process capability indices. MPCI aims to provide a useful tool for dealing with multivariate capability assessment problems. We illustrate the use of MPCI package through both simulated and real examples.

Modified (MCp) in the presence of measurement errors

PurposeThe purpose of this paper is to introduce a modified MCp by considering the effect of gauge measurement error on the multivariate process capability index.Design/methodology/approachIn this paper, the effect of measurement system on the quality characteristics appears as coefficient matrix A which changes the variance‐covariance matrix of quality characteristics ∑, into A ∑. In this case, the authors investigate the properties of multivariate index and present adjusted confidence intervals and critical values for capability testing purpose of this index.FindingsThe results show that the simplicity of the obtained index, calculating the true process capability by using empirical process capability and also computing the critical value and power of the process capability testing is simpler in this modified approach.Originality/valueThe results presented in this paper would help practitioners to determine their actual process capability and see if their processes meet the preset capability requirement, and then make reliable decisions.

Comparing Confidence Intervals for Multivariate Process Capability Indices

Multivariate process capability indices (MPCIs) are needed for process capability analysis when the quality of a process is determined by several univariate quality characteristics that are correlated. There are several different MPCIs described in the literature, but confidence intervals have been derived for only a handful of these. In practice, the conclusion about process capability must be drawn from a random sample. Hence, confidence intervals or tests for MPCIs are important. With a case study as a start and under the assumption of multivariate normality, we review and compare four different available methods for calculating confidence intervals of MPCIs that generalize the univariate index Cp. Two of the methods are based on the ratio of a tolerance region to a process region, and two are based on the principal component analysis. For two of the methods, we derive approximate confidence intervals, which are easy to calculate and can be used for moderate sample sizes. We discuss issues that need to be solved before the studied methods can be applied more generally in practice. For instance, three of the methods have approximate confidence levels only, but no investigation has been carried out on how good these approximations are. Furthermore, we highlight the problem with the correspondence between the index value and the probability of nonconformance. We also elucidate a major drawback with the existing MPCIs on the basis of the principal component analysis. Our investigation shows the need for more research to obtain an MPCI with confidence interval such that conclusions about the process capability can be drawn at a known confidence level and that a stated value of the MPCI limits the probability of nonconformance in a known way. Copyright © 2011 John Wiley & Sons, Ltd.

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.

A multivariate process capability index model system

This paper presents a systematic multivariate process capability index (MPCI) method, which may provide references for assuring and improving process quality levels while achieving an overall evaluation of process quality. The system method includes a spatial MPCI model for multivariate normal distribution data, MPCI model based on factor weight for multivariate no-normal distribution application, and MPCI model based on yield for yield application. At last, examples for calculating MPCI are given, and the experimental results show that this systematic method is effective and practical.

A note on multivariate non-normal process capability analysis using “mvtnorm” library in R

In this note we examine using Genzbretz and Miwa algorithms to improve estimation of proposition of non-conformance in multivariate normal distributions. This estimation is required in the procedure outlined in Abbasi and Niaki (Int J Adv Manuf Technol 50(5–8):823–830, 2010) to determine process capability index of multivariate non-normal processes.

Process Capability Sensitivity Analysis for Design Evaluation of Multistage Assembly Processes

Yield-based sensitivity analysis methods and algorithms are developed for process capability evaluation in this paper. Yield, the conformity to product specifications, is subject to critical design parameters such as dimensions, tolerances, and specification limits. The uncertainties in determining these parameters in design and manufacturing affect the process capability of producing high-quality products. Yield is a transparent and thus a desirable index, especially for multivariate process capability evaluation. Thus, yield sensitivity with respect to these design parameters indicates key contributors to final product quality, providing valuable information in design for quality control. Yield is formulated as a high dimension probability integral over a specification region. In multistage assembly processes, an assembly variation model links the quality characteristics to design parameters. This model is adopted in yield sensitivity analysis. Derivatives of yield with respect to the design parameters are developed using matrix calculus. Three sensitivity analysis algorithms, i.e., finite difference, yield derivative, and regression modeling, are implemented. Monte Carlo simulation is used for yield estimation in the three algorithms. A case study using floor pan assembly in automotive body manufacturing is presented for the validation of the proposed methodology.

Capability evaluation for a product family composed of unilateral and multi-process quality characteristics

Quality is a key indicator for customer’s satisfaction on a product or service. It is also one of the most important factors that a customer takes into account in making selection among competing products and services. Generally, a product is featured by many quality characteristics on performance. A product family is composed of numerous types of products. Customers are satisfied when all performance quality characteristics of an entire product meet the preset specification. Obviously, neither unilateral process capability nor multivariate process capability indices can fulfill the above requirements. Therefore, this study will propose a novel process capability index, which exhibits one-to-one mathematical relation with process yield. Meanwhile, the relationship between six-sigma quality level and capability index will be discussed. In addition, the study will also evaluate the quality of a multi-process product with numerous unilateral specifications. Furthermore, because of the difference in production volume for the products of every type, an index based on the weight of production volume of individual product will be developed so that the product quality of the entire product family can be evaluated.

Multiresponse optimization using multivariate process capability index

AbstractRobust design is an efficient method for product and process improvement which combines experimentation with optimization to create a system that is less sensitive to uncontrollable variation. In this article, a simple and integrated modeling methodology for robust design is proposed. This methodology achieves the robustness objective function and input variables constraints simultaneously. The objective function is written in terms of the multivariate process capability vector (MCpm) of several competing features of the system under study. The proposed methodology is applicable to general functions of the system performance with random variables. The effectiveness of the methodology is verified using two real‐world examples which are compared with those of other robust design methods. Copyright © 2010 John Wiley & Sons, Ltd.

Properties of multivariate process capability in the presence of gauge measurement errors and dependency measure of process variables

Nowadays process capability has been extensively studied as a mean of summarizing process performance relative to a set of specification limits. Most of the research papers related to process capability indices have assumed that there are no gauge measurement errors. Unfortunately, such an assumption does not reflect real situations accurately even with highly sophisticated advanced measuring instruments. In this paper, we study the behavior of multivariate process capability index ( M C p ) in the presence of gauge measurement errors. In addition, the effect of correlation coefficient and measurement capability on the statistical properties of the estimated M C p are also studied. The results indicate that correlation coefficients of process variables will notability change the statistical behavior of the multivariate process capability in the presence of gauge measurement errors. With the adjusted confidence intervals and suitable estimation method the more accurate testing procedure for M C p by considering different values of correlation coefficient is obtained.

Quality performance analysis for the whole product family with multiple characteristics: apply the cooling pump as an example

The function of cooling pump is to release and cool off the heat caused by the operation process of cutting tools and their related cutting materials, through cooling oil cycle effects, therefore to avoid damage to the tools and their materials. Three quality characteristics of cooling pump: function, safety and energy-saving, must meet the requirement of CNS-5703 product safety standard. Through multi-process capability analysis chart, we can improve and control those poor qualities of various pump products. Process Capability Indices (PCIs) are powerful tools to study the process ability for manufacturing product that meets specifications. A larger process capability index implies a higher process yield, and lower expected process loss. In the past references based on Cp , Cpu , Cpl , and Cps , the approach of this research is to develop one integrated process capability analysis that can precisely measure product with larger-the-better and smaller-the-better multi-process, and nominal-the-best characteristics. However, Cps suffers from the weakness of being unable to reflect process biased. Therefore, the paper selects index Cpn to replace Cps . Meanwhile, an integrated PCI for the whole pump product family is proposed, and the relationship between the PCI and process yield is described. We could find and improve unusual quality of the pump product family by multi-process capability analysis chart.