Multivariate Capability Indices Research Articles

Application of a proposed reliability analysis multivariate capability index on manufacturing processes

n the manufacturing process of any product intervene several factors associated with the monitored quality characteristics. These factors must be examined with adequate Statistical Quality Control tools to provide estimates of product specification compliance. For this purpose, Process Capability Indices are commonly applied to manufacturing processes, assuming that the quality variables are independent, and the data is normally distributed. Additionally, an increasingly prominent quality characteristic across manufacturing industries is product reliability. This paper explores the advantages of using multivariate capability indices that include product reliability as a key quality criterion.

Un índice de capacidad de procesos para distribuciones multivariadas normales y no normales, de variables correlacionadas y no correlacionadas

DOI: https://doi.org/10.26439/ing.ind2020.n038.4814 The multivariate process capability analysis includes many indices that are only used when the data is normal and others, when the data is not normal. The same occurs with correlated and uncorrelated quality variables. In this research work, a CPME multivariate capability index was developed by initially using a univariate index—depending on whether the data was o not normal—and any correlation between variables, and then through a characteristic function for the multivariate data. This index may be used in all the aforementioned cases. Some examples of this alternative are presented on a set of real and simulated data, where a broad performance of our proposed index was found against other similar capability indices.

Multivariate capability indices in inventory control

Inventory control in the industry has an important role in supply chain management to produce products according to management standards or quality standards based on consumer demand. In the apparel industry, inventory control is adjusted to the type of apparel product according to consumer demand. The research aims are to derive a multivariate inventory control model with multivariate capability analysis based on several types of apparel products as quality characteristics. The method in the research uses the T2 Hotelling multivariate control chart and the multivariate capability indices for analysis apparel inventory control by management specification standards. We derived the T2 Hotelling multivariate control chart and the multivariate capability indices using data with the assumption that multivariate data are controlled and has multivariate normal distribution Variable research uses quality characteristics that are grouped into two types of apparel products. This model is useful as a multivariate inventory control model that can manage the amount of production and supply of products based on probabilistic consumer demand.

Integrating Multivariate Statistical Analysis Into Six Sigma DMAIC Projects: A Case Study on AISI 52100 Hardened Steel Turning

DMAIC (define, measure, analyze, improve and control) is one of the most utilized methods for guiding practitioners in the decision-making process of quality improvement projects. Industrial processes commonly deal with multiple critical-to-quality (CTQ) characteristics. When these characteristics are correlated, multivariate statistical techniques should be applied. This paper aims to propose a domain-specific Six Sigma method, the MDMAIC (multivariate DMAIC). The new stepwise procedure helps practitioners not only to reduce problem dimension but also to take account of the correlation structure among CTQs during the decision-making process. Principal component analysis has been applied for assessing the measurement system, analyzing process stability and capability, as well as modeling and optimizing multivariate manufacturing processes. A hardened steel turning case has been presented for proposal validation. The result analysis has shown that the MDMAIC was very successful in leading the practitioner during the steps and phases of the quality improvement project. The multivariate capability index of the enhanced process emphasized the substantial economic improvement.

Estimating Non-Conformance Using the Modified Tolerance Region Method and the Target Distance Method

In many cases, the quality of a manufactured product is determined by more than one characteristic and often, these quality characteristics are correlated. A number of methods for dealing with quality evaluation of multivariate processes have been proposed in the literature. However, some of these studies do not consider correlation among quality characteristics. In this paper, two new approaches for estimating the proportion of non-conformance for correlated multivariate quality characteristics with nominal specifications are proposed: (i) the modified tolerance region approach and (ii) the target distance approach. In the first approach, the p number of correlated variables are analysed based on the projected shadow of the p-dimensional hyper ellipsoid so that the ability to visualise the tolerance region and the process region for is preserved. In the second approach, the correlated variables are combined and a new variable called the target distance is introduced. The proportion of non-conformance results estimated using both methods were used to compute the multivariate capability index and the total expected quality cost. This study also suggest modification to the NMCp index as proposed in Pan and Lee (2010) such that the process capability for can be measured correctly. The application of both approaches is demonstrated using two examples and it is shown that both methods i.e. the modified tolerance region and the target distance methods are capable of estimating the capability of multivariate processes.

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

Distributional Properties of Multivariate Process Capability Indices under Normal and Non-normal Distributions

Multivariate capability analysis has been the focus of study in recent years, during which many authors have proposed different multivariate capability indices. In the operative context, capability indices are used as measures of the ability of the process to operate according to specifications. Because the numerical value of the index is used to conclude about the capability of the process, it is essential to bear in mind that almost always that value is obtained from a sample of process units. Therefore, it is really necessary to know the properties that the indices have when they are calculated on sampling information, in order to assess the goodness of the inferences made from them. In this work, we conduct a simulation study to investigate distributional properties of two existing indices: NMCpm index based on ratio of volumes and Mp2 index based on principal component analysis. We analyze the relative bias and the mean square error of the estimators of the indices, and we also obtain their empirical distributions that are used to estimate the probability that the indices classify correctly a process as capable or as incapable. The results allow us to recommend the use of one of these indices, as it has shown better properties. Copyright © 2016 John Wiley & Sons, Ltd.

Fuzzy multivariate process capability vector

Process capability indices are numerical measures used to determine whether the process products meet the quality requirement dedicated by the customers and producers or not. When the products' quality counts on two or more correlated characteristics, multivariate process capability indices are applied. In this paper, we focus on two indices proposed by Shahriari and Abdollahzadeh (2009) and Taam et al. (1993). The traditional multivariate capability indices can't be applied if the specification limits and target value of each characteristic are presented in a linguistic form. Instead, the fuzzy set theory is used to deal with these circumstances. Here, the fuzzy formulations of the indices are developed. Furthermore, numerical examples are presented to illustrate the applicability of the proposed methods and we make a comparison between them.

Performance of Multivariate Process Capability Indices Under Normal and Non‐Normal Distributions

In the context of process capability analysis, the results of most processes are dominated by two or even more quality characteristics, so that the assessment of process capability requires that all of them are considered simultaneously. In recent years, many researchers have developed different alternatives of multivariate capability indices using different approaches of construction.In this paper, four of them are compared through the study of their ability to correctly distinguish capable processes from incapable processes under a diversity of simulated scenarios, defining suitable minimum desirable values that allow to decide whether the process meets or does not meet specifications. In this sense, properties analyzed can be seen as sensitivity and specificity, assuming that a measure is sensitive if it can detect the lack of capability when it actually exists and specific if it correctly identifies capable processes. Two indices based on ratios of regions and two based on the principal component analysis have been selected for the study. The scenarios take into account several joint distributions for the quality variables, normal and non‐normal, several numbers of variables, and different levels of correlation between them, covering a wide range of possible situations.The results showed that one of the indices has better properties across most scenarios, leading to right conclusions about the state of capability of processes and making it a recommendable option for its use in real‐world practice. Copyright © 2015 John Wiley & Sons, Ltd.

A modified multivariate process capability vector

Process capability indices have been widely used in industries to assess the performance of the manufacturing processes. Various different multivariate capability indices have been introduced. In this paper, a new multivariate capability vector is proposed under the assumption of multivariate normality, to assess the production capability of the processes that involve multiple product quality characteristics. Also, we investigate the relation between this index and process centering, as well as the relation between this index and the lower and upper bounds of percentage of non-conforming items manufactured. Two real manufacturing data set are used to demonstrate the effectiveness of the proposed index.

The Study of Relationship between Multivariate Capability Indices and Sampling Number

ultivariate process capability indices (MPCI), as an important means of statistical process control (SPC), can be used to ensure the high reliability of semiconductors manufacturing process. However, the reasonable sampling number is an important factor when considering MPCI values. As general, the large sample number requires much effort and time, or even cannot be achieved. In this paper, we evaluated the impact of different sample size on the calculations of multivariate process capability indices using simulation and analyses. After getting enough data and choosing disparate sample numbers, corresponding multivariate process capability indices can be obtained, which demonstrate the relationship between sampling numbers and calculation results. The conclusions have critical guiding significance for manufacturing semiconductors with high reliability requirement.

Six Sigma management and grey relational analysis to improve performance of tableting process

This research implements the well–known Six Sigma approach define–measure–analyse improve–control (DMAIC) to improve the performance of direct compression process with two quality responses: tablet's weight and hardness. At define phase, the x–bar and s charts are established and judged in–control for both responses. To measure performance, the values of the actual capability indices for hardness and weight are estimated 1.5 and 0.587, respectively, which indicate that the tableting process is capable for tablet's hardness but it is incapable for weight. At the analyse and improve phases, designed experiments utilising the Taguchi's L27 array followed by the grey relational analysis (GRA) technique are implemented to determine the combination of factor levels that enhance process performance. After process improvement, the capability index values are found equal to 3.31 and 0.848. To control performance, the value of the multivariate capability index is found equal to 0.938 at initial factor settings, while it is enhanced using optimal factor settings to 1.68. Finally, the multivariate control charts for Hotelling T² and generalised variance are employed to monitor future production. In conclusion, the DMAIC approach including GRA techniques is found effective for improving the performance of direct compression with tablet's weight and hardness.

Multivariate Process Capability Vector Based on One‐Sided Model

To evaluate the capability of manufacturing processes in satisfying the customer's needs, a variety of indices has been developed. Some of them are introduced by researchers to analyse the processes with multivariate quality characteristics. Most of the proposed in the literature multivariate capability indices are defined under assumption of normality distribution of the quality characteristics. Thus, the process region describing the variation of the data has an elliptical shape. In this paper, a multivariate process capability vector with three components is introduced, which allows to access the capability of a process with both normally and non‐normally quality characteristics due to application of a pair of one‐sided models as the process region shape. At the beginning, one‐sided models are defined, next the proposed vector components are proposed and the methodology of their evaluation is presented. The methodology (which in fact could be also applied to both the correlated and non‐correlated characteristics) is verified by applying simulation and real problems. The obtained results show that the proposed methodology performs satisfactorily in all considered cases. Copyright © 2014 John Wiley & Sons, Ltd.

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 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.

On Capability Indices for Multivariate Autocorrelated Processes

In this paper the effects of the autocorrelation on some multivariate capability indices commonly used for independent processes are discussed and a correction is proposed. Some results are shown for VARMA(1,1) and VAR(1) time series processes under the multivariate normality assumption and the proportion of non-conforming units is calculated for some bivariate VAR(1) models. An extension of Veevers capability index for non-centered processes is also a subject addressed in this paper. An example of application in blast charcoal furnace pig iron process is presented and bootstrap is used to build confidence intervals for its true capability value as well as to evaluate the performance of the capability estimators. Similar as to what is already known for univariate processes the results showed that autocorrelation has a large impact in the multivariate capabilities indices. This paper also shows that some care should be taken when using Niverthi and Dey’s capabilities indices since they are very sensitive to any deviations from the process means to the specification means up to a point that a capable process might be considered non-capable.

Univariate and multivariate approaches for evaluating the capability of dynamic-behavior processes (case study)

The majority of classic SPC methodologies assume a steady-state (static) process behavior (i.e., the process mean and variance are constant) without the influence of the dynamic behavior (i.e., an intended or unintended drift in the process mean or variance). Traditional SPC methods have been successfully used in steady-state manufacturing processes, but these approaches are not valid for use in dynamic behavior environments. The standard assumptions in SPC are that the observed process characteristics are normally, independently and identically distributed (IID) with fixed mean μ and standard deviation σ when the process is in control. Due to the dynamic behavior, these assumptions are not always valid. This study provides a scientific approach for evaluating the capability of cold rolling processes (as an example of manufacturing processes that undergo many disturbances and dynamic behavior) so that quality improvement may be attained because of the good understanding of the nature of the processes. The paper proposes the appropriate procedures for evaluating the capability of the manufacturing processes, especially for those in a dynamic behavior mode, with a comparison between the univariate and multivariate capability indices.

Multivariate nonnormal process capability analysis

There is a great deal of interest in the manufacturing industry for quantitative measures of process performance with multiple quality characteristics. Unfortunately, multivariate process capability indices that are currently employed, except for a handful of cases, depend intrinsically on the underlying data being normally distributed. In this paper, we propose a general multivariate capability index based on the Mahanalobis distance, which is very easy to use. We also approximate the distribution of these distances by the Burr XII distribution and then estimate its parameters using a simulated annealing search algorithm. Finally, we give an example, based on real manufacturing process data, which demonstrates that the proportion of nonconformance (PNC) using our proposed method is very close to the actual PNC value, which also justifies its adoption in this paper.

Capability indices and nonconforming proportion in univariate and multivariate processes

The main usefulness of a capability index is to relate the actual variability of the process with the admissible one. This admissible variability is, in turn, related with the nonconforming proportion. Hence, the capability index should be closely related to the nonconforming proportion. In univariate and centered processes, the classical Cp index explicitly admits this interpretation. For instance, if Cp = 0.5, the standard deviation should be reduced to 50% to attain Cp = 1. However, for noncentered processes and multivariate processes, there is a lack of capability indices that admit such an interpretation. This article fills this gap in the literature and proposes univariate and multivariate capability indices that have a direct interpretation of how much the variability of the process should increase or decrease to attain a unitary index. Some numerical examples are used to compare the proposed indices with the existing ones, showing the advantages of the proposals.

Multivariate process capability using principal component analysis

AbstractWang and Chen (Qual. Eng. 1998; 11:21–27) have defined process capability indices (PCIs) for multivariate normal processes data using principal component analysis (PCA). Veevers (Statistical Process Monitoring and Optimization. Marcel Dekker: New York, NY, 1999; 241–256) has suggested a multivariate capability index based on the first principal component (PC). In this paper we demonstrate the problem in the definition of PCIs given by Wang and Chen (Qual. Eng. 1998; 11:21–27) and the non‐suitability of PCI given by Veevers (Statistical Process Monitoring and Optimization. Marcel Dekker: New York, NY, 1999; 241–256) through some examples. We also suggest an alternative method for assessing multivariate process capability based on the empirical probability distribution of PCs. This method has been performed on industrial and simulated data. Copyright © 2008 John Wiley & Sons, Ltd.